diff --git a/parse/train/H1ecDoR5Y7/H1ecDoR5Y7.md b/parse/train/H1ecDoR5Y7/H1ecDoR5Y7.md new file mode 100644 index 0000000000000000000000000000000000000000..f1dc05165b05df7d1023d3d325a757ccbfe0396c --- /dev/null +++ b/parse/train/H1ecDoR5Y7/H1ecDoR5Y7.md @@ -0,0 +1,515 @@ +# LOCAL STABILITY AND PERFORMANCE OF SIMPLE GRADIENT PENALTY $\mu$ -WASSERSTEIN GAN + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance between a data distribution and sample distribution. Recent studies have proposed stabilizing the training process for the WGAN and implementing the Lipschitz constraint. In this study, we prove the local stability of optimizing the simple gradient penalty $\mu$ -WGAN(SGP $\mu$ -WGAN) under suitable assumptions regarding the equilibrium and penalty measure $\mu$ . The measure valued differentiation concept is employed to deal with the derivative of the penalty terms, which is helpful for handling abstract singular measures with lower dimensional support. Based on this analysis, we claim that penalizing the data manifold or sample manifold is the key to regularizing the original WGAN with a gradient penalty. Experimental results obtained with unintuitive penalty measures that satisfy our assumptions are also provided to support our theoretical results. + +# 1 INTRODUCTION + +Deep generative models reached a turning point after generative adversarial networks (GANs) were proposed by Goodfellow et al. (2014). GANs are capable of modeling data with complex structures. For example, DCGAN can sample realistic images using a convolutional neural network (CNN) structure(Radford et al., 2015). GANs have been implemented in many applications in the field of computer vision with good results, such as super-resolution, image translation, and text-to-image generation(Ledig et al., 2017; Isola et al., 2017; Zhang et al., 2017; Reed et al., 2016). + +However, despite these successes, GANs are affected by training instability and mode collapse problems. GANs often fail to converge, which can result in unrealistic fake samples. Furthermore, even if GANs successfully synthesize realistic data, the fake samples exhibit little variability. + +A common solution to this instability problem is injecting an instance noise and finding different divergences. The injection of instance noise into real and fake samples during the training procedure was proposed by Sønderby et al. (2017), where its positive impact on the low dimensional support for the data distribution was shown to be a regularizing factor based on the Wasserstein distance, as demonstrated analytically by Arjovsky & Bottou (2017). In $f$ -GAN, $f$ -divergence between the target and generator distributions was suggested which generalizes the divergence between two distributions(Nowozin et al., 2016). In addition, a gradient penalty term which is related with Sobolev IPM(Integral Probability Metric) between data distribution and sample distribution was suggested by Mroueh et al. (2018). + +The Wasserstein GAN (WGAN) is known to resolve the problems of generic GANs by selecting the Wasserstein distance as the divergence(Arjovsky et al., 2017). However, WGAN often fails with simple examples because the Lipschitz constraint on discriminator is rarely achieved during the optimization process and weight clipping. Thus, mimicking the Lipschitz constraint on the discriminator by using a gradient penalty was proposed by Gulrajani et al. (2017). + +Noise injection and regularizing with a gradient penalty appear to be equivalent. The addition of instance noise in $f$ -GAN can be approximated to adding a zero centered gradient penalty(Roth et al., 2017). Thus, regularizing GAN with a simple gradient penalty term was suggested by Mescheder et al. (2018) who provided a proof of its stability. + +Based on a theoretical analysis of the dynamic system, Nagarajan & Kolter (2017) proved the local exponential stability of the gradient-based optimization dynamics in GANs by treating the simultaneous gradient descent algorithm with a dynamic system approach. These previous studies were useful because they showed that the local behavior of GANs can be explained using dynamic system tools and the related Jacobian’s eigenvalues. + +In this study, we aim to prove the convergence property of the simple gradient penalty $\mu$ -Wasserstein GAN(SGP $\mu$ -WGAN) dynamic system under general gradient penalty measures $\mu$ . To the best of our knowledge, our study is the first theoretical approach to GAN stability analysis which deals with abstract singular penalty measure. In addition, measure valued differentiation(Heidergott & Vazquez-Abad, 2008) is applied to take the derivative on the integral with a parametric measure, ´ which is helpful for handling an abstract measure and its integral in our proof. + +The main contributions of this study are as follows. + +• We prove the regularized effect and local stability of the dynamic system for a general penalty measure under suitable assumptions. The assumptions are written as both a tractable strong version and intractable weak version. To prove the main theorem, we also introduce the measure valued differentiation concept to handle the parametric measure. Based on the proof of the stability, we explain the reason for the success of previous penalty measures. We claim that the support of a penalty measure will be strongly related to the stability, where the weight on the limiting penalty measure might affect the speed of convergence. • We experimentally examined the general convergence results by applying two test penalty measures to several examples. The proposed test measures are unintuitive but they still satisfy the assumptions and similar convergence results were obtained in the experiment. + +# 2 PRELIMINARIES + +First, we introduce our notations and basic measure-theoretic concepts. Second, we define our SGP $\mu$ -WGAN optimization problem and treat this problem as a continuous dynamic system. Preliminary measure theoretic concepts are required to justify that the dynamic system changes in a sufficiently smooth manner as the parameter changes, so it is possible to use linearization theorem. They are also important for dealing with the parametric measure and its derivative. The problem setting with a simple gradient term is also discussed. The squared gradient size and simple gradient penalty term are used to build a differentiable dynamic system and to apply soft regularization as a resolving constraint, respectively. The continuous dynamic system approach, which is a so-called ODE method, is used to analyze the GAN optimization problem with the simultaneous gradient descent algorithm, as described by Nagarajan & Kolter (2017). + +# 2.1 NOTATIONS AND PRELIMINARIES REGARDING MEASURE THEORY + +$D ( x ; \psi ) : \mathcal { X } \to \mathbb { R }$ is a discriminator function with its parameter $\psi$ and $G ( z ; \theta ) : \mathcal { Z } \to \mathcal { X }$ is a generator function with its parameter $\theta$ . $p _ { d }$ is the distribution of real data and $p _ { g } ~ = ~ p _ { \theta }$ is the distribution of the generated samples in $\mathcal { X }$ , which is induced from the generator function $G ( z ; \theta )$ and a known initial distribution $p _ { l a t e n t } ( z )$ in the latent space $\mathcal { Z } . \ \Vert \cdot \Vert$ denotes the $L ^ { 2 }$ Euclidean norm if no special subscript is present. + +The concept of weak convergence for finite measures is used to ensure the continuity of the integral term over the measure in the dynamic system, which must be checked before applying the theorems related to stability. Throughout this study, we assume that the measures in the sample space are all finite and bounded. + +Definition 1. For a set of finite measures $\{ \mu _ { i } \} _ { i \in \mathcal { I } }$ in $( \mathbb { R } ^ { n } , d )$ with euclidean distance $d _ { \mathrm { { z } } }$ , $\{ \mu _ { i } \} _ { i \in \mathbb { Z } }$ is referred to as bounded if there exists some $M > 0$ such that for all $i \in \mathcal { Z }$ , + +$$ +\mu _ { i } ( \mathbb { R } ^ { n } ) \leq M +$$ + +For instance, $M$ can be set as 1 if $\{ \mu _ { i } \}$ are probability measures on $\mathbb { R } ^ { n }$ . Assuming that the penalty measures are bounded, Portmanteau theorem offers the equivalent definition of the weak convergence for finite measures. This definition is important for ensuring that the integrals over $p _ { \theta }$ and $\mu$ in the dynamic system change continuously. + +Definition 2. (Portmanteau Theorem) For a bounded sequence of finite measures $\{ \mu _ { n } \} _ { n \in \mathbb { N } }$ on the Euclidean space $\mathbb { R } ^ { n }$ with a $\sigma$ -field of Borel subsets $B ( \mathbb { R } ^ { n } )$ , $\mu _ { n }$ converges weakly to $\mu$ if and only $i f$ for every continuous bounded function $\phi$ on $\mathbb { R } ^ { n }$ , its integrals with respect to $\mu _ { n }$ converge to $\int \phi \dot { d \mu }$ , i.e., + +$$ +\mu _ { n } \mu \Longleftrightarrow \int \phi d \mu _ { n } \int \phi d \mu +$$ + +The most challenging problem in our analysis with the general penalty measure is taking the derivative of the integral, where the measure depends on the variable that we want to differentiate. If our penalty measure is either absolutely continuous or discrete, then it is easy to deal with the integral. However, in the case of singular penalty measure, dealing with the integral term is not an easy task. Therefore, we introduce the concept of a weak derivative of a probability measure in the following(Heidergott & Vazquez-Abad, 2008). The weak derivative of a measure is useful for handling a ´ parametric measure that is not absolutely continuous with low dimensional support. + +Definition 3. (Weak Derivatives of a Probability Measure) Consider the Euclidean space and its $\sigma$ -field of Borel subsets $( \mathbb { R } ^ { d } , B ( \mathbb { R } ^ { d } ) )$ . The probability measure $P _ { \theta }$ is called weakly differentiable at $\theta$ if a signed finite measure $P _ { \theta } ^ { \prime }$ exists where + +$$ +\frac { d } { d \theta } \int \phi ( x ) d P _ { \theta } = \operatorname * { l i m } _ { \Delta 0 } \frac { 1 } { \Delta } \{ \int \phi ( x ) d P _ { \theta + \Delta } - \int \phi ( x ) d P _ { \theta } \} = \int \phi ( x ) d P _ { \theta } ^ { \prime } +$$ + +is satisfied for every continuous bounded function $\phi$ on $\mathbb { R } ^ { n }$ . For the multidimensional parameter $\theta$ , this can be defined similar manner. + +We can show that the positive part and negative part of $P _ { \theta } ^ { \prime }$ have the same mass by putting $\phi ( x ) = 1$ and the Hahn–Jordan decomposition on $P _ { \theta } ^ { \prime }$ . Therefore, the following triple $( c _ { \theta } , P _ { \theta } ^ { + } , P _ { \theta } ^ { - } )$ is called a weak derivative of $P _ { \theta }$ , where $P _ { \theta } ^ { \pm }$ are probability measures and $P _ { \theta } ^ { \prime }$ is rewritten as: + +$$ +P _ { \theta } ^ { \prime } = c _ { \theta } P _ { \theta } ^ { + } - c _ { \theta } P _ { \theta } ^ { - } +$$ + +Therefore, + +$$ +\frac { d } { d \theta } \int \phi ( x ) d P _ { \theta } = \int \phi ( x ) d P _ { \theta } ^ { \prime } = c _ { \theta } ( \int \phi ( x ) d P _ { \theta } ^ { + } - \int \phi ( x ) d P _ { \theta } ^ { - } ) +$$ + +holds for every continuous bounded function $\phi$ on $\mathbb { R } ^ { n }$ . It is known that the representation of $( c _ { \theta } , P _ { \theta } ^ { + } , P _ { \theta } ^ { - } )$ for $P _ { \theta } ^ { \prime }$ is not unique because $( c _ { \theta } + C _ { \theta } , P _ { \theta } ^ { + } + q _ { \theta } , P _ { \theta } ^ { - } + q _ { \theta } )$ is also another representation of $P _ { \theta } ^ { \prime }$ . + +For the general finite measure $Q _ { \theta }$ , a normalizing coefficient $M ( \theta ) < \infty$ can be introduced. The product rule for differentiating can also be applied in a similar manner to calculus. + +$$ +\frac { d } { d \theta } \int \phi ( x ; \theta ) d P _ { \theta } = \int \nabla _ { \theta } \phi ( x ; \theta ) d P _ { \theta } + \int \phi ( x ; \theta ) d P _ { \theta } ^ { \prime } +$$ + +Therefore, for the general finite measure $Q _ { \theta } = M ( \theta ) P _ { \theta }$ , its derivative $Q _ { \theta } ^ { \prime }$ can be represented as below. + +$$ +Q _ { \theta } ^ { \prime } = M ^ { \prime } ( \theta ) P _ { \theta } + M ( \theta ) P _ { \theta } ^ { \prime } = M ^ { \prime } ( \theta ) P _ { \theta } + c _ { \theta } M ( \theta ) P _ { \theta } ^ { + } - c _ { \theta } M ( \theta ) P _ { \theta } ^ { - } +$$ + +# 2.2 PROBLEM SETTING AS A DYNAMIC SYSTEM + +Previous work of Mescheder et al. (2018) showed that the dynamic system of WGAN-GP is not necessarily stable at equilibrium by demonstrating that the sequence of parameters is not Cauchy sequence. This is mainly due to the term $\| x \|$ in the dynamic system which has a derivative $\frac { x } { \| x \| }$ that is not defined at $x = 0$ . WGAN-GP has a penalty term $\mathbb { E } _ { \mu _ { G P } } [ ( \| \nabla _ { x } D ( x ; \psi ) \| - 1 ) ^ { 2 } ]$ that can lead to a discontinuity in its dynamic system. + +These problems can be avoided by using the squared value of the gradient’s norm $\| \nabla _ { x } D \| ^ { 2 }$ , which is a differentiable function. In contrast to the WGAN-GP, recent methods based on a gradient penalty such as the simple gradient penalty employed by Mescheder et al. (2018) and the Sobolev GAN used the average of the squared values for the penalty area, whereas the WGAN-GP penalizes the size of the discriminator’s gradient $\lVert \nabla _ { x } D \rVert$ away from 1 in a pointwise manner. + +This advantage of squared gradient term1, $\mathbb { E } _ { \mu } [ \| \nabla _ { x } D \| ^ { 2 } ]$ , makes the dynamic system differentiable and we define the WGAN problem with the square of the gradient’s norm as a simple gradient penalty. This simple gradient penalty can be treated as soft regularization based on the size of the discriminator’s gradient, especially in case where $\mu$ is the probability measure (Roth et al., 2017). It is convenient to determine whether the system is stable by observing the spectrum of the Jacobian matrix. In the following, $( D ( x ; \psi ) , p _ { d } , p _ { \theta } , \mu )$ is defined as an SGP $\mu$ -WGAN optimization problem (SGP-form) with a simple gradient penalty term on the penalty measure $\mu$ . + +Definition 4. The WGAN optimization problem with a simple gradient penalty term $\| \nabla _ { x } D \| ^ { 2 }$ , penalty measure $\mu _ { ; }$ , and penalty weight hyperparameter $\rho > 0$ is given as follows, where the penalty term is only introduced to update the discriminator. + +$$ +\begin{array} { r l } & { \displaystyle \operatorname* { m a x } _ { \psi } : \mathbb { E } _ { p _ { d } } [ D ( x ; \psi ) ] - \mathbb { E } _ { p _ { \theta } } [ D ( x ; \psi ) ] - \frac { \rho } { 2 } \mathbb { E } _ { \mu } [ \| \nabla _ { x } D ( x ; \psi ) \| ^ { 2 } ] } \\ & { \displaystyle \operatorname* { m i n } _ { \theta } : \mathbb { E } _ { p _ { d } } [ D ( x ; \psi ) ] - \mathbb { E } _ { p _ { \theta } } [ D ( x ; \psi ) ] } \end{array} +$$ + +According to Nagarajan & Kolter (2017) and many other optimization problem studies, the simultaneous gradient descent algorithm for GAN updating can be viewed as an autonomous dynamic system of discriminator parameters and generator parameters, which we denote as $\psi$ and $\theta$ . As a result, the related dynamic system is given as follows. + +$$ +\begin{array} { r l } & { \dot { \boldsymbol { \psi } } = \mathbb { E } _ { \boldsymbol { p } _ { d } } [ \nabla _ { \boldsymbol { \psi } } D ] - \mathbb { E } _ { \boldsymbol { p } _ { \boldsymbol { \theta } } } [ \nabla _ { \boldsymbol { \psi } } D ] - \frac { \rho } { 2 } \nabla _ { \boldsymbol { \psi } } \mathbb { E } _ { \boldsymbol { \mu } } [ \nabla _ { \boldsymbol { x } } ^ { T } D \nabla _ { \boldsymbol { x } } D ] } \\ & { \dot { \boldsymbol { \theta } } = \nabla _ { \boldsymbol { \theta } } \mathbb { E } _ { \boldsymbol { p } _ { \boldsymbol { \theta } } } [ D ] } \end{array} +$$ + +# 3 TOY EXAMPLES + +We investigate two examples considered in previous studies by Mescheder et al. (2018) and Nagarajan & Kolter (2017). We then generalize the results to a finite measure case. The first example is the univariate Dirac GAN, which was introduced by Mescheder et al. (2018). + +Definition 5. (Dirac GAN) The Dirac GAN comprises a linear discriminator $D ( x ; \psi ) = \psi x$ , data distribution $p _ { d } = \delta _ { 0 }$ , and sample distribution $p _ { \theta } = \delta _ { \theta }$ . + +The Dirac GAN with a gradient penalty with an arbitrary probability measure is known to be globally convergent(Mescheder et al., 2018). We argue that this result can be generalized to a finite penalty measure case. + +Lemma 1. Consider the Dirac GAN problem with SGP form $\begin{array} { r } { ( D ( x ; \psi ) = \psi x , \delta _ { 0 } , \delta _ { \theta } , \mu _ { \psi , \theta } ) } \end{array}$ . Suppose that some small $\eta > 0$ exists such that its finite penalty measure $\mu _ { \psi , \theta }$ with mass $M ( \psi , \theta ) =$ $\mathbf { \bar { \rho } } _ { \int 1 d \mu _ { \psi , \theta } } \geq 0$ satisfies either + +$$ +M ( \psi , \theta ) > 0 f o r \left( \psi , \theta \right) \in B _ { \eta } ( ( 0 , 0 ) ) o +$$ + +$$ +^ { \prime } \psi \nabla _ { \psi } M ( \psi , \theta ) \geq 0 f o r ( \psi , \theta ) \in B _ { \eta } ( ( 0 , 0 ) ) . +$$ + +Then, the SGP $\mu$ -WGAN optimization dynamics with $\begin{array} { r } { ( D ( x ; \psi ) = \psi x , \delta _ { 0 } , \delta _ { \theta } , \mu _ { \psi , \theta } ) } \end{array}$ are locally stable at the origin and the basin of attraction $B = B _ { R } ( ( 0 , 0 ) )$ is open ball with radius $R$ . Its radius is given as follows. + +Motivated by this example, we can extend this idea to the other toy example given by Nagarajan & Kolter (2017), where WGAN fails to converge to the equilibrium points $( \psi , \theta ) = ( 0 , \pm 1 )$ . + +Lemma 2. Consider the toy example $\begin{array} { r c l } { ( D ( x ; \psi ) } & { = } & { \psi x ^ { 2 } , U ( - 1 , 1 ) , U ( - | \theta | , | \theta | ) , \mu _ { \theta } ) } \end{array}$ where $U ( 0 , 0 ) = \delta _ { 0 }$ and the ideal equilibrium points are given by $( \psi ^ { * } , \theta ^ { * } ) = ( 0 , \pm 1 )$ . For a finite measure $\mu = \mu _ { \theta }$ on $\mathbb { R }$ which is independent of $\psi$ , suppose that $\mu _ { \boldsymbol { \theta } } \to \mu ^ { * }$ with $\mu ^ { * } \neq C \delta _ { 0 }$ for $C \geq 0$ . The dynamic system is locally stable near the desired equilibrium $( 0 , \pm 1 )$ , where the spectrum of the + +Jacobian at $( 0 , \pm 1 )$ is given by $\begin{array} { r } { \lambda = - 2 \rho \mathbb { E } _ { \mu ^ { * } } [ x ^ { 2 } ] \pm \sqrt { 4 \rho ^ { 2 } \mathbb { E } _ { \mu ^ { * } } [ x ^ { 2 } ] ^ { 2 } - \frac { 4 } { 9 } } } \end{array}$ + +# 4 MAIN CONVERGENCE THEOREM + +We propose the convergence property of WGAN with a simple gradient penalty on an arbitrary penalty measure $\mu$ for a realizable case: $\theta = \theta ^ { * }$ with $p _ { d } = p _ { \theta ^ { \ast } }$ exists. In subsection 4.1, we provide the necessary assumptions, which comprise our main convergence theorem. In subsection 4.2, we give the main convergence theorem with a sketch of the proof. A more rigorous analysis is given in the Appendix. + +# 4.1 ASSUMPTIONS + +The first assumption is made regarding the equilibrium condition for GANs, where we state the ideal conditions for the discriminator parameter and generator parameter. As the parameters converge to the ideal equilibrium, the sample distribution $\left( p _ { \theta } \right)$ converges to the real data distribution $\left( p _ { d } \right)$ and the discriminator cannot distinguish the generated sample and the real data. + +Assumption 1. $p _ { \theta } p _ { d }$ as $\theta ~ \to ~ \theta ^ { * }$ and $D ( x ; \psi ^ { * } ) ~ = ~ 0$ on supp $\left( p _ { d } \right)$ and its small open neighborhood, i.e., $x \in \cup _ { x ^ { \prime } \in s u p p ( p _ { d } ) } B _ { \epsilon _ { x ^ { \prime } } } ( x ^ { \prime } )$ implies $D ( x ; \psi ^ { * } ) = 0$ . For simplicity, we denote $\cup _ { x ^ { \prime } \in s u p p ( p _ { d } ) } B _ { \epsilon _ { x ^ { \prime } } } ( x ^ { \prime } )$ as $B ( s u p p ( p _ { d } ) )$ . + +The second assumption ensures that the higher order terms cannot affect the stability of the SGP $\mu$ -WGAN. In the Appendix, we consider the case where the WGAN fails to converge when Assumption 2 is not satisfied. Compared with the previous study by Nagarajan & Kolter (2017), the conditions for the discriminator parameter are slightly modified. + +# Assumption 2. + +$$ +\begin{array} { r } { g ( \theta ) = \| \mathbb { E } _ { p _ { d } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } ) ] - \mathbb { E } _ { p _ { \theta } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } ) ] \| ^ { 2 } , h ( \psi ) = \mathbb { E } _ { \mu _ { \psi , \theta ^ { * } } } [ \| \nabla _ { x } D ( x ; \psi ) \| ^ { 2 } ] } \end{array} +$$ + +are locally constant along the nullspace of the Hessian matrix. + +The third assumption allows us to extend our results to discrete probability distribution cases, as described by Mescheder et al. (2018). + +Assumption 3. $\exists \epsilon _ { g } > 0$ such that $D ( x ; \psi ^ { * } ) = 0$ on $\cup _ { | \theta - \theta ^ { * } | < \epsilon _ { g } } s u p p ( p _ { \theta } ) .$ . + +The fourth assumption indicates that there are no other “bad” equilibrium points near $( \psi ^ { * } , \theta ^ { * } )$ , which justifies the projection along the axis perpendicular to the null space. + +Assumption 4. A bad equilibrium does not exist near the desired equilibrium point. Thus, $( \psi ^ { * } , \theta ^ { * } )$ is an isolated equilibrium or there exist $\delta _ { d } , \delta _ { g } > 0$ such that all equilibrium points in $B _ { \delta _ { d } } ( \psi ^ { * } ) \times$ $B _ { \delta _ { g } } ( \theta ^ { * } )$ satisfy the other assumptions. + +The last assumption is related to the necessary conditions for the penalty measure. A calculation of the gradient penalty based on samples from the data manifold and generator manifold or the interpolation of both was introduced in recent studies (Gulrajani et al., 2017; Roth et al., 2017; Mescheder et al., 2018). First, we propose strong conditions for the penalty measure. + +Assumption 5. The finite penalty measure $\mu = \mu _ { \theta }$ satisfies the followings: + +a $\mu _ { \theta } \to \mu _ { \theta ^ { * } } = \mu ^ { * }$ and $\mu _ { \theta }$ is independent of the discriminator parameter $\psi$ . $\begin{array} { r l } & { b \ s u p p ( p _ { d } ) \subset s u p p ( \mu ^ { * } ) } \\ & { c \ \exists \epsilon _ { \mu } > 0 \ s u c h \ t h a t \ s u p p ( \mu _ { \theta } ) \subset B ( s u p p ( p _ { d } ) ) f o r \vert \theta - \theta ^ { * } \vert < \epsilon _ { \mu } . } \end{array}$ + +The assumption given above means that the support of the penalty measure $\mu _ { \theta }$ should approach the data manifolds smoothly as $\theta \to \theta ^ { * }$ . However, the penalty measure from WGAN-GP with a simple gradient penalty still reaches equilibrium without satisfying Assumption 5c. Therefore, we suggest Assumption 6, which is a weak version of Assumption 5. Assumption $6 \mathrm { a } ^ { 2 }$ is technically required to take the derivative of the integral $\mathbb { E } _ { \mu _ { \psi , \theta } } [ \| \nabla _ { x } D ( x ; \psi ) \| ^ { 2 } ]$ with respect to $\psi$ . + +Assumption 6. (Weak version of Assumption 5) The finite penalty measure $\mu = \mu _ { \psi , \theta }$ satisfies the following. + +a $\mu _ { \psi , \theta } \mu _ { \psi ^ { * } , \theta ^ { * } } = \mu ^ { * }$ , where supp $( \mu _ { \psi , \theta } )$ only depends on $\theta$ . Near the equilibrium, $\mu _ { \psi , \theta }$ can be weakly differentiated twice with respect to $\psi$ . In addition, its mass $M ( \psi , \theta ) \stackrel { \cdot } { = }$ $\int 1 d \mu _ { \psi , \theta }$ is a twice-differentiable function of $\psi$ and bounded near the equilibrium. + +b $E _ { \mu ^ { * } } [ \nabla _ { \psi x } D \nabla _ { \psi x } ^ { T } D ]$ is positive definite or $s u p p ( p _ { d } ) \subset s u p p ( \mu ^ { * } ) .$ + +c $\ : \exists \epsilon _ { \mu } > 0 \ :$ such that $\operatorname { s u p p } ( \mu _ { \theta } ) \subset V$ for $| \theta - \theta ^ { * } | < \epsilon _ { \mu }$ , where $V = \{ x | \nabla _ { x } D ( x ; \psi ^ { * } ) = 0 \}$ . + +The assumption above implies the following situations; The penalty measure’s support approaches to data manifold and its weight changes smoothly with respect to $\psi$ and $\theta$ . At the equilibrium, penalty measure’s support contains data manifold. Also, ideal discriminator will remain flat on the penalty area. + +In summary, the gradient penalty regularization term with any penalty measure where the support approaches $B ( s u p p ( p _ { d } ) )$ in a smooth manner works well and this main result can explain the regularization effect of previously proposed penalty measures such as $\mu _ { G P } , p _ { d } , p _ { \ell }$ , and their mixtures. + +# 4.2 MAIN CONVERGENCE THEOREM + +According to the modified assumptions given above, we prove that the related dynamic system is locally stable near the equilibrium. The tools used for analyzing stability are mainly based on those described by Nagarajan & Kolter (2017). Our main contributions comprise proposing the necessary conditions for the penalty measure and proving the local stability for all penalty measures that satisfy Assumption 6. + +Theorem 1. Suppose that our SGP $\mu$ -WGAN optimization problem $( D , p _ { d } , p _ { \theta } , \mu )$ with equilibrium point $( \psi ^ { * } , \theta ^ { * } )$ satisfies the assumptions given above. Then, the related dynamic system is locally stable at the equilibrium. + +A detailed proof of the main convergence theorem is given in the Appendix. A sketch of the proof is given in three steps. First, the undesired terms in the Jacobian matrix of the system at the equilibrium are cancelled out. Next, the Jacobian matrix at equilibrium is given by $\left[ \begin{array} { c c } { \therefore } & { \mathbf { 0 } } \\ { R ^ { T } } & { 0 } \end{array} \right]$ , where $Q =$ $\mathbb { E } _ { \mu ^ { * } } [ \nabla _ { \psi x } D \nabla _ { \psi x } ^ { T } D ]$ and $R = \nabla _ { \theta } \mathbb { E } _ { p _ { \theta } } [ \nabla _ { \psi } D ] | _ { \theta = \theta ^ { * } }$ . The system is locally stable when both $Q$ and $R ^ { T } R$ are positive definite. We can complete the proof by dealing with zero eigenvalues by showing that ${ \cal N } ( Q ^ { \hat { T } } ) \subset { \cal N } ( R ^ { T } )$ and the projected system’s stability implies the original system’s stability. + +Our analysis mainly focuses on WGAN, which is the simplest case of general GAN minimax optimization + +$$ +\begin{array} { r l } & { \underset { \psi } { \operatorname* { m a x } } : \mathbb { E } _ { p _ { d } } [ f ( D ( x ; \psi ) ) ] + \mathbb { E } _ { p _ { \theta } } [ f ( - D ( x ; \psi ) ) ] - \frac { \rho } { 2 } \mathbb { E } _ { \mu } [ \| \nabla _ { x } D ( x ; \psi ) \| ^ { 2 } ] } \\ & { \underset { \theta } { \operatorname* { m i n } } : \mathbb { E } _ { p _ { d } } [ f ( D ( x ; \psi ) ) ] + \mathbb { E } _ { p _ { \theta } } [ f ( - D ( x ; \psi ) ) ] } \end{array} +$$ + +with $f ( x ) = x$ . Similar approach is still valid for general GANs with concave function $f$ with $f ^ { \prime \prime } ( x ) < 0$ and $f ^ { \prime } ( 0 ) \neq 0$ . + +# 5 EXPERIMENTAL RESULTS + +We claim that every penalty measure that satisfies the assumptions can regularize the WGAN and generate similar results to the recently proposed gradient penalty methods. Several penalty measures were tested based on two-dimensional problems (mixture of 8 Gaussians, mixture of 25 Gaussians, and swissroll), MNIST and CIFAR-10 datasets using a simple gradient penalty term. In the comparisons with WGAN, the recently proposed penalty measures and our test penalty measures used the same network settings and hyperparameters. The penalty measures and its detailed sampling methods are listed in Table 1, where $x _ { d } \sim p _ { d } , x _ { g } \sim p _ { \theta }$ , and $\alpha \sim U ( 0 , 1 )$ . $\mathcal { A }$ indicates fixed anchor point in $\mathcal { X }$ . + +Table 1: List of benchmark WGANs (WGAN and WGAN-GP with non-zero centered gradient penalty) and 5 penalty measures with a simple gradient penalty term. In this table, WGAN-GP represents the previous model proposed by (Gulrajani et al., 2017), which penalizes the WGAN with non-zero centered gradient penalty terms, whereas $\mu _ { G P }$ represents the simple method. In our experiment, no additional weights are applied on 5 penalty measures and they are all probability distributions. + +
PenaltyPenalty termPenalty measure, sampling method
WGAN WGAN-GPNone(Weight Clipping) Eμ[(IVxD|-1)2]None x=axd+(1-α)xg
Eμ[VxD|2]
Pg PdEμ[VxDii2]x=xg x=xd
μGPEμVDii2]x=axd+(1-α)xg
μmidEμ[VxDii2]x= 0.5xd+0.5xg
μg,ancEμVDii2]x=αA+(1-α)xg
+ +By setting the previously proposed WGAN with weight-clipping(Arjovsky et al., 2017) and WGANGP(Gulrajani et al., 2017) as the baseline models, $\operatorname { S G P } \mu$ -WGAN was examined with various penalty measures comprising three recently proposed measures and two artificially generated measures. $p _ { \theta }$ and $p _ { d }$ were suggested by Mescheder et al. (2018) and $\mu _ { G P }$ was introduced from the WGAN-GP. We analyzed the artificial penalty measures $\mu _ { m i d }$ and $\mu _ { g , a n c }$ as the test penalty measures. + +The experiments were conducted based on the implementation of the Gulrajani et al. (2017). The hyperparameters, generator/discriminator structures, and related TensorFlow implementations can be found at https://github.com/igul222/improved_wgan_training (Gulrajani et al., 2017). Only the loss function was modified slightly from a non-zero centered gradient penalty to a simple penalty. For the CIFAR-10 image generation tasks, the inception score(Salimans et al., 2016) and FID(Heusel et al., 2017) were used as benchmark scores to evaluate the generated images. + +# 5.1 2D EXAMPLES AND MNIST + +We checked the convergence of $p _ { \theta }$ for the 2D examples (8 Gaussians, swissroll data, and 25 Gaussians) and MNIST digit generation for the SGP-WGANs with five penalty measures. MNIST and 25 Gaussians were trained over 200K iterations, the 8 Gaussians were trained for 30K iterations, and the Swiss Roll data were trained for 100K iterations. The anchor $\mathcal { A }$ for $\mu _ { g , a n c }$ was set as $( 2 , - 1 )$ for the 2D examples and 784 gray pixels for MNIST. We only present the results obtained for the MNIST dataset with the penalty measures comprising $\mu _ { m i d }$ and $\mu _ { g , a n c }$ in Figure 1. The others are presented in the Appendix. + +![](images/07f5e9693ee7d3a76d2c35b0604f32d9f8eef9b1b58131c18cc2525916acb8b1.jpg) +Figure 1: MNIST example. Images generated with $\mu _ { m i d }$ (left) and $\mu _ { g , a n c } ( \mathrm { r i g h t } )$ . + +# 5.2 CIFAR-10 + +DCGAN and ResNet architectures were tested on the CIFAR-10 dataset. The generators were trained for 200K iterations. The anchor $\mathcal { A }$ for $\mu _ { g , a n c }$ during CIFAR-10 generation was set as fixed random pixels. The WGAN, WGAN-GP, and five penalty measures were evaluated based on the inception score and FID, as shown in Table 2, which are useful tools for scoring the quality of generated images. The images generated from $\mu _ { m i d }$ and $\mu _ { g , a n c }$ with ResNet are shown in Figure 2. The others are presented in the Appendix. + +Table 2: Benchmark score results obtained based on the CIFAR-10 dataset under DCGAN and ResNet architectures. The higher inception score and lower FID indicate the good quality of the generated images. + +
PenaltyDCGAN Inception FIDResNet InceptionFID
WGAN 35.64 ± 0.09 48.7==
WGAN-GP6.48 ± 0.10 35.07.82 ± 0.0918.1
Pg6.46 ± 0.09 38.07.63 ± 0.1020.9
pd6.33 ± 0.0738.9 7.63 ± 0.0920.3
μGP6.40 ±0.0835.4 7.60 ± 0.0918.3
μmid6.60 ± 0.0733.9 7.86 ± 0.0716.4
μg,anc6.45 ± 0.0733.7 7.36 ± 0.0922.4
+ +![](images/af56d7fca60411caad1f41e50993c7c1651b0890d3dce2a1e967749ed2d6734d.jpg) +Figure 2: CIFAR-10 example. Images generated with $\mu _ { m i d }$ (left) and $\mu _ { g , a n c }$ (right) under the ResNet architecture. + +# 6 CONCLUSION + +In this study, we proved the local stability of simple gradient penalty $\mu$ -WGAN optimization for a general class of finite measure $\mu$ . This proof provides insight into the success of regularization with previously proposed penalty measures. We explored previously proposed analyses based on various gradient penalty methods. Furthermore, our theoretical approach was supported by experiments using unintuitive penalty measures. In future research, our works can be extended to alternative gradient descent algorithm and its related optimal hyperparameters. Stability at non-realizable equilibrium points is one of the important topics on stability of GANs. Optimal penalty measure for achieving the best convergence speed can be also investigated using a spectral theory, which provides the mathematical analysis on stability of GAN with a precise information on the convergence theory. + +# REFERENCES + +Mart´ın Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial ´ networks. In International Conference on Learning Representations, 2017. + +Mart´ın Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks. ´ In Proceedings of the 34th International Conference on Machine Learning, pp. 214–223, 2017. + +Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron C. Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems, pp. 2672–2680, 2014. + +Ishaan Gulrajani, Faruk Ahmed, Mart´ın Arjovsky, Vincent Dumoulin, and Aaron C. Courville. Improved training of wasserstein gans. In Advances in Neural Information Processing Systems, pp. 5769–5779, 2017. + +B. Heidergott and F. J. Vazquez-Abad. Measure-valued differentiation for markov chains. ´ Journal of Optimization Theory and Applications, 136:187–209, 2008. ISSN 1573-2878. doi: 10.1007/ s10957-007-9297-7. + +Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in Neural Information Processing Systems, pp. 6629–6640, 2017. + +Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A. Efros. Image-to-image translation with conditional adversarial networks. In 2017 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 5967–5976, 2017. + +Christian Ledig, Lucas Theis, Ferenc Huszar, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew P. Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, and Wenzhe Shi. Photorealistic single image super-resolution using a generative adversarial network. In 2017 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017, Honolulu, HI, USA, July 21-26, 2017, pp. 105–114, 2017. + +Lars M. Mescheder, Andreas Geiger, and Sebastian Nowozin. Which training methods for gans do actually converge? In Proceedings of the 35th International Conference on Machine Learning, pp. 3478–3487, 2018. + +Youssef Mroueh, Chun-Liang Li, Tom Sercu, Anant Raj, and Yu Cheng. Sobolev GAN. In International Conference on Learning Representations, 2018. + +Vaishnavh Nagarajan and J. Zico Kolter. Gradient descent GAN optimization is locally stable. In Advances in Neural Information Processing Systems, pp. 5591–5600, 2017. + +Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems, pp. 271–279, 2016. + +Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. CoRR, abs/1511.06434, 2015. URL http:// arxiv.org/abs/1511.06434. + +Scott E. Reed, Zeynep Akata, Xinchen Yan, Lajanugen Logeswaran, Bernt Schiele, and Honglak Lee. Generative adversarial text to image synthesis. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pp. 1060– 1069, 2016. URL http://jmlr.org/proceedings/papers/v48/reed16.html. + +Kevin Roth, Aurelien Lucchi, Sebastian Nowozin, and Thomas Hofmann. Stabilizing training of ´ generative adversarial networks through regularization. In Advances in Neural Information Processing Systems, pp. 2015–2025, 2017. + +Tim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pp. 2226–2234, 2016. + +Casper Kaae Sønderby, Jose Caballero, Lucas Theis, Wenzhe Shi, and Ferenc Huszar. Amortised ´ MAP inference for image super-resolution. International Conference on Learning Representations, 2017. + +Han Zhang, Tao Xu, and Hongsheng Li. Stackgan: Text to photo-realistic image synthesis with stacked generative adversarial networks. In IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017, pp. 5908–5916, 2017. + +# APPENDIX A : PROOF OF LEMMAS BASED ON TOY EXAMPLES + +Proof of Lemma 1. The related dynamic system of $\begin{array} { r } { ( D ( x ; \psi ) = \psi x , \delta _ { 0 } , \delta _ { \theta } , \mu _ { \psi , \theta } ) } \end{array}$ can be written as follows. + +$$ +\begin{array} { l } { \displaystyle \dot { \psi } = - \theta - \frac { \rho } { 2 } \nabla \psi \mathbb { E } _ { \mu _ { \psi , \theta } } [ \psi ^ { 2 } ] } \\ { \displaystyle \dot { \theta } = \psi } \end{array} +$$ + +First, the only equilibrium point is given by $( \psi ^ { * } , \theta ^ { * } ) = ( 0 , 0 )$ from + +$$ +\begin{array} { l } { 0 = - \theta - 2 \psi M ( \psi , \theta ) - \psi ^ { 2 } \nabla _ { \psi } M ( \psi , \theta ) } \\ { \quad 0 = \psi } \end{array} +$$ + +The corresponding Jacobian matrix for the dynamic system is written as: + +$$ +J = \left[ { \begin{array} { r r } { Z } & { - 1 } \\ { 1 } & { 0 } \end{array} } \right] +$$ + +where + +$$ +Z = - \frac { \rho } { 2 } \nabla _ { \psi \psi } \mathbb { E } _ { \mu _ { \psi , \theta } } [ \psi ^ { 2 } ] \bigg | _ { \psi = 0 , \theta = 0 } +$$ + +$\nabla _ { \psi } D ( x ; \psi ) = \psi$ does not depend on $x$ , so this can be rewritten as: + +$$ +Z = - \frac { \rho } { 2 } \nabla _ { \psi \psi } ( \psi ^ { 2 } \mathbb { E } _ { \mu _ { \psi } , \theta } [ 1 ] ) = - \frac { \rho } { 2 } ( 2 M ( \psi , \theta ) + 4 \psi \nabla _ { \psi } M ( \psi , \theta ) + \psi ^ { 2 } M _ { \psi \psi } ( \psi , \theta ) ) \bigg | _ { \psi = 0 , \theta = 0 } +$$ + +Therefore, if $M ( 0 , 0 ) > 0$ , then the given system is locally stable because the eigenvalues of its linearized system have negative real parts. If $M ( 0 , 0 ) = 0$ , then the stability of the system cannot be proved by the linearization theorem. In this case, we consider the following Lyapunov function. + +$$ +L ( \psi ( t ) , \theta ( t ) ) = \psi ( t ) ^ { 2 } + \theta ( t ) ^ { 2 } +$$ + +By differentiating with $t$ , we obtain + +$$ +\begin{array} { r l } & { \dot { L } = 2 ( \psi \psi ^ { \prime } + \theta \theta ^ { \prime } ) = - \rho \psi \nabla _ { \psi } ( \psi ^ { 2 } M ( \psi , \theta ) ) = - \rho \psi ( 2 \psi M ( \psi , \theta ) + \psi ^ { 2 } \nabla _ { \psi } M ( \psi , \theta ) ) } \\ & { \quad = - \rho \psi ^ { 2 } ( 2 M ( \psi , \theta ) + \psi \nabla _ { \psi } M ( \psi , \theta ) ) \leq 0 } \end{array} +$$ + +Clearly, $L ( \psi , \theta ) \geq 0$ and the equality holds iff $\psi = \theta = 0$ . In addition, $\dot { L } \leq 0$ since $M ( \psi , \theta ) \geq$ 0 and $\psi \nabla _ { \psi } M ( \psi , \theta ) \geq 0$ from the assumption. Furthermore, it is clear that if $( \psi ( 0 ) , \theta ( 0 ) ) \ \in \qquad $ $B _ { \eta } ( ( 0 , 0 ) )$ , then $( \psi ( \tau ) , \theta ( \tau ) ) \in B _ { \eta } ( ( 0 , 0 ) )$ for all $\tau \geq 0$ because the Lyapunov function (square of the distance between the origin and $( \psi ( \tau ) , \theta ( \tau ) ) )$ always decreases as $\tau \infty$ . Therefore, the given system is stable according to the Lyapunov stability theorem. + +Again, we can check that if $\mu _ { \psi , \theta }$ is a probability measure, then the system is globally stable, as shown by Mescheder et al. (2018). The basin of attraction is given by the whole $\mathbb { R } ^ { 2 }$ plane since $M ( \psi , \theta ) = 1$ , so $\dot { L } = - \rho \psi ^ { 2 } ( 2 M + \psi \nabla _ { \psi } M ) = - 2 \rho \psi ^ { 2 } \leq 0$ for every $( \psi , \theta ) \in \mathbb { R } ^ { 2 }$ . □ + +Proof of Lemma 2. From the general setup of the $\operatorname { S G P } \mu$ -WGAN optimization problem, the dynamic system corresponding to the simple-GAN in Definition 6 can be written as follows. + +$$ +\begin{array} { l } { { \displaystyle { \dot { \psi } = \frac { 1 } { 3 } - \frac { \theta ^ { 2 } } { 3 } - 4 \rho \psi \mathbb { E } _ { \mu } [ x ^ { 2 } ] } } } \\ { { \displaystyle { \dot { \theta } = \frac { 2 \psi \theta } { 3 } } } } \end{array} +$$ + +If we let $\mathbb { E } _ { \mu ^ { * } } [ x ^ { 2 } ] = A ^ { 2 }$ , then the Jacobian matrix at the equilibrium $( 0 , \pm 1 )$ is given by ${ \boldsymbol { J } } =$ $\left[ { \begin{array} { c c } { - 4 \rho A ^ { 2 } } & { { \mp } { \frac { 2 } { 3 } } } \\ { \pm { \frac { 2 } { 3 } } } & { 0 } \end{array} } \right]$ Therefore, the given system is locally stable when $A \neq 0$ . + +# APPENDIX B : PROOF OF LEMMA RELATED WITH ASSUMPTION 2 + +Lemma 3. Consider the Dirac-GAN setup and SGP $\mu$ -WGAN optimization system with a slightly changed discriminator function $D _ { 2 } ( x ; \psi ) \stackrel { \textstyle - } { = } \psi x ^ { 2 }$ . The system $( D _ { 2 } , \delta _ { 0 } , \delta _ { \theta } , \mu _ { G P } )$ does not converge to $( 0 , 0 )$ but for any point $( a , 0 )$ with $a < 0$ , the system has equilibrium points on the whole $\psi$ -axis and it violates Assumption 2. + +Proof of Lemma 3. For the SGP $\mu$ -WGAN optimization problem $( D _ { 2 } , \delta _ { 0 } , \delta _ { \theta } , \mu _ { G P } )$ , the dynamic system can be written as follows. + +$$ +\begin{array} { l } { { \dot { \psi } = - \theta ^ { 2 } - \frac { 4 } { 3 } \rho \psi \theta ^ { 2 } } } \\ { { \dot { \theta } = 2 \psi \theta } } \end{array} +$$ + +$2 \psi \theta = 0$ and $\begin{array} { r } { \theta ^ { 2 } ( 1 + \frac { 4 } { 3 } \rho \psi ) = 0 } \end{array}$ implies that $\theta = 0$ , so the $\psi$ -axis is the set of all equilibrium points. By drawing the nullclines $\psi = 0$ and $\begin{array} { r } { \psi = - \frac { 3 } { 4 \rho } } \end{array}$ in the $\psi \theta$ -plane, it is clear that no solution curve converges to $( b , 0 )$ with $b \geq 0$ , as shown in Figure 3. □ + +![](images/c43c5cb4133543240663135cefcb52ec5abc17c3330a731f348c08a6b7589f1c.jpg) +Figure 3: Phase portrait of the SGP $\mu$ -WGAN optimization problem $( D _ { 2 } , \delta _ { 0 } , \delta _ { \theta } , \mu _ { G P } )$ with $\textstyle \rho = { \frac { 3 } { 8 } }$ . Along the line $\theta = 0$ , the system is stable so no updating will occur. Every solution curve that passes the nullcline $\psi = 0$ has $\dot { \theta } = 0$ . For the nullcline $\begin{array} { r } { \psi = - \frac { 3 } { 4 \rho } = - 2 } \end{array}$ , no updating on $\psi$ will occur and only $\theta$ will be updated. Given that the solution curves do not intersect with each other, every solution curve is exactly one of the followinstays in area A. (2) Solution curve converges to $\begin{array} { r } { \bar { ( \psi , \theta ) } \overset { * } { = } ( - \frac { 3 } { 4 \rho } , 0 ) } \end{array}$ trivial cases; (1) Solalong the nullcline $\begin{array} { r } { \psi = - \frac { 3 } { 4 \rho } } \end{array}$ e. (3) Solution curve stays in area B. (4) Solution curve starts from area C, crosses the nullcline $\psi = 0$ perpendicularly, and converges to $( b , 0 )$ with $b < 0$ . Therefore, no solution curve converges to $( 0 , 0 )$ . + +# APPENDIX C : PROOF OF THE MAIN CONVERGENCE THEOREM + +Proof. Let us consider the Jacobian matrix $J = \left[ \begin{array} { l l } { K _ { D D } } & { K _ { D G } } \\ { K _ { G D } } & { K _ { G G } } \end{array} \right]$ at the first equilibrium $( \psi ^ { * } , \theta ^ { * } ) ^ { 4 }$ + +$$ +\begin{array} { r l } { J = \left[ \mathbb { E } _ { p _ { d } } [ \nabla _ { \psi \psi } D ] - \mathbb { E } _ { p _ { \theta ^ { \star } } } [ \nabla _ { \psi \psi } D ] - \frac { \rho } { 2 } \nabla _ { \psi \psi } \mathbb { E } _ { \mu } [ \| \nabla _ { x } D \| ^ { 2 } ] } & { - \nabla _ { \theta \psi } \mathbb { E } _ { p _ { \theta } } [ D ] - \frac { \rho } { 2 } \nabla _ { \theta \psi } \mathbb { E } _ { \mu } [ \| \nabla _ { x } D \| ^ { 2 } ] \right] } \\ & { \qquad \nabla _ { \psi \theta } \mathbb { E } _ { p _ { \theta } } [ D ] ^ { 2 } } \end{array} +$$ + +First, Assumption 1 implies that $\mathbb { E } _ { p _ { d } } [ \nabla _ { \psi \psi } D ] - \mathbb { E } _ { p _ { \theta ^ { * } } } [ \nabla _ { \psi \psi } D ] = 0$ since $p _ { \theta } p _ { d }$ as $\theta \to \theta ^ { * }$ . From Assumption 3, $\mathbb { E } _ { p _ { \theta } } [ D ( x ; \psi ^ { * } ) ]$ is locally zero near the equilibrium $\theta ^ { * }$ , which implies that + +$$ +K _ { G G } = \nabla _ { \theta \theta } \mathbb { E } _ { p _ { \theta } } [ D ( x ; \psi ^ { * } ) ] \bigg \rvert _ { \theta = \theta ^ { * } } = 0 +$$ + +We still need to evaluate $\nabla _ { \psi \psi } \mathbb { E } _ { \mu } [ \| \nabla _ { x } D \| ^ { 2 } ]$ and $\nabla _ { \theta \psi } \mathbb { E } _ { \mu } [ \| \nabla _ { x } D \| ^ { 2 } ]$ . According to Assumption 6a, finite signed measures $\mu _ { \psi , \theta } ^ { \prime }$ and $\mu _ { \psi , \theta } ^ { \prime \prime }$ exist5, so they are the first and second weak derivatives of $\mu _ { \psi , \theta }$ with respect to the parameter $\psi$ at $\left( \psi ^ { * } , \theta ^ { * } \right)$ . Therefore, the expectations given above can be rewritten as below. + +$$ +\begin{array} { l } { I = \nabla _ { \psi \psi } \displaystyle \int _ { s u p p ( \mu _ { \psi } , \theta ) } \| \nabla _ { x } D \| ^ { 2 } d \mu _ { \psi , \theta } } \\ { = \displaystyle \int _ { s u p p ( \mu _ { \psi , \theta } ) } ( 2 \nabla _ { \psi x } ^ { T } D \nabla _ { \psi x } D + 2 K _ { 0 } ) d \mu _ { \psi , \theta } + \displaystyle \int _ { s u p p ( \mu _ { \psi , \theta } ) } 2 ( \nabla _ { \psi x } ^ { T } D \nabla _ { x } D ) d \mu _ { \psi , \theta } ^ { \prime } + \displaystyle \int _ { s u p p ( \mu _ { \psi , \theta } ) } \| \nabla _ { x } \nabla _ { \psi } D \nabla _ { x } \| ^ { 2 } d \mu _ { \psi , \theta } ^ { \prime } } \\ { \cdot I = \nabla _ { \theta \psi } \displaystyle \int _ { s u p p ( \mu _ { \psi , \theta } ) } \| \nabla _ { x } D \| ^ { 2 } d \mu _ { \psi , \theta } } \\ { = \nabla _ { \theta } ( \displaystyle \int _ { s u p p ( \mu _ { \psi , \theta } ) } 2 ( \nabla _ { \psi x } ^ { T } D \nabla _ { x } D ) d \mu _ { \psi , \theta } + \displaystyle \int _ { s u p p ( \mu _ { \psi , \theta } ) } \| \nabla _ { x } D \| ^ { 2 } d \mu _ { \psi , \theta } ^ { \prime } ) } \end{array} +$$ + +where + +$$ +\begin{array} { r } { K _ { 0 } ( x ; \psi ) = \left[ \sum _ { k } \frac { \partial ^ { 3 } } { \partial \psi _ { i } \partial \psi _ { j } \partial x _ { k } } D ( x ; \psi ) \frac { \partial } { \partial x _ { k } } D ( x ; \psi ) \right] _ { i j } } \end{array} +$$ + +From Assumption 6c and the fact that the weak derivative of $\mu _ { \psi , \theta }$ vanishes outside of $s u p p ( \mu _ { \psi , \theta } )$ , $\nabla _ { x } D ( x ; \psi ^ { * } ) = 0$ on $s u p p ( \mu _ { \psi , \theta } ) \subset V$ for all $\theta$ with $\left| \theta - \theta ^ { * } \right| < \epsilon _ { \mu }$ and $\mu _ { \psi , \theta } ^ { \prime } = \mu _ { \psi , \theta } ^ { \prime \prime } = 0$ on the outside of $s u p p ( \mu _ { \psi , \theta } )$ , which leads to the desired results: + +$$ +\begin{array} { c } { \displaystyle { I = \int _ { s u p p ( \mu ^ { * } ) } 2 ( \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } ) \nabla _ { \psi x } D ( x ; \psi ^ { * } ) ) d \mu ^ { * } } } \\ { \displaystyle { I I = 0 } } \end{array} +$$ + +After cancelling the undesired terms, the Jacobian matrix at the equilibrium $( \psi ^ { * } , \theta ^ { * } )$ is given as: + +$$ +J = \left[ \begin{array} { c c } { - \rho Q } & { - R } \\ { R ^ { T } } & { 0 } \end{array} \right] +$$ + +where + +$$ +\begin{array} { r } { Q = \mathbb { E } _ { \mu ^ { * } } [ \nabla _ { \psi x } ^ { T } D \nabla _ { \psi x } D ] } \\ { \boldsymbol { R } = \nabla _ { \theta } \mathbb { E } _ { p _ { \theta } } [ \nabla _ { \psi } D ] \bigg | _ { \theta = \theta ^ { * } } } \end{array} +$$ + +From the definition of $Q$ , it is easy to check that $Q$ is at least positive semi-definite. It is known that for a negative definite matrix $A$ and full column rank matrix $B$ , the block matrix $\left[ \begin{array} { l l } { A } & { B } \\ { - B ^ { T } } & { 0 } \end{array} \right]$ is Hurwitz, i.e., all eigenvalues of the matrix have a negative real part. Therefore, if $\bar { Q }$ is positive definite and $R$ is full column rank, the proof is complete. We consider the complementary case. + +Suppose that $Q$ or $R ^ { T } R$ have some zero eigenvalues. Let $Q = U _ { D } \Lambda _ { D } U _ { D } ^ { T }$ and $R ^ { T } R = U _ { G } \Lambda _ { G } U _ { G } ^ { T }$ with $U _ { D } ~ = ~ \lbrack T _ { D } ~ S _ { D } \rbrack$ and $U _ { G } ~ = ~ [ T _ { G } ^ { ~ - } ~ S _ { G } ]$ , where $T _ { D }$ and $T _ { G }$ are the eigenvectors of $\bar { Q }$ and $R ^ { T } R$ that correspond to non-zero eigenvalues. First, we assume that $T _ { D }$ and $T _ { G }$ are not empty. We can show that $( \psi ^ { * } + \xi v , \theta ^ { * } \overset { \cdot } { + } \nu w )$ is also an equilibrium point for a sufficiently small $\xi , \nu$ and $v \in N ( Q ) , w \in N ( R ^ { T } R )$ by using the techniques given by Nagarajan & Kolter (2017). If the system does not update at the equilibrium point $( \psi ^ { * } , \theta ^ { * } )$ and its small neighborhood $\left( \psi ^ { * } + \xi v , \theta ^ { * } + \nu w \right)$ is perturbed along $N ( Q )$ and $N ( \bar { R } ^ { T } R )$ , then it is reasonable to project the system orthogonal to $N ( Q )$ and $N ( R ^ { T } { \bar { R } } )$ . + +First, we assume that $v \in N ( Q )$ . By Assumption 2, $h ( \psi ^ { * } + \xi v ) = h ( \psi ^ { * } ) = 0$ for $| \xi | < \xi _ { d }$ , which implies that $\nabla _ { \boldsymbol { x } } D ( \boldsymbol { x } ; \psi ^ { * } + \xi \boldsymbol { v } ) = 0$ for $x \in s u p p ( \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } ) = s u p p ( \mu ^ { * } )$ and $| \xi | < \xi _ { d }$ . Thus, we obtain + +$$ +\mathbb { E } _ { \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } } [ \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } + \xi v ) \nabla _ { x } D ( x ; \psi ^ { * } + \xi v ) ] = 0 +$$ + +and + +$$ +\int _ { s u p p ( \mu ^ { * } ) } \left\| \nabla _ { x } D ( x ; \psi ^ { * } + \xi v ) \right\| ^ { 2 } d \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } ^ { \prime } = 0 +$$ + +By Assumption 4, $\begin{array} { r } { \mathbb { E } _ { p _ { d } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } + \xi v ) ] - \mathbb { E } _ { p _ { \theta ^ { * } } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } + \xi v ) ] = 0 } \end{array}$ since $p _ { d } = p _ { \theta ^ { \ast } }$ . By adding these equations, we obtain + +$$ +\begin{array} { l } { \displaystyle \dot { \psi } = \mathbb { E } _ { p _ { d } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } + \xi v ) ] - \mathbb { E } _ { p _ { \theta ^ { * } } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } + \xi v ) ] } \\ { \displaystyle - \frac { \rho } { 2 } \int _ { s u p p ( \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } ) } 2 \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } + \xi v ) \nabla _ { x } D ( x ; \psi ^ { * } + \xi v ) d \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } } \\ { \displaystyle - \frac { \rho } { 2 } \int _ { s u p p ( \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } ) } \| \nabla _ { x } D ( x ; \psi ^ { * } + \xi v ) \| ^ { 2 } d \mu _ { \psi ^ { * } + \xi v , \theta ^ { * } } ^ { \prime } } \\ { \displaystyle = 0 } \end{array} +$$ + +In addition, + +$$ +\begin{array} { l } { \displaystyle \dot { \theta } = \frac { \partial } { \partial \theta } \int _ { \mathcal X } D ( x ; \psi ^ { * } + \xi v ) d p _ { \theta } \bigg \vert _ { \theta = \theta ^ { * } } } \\ { \displaystyle = \int _ { \mathcal Z } \nabla _ { \theta } ^ { T } G ( z ; \theta ^ { * } ) \nabla _ { x } D ( G ( z ; \theta ^ { * } ) ; \psi ^ { * } + \xi v ) p _ { l a t e n t } ( z ) d z = 0 . } \end{array} +$$ + +Therefore, the point $( \psi ^ { * } + \xi v , \theta ^ { * } )$ with $| \xi | < \xi _ { d }$ is an equilibrium point. According to Assumption 4, $D ( x ; \psi ^ { * } + \xi v )$ is an equilibrium discriminator for $| \xi | < \delta _ { d }$ , and thus $D ( x ; \psi ^ { * } + \bar { \xi } v )$ is already an optimal discriminator for $| \xi | < \operatorname* { m i n } ( \xi _ { d } , \delta _ { d } )$ . + +Suppose that $w \ \in \ N ( R ^ { T } R )$ . By Assumption 2, $g ( \theta ^ { * } ) ~ = ~ g ( \theta ^ { * } + \nu w ) ~ = ~ 0$ for $| \nu | < \nu _ { g }$ , and thus $\begin{array} { r l } { \mathbb { E } _ { p _ { d } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } ) ] - \mathbb { E } _ { p _ { \theta ^ { * } + \nu w } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } ) ] ~ = ~ 0 } \end{array}$ for $| \nu | ~ < ~ \nu _ { g }$ . Furthermore, Assumption 3 gives $\mathbb { E } _ { p _ { \theta ^ { * } + \nu w } } [ D ( x ; \psi ^ { * } ) ] = 0$ for a sufficiently close $| \nu | < \epsilon _ { g }$ , which implies that $\begin{array} { r } { \dot { \theta } = \nabla _ { \theta } \mathbb { E } _ { p _ { \theta } } [ D ( x ; \psi ^ { * } ) ] \bigg \rvert _ { \theta = \theta ^ { * } + \nu w } = 0 } \end{array}$ for $| \nu | < \epsilon _ { g }$ . Finally, + +$$ +\int _ { s u p p ( \mu _ { \psi ^ { * } , \theta ^ { * } + s w } ) } 2 \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } ) \nabla _ { x } D ( x ; \psi ^ { * } ) d \mu _ { i \psi ^ { * } , \theta ^ { * } + \nu w } + \int _ { s u p p ( \mu _ { \psi ^ { * } , \theta ^ { * } + s w } ) } \| \nabla _ { x } D ( x ; \psi ^ { * } ) \| ^ { 2 } d \mu _ { i \psi ^ { * } , \theta ^ { * } + \nu w } ^ { \prime } +$$ + +since $s u p p ( \mu _ { \psi ^ { * } , \theta ^ { * } + \nu w } ) \subset V$ and $\nabla _ { x } D ( x ; \psi ^ { * } ) = 0$ on $V$ for a sufficiently small $| \nu | < \epsilon _ { \mu }$ (Assumption 6c). By adding these results, we obtain + +$$ +\begin{array} { l } { \displaystyle \dot { \psi } = \mathbb { E } _ { p _ { d } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } ) ] - \mathbb { E } _ { p _ { \theta ^ { * } + \nu w } } [ \nabla _ { \psi } D ( x ; \psi ^ { * } ) ] } \\ { \displaystyle \quad - \frac { \rho } { 2 } \int _ { s u p p ( \mu _ { \psi ^ { * } , \theta ^ { * } + \nu w } ) } 2 \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } ) \nabla _ { x } D ( x ; \psi ^ { * } ) d \mu _ { \psi ^ { * } , \theta ^ { * } + \nu w } } \\ { \displaystyle \quad - \frac { \rho } { 2 } \int _ { s u p p ( \mu _ { \psi ^ { * } , \theta ^ { * } + \nu w } ) } \left\| \nabla _ { x } D ( x ; \psi ^ { * } ) \right\| ^ { 2 } d \mu _ { \psi ^ { * } , \theta ^ { * } + \nu w } ^ { \prime } } \\ { \displaystyle = 0 } \end{array} +$$ + +Therefore, the point $( \psi ^ { * } , \theta ^ { * } + \nu w )$ with $| \nu | < \operatorname* { m i n } ( \epsilon _ { \mu } , \epsilon _ { g } , \nu _ { g } , \delta _ { g } )$ is an equilibrium point, which implies that $p _ { \theta ^ { * } + \nu w } = p _ { d }$ according to Assumption 4. + +If we consider the projected system $( \alpha , \beta ) \ : = \ : ( T _ { D } ^ { T } \psi , T _ { G } ^ { T } \theta )$ , then the projected dynamic system’s Jacobian at $( T _ { D } ^ { T } \psi ^ { * } , T _ { G } ^ { T } \theta ^ { * } )$ is given as follows. + +$$ +J ^ { \prime } = \left[ \begin{array} { c c } { { - \rho T _ { D } ^ { T } Q T _ { D } } } & { { - T _ { D } ^ { T } R T _ { G } } } \\ { { T _ { G } ^ { T } R ^ { T } T _ { D } } } & { { 0 } } \end{array} \right] = \left[ \begin{array} { c c } { { - \rho \Lambda _ { D } ^ { ( + ) } } } & { { - T _ { D } ^ { T } R T _ { G } } } \\ { { T _ { G } ^ { T } R ^ { T } T _ { D } } } & { { 0 } } \end{array} \right] +$$ + +Therefore, we only need to prove that $T _ { D } ^ { T } R T _ { G }$ is of full column rank. Suppose that $u \in N ( Q ^ { T } ) =$ $N ( Q )$ . According to Assumption 2, $h ( \psi )$ is locally constant at $\psi ^ { * }$ along the direction $u$ . Therefore, for a sufficiently small scalar $\xi$ with $| \xi | < \xi _ { u }$ , + +$$ +h ( \psi ^ { * } + \xi u ) = h ( \psi ^ { * } ) = 0 +$$ + +where the last equality comes from the Assumption 6. This implies that $\nabla _ { x } D ( x ; \psi ^ { * } + \xi u ) = 0$ on $x \in s u p p ( \mu ^ { * } )$ for a small value of $| \xi | < \epsilon _ { u }$ . By taking directional derivative w.r.t. $\psi$ along the direction $u$ , we obtain: + +$$ +u ^ { T } \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } ) = 0 , x \in s u p p ( \mu _ { \psi ^ { * } + \xi u , \theta ^ { * } } ) = s u p p ( \mu ^ { * } ) +$$ + +and thus + +$$ +u ^ { T } \nabla _ { \psi x } ^ { T } D ( x ; \psi ^ { * } ) = u ^ { T } \nabla _ { x \psi } D ( x ; \psi ^ { * } ) = 0 , x \in s u p p ( p _ { \theta ^ { + } } ) = s u p p ( p _ { d } ) +$$ + +according to Assumption $^ \mathrm { 6 b }$ (the inclusion condition that $s u p p ( p _ { d } ) = s u p p ( p _ { \theta ^ { * } } ) \subset s u p p ( \mu ^ { * } )$ is required). By calculating $u ^ { T } R$ directly, we obtain + +$$ +\begin{array} { r l } & { \displaystyle \boldsymbol { u } ^ { T } \boldsymbol { R } = u ^ { T } \frac { \partial } { \partial \boldsymbol { \theta } } \int _ { \mathcal { X } } \nabla _ { \boldsymbol { \psi } } D ( \boldsymbol { x } ; \boldsymbol { \psi } ^ { * } ) d p _ { \boldsymbol { \theta } } \bigg \rvert _ { \boldsymbol { \theta = \theta } ^ { = } } } \\ & { \qquad = u ^ { T } \frac { \partial } { \partial \boldsymbol { \theta } } \int _ { \mathcal { X } } \nabla _ { \boldsymbol { \psi } } D ( G ( \boldsymbol { z } ; \boldsymbol { \theta } ) ; \boldsymbol { \psi } ^ { * } ) p _ { l a t e n t } ( \boldsymbol { z } ) d \boldsymbol { z } \bigg \rvert _ { \boldsymbol { \theta = \theta } ^ { * } } } \\ & { \quad = \int _ { \mathcal { X } } u ^ { T } \nabla _ { \boldsymbol { x } \boldsymbol { \psi } } D ( G ( \boldsymbol { z } ; \boldsymbol { \theta } ^ { * } ) ; \boldsymbol { \psi } ^ { * } ) \nabla _ { \boldsymbol { \theta } } G ( \boldsymbol { z } ; \boldsymbol { \theta } ^ { * } ) p _ { l a t e n t } ( \boldsymbol { z } ) d \boldsymbol { z } = 0 } \end{array} +$$ + +Thus, we obtain $u \in N ( R ^ { T } )$ , which implies that $N ( Q ^ { T } ) \subset N ( R ^ { T } )$ and $C ( R ) \subset C ( Q )$ . Now, we can check that $R T _ { G }$ is of full column rank since $T _ { G } ^ { T } R ^ { T } R T _ { G } = \Lambda _ { G } ^ { ( + ) }$ is positive definite. Therefore, + +$$ +R T _ { G } w = 0 \Rightarrow w = 0 +$$ + +We note that the projection matrix on $C ( Q )$ is given by $T _ { D } ( T _ { D } ^ { T } T _ { D } ) ^ { - 1 } T _ { D } ^ { T } = T _ { D } T _ { D } ^ { T }$ . In addition, we know that $C ( R T _ { G } ) \subset C ( R ) \subset C ( Q )$ . Therefore, + +$$ +\begin{array} { r l } & { T _ { D } ^ { T } R T _ { G } w = 0 } \\ & { \Rightarrow T _ { D } T _ { D } ^ { T } R T _ { G } w = 0 } \\ & { \Rightarrow T _ { D } T _ { D } ^ { T } w ^ { \prime } = 0 , w ^ { \prime } = R T _ { G } w \in C ( R T _ { G } ) } \end{array} +$$ + +$$ +\begin{array} { l } { { \Rightarrow w ^ { \prime } = R T _ { G } w = 0 } } \\ { { \Rightarrow w = 0 } } \end{array} +$$ + +which completes the proof that $T _ { D } ^ { T } R T _ { G }$ is a full column rank matrix. + +Now, we only need to obtain proofs for the trivial cases where either one of $T _ { D }$ or $T _ { G }$ is empty. First, suppose that $T _ { G }$ is empty. Similar to the analysis given above, we can find that the point $( \psi ^ { * } , \theta )$ with $| \theta - \theta ^ { * } | < \mathrm { m i n } ( \epsilon _ { \mu } , \epsilon _ { g } , \delta _ { g } , \nu )$ is an equilibrium point, where $g ( \theta ^ { * } ) = g ( \theta )$ for a sufficiently small $\lvert \theta - \theta ^ { * } \rvert < \nu$ . We conclude that $p _ { \theta } = p _ { d }$ for $| \theta - \theta ^ { * } | < \operatorname * { m i n } ( \epsilon _ { \mu } , \epsilon _ { g } , \delta _ { g } , \nu )$ . Under the generator initialization that is sufficiently close according to $\theta ^ { * }$ , we can only observe the discriminator update + +$$ +\dot { \psi } = - \frac { \rho } { 2 } \nabla _ { \psi } \mathbb { E } _ { \mu _ { \psi , \theta } } [ \| \nabla _ { x } D ( x ; \psi ) \| ^ { 2 } ] +$$ + +since $\mathbb { E } _ { p _ { d } } [ D ( x ; \psi ) ] - \mathbb { E } _ { p _ { \theta } } [ D ( x ; \psi ) ] = 0$ for any $\psi$ and $| \theta - \theta ^ { * } | < \mathrm { m i n } ( \epsilon _ { \mu } , \epsilon _ { g } , \delta _ { g } , \nu )$ . The discriminator update described above is locally stable system near the equilibrium $\psi = \psi ^ { * }$ since the Jacobian of the update on $\psi$ is given as $- \rho Q$ and the zero eigenvalues can be ignored in a similar manner to the previous step. Therefore, the given system is stable near the equilibrium. + +Suppose that $T _ { D }$ is empty. Given that $N ( Q ^ { T } ) \subset N ( R ^ { T } ) .$ , $R = 0$ , then the results are similar to those presented above, but our goal is to show that $( \psi , \theta )$ is an equilibrium point, where $( \psi , \theta )$ is sufficiently close to the original equilibrium point. We note that $( \psi ^ { * } , \theta )$ is also an equilibrium point that satisfies the assumptions. + +By Assumption 2, $h ( \psi ) = h ( \psi ^ { * } ) = 0$ for $| \psi - \psi ^ { * } | < \xi$ , which implies that $\nabla _ { x } D ( x ; \psi ) = 0$ for $x \in s u p p ( \mu _ { \psi , \theta ^ { * } } ) = s u p p ( \mu ^ { * } )$ and $| \psi - \psi ^ { * } | < \xi$ . Thus, we obtain + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } _ { \mu _ { \psi , \theta ^ { * } } } \big [ \nabla _ { \psi x } ^ { T } D ( \boldsymbol { x } ; \psi ) \nabla _ { x } D ( \boldsymbol { x } ; \psi ) \big ] = 0 } \\ { \displaystyle \frac { \rho } { 2 } \int _ { s u p p ( \mu ^ { * } ) } \big \| \nabla _ { x } D \big \| ^ { 2 } d \mu _ { \psi , \theta ^ { * } } ^ { \prime } d \boldsymbol { x } = 0 } \end{array} +$$ + +By Assumption 4, $\mathbb { E } _ { p _ { d } } [ \nabla _ { \psi } D ( x ; \psi ) ] - \mathbb { E } _ { p _ { \theta ^ { * } } } [ \nabla _ { \psi } D ( x ; \psi ) ] = 0$ since $p _ { d } = p _ { \theta ^ { \ast } }$ . In addition, + +$$ +\dot { \theta } = \frac { \partial } { \partial \theta } \int _ { \mathcal { X } } D ( x ; \psi ) d p _ { \theta } \bigg | _ { \theta = \theta ^ { * } } = \int _ { \mathcal { Z } } \nabla _ { \theta } ^ { T } G ( z ; \theta ^ { * } ) \nabla _ { x } D ( G ( z ; \theta ^ { * } ) ; \psi ) p _ { l a t e n t } ( z ) d z = 0 +$$ + +Therefore, the point $( \psi , \theta ^ { * } )$ with $| \psi - \psi ^ { * } | < \operatorname* { m i n } ( \xi , \delta _ { d } )$ is an equilibrium point. From Assumption 4, $D ( x ; \psi )$ is an equilibrium discriminator, and thus $D ( x ; \psi )$ is already an optimal discriminator for $| \psi - \psi ^ { * } | < \operatorname* { m i n } ( \xi , \delta _ { d } )$ and $p _ { \theta }$ coincides with the data distribution $p _ { d }$ for $| \theta - \bar { \theta } ^ { * } | < \operatorname* { m i n } ( \epsilon _ { \mu } , \epsilon _ { g } , \delta _ { g } )$ , which indicates that every discriminator and generator near $( \psi ^ { * } , \theta ^ { * } )$ is an equilibrium point and this completes the proof of the main theorem. □ + +# APPENDIX D : DETAILED EXPERIMENTAL RESULTS + +![](images/c27519b391c821a8a732501033759f5a0bf8d62bd8c66e63574a2babdc1debb6.jpg) + +Figure 4: 2D example on 8 Gaussians, swissroll, 25 Gaussians datasets. Images generated with 5 penalty measures: $\mu _ { G P } , \mu _ { m i d } , p _ { g } , p _ { d } , \mu _ { g , a n c }$ . + +![](images/e9add33f13a815a66a7c30fd8fcb4008bda276bf58cd9026c58fd8877cc49d1a.jpg) +Figure 5: MNIST example. Images generated with $\mu _ { G P } , \mu _ { m i d } , p _ { g } , p _ { d } , \mu _ { g , a n c }$ + +![](images/2dd27c24c109293e33374a284d06f390455d2abd1929b0a3508768fbd5710adf.jpg) +Figure 6: CIFAR-10 example. Images generated with WGAN, WGAN-GP, $\mu _ { G P } , \mu _ { m i d } , p _ { g } , p _ { d } , \mu _ { g , a n c }$ under the DCGAN architecture. + +![](images/d9c4fa273546a3058613783e5118feaf32c9be46e77bdd5be9b4add130cb07b7.jpg) +Figure 7: CIFAR-10 example. Images generated with WGAN, WGAN-GP, $\mu _ { G P } , \mu _ { m i d } , p _ { g } , p _ { d } , \mu _ { g , a n c }$ under the ResNet architecture. \ No newline at end of file diff --git a/parse/train/H1ecDoR5Y7/H1ecDoR5Y7_content_list.json b/parse/train/H1ecDoR5Y7/H1ecDoR5Y7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..702e17631dd794a47962ef1c422d4c29a7736993 --- /dev/null +++ b/parse/train/H1ecDoR5Y7/H1ecDoR5Y7_content_list.json @@ -0,0 +1,2471 @@ +[ + { + "type": "text", + "text": "LOCAL STABILITY AND PERFORMANCE OF SIMPLE GRADIENT PENALTY $\\mu$ -WASSERSTEIN GAN ", + "text_level": 1, + "bbox": [ + 176, + 99, + 803, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 174, + 398, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 238, + 544, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance between a data distribution and sample distribution. Recent studies have proposed stabilizing the training process for the WGAN and implementing the Lipschitz constraint. In this study, we prove the local stability of optimizing the simple gradient penalty $\\mu$ -WGAN(SGP $\\mu$ -WGAN) under suitable assumptions regarding the equilibrium and penalty measure $\\mu$ . The measure valued differentiation concept is employed to deal with the derivative of the penalty terms, which is helpful for handling abstract singular measures with lower dimensional support. Based on this analysis, we claim that penalizing the data manifold or sample manifold is the key to regularizing the original WGAN with a gradient penalty. Experimental results obtained with unintuitive penalty measures that satisfy our assumptions are also provided to support our theoretical results. ", + "bbox": [ + 233, + 275, + 764, + 443 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 483, + 336, + 498 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep generative models reached a turning point after generative adversarial networks (GANs) were proposed by Goodfellow et al. (2014). GANs are capable of modeling data with complex structures. For example, DCGAN can sample realistic images using a convolutional neural network (CNN) structure(Radford et al., 2015). GANs have been implemented in many applications in the field of computer vision with good results, such as super-resolution, image translation, and text-to-image generation(Ledig et al., 2017; Isola et al., 2017; Zhang et al., 2017; Reed et al., 2016). ", + "bbox": [ + 174, + 518, + 825, + 603 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, despite these successes, GANs are affected by training instability and mode collapse problems. GANs often fail to converge, which can result in unrealistic fake samples. Furthermore, even if GANs successfully synthesize realistic data, the fake samples exhibit little variability. ", + "bbox": [ + 176, + 609, + 825, + 651 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A common solution to this instability problem is injecting an instance noise and finding different divergences. The injection of instance noise into real and fake samples during the training procedure was proposed by Sønderby et al. (2017), where its positive impact on the low dimensional support for the data distribution was shown to be a regularizing factor based on the Wasserstein distance, as demonstrated analytically by Arjovsky & Bottou (2017). In $f$ -GAN, $f$ -divergence between the target and generator distributions was suggested which generalizes the divergence between two distributions(Nowozin et al., 2016). In addition, a gradient penalty term which is related with Sobolev IPM(Integral Probability Metric) between data distribution and sample distribution was suggested by Mroueh et al. (2018). ", + "bbox": [ + 174, + 659, + 825, + 784 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The Wasserstein GAN (WGAN) is known to resolve the problems of generic GANs by selecting the Wasserstein distance as the divergence(Arjovsky et al., 2017). However, WGAN often fails with simple examples because the Lipschitz constraint on discriminator is rarely achieved during the optimization process and weight clipping. Thus, mimicking the Lipschitz constraint on the discriminator by using a gradient penalty was proposed by Gulrajani et al. (2017). ", + "bbox": [ + 174, + 791, + 823, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Noise injection and regularizing with a gradient penalty appear to be equivalent. The addition of instance noise in $f$ -GAN can be approximated to adding a zero centered gradient penalty(Roth et al., 2017). Thus, regularizing GAN with a simple gradient penalty term was suggested by Mescheder et al. (2018) who provided a proof of its stability. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Based on a theoretical analysis of the dynamic system, Nagarajan & Kolter (2017) proved the local exponential stability of the gradient-based optimization dynamics in GANs by treating the simultaneous gradient descent algorithm with a dynamic system approach. These previous studies were useful because they showed that the local behavior of GANs can be explained using dynamic system tools and the related Jacobian’s eigenvalues. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this study, we aim to prove the convergence property of the simple gradient penalty $\\mu$ -Wasserstein GAN(SGP $\\mu$ -WGAN) dynamic system under general gradient penalty measures $\\mu$ . To the best of our knowledge, our study is the first theoretical approach to GAN stability analysis which deals with abstract singular penalty measure. In addition, measure valued differentiation(Heidergott & Vazquez-Abad, 2008) is applied to take the derivative on the integral with a parametric measure, ´ which is helpful for handling an abstract measure and its integral in our proof. ", + "bbox": [ + 174, + 180, + 825, + 263 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main contributions of this study are as follows. ", + "bbox": [ + 176, + 271, + 509, + 285 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We prove the regularized effect and local stability of the dynamic system for a general penalty measure under suitable assumptions. The assumptions are written as both a tractable strong version and intractable weak version. To prove the main theorem, we also introduce the measure valued differentiation concept to handle the parametric measure. Based on the proof of the stability, we explain the reason for the success of previous penalty measures. We claim that the support of a penalty measure will be strongly related to the stability, where the weight on the limiting penalty measure might affect the speed of convergence. • We experimentally examined the general convergence results by applying two test penalty measures to several examples. The proposed test measures are unintuitive but they still satisfy the assumptions and similar convergence results were obtained in the experiment. ", + "bbox": [ + 215, + 297, + 825, + 460 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 479, + 339, + 496 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "First, we introduce our notations and basic measure-theoretic concepts. Second, we define our SGP $\\mu$ -WGAN optimization problem and treat this problem as a continuous dynamic system. Preliminary measure theoretic concepts are required to justify that the dynamic system changes in a sufficiently smooth manner as the parameter changes, so it is possible to use linearization theorem. They are also important for dealing with the parametric measure and its derivative. The problem setting with a simple gradient term is also discussed. The squared gradient size and simple gradient penalty term are used to build a differentiable dynamic system and to apply soft regularization as a resolving constraint, respectively. The continuous dynamic system approach, which is a so-called ODE method, is used to analyze the GAN optimization problem with the simultaneous gradient descent algorithm, as described by Nagarajan & Kolter (2017). ", + "bbox": [ + 173, + 511, + 825, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 NOTATIONS AND PRELIMINARIES REGARDING MEASURE THEORY ", + "text_level": 1, + "bbox": [ + 176, + 667, + 673, + 681 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "$D ( x ; \\psi ) : \\mathcal { X } \\to \\mathbb { R }$ is a discriminator function with its parameter $\\psi$ and $G ( z ; \\theta ) : \\mathcal { Z } \\to \\mathcal { X }$ is a generator function with its parameter $\\theta$ . $p _ { d }$ is the distribution of real data and $p _ { g } ~ = ~ p _ { \\theta }$ is the distribution of the generated samples in $\\mathcal { X }$ , which is induced from the generator function $G ( z ; \\theta )$ and a known initial distribution $p _ { l a t e n t } ( z )$ in the latent space $\\mathcal { Z } . \\ \\Vert \\cdot \\Vert$ denotes the $L ^ { 2 }$ Euclidean norm if no special subscript is present. ", + "bbox": [ + 174, + 693, + 825, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The concept of weak convergence for finite measures is used to ensure the continuity of the integral term over the measure in the dynamic system, which must be checked before applying the theorems related to stability. Throughout this study, we assume that the measures in the sample space are all finite and bounded. ", + "bbox": [ + 174, + 770, + 825, + 825 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Definition 1. For a set of finite measures $\\{ \\mu _ { i } \\} _ { i \\in \\mathcal { I } }$ in $( \\mathbb { R } ^ { n } , d )$ with euclidean distance $d _ { \\mathrm { { z } } }$ , $\\{ \\mu _ { i } \\} _ { i \\in \\mathbb { Z } }$ is referred to as bounded if there exists some $M > 0$ such that for all $i \\in \\mathcal { Z }$ , ", + "bbox": [ + 169, + 829, + 825, + 858 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/1f3e3bf2c0b71234bc7053e9d719032d7c76614e8813c18b3c6d88dfe18599ba.jpg", + "text": "$$\n\\mu _ { i } ( \\mathbb { R } ^ { n } ) \\leq M\n$$", + "text_format": "latex", + "bbox": [ + 454, + 864, + 544, + 882 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For instance, $M$ can be set as 1 if $\\{ \\mu _ { i } \\}$ are probability measures on $\\mathbb { R } ^ { n }$ . Assuming that the penalty measures are bounded, Portmanteau theorem offers the equivalent definition of the weak convergence for finite measures. This definition is important for ensuring that the integrals over $p _ { \\theta }$ and $\\mu$ in the dynamic system change continuously. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2. (Portmanteau Theorem) For a bounded sequence of finite measures $\\{ \\mu _ { n } \\} _ { n \\in \\mathbb { N } }$ on the Euclidean space $\\mathbb { R } ^ { n }$ with a $\\sigma$ -field of Borel subsets $B ( \\mathbb { R } ^ { n } )$ , $\\mu _ { n }$ converges weakly to $\\mu$ if and only $i f$ for every continuous bounded function $\\phi$ on $\\mathbb { R } ^ { n }$ , its integrals with respect to $\\mu _ { n }$ converge to $\\int \\phi \\dot { d \\mu }$ , i.e., ", + "bbox": [ + 173, + 136, + 826, + 191 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/58331e1cb1266600e341916176898473cb2ee476f4e35904b63eb0d3cdc49479.jpg", + "text": "$$\n\\mu _ { n } \\mu \\Longleftrightarrow \\int \\phi d \\mu _ { n } \\int \\phi d \\mu\n$$", + "text_format": "latex", + "bbox": [ + 385, + 189, + 612, + 222 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The most challenging problem in our analysis with the general penalty measure is taking the derivative of the integral, where the measure depends on the variable that we want to differentiate. If our penalty measure is either absolutely continuous or discrete, then it is easy to deal with the integral. However, in the case of singular penalty measure, dealing with the integral term is not an easy task. Therefore, we introduce the concept of a weak derivative of a probability measure in the following(Heidergott & Vazquez-Abad, 2008). The weak derivative of a measure is useful for handling a ´ parametric measure that is not absolutely continuous with low dimensional support. ", + "bbox": [ + 173, + 231, + 825, + 330 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 3. (Weak Derivatives of a Probability Measure) Consider the Euclidean space and its $\\sigma$ -field of Borel subsets $( \\mathbb { R } ^ { d } , B ( \\mathbb { R } ^ { d } ) )$ . The probability measure $P _ { \\theta }$ is called weakly differentiable at $\\theta$ if a signed finite measure $P _ { \\theta } ^ { \\prime }$ exists where ", + "bbox": [ + 173, + 333, + 825, + 376 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/83da5b6b83ba46d428f1dd89bdb2d938718e108c591a95ee8df153ad8bccd240.jpg", + "text": "$$\n\\frac { d } { d \\theta } \\int \\phi ( x ) d P _ { \\theta } = \\operatorname * { l i m } _ { \\Delta 0 } \\frac { 1 } { \\Delta } \\{ \\int \\phi ( x ) d P _ { \\theta + \\Delta } - \\int \\phi ( x ) d P _ { \\theta } \\} = \\int \\phi ( x ) d P _ { \\theta } ^ { \\prime }\n$$", + "text_format": "latex", + "bbox": [ + 256, + 383, + 741, + 416 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "is satisfied for every continuous bounded function $\\phi$ on $\\mathbb { R } ^ { n }$ . For the multidimensional parameter $\\theta$ , this can be defined similar manner. ", + "bbox": [ + 173, + 422, + 825, + 450 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can show that the positive part and negative part of $P _ { \\theta } ^ { \\prime }$ have the same mass by putting $\\phi ( x ) = 1$ and the Hahn–Jordan decomposition on $P _ { \\theta } ^ { \\prime }$ . Therefore, the following triple $( c _ { \\theta } , P _ { \\theta } ^ { + } , P _ { \\theta } ^ { - } )$ is called a weak derivative of $P _ { \\theta }$ , where $P _ { \\theta } ^ { \\pm }$ are probability measures and $P _ { \\theta } ^ { \\prime }$ is rewritten as: ", + "bbox": [ + 174, + 462, + 825, + 510 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dd940bf111d2d8c8bc401d42f66f41f8c2e044cd1a9cda54b8404eaade452f17.jpg", + "text": "$$\nP _ { \\theta } ^ { \\prime } = c _ { \\theta } P _ { \\theta } ^ { + } - c _ { \\theta } P _ { \\theta } ^ { - }\n$$", + "text_format": "latex", + "bbox": [ + 428, + 517, + 568, + 536 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 174, + 542, + 245, + 556 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/22037418bc3a27bdbec10a84f50e1624a4d03366c367e9ae5e913e859d53db40.jpg", + "text": "$$\n\\frac { d } { d \\theta } \\int \\phi ( x ) d P _ { \\theta } = \\int \\phi ( x ) d P _ { \\theta } ^ { \\prime } = c _ { \\theta } ( \\int \\phi ( x ) d P _ { \\theta } ^ { + } - \\int \\phi ( x ) d P _ { \\theta } ^ { - } )\n$$", + "text_format": "latex", + "bbox": [ + 281, + 554, + 717, + 587 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "holds for every continuous bounded function $\\phi$ on $\\mathbb { R } ^ { n }$ . It is known that the representation of $( c _ { \\theta } , P _ { \\theta } ^ { + } , P _ { \\theta } ^ { - } )$ for $P _ { \\theta } ^ { \\prime }$ is not unique because $( c _ { \\theta } + C _ { \\theta } , P _ { \\theta } ^ { + } + q _ { \\theta } , P _ { \\theta } ^ { - } + q _ { \\theta } )$ is also another representation of $P _ { \\theta } ^ { \\prime }$ . ", + "bbox": [ + 176, + 588, + 821, + 632 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For the general finite measure $Q _ { \\theta }$ , a normalizing coefficient $M ( \\theta ) < \\infty$ can be introduced. The product rule for differentiating can also be applied in a similar manner to calculus. ", + "bbox": [ + 173, + 637, + 823, + 666 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/79a5b8689162a000bbbbc5d53e4cdca2650cd9d1d4cff86ee28804ef8f7b217b.jpg", + "text": "$$\n\\frac { d } { d \\theta } \\int \\phi ( x ; \\theta ) d P _ { \\theta } = \\int \\nabla _ { \\theta } \\phi ( x ; \\theta ) d P _ { \\theta } + \\int \\phi ( x ; \\theta ) d P _ { \\theta } ^ { \\prime }\n$$", + "text_format": "latex", + "bbox": [ + 315, + 672, + 681, + 707 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Therefore, for the general finite measure $Q _ { \\theta } = M ( \\theta ) P _ { \\theta }$ , its derivative $Q _ { \\theta } ^ { \\prime }$ can be represented as below. ", + "bbox": [ + 171, + 712, + 825, + 739 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a55fa1d1447265063e8b6151dc2adfe2dd6f743e19268f0965e13751407c290c.jpg", + "text": "$$\nQ _ { \\theta } ^ { \\prime } = M ^ { \\prime } ( \\theta ) P _ { \\theta } + M ( \\theta ) P _ { \\theta } ^ { \\prime } = M ^ { \\prime } ( \\theta ) P _ { \\theta } + c _ { \\theta } M ( \\theta ) P _ { \\theta } ^ { + } - c _ { \\theta } M ( \\theta ) P _ { \\theta } ^ { - }\n$$", + "text_format": "latex", + "bbox": [ + 269, + 738, + 728, + 757 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 PROBLEM SETTING AS A DYNAMIC SYSTEM ", + "text_level": 1, + "bbox": [ + 174, + 772, + 522, + 787 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Previous work of Mescheder et al. (2018) showed that the dynamic system of WGAN-GP is not necessarily stable at equilibrium by demonstrating that the sequence of parameters is not Cauchy sequence. This is mainly due to the term $\\| x \\|$ in the dynamic system which has a derivative $\\frac { x } { \\| x \\| }$ that is not defined at $x = 0$ . WGAN-GP has a penalty term $\\mathbb { E } _ { \\mu _ { G P } } [ ( \\| \\nabla _ { x } D ( x ; \\psi ) \\| - 1 ) ^ { 2 } ]$ that can lead to a discontinuity in its dynamic system. ", + "bbox": [ + 173, + 797, + 825, + 873 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "These problems can be avoided by using the squared value of the gradient’s norm $\\| \\nabla _ { x } D \\| ^ { 2 }$ , which is a differentiable function. In contrast to the WGAN-GP, recent methods based on a gradient penalty such as the simple gradient penalty employed by Mescheder et al. (2018) and the Sobolev GAN used the average of the squared values for the penalty area, whereas the WGAN-GP penalizes the size of the discriminator’s gradient $\\lVert \\nabla _ { x } D \\rVert$ away from 1 in a pointwise manner. ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This advantage of squared gradient term1, $\\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ]$ , makes the dynamic system differentiable and we define the WGAN problem with the square of the gradient’s norm as a simple gradient penalty. This simple gradient penalty can be treated as soft regularization based on the size of the discriminator’s gradient, especially in case where $\\mu$ is the probability measure (Roth et al., 2017). It is convenient to determine whether the system is stable by observing the spectrum of the Jacobian matrix. In the following, $( D ( x ; \\psi ) , p _ { d } , p _ { \\theta } , \\mu )$ is defined as an SGP $\\mu$ -WGAN optimization problem (SGP-form) with a simple gradient penalty term on the penalty measure $\\mu$ . ", + "bbox": [ + 173, + 138, + 825, + 239 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 4. The WGAN optimization problem with a simple gradient penalty term $\\| \\nabla _ { x } D \\| ^ { 2 }$ , penalty measure $\\mu _ { ; }$ , and penalty weight hyperparameter $\\rho > 0$ is given as follows, where the penalty term is only introduced to update the discriminator. ", + "bbox": [ + 174, + 244, + 825, + 287 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b912dbb6e2bbe4f8ba417c3fb9e6808043d8e5aaa17dc4700d78e6e7cd0d8d1a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\operatorname* { m a x } _ { \\psi } : \\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] - \\frac { \\rho } { 2 } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\\\ & { \\displaystyle \\operatorname* { m i n } _ { \\theta } : \\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 295, + 702, + 349 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "According to Nagarajan & Kolter (2017) and many other optimization problem studies, the simultaneous gradient descent algorithm for GAN updating can be viewed as an autonomous dynamic system of discriminator parameters and generator parameters, which we denote as $\\psi$ and $\\theta$ . As a result, the related dynamic system is given as follows. ", + "bbox": [ + 174, + 362, + 825, + 417 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0644b4f346d433451357f27f989dccf843641a706b9f912012549e316225522f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\dot { \\boldsymbol { \\psi } } = \\mathbb { E } _ { \\boldsymbol { p } _ { d } } [ \\nabla _ { \\boldsymbol { \\psi } } D ] - \\mathbb { E } _ { \\boldsymbol { p } _ { \\boldsymbol { \\theta } } } [ \\nabla _ { \\boldsymbol { \\psi } } D ] - \\frac { \\rho } { 2 } \\nabla _ { \\boldsymbol { \\psi } } \\mathbb { E } _ { \\boldsymbol { \\mu } } [ \\nabla _ { \\boldsymbol { x } } ^ { T } D \\nabla _ { \\boldsymbol { x } } D ] } \\\\ & { \\dot { \\boldsymbol { \\theta } } = \\nabla _ { \\boldsymbol { \\theta } } \\mathbb { E } _ { \\boldsymbol { p } _ { \\boldsymbol { \\theta } } } [ D ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 424, + 678, + 474 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 TOY EXAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 491, + 341, + 507 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We investigate two examples considered in previous studies by Mescheder et al. (2018) and Nagarajan & Kolter (2017). We then generalize the results to a finite measure case. The first example is the univariate Dirac GAN, which was introduced by Mescheder et al. (2018). ", + "bbox": [ + 174, + 522, + 825, + 565 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 5. (Dirac GAN) The Dirac GAN comprises a linear discriminator $D ( x ; \\psi ) = \\psi x$ , data distribution $p _ { d } = \\delta _ { 0 }$ , and sample distribution $p _ { \\theta } = \\delta _ { \\theta }$ . ", + "bbox": [ + 173, + 569, + 823, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Dirac GAN with a gradient penalty with an arbitrary probability measure is known to be globally convergent(Mescheder et al., 2018). We argue that this result can be generalized to a finite penalty measure case. ", + "bbox": [ + 173, + 609, + 825, + 652 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 1. Consider the Dirac GAN problem with SGP form $\\begin{array} { r } { ( D ( x ; \\psi ) = \\psi x , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { \\psi , \\theta } ) } \\end{array}$ . Suppose that some small $\\eta > 0$ exists such that its finite penalty measure $\\mu _ { \\psi , \\theta }$ with mass $M ( \\psi , \\theta ) =$ $\\mathbf { \\bar { \\rho } } _ { \\int 1 d \\mu _ { \\psi , \\theta } } \\geq 0$ satisfies either ", + "bbox": [ + 173, + 656, + 825, + 700 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/97151248a5cf3b75961b1ebc9810cb7fca87ca8ac828a46a4a5e87861980837d.jpg", + "text": "$$\nM ( \\psi , \\theta ) > 0 f o r \\left( \\psi , \\theta \\right) \\in B _ { \\eta } ( ( 0 , 0 ) ) o\n$$", + "text_format": "latex", + "bbox": [ + 227, + 710, + 488, + 728 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e8459292a266269810d80e7a360306c2daa733e146efd3409cb921ea77b3acc3.jpg", + "text": "$$\n^ { \\prime } \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) \\geq 0 f o r ( \\psi , \\theta ) \\in B _ { \\eta } ( ( 0 , 0 ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 734, + 630, + 752 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then, the SGP $\\mu$ -WGAN optimization dynamics with $\\begin{array} { r } { ( D ( x ; \\psi ) = \\psi x , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { \\psi , \\theta } ) } \\end{array}$ are locally stable at the origin and the basin of attraction $B = B _ { R } ( ( 0 , 0 ) )$ is open ball with radius $R$ . Its radius is given as follows. ", + "bbox": [ + 173, + 762, + 826, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Motivated by this example, we can extend this idea to the other toy example given by Nagarajan & Kolter (2017), where WGAN fails to converge to the equilibrium points $( \\psi , \\theta ) = ( 0 , \\pm 1 )$ . ", + "bbox": [ + 176, + 843, + 823, + 872 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 2. Consider the toy example $\\begin{array} { r c l } { ( D ( x ; \\psi ) } & { = } & { \\psi x ^ { 2 } , U ( - 1 , 1 ) , U ( - | \\theta | , | \\theta | ) , \\mu _ { \\theta } ) } \\end{array}$ where $U ( 0 , 0 ) = \\delta _ { 0 }$ and the ideal equilibrium points are given by $( \\psi ^ { * } , \\theta ^ { * } ) = ( 0 , \\pm 1 )$ . For a finite measure $\\mu = \\mu _ { \\theta }$ on $\\mathbb { R }$ which is independent of $\\psi$ , suppose that $\\mu _ { \\boldsymbol { \\theta } } \\to \\mu ^ { * }$ with $\\mu ^ { * } \\neq C \\delta _ { 0 }$ for $C \\geq 0$ . The dynamic system is locally stable near the desired equilibrium $( 0 , \\pm 1 )$ , where the spectrum of the ", + "bbox": [ + 173, + 102, + 825, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Jacobian at $( 0 , \\pm 1 )$ is given by $\\begin{array} { r } { \\lambda = - 2 \\rho \\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] \\pm \\sqrt { 4 \\rho ^ { 2 } \\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] ^ { 2 } - \\frac { 4 } { 9 } } } \\end{array}$ ", + "bbox": [ + 173, + 161, + 648, + 183 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 MAIN CONVERGENCE THEOREM ", + "text_level": 1, + "bbox": [ + 174, + 200, + 478, + 217 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We propose the convergence property of WGAN with a simple gradient penalty on an arbitrary penalty measure $\\mu$ for a realizable case: $\\theta = \\theta ^ { * }$ with $p _ { d } = p _ { \\theta ^ { \\ast } }$ exists. In subsection 4.1, we provide the necessary assumptions, which comprise our main convergence theorem. In subsection 4.2, we give the main convergence theorem with a sketch of the proof. A more rigorous analysis is given in the Appendix. ", + "bbox": [ + 173, + 232, + 825, + 303 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 319, + 316, + 333 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first assumption is made regarding the equilibrium condition for GANs, where we state the ideal conditions for the discriminator parameter and generator parameter. As the parameters converge to the ideal equilibrium, the sample distribution $\\left( p _ { \\theta } \\right)$ converges to the real data distribution $\\left( p _ { d } \\right)$ and the discriminator cannot distinguish the generated sample and the real data. ", + "bbox": [ + 173, + 344, + 825, + 401 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 1. $p _ { \\theta } p _ { d }$ as $\\theta ~ \\to ~ \\theta ^ { * }$ and $D ( x ; \\psi ^ { * } ) ~ = ~ 0$ on supp $\\left( p _ { d } \\right)$ and its small open neighborhood, i.e., $x \\in \\cup _ { x ^ { \\prime } \\in s u p p ( p _ { d } ) } B _ { \\epsilon _ { x ^ { \\prime } } } ( x ^ { \\prime } )$ implies $D ( x ; \\psi ^ { * } ) = 0$ . For simplicity, we denote $\\cup _ { x ^ { \\prime } \\in s u p p ( p _ { d } ) } B _ { \\epsilon _ { x ^ { \\prime } } } ( x ^ { \\prime } )$ as $B ( s u p p ( p _ { d } ) )$ . ", + "bbox": [ + 173, + 404, + 825, + 450 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The second assumption ensures that the higher order terms cannot affect the stability of the SGP $\\mu$ -WGAN. In the Appendix, we consider the case where the WGAN fails to converge when Assumption 2 is not satisfied. Compared with the previous study by Nagarajan & Kolter (2017), the conditions for the discriminator parameter are slightly modified. ", + "bbox": [ + 174, + 459, + 825, + 516 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 2. ", + "text_level": 1, + "bbox": [ + 174, + 520, + 276, + 535 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/57cde38dc1d5fe809648c3b6e3c15199978983fa1d367eb6377e6bd493e47cd9.jpg", + "text": "$$\n\\begin{array} { r } { g ( \\theta ) = \\| \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] - \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] \\| ^ { 2 } , h ( \\psi ) = \\mathbb { E } _ { \\mu _ { \\psi , \\theta ^ { * } } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 540, + 777, + 560 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "are locally constant along the nullspace of the Hessian matrix. ", + "bbox": [ + 173, + 564, + 591, + 579 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The third assumption allows us to extend our results to discrete probability distribution cases, as described by Mescheder et al. (2018). ", + "bbox": [ + 173, + 590, + 823, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 3. $\\exists \\epsilon _ { g } > 0$ such that $D ( x ; \\psi ^ { * } ) = 0$ on $\\cup _ { | \\theta - \\theta ^ { * } | < \\epsilon _ { g } } s u p p ( p _ { \\theta } ) .$ . ", + "bbox": [ + 176, + 622, + 658, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The fourth assumption indicates that there are no other “bad” equilibrium points near $( \\psi ^ { * } , \\theta ^ { * } )$ , which justifies the projection along the axis perpendicular to the null space. ", + "bbox": [ + 174, + 650, + 823, + 679 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 4. A bad equilibrium does not exist near the desired equilibrium point. Thus, $( \\psi ^ { * } , \\theta ^ { * } )$ is an isolated equilibrium or there exist $\\delta _ { d } , \\delta _ { g } > 0$ such that all equilibrium points in $B _ { \\delta _ { d } } ( \\psi ^ { * } ) \\times$ $B _ { \\delta _ { g } } ( \\theta ^ { * } )$ satisfy the other assumptions. ", + "bbox": [ + 174, + 683, + 825, + 724 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The last assumption is related to the necessary conditions for the penalty measure. A calculation of the gradient penalty based on samples from the data manifold and generator manifold or the interpolation of both was introduced in recent studies (Gulrajani et al., 2017; Roth et al., 2017; Mescheder et al., 2018). First, we propose strong conditions for the penalty measure. ", + "bbox": [ + 174, + 736, + 825, + 792 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumption 5. The finite penalty measure $\\mu = \\mu _ { \\theta }$ satisfies the followings: ", + "bbox": [ + 176, + 796, + 665, + 811 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "a $\\mu _ { \\theta } \\to \\mu _ { \\theta ^ { * } } = \\mu ^ { * }$ and $\\mu _ { \\theta }$ is independent of the discriminator parameter $\\psi$ . $\\begin{array} { r l } & { b \\ s u p p ( p _ { d } ) \\subset s u p p ( \\mu ^ { * } ) } \\\\ & { c \\ \\exists \\epsilon _ { \\mu } > 0 \\ s u c h \\ t h a t \\ s u p p ( \\mu _ { \\theta } ) \\subset B ( s u p p ( p _ { d } ) ) f o r \\vert \\theta - \\theta ^ { * } \\vert < \\epsilon _ { \\mu } . } \\end{array}$ ", + "bbox": [ + 212, + 820, + 712, + 887 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The assumption given above means that the support of the penalty measure $\\mu _ { \\theta }$ should approach the data manifolds smoothly as $\\theta \\to \\theta ^ { * }$ . However, the penalty measure from WGAN-GP with a simple gradient penalty still reaches equilibrium without satisfying Assumption 5c. Therefore, we suggest Assumption 6, which is a weak version of Assumption 5. Assumption $6 \\mathrm { a } ^ { 2 }$ is technically required to take the derivative of the integral $\\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ]$ with respect to $\\psi$ . ", + "bbox": [ + 171, + 895, + 821, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 148 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assumption 6. (Weak version of Assumption 5) The finite penalty measure $\\mu = \\mu _ { \\psi , \\theta }$ satisfies the following. ", + "bbox": [ + 174, + 150, + 825, + 179 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "a $\\mu _ { \\psi , \\theta } \\mu _ { \\psi ^ { * } , \\theta ^ { * } } = \\mu ^ { * }$ , where supp $( \\mu _ { \\psi , \\theta } )$ only depends on $\\theta$ . Near the equilibrium, $\\mu _ { \\psi , \\theta }$ can be weakly differentiated twice with respect to $\\psi$ . In addition, its mass $M ( \\psi , \\theta ) \\stackrel { \\cdot } { = }$ $\\int 1 d \\mu _ { \\psi , \\theta }$ is a twice-differentiable function of $\\psi$ and bounded near the equilibrium. ", + "bbox": [ + 214, + 188, + 825, + 232 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "b $E _ { \\mu ^ { * } } [ \\nabla _ { \\psi x } D \\nabla _ { \\psi x } ^ { T } D ]$ is positive definite or $s u p p ( p _ { d } ) \\subset s u p p ( \\mu ^ { * } ) .$ ", + "bbox": [ + 214, + 239, + 655, + 258 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "c $\\ : \\exists \\epsilon _ { \\mu } > 0 \\ :$ such that $\\operatorname { s u p p } ( \\mu _ { \\theta } ) \\subset V$ for $| \\theta - \\theta ^ { * } | < \\epsilon _ { \\mu }$ , where $V = \\{ x | \\nabla _ { x } D ( x ; \\psi ^ { * } ) = 0 \\}$ . ", + "bbox": [ + 212, + 265, + 803, + 282 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The assumption above implies the following situations; The penalty measure’s support approaches to data manifold and its weight changes smoothly with respect to $\\psi$ and $\\theta$ . At the equilibrium, penalty measure’s support contains data manifold. Also, ideal discriminator will remain flat on the penalty area. ", + "bbox": [ + 173, + 291, + 825, + 348 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In summary, the gradient penalty regularization term with any penalty measure where the support approaches $B ( s u p p ( p _ { d } ) )$ in a smooth manner works well and this main result can explain the regularization effect of previously proposed penalty measures such as $\\mu _ { G P } , p _ { d } , p _ { \\ell }$ , and their mixtures. ", + "bbox": [ + 174, + 353, + 825, + 397 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 MAIN CONVERGENCE THEOREM ", + "text_level": 1, + "bbox": [ + 176, + 412, + 442, + 426 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "According to the modified assumptions given above, we prove that the related dynamic system is locally stable near the equilibrium. The tools used for analyzing stability are mainly based on those described by Nagarajan & Kolter (2017). Our main contributions comprise proposing the necessary conditions for the penalty measure and proving the local stability for all penalty measures that satisfy Assumption 6. ", + "bbox": [ + 174, + 438, + 825, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 1. Suppose that our SGP $\\mu$ -WGAN optimization problem $( D , p _ { d } , p _ { \\theta } , \\mu )$ with equilibrium point $( \\psi ^ { * } , \\theta ^ { * } )$ satisfies the assumptions given above. Then, the related dynamic system is locally stable at the equilibrium. ", + "bbox": [ + 173, + 510, + 825, + 553 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A detailed proof of the main convergence theorem is given in the Appendix. A sketch of the proof is given in three steps. First, the undesired terms in the Jacobian matrix of the system at the equilibrium are cancelled out. Next, the Jacobian matrix at equilibrium is given by $\\left[ \\begin{array} { c c } { \\therefore } & { \\mathbf { 0 } } \\\\ { R ^ { T } } & { 0 } \\end{array} \\right]$ , where $Q =$ $\\mathbb { E } _ { \\mu ^ { * } } [ \\nabla _ { \\psi x } D \\nabla _ { \\psi x } ^ { T } D ]$ and $R = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ] | _ { \\theta = \\theta ^ { * } }$ . The system is locally stable when both $Q$ and $R ^ { T } R$ are positive definite. We can complete the proof by dealing with zero eigenvalues by showing that ${ \\cal N } ( Q ^ { \\hat { T } } ) \\subset { \\cal N } ( R ^ { T } )$ and the projected system’s stability implies the original system’s stability. ", + "bbox": [ + 173, + 561, + 826, + 670 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our analysis mainly focuses on WGAN, which is the simplest case of general GAN minimax optimization ", + "bbox": [ + 174, + 674, + 823, + 702 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c9cfe94c0eec51cce17d86c66b5deff479567e4073bef83271dac237aa8ce112.jpg", + "text": "$$\n\\begin{array} { r l } & { \\underset { \\psi } { \\operatorname* { m a x } } : \\mathbb { E } _ { p _ { d } } [ f ( D ( x ; \\psi ) ) ] + \\mathbb { E } _ { p _ { \\theta } } [ f ( - D ( x ; \\psi ) ) ] - \\frac { \\rho } { 2 } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\\\ & { \\underset { \\theta } { \\operatorname* { m i n } } : \\mathbb { E } _ { p _ { d } } [ f ( D ( x ; \\psi ) ) ] + \\mathbb { E } _ { p _ { \\theta } } [ f ( - D ( x ; \\psi ) ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 702, + 732, + 756 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "with $f ( x ) = x$ . Similar approach is still valid for general GANs with concave function $f$ with $f ^ { \\prime \\prime } ( x ) < 0$ and $f ^ { \\prime } ( 0 ) \\neq 0$ . ", + "bbox": [ + 174, + 757, + 820, + 787 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 805, + 419, + 821 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We claim that every penalty measure that satisfies the assumptions can regularize the WGAN and generate similar results to the recently proposed gradient penalty methods. Several penalty measures were tested based on two-dimensional problems (mixture of 8 Gaussians, mixture of 25 Gaussians, and swissroll), MNIST and CIFAR-10 datasets using a simple gradient penalty term. In the comparisons with WGAN, the recently proposed penalty measures and our test penalty measures used the same network settings and hyperparameters. The penalty measures and its detailed sampling methods are listed in Table 1, where $x _ { d } \\sim p _ { d } , x _ { g } \\sim p _ { \\theta }$ , and $\\alpha \\sim U ( 0 , 1 )$ . $\\mathcal { A }$ indicates fixed anchor point in $\\mathcal { X }$ . ", + "bbox": [ + 174, + 835, + 823, + 864 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f69d536cffc9776fe39e23c67e6883d4867b3d70e47546aaf0d04cc93fb1011c.jpg", + "table_caption": [ + "Table 1: List of benchmark WGANs (WGAN and WGAN-GP with non-zero centered gradient penalty) and 5 penalty measures with a simple gradient penalty term. In this table, WGAN-GP represents the previous model proposed by (Gulrajani et al., 2017), which penalizes the WGAN with non-zero centered gradient penalty terms, whereas $\\mu _ { G P }$ represents the simple method. In our experiment, no additional weights are applied on 5 penalty measures and they are all probability distributions. " + ], + "table_footnote": [], + "table_body": "
PenaltyPenalty termPenalty measure, sampling method
WGAN WGAN-GPNone(Weight Clipping) Eμ[(IVxD|-1)2]None x=axd+(1-α)xg
Eμ[VxD|2]
Pg PdEμ[VxDii2]x=xg x=xd
μGPEμVDii2]x=axd+(1-α)xg
μmidEμ[VxDii2]x= 0.5xd+0.5xg
μg,ancEμVDii2]x=αA+(1-α)xg
", + "bbox": [ + 204, + 303, + 794, + 448 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "By setting the previously proposed WGAN with weight-clipping(Arjovsky et al., 2017) and WGANGP(Gulrajani et al., 2017) as the baseline models, $\\operatorname { S G P } \\mu$ -WGAN was examined with various penalty measures comprising three recently proposed measures and two artificially generated measures. $p _ { \\theta }$ and $p _ { d }$ were suggested by Mescheder et al. (2018) and $\\mu _ { G P }$ was introduced from the WGAN-GP. We analyzed the artificial penalty measures $\\mu _ { m i d }$ and $\\mu _ { g , a n c }$ as the test penalty measures. ", + "bbox": [ + 174, + 468, + 825, + 540 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The experiments were conducted based on the implementation of the Gulrajani et al. (2017). The hyperparameters, generator/discriminator structures, and related TensorFlow implementations can be found at https://github.com/igul222/improved_wgan_training (Gulrajani et al., 2017). Only the loss function was modified slightly from a non-zero centered gradient penalty to a simple penalty. For the CIFAR-10 image generation tasks, the inception score(Salimans et al., 2016) and FID(Heusel et al., 2017) were used as benchmark scores to evaluate the generated images. ", + "bbox": [ + 174, + 545, + 825, + 628 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 2D EXAMPLES AND MNIST ", + "text_level": 1, + "bbox": [ + 176, + 646, + 410, + 660 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We checked the convergence of $p _ { \\theta }$ for the 2D examples (8 Gaussians, swissroll data, and 25 Gaussians) and MNIST digit generation for the SGP-WGANs with five penalty measures. MNIST and 25 Gaussians were trained over 200K iterations, the 8 Gaussians were trained for 30K iterations, and the Swiss Roll data were trained for 100K iterations. The anchor $\\mathcal { A }$ for $\\mu _ { g , a n c }$ was set as $( 2 , - 1 )$ for the 2D examples and 784 gray pixels for MNIST. We only present the results obtained for the MNIST dataset with the penalty measures comprising $\\mu _ { m i d }$ and $\\mu _ { g , a n c }$ in Figure 1. The others are presented in the Appendix. ", + "bbox": [ + 174, + 671, + 825, + 770 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/07f5e9693ee7d3a76d2c35b0604f32d9f8eef9b1b58131c18cc2525916acb8b1.jpg", + "image_caption": [ + "Figure 1: MNIST example. Images generated with $\\mu _ { m i d }$ (left) and $\\mu _ { g , a n c } ( \\mathrm { r i g h t } )$ . " + ], + "image_footnote": [], + "bbox": [ + 267, + 784, + 730, + 875 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 103, + 290, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "DCGAN and ResNet architectures were tested on the CIFAR-10 dataset. The generators were trained for 200K iterations. The anchor $\\mathcal { A }$ for $\\mu _ { g , a n c }$ during CIFAR-10 generation was set as fixed random pixels. The WGAN, WGAN-GP, and five penalty measures were evaluated based on the inception score and FID, as shown in Table 2, which are useful tools for scoring the quality of generated images. The images generated from $\\mu _ { m i d }$ and $\\mu _ { g , a n c }$ with ResNet are shown in Figure 2. The others are presented in the Appendix. ", + "bbox": [ + 174, + 130, + 825, + 214 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/1179439b10d45ea6e0e262af20d6e0efa166918659e273b27572a8c49f7b9646.jpg", + "table_caption": [ + "Table 2: Benchmark score results obtained based on the CIFAR-10 dataset under DCGAN and ResNet architectures. The higher inception score and lower FID indicate the good quality of the generated images. " + ], + "table_footnote": [], + "table_body": "
PenaltyDCGAN Inception FIDResNet InceptionFID
WGAN 35.64 ± 0.09 48.7==
WGAN-GP6.48 ± 0.10 35.07.82 ± 0.0918.1
Pg6.46 ± 0.09 38.07.63 ± 0.1020.9
pd6.33 ± 0.0738.9 7.63 ± 0.0920.3
μGP6.40 ±0.0835.4 7.60 ± 0.0918.3
μmid6.60 ± 0.0733.9 7.86 ± 0.0716.4
μg,anc6.45 ± 0.0733.7 7.36 ± 0.0922.4
", + "bbox": [ + 312, + 287, + 684, + 436 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/af56d7fca60411caad1f41e50993c7c1651b0890d3dce2a1e967749ed2d6734d.jpg", + "image_caption": [ + "Figure 2: CIFAR-10 example. Images generated with $\\mu _ { m i d }$ (left) and $\\mu _ { g , a n c }$ (right) under the ResNet architecture. " + ], + "image_footnote": [], + "bbox": [ + 269, + 465, + 728, + 642 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 727, + 318, + 742 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this study, we proved the local stability of simple gradient penalty $\\mu$ -WGAN optimization for a general class of finite measure $\\mu$ . This proof provides insight into the success of regularization with previously proposed penalty measures. We explored previously proposed analyses based on various gradient penalty methods. Furthermore, our theoretical approach was supported by experiments using unintuitive penalty measures. In future research, our works can be extended to alternative gradient descent algorithm and its related optimal hyperparameters. Stability at non-realizable equilibrium points is one of the important topics on stability of GANs. Optimal penalty measure for achieving the best convergence speed can be also investigated using a spectral theory, which provides the mathematical analysis on stability of GAN with a precise information on the convergence theory. ", + "bbox": [ + 173, + 758, + 825, + 897 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Mart´ın Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial ´ networks. In International Conference on Learning Representations, 2017. 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", + "bbox": [ + 174, + 830, + 823, + 873 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pp. 2226–2234, 2016. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Casper Kaae Sønderby, Jose Caballero, Lucas Theis, Wenzhe Shi, and Ferenc Huszar. Amortised ´ MAP inference for image super-resolution. International Conference on Learning Representations, 2017. ", + "bbox": [ + 173, + 103, + 823, + 145 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Han Zhang, Tao Xu, and Hongsheng Li. Stackgan: Text to photo-realistic image synthesis with stacked generative adversarial networks. In IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017, pp. 5908–5916, 2017. ", + "bbox": [ + 176, + 155, + 823, + 196 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "APPENDIX A : PROOF OF LEMMAS BASED ON TOY EXAMPLES ", + "text_level": 1, + "bbox": [ + 173, + 102, + 687, + 119 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proof of Lemma 1. The related dynamic system of $\\begin{array} { r } { ( D ( x ; \\psi ) = \\psi x , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { \\psi , \\theta } ) } \\end{array}$ can be written as follows. ", + "bbox": [ + 174, + 132, + 823, + 162 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/73bdd30368acc5ebfd92175ccbcc4e3aa653d0a11d2e143c01ca06dae90d2a83.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\dot { \\psi } = - \\theta - \\frac { \\rho } { 2 } \\nabla \\psi \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\psi ^ { 2 } ] } \\\\ { \\displaystyle \\dot { \\theta } = \\psi } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 408, + 166, + 588, + 215 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "First, the only equilibrium point is given by $( \\psi ^ { * } , \\theta ^ { * } ) = ( 0 , 0 )$ from ", + "bbox": [ + 171, + 220, + 612, + 237 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/11feeb6b3820e36c142d1e8b6aef9276fb59080ccfe9959982cc90756eaaf513.jpg", + "text": "$$\n\\begin{array} { l } { 0 = - \\theta - 2 \\psi M ( \\psi , \\theta ) - \\psi ^ { 2 } \\nabla _ { \\psi } M ( \\psi , \\theta ) } \\\\ { \\quad 0 = \\psi } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 361, + 243, + 633, + 281 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The corresponding Jacobian matrix for the dynamic system is written as: ", + "bbox": [ + 174, + 286, + 650, + 303 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/a595bbd7f831617f8cd58599e2e1aff778bc300bdd3058a707ea5231d7ddabba.jpg", + "text": "$$\nJ = \\left[ { \\begin{array} { r r } { Z } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 447, + 308, + 550, + 344 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 351, + 217, + 364 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/8ca3fa140aa723f934b3ad2afb4750c4a94f366b1468f44a121b9153fa19e56d.jpg", + "text": "$$\nZ = - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\psi ^ { 2 } ] \\bigg | _ { \\psi = 0 , \\theta = 0 }\n$$", + "text_format": "latex", + "bbox": [ + 388, + 361, + 609, + 398 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "$\\nabla _ { \\psi } D ( x ; \\psi ) = \\psi$ does not depend on $x$ , so this can be rewritten as: ", + "bbox": [ + 173, + 402, + 611, + 417 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/fbaf1589dc439a233d0447e16c827632565803a15d7716a00ca39b2bd8ab939e.jpg", + "text": "$$\nZ = - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } ( \\psi ^ { 2 } \\mathbb { E } _ { \\mu _ { \\psi } , \\theta } [ 1 ] ) = - \\frac { \\rho } { 2 } ( 2 M ( \\psi , \\theta ) + 4 \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) + \\psi ^ { 2 } M _ { \\psi \\psi } ( \\psi , \\theta ) ) \\bigg | _ { \\psi = 0 , \\theta = 0 }\n$$", + "text_format": "latex", + "bbox": [ + 199, + 425, + 799, + 470 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Therefore, if $M ( 0 , 0 ) > 0$ , then the given system is locally stable because the eigenvalues of its linearized system have negative real parts. If $M ( 0 , 0 ) = 0$ , then the stability of the system cannot be proved by the linearization theorem. In this case, we consider the following Lyapunov function. ", + "bbox": [ + 176, + 488, + 823, + 530 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/c80231cde598e57e89c34285fc160a222055cae6b66ba8d6e44bdd43023c446e.jpg", + "text": "$$\nL ( \\psi ( t ) , \\theta ( t ) ) = \\psi ( t ) ^ { 2 } + \\theta ( t ) ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 397, + 536, + 599, + 555 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "By differentiating with $t$ , we obtain ", + "bbox": [ + 174, + 561, + 408, + 577 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/a07979e4d63cf803a5e69d8098d7bec2cbff2467fae906f1e2dccac861941cb3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\dot { L } = 2 ( \\psi \\psi ^ { \\prime } + \\theta \\theta ^ { \\prime } ) = - \\rho \\psi \\nabla _ { \\psi } ( \\psi ^ { 2 } M ( \\psi , \\theta ) ) = - \\rho \\psi ( 2 \\psi M ( \\psi , \\theta ) + \\psi ^ { 2 } \\nabla _ { \\psi } M ( \\psi , \\theta ) ) } \\\\ & { \\quad = - \\rho \\psi ^ { 2 } ( 2 M ( \\psi , \\theta ) + \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) ) \\leq 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 222, + 583, + 776, + 626 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Clearly, $L ( \\psi , \\theta ) \\geq 0$ and the equality holds iff $\\psi = \\theta = 0$ . In addition, $\\dot { L } \\leq 0$ since $M ( \\psi , \\theta ) \\geq$ 0 and $\\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) \\geq 0$ from the assumption. Furthermore, it is clear that if $( \\psi ( 0 ) , \\theta ( 0 ) ) \\ \\in \\qquad $ $B _ { \\eta } ( ( 0 , 0 ) )$ , then $( \\psi ( \\tau ) , \\theta ( \\tau ) ) \\in B _ { \\eta } ( ( 0 , 0 ) )$ for all $\\tau \\geq 0$ because the Lyapunov function (square of the distance between the origin and $( \\psi ( \\tau ) , \\theta ( \\tau ) ) )$ always decreases as $\\tau \\infty$ . Therefore, the given system is stable according to the Lyapunov stability theorem. ", + "bbox": [ + 173, + 632, + 826, + 704 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Again, we can check that if $\\mu _ { \\psi , \\theta }$ is a probability measure, then the system is globally stable, as shown by Mescheder et al. (2018). The basin of attraction is given by the whole $\\mathbb { R } ^ { 2 }$ plane since $M ( \\psi , \\theta ) = 1$ , so $\\dot { L } = - \\rho \\psi ^ { 2 } ( 2 M + \\psi \\nabla _ { \\psi } M ) = - 2 \\rho \\psi ^ { 2 } \\leq 0$ for every $( \\psi , \\theta ) \\in \\mathbb { R } ^ { 2 }$ . □ ", + "bbox": [ + 173, + 709, + 825, + 757 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Proof of Lemma 2. From the general setup of the $\\operatorname { S G P } \\mu$ -WGAN optimization problem, the dynamic system corresponding to the simple-GAN in Definition 6 can be written as follows. ", + "bbox": [ + 173, + 772, + 823, + 803 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/4e998f6091eba7f6033f5fd09344d95a68f943a8409f4336df57850b4d236018.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\dot { \\psi } = \\frac { 1 } { 3 } - \\frac { \\theta ^ { 2 } } { 3 } - 4 \\rho \\psi \\mathbb { E } _ { \\mu } [ x ^ { 2 } ] } } } \\\\ { { \\displaystyle { \\dot { \\theta } = \\frac { 2 \\psi \\theta } { 3 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 408, + 809, + 589, + 875 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "If we let $\\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] = A ^ { 2 }$ , then the Jacobian matrix at the equilibrium $( 0 , \\pm 1 )$ is given by ${ \\boldsymbol { J } } =$ $\\left[ { \\begin{array} { c c } { - 4 \\rho A ^ { 2 } } & { { \\mp } { \\frac { 2 } { 3 } } } \\\\ { \\pm { \\frac { 2 } { 3 } } } & { 0 } \\end{array} } \\right]$ Therefore, the given system is locally stable when $A \\neq 0$ . ", + "bbox": [ + 173, + 878, + 825, + 929 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX B : PROOF OF LEMMA RELATED WITH ASSUMPTION 2 ", + "text_level": 1, + "bbox": [ + 173, + 102, + 717, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Lemma 3. Consider the Dirac-GAN setup and SGP $\\mu$ -WGAN optimization system with a slightly changed discriminator function $D _ { 2 } ( x ; \\psi ) \\stackrel { \\textstyle - } { = } \\psi x ^ { 2 }$ . The system $( D _ { 2 } , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { G P } )$ does not converge to $( 0 , 0 )$ but for any point $( a , 0 )$ with $a < 0$ , the system has equilibrium points on the whole $\\psi$ -axis and it violates Assumption 2. ", + "bbox": [ + 173, + 132, + 825, + 190 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof of Lemma 3. For the SGP $\\mu$ -WGAN optimization problem $( D _ { 2 } , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { G P } )$ , the dynamic system can be written as follows. ", + "bbox": [ + 173, + 204, + 823, + 233 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/ff74a298902e83262a0f40cf2a6a00238dd792d670717d5025daa416e44859a6.jpg", + "text": "$$\n\\begin{array} { l } { { \\dot { \\psi } = - \\theta ^ { 2 } - \\frac { 4 } { 3 } \\rho \\psi \\theta ^ { 2 } } } \\\\ { { \\dot { \\theta } = 2 \\psi \\theta } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 431, + 239, + 565, + 291 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "$2 \\psi \\theta = 0$ and $\\begin{array} { r } { \\theta ^ { 2 } ( 1 + \\frac { 4 } { 3 } \\rho \\psi ) = 0 } \\end{array}$ implies that $\\theta = 0$ , so the $\\psi$ -axis is the set of all equilibrium points. By drawing the nullclines $\\psi = 0$ and $\\begin{array} { r } { \\psi = - \\frac { 3 } { 4 \\rho } } \\end{array}$ in the $\\psi \\theta$ -plane, it is clear that no solution curve converges to $( b , 0 )$ with $b \\geq 0$ , as shown in Figure 3. □ ", + "bbox": [ + 173, + 297, + 825, + 347 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/c43c5cb4133543240663135cefcb52ec5abc17c3330a731f348c08a6b7589f1c.jpg", + "image_caption": [ + "Figure 3: Phase portrait of the SGP $\\mu$ -WGAN optimization problem $( D _ { 2 } , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { G P } )$ with $\\textstyle \\rho = { \\frac { 3 } { 8 } }$ . Along the line $\\theta = 0$ , the system is stable so no updating will occur. Every solution curve that passes the nullcline $\\psi = 0$ has $\\dot { \\theta } = 0$ . For the nullcline $\\begin{array} { r } { \\psi = - \\frac { 3 } { 4 \\rho } = - 2 } \\end{array}$ , no updating on $\\psi$ will occur and only $\\theta$ will be updated. Given that the solution curves do not intersect with each other, every solution curve is exactly one of the followinstays in area A. (2) Solution curve converges to $\\begin{array} { r } { \\bar { ( \\psi , \\theta ) } \\overset { * } { = } ( - \\frac { 3 } { 4 \\rho } , 0 ) } \\end{array}$ trivial cases; (1) Solalong the nullcline $\\begin{array} { r } { \\psi = - \\frac { 3 } { 4 \\rho } } \\end{array}$ e. (3) Solution curve stays in area B. (4) Solution curve starts from area C, crosses the nullcline $\\psi = 0$ perpendicularly, and converges to $( b , 0 )$ with $b < 0$ . Therefore, no solution curve converges to $( 0 , 0 )$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 363, + 823, + 632 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX C : PROOF OF THE MAIN CONVERGENCE THEOREM ", + "text_level": 1, + "bbox": [ + 173, + 101, + 699, + 119 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. Let us consider the Jacobian matrix $J = \\left[ \\begin{array} { l l } { K _ { D D } } & { K _ { D G } } \\\\ { K _ { G D } } & { K _ { G G } } \\end{array} \\right]$ at the first equilibrium $( \\psi ^ { * } , \\theta ^ { * } ) ^ { 4 }$ ", + "bbox": [ + 174, + 128, + 818, + 162 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e7c9c27b2a41ee6f9e13ea7a81eab9b1746accc232b6bf11ad215c5545de1b58.jpg", + "text": "$$\n\\begin{array} { r l } { J = \\left[ \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi \\psi } D ] - \\mathbb { E } _ { p _ { \\theta ^ { \\star } } } [ \\nabla _ { \\psi \\psi } D ] - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] } & { - \\nabla _ { \\theta \\psi } \\mathbb { E } _ { p _ { \\theta } } [ D ] - \\frac { \\rho } { 2 } \\nabla _ { \\theta \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] \\right] } \\\\ & { \\qquad \\nabla _ { \\psi \\theta } \\mathbb { E } _ { p _ { \\theta } } [ D ] ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 167, + 823, + 203 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "First, Assumption 1 implies that $\\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi \\psi } D ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi \\psi } D ] = 0$ since $p _ { \\theta } p _ { d }$ as $\\theta \\to \\theta ^ { * }$ . From Assumption 3, $\\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ^ { * } ) ]$ is locally zero near the equilibrium $\\theta ^ { * }$ , which implies that ", + "bbox": [ + 173, + 213, + 823, + 244 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/81facc905734a786b16b7c75eef91b4b43ca971f787d99658755482c45df4310.jpg", + "text": "$$\nK _ { G G } = \\nabla _ { \\theta \\theta } \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ^ { * } ) ] \\bigg \\rvert _ { \\theta = \\theta ^ { * } } = 0\n$$", + "text_format": "latex", + "bbox": [ + 374, + 250, + 624, + 285 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We still need to evaluate $\\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ]$ and $\\nabla _ { \\theta \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ]$ . According to Assumption 6a, finite signed measures $\\mu _ { \\psi , \\theta } ^ { \\prime }$ and $\\mu _ { \\psi , \\theta } ^ { \\prime \\prime }$ exist5, so they are the first and second weak derivatives of $\\mu _ { \\psi , \\theta }$ with respect to the parameter $\\psi$ at $\\left( \\psi ^ { * } , \\theta ^ { * } \\right)$ . Therefore, the expectations given above can be rewritten as below. ", + "bbox": [ + 173, + 297, + 826, + 359 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/873c6e40f75e2fb970d90b1797bb7a55fb68d12044dcabd0e11179e0dc5c94ca.jpg", + "text": "$$\n\\begin{array} { l } { I = \\nabla _ { \\psi \\psi } \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi } , \\theta ) } \\| \\nabla _ { x } D \\| ^ { 2 } d \\mu _ { \\psi , \\theta } } \\\\ { = \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi , \\theta } ) } ( 2 \\nabla _ { \\psi x } ^ { T } D \\nabla _ { \\psi x } D + 2 K _ { 0 } ) d \\mu _ { \\psi , \\theta } + \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi , \\theta } ) } 2 ( \\nabla _ { \\psi x } ^ { T } D \\nabla _ { x } D ) d \\mu _ { \\psi , \\theta } ^ { \\prime } + \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi , \\theta } ) } \\| \\nabla _ { x } \\nabla _ { \\psi } D \\nabla _ { x } \\| ^ { 2 } d \\mu _ { \\psi , \\theta } ^ { \\prime } } \\\\ { \\cdot I = \\nabla _ { \\theta \\psi } \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi , \\theta } ) } \\| \\nabla _ { x } D \\| ^ { 2 } d \\mu _ { \\psi , \\theta } } \\\\ { = \\nabla _ { \\theta } ( \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi , \\theta } ) } 2 ( \\nabla _ { \\psi x } ^ { T } D \\nabla _ { x } D ) d \\mu _ { \\psi , \\theta } + \\displaystyle \\int _ { s u p p ( \\mu _ { \\psi , \\theta } ) } \\| \\nabla _ { x } D \\| ^ { 2 } d \\mu _ { \\psi , \\theta } ^ { \\prime } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 362, + 839, + 508 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 530, + 217, + 544 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ee71c9c77bf1d82ca5bf10330df4efef6b547e38d33d2fe051aef03524d5ac96.jpg", + "text": "$$\n\\begin{array} { r } { K _ { 0 } ( x ; \\psi ) = \\left[ \\sum _ { k } \\frac { \\partial ^ { 3 } } { \\partial \\psi _ { i } \\partial \\psi _ { j } \\partial x _ { k } } D ( x ; \\psi ) \\frac { \\partial } { \\partial x _ { k } } D ( x ; \\psi ) \\right] _ { i j } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 540, + 671, + 569 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "From Assumption 6c and the fact that the weak derivative of $\\mu _ { \\psi , \\theta }$ vanishes outside of $s u p p ( \\mu _ { \\psi , \\theta } )$ , $\\nabla _ { x } D ( x ; \\psi ^ { * } ) = 0$ on $s u p p ( \\mu _ { \\psi , \\theta } ) \\subset V$ for all $\\theta$ with $\\left| \\theta - \\theta ^ { * } \\right| < \\epsilon _ { \\mu }$ and $\\mu _ { \\psi , \\theta } ^ { \\prime } = \\mu _ { \\psi , \\theta } ^ { \\prime \\prime } = 0$ on the outside of $s u p p ( \\mu _ { \\psi , \\theta } )$ , which leads to the desired results: ", + "bbox": [ + 173, + 578, + 826, + 623 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c9ce5a9128efc889ba76694e6ff639d55872f87fed9b2ba5be125d0d40d2fbf6.jpg", + "text": "$$\n\\begin{array} { c } { \\displaystyle { I = \\int _ { s u p p ( \\mu ^ { * } ) } 2 ( \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) \\nabla _ { \\psi x } D ( x ; \\psi ^ { * } ) ) d \\mu ^ { * } } } \\\\ { \\displaystyle { I I = 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 330, + 628, + 669, + 683 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "After cancelling the undesired terms, the Jacobian matrix at the equilibrium $( \\psi ^ { * } , \\theta ^ { * } )$ is given as: ", + "bbox": [ + 171, + 694, + 803, + 710 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/6a1b89e015cb05c70482ad4e9599b6ec7e03937eb85552532e9a0aa00e4856bc.jpg", + "text": "$$\nJ = \\left[ \\begin{array} { c c } { - \\rho Q } & { - R } \\\\ { R ^ { T } } & { 0 } \\end{array} \\right]\n$$", + "text_format": "latex", + "bbox": [ + 434, + 714, + 563, + 750 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 755, + 217, + 768 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/5e0f85f8ea237fdb1e27b313d050f3bedef57ebedaa50b5a8267342a528e73cf.jpg", + "text": "$$\n\\begin{array} { r } { Q = \\mathbb { E } _ { \\mu ^ { * } } [ \\nabla _ { \\psi x } ^ { T } D \\nabla _ { \\psi x } D ] } \\\\ { \\boldsymbol { R } = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ] \\bigg | _ { \\theta = \\theta ^ { * } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 415, + 771, + 581, + 830 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "From the definition of $Q$ , it is easy to check that $Q$ is at least positive semi-definite. It is known that for a negative definite matrix $A$ and full column rank matrix $B$ , the block matrix $\\left[ \\begin{array} { l l } { A } & { B } \\\\ { - B ^ { T } } & { 0 } \\end{array} \\right]$ is Hurwitz, i.e., all eigenvalues of the matrix have a negative real part. Therefore, if $\\bar { Q }$ is positive definite and $R$ is full column rank, the proof is complete. We consider the complementary case. ", + "bbox": [ + 173, + 102, + 825, + 176 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Suppose that $Q$ or $R ^ { T } R$ have some zero eigenvalues. Let $Q = U _ { D } \\Lambda _ { D } U _ { D } ^ { T }$ and $R ^ { T } R = U _ { G } \\Lambda _ { G } U _ { G } ^ { T }$ with $U _ { D } ~ = ~ \\lbrack T _ { D } ~ S _ { D } \\rbrack$ and $U _ { G } ~ = ~ [ T _ { G } ^ { ~ - } ~ S _ { G } ]$ , where $T _ { D }$ and $T _ { G }$ are the eigenvectors of $\\bar { Q }$ and $R ^ { T } R$ that correspond to non-zero eigenvalues. First, we assume that $T _ { D }$ and $T _ { G }$ are not empty. We can show that $( \\psi ^ { * } + \\xi v , \\theta ^ { * } \\overset { \\cdot } { + } \\nu w )$ is also an equilibrium point for a sufficiently small $\\xi , \\nu$ and $v \\in N ( Q ) , w \\in N ( R ^ { T } R )$ by using the techniques given by Nagarajan & Kolter (2017). If the system does not update at the equilibrium point $( \\psi ^ { * } , \\theta ^ { * } )$ and its small neighborhood $\\left( \\psi ^ { * } + \\xi v , \\theta ^ { * } + \\nu w \\right)$ is perturbed along $N ( Q )$ and $N ( \\bar { R } ^ { T } R )$ , then it is reasonable to project the system orthogonal to $N ( Q )$ and $N ( R ^ { T } { \\bar { R } } )$ . ", + "bbox": [ + 173, + 194, + 825, + 310 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "First, we assume that $v \\in N ( Q )$ . By Assumption 2, $h ( \\psi ^ { * } + \\xi v ) = h ( \\psi ^ { * } ) = 0$ for $| \\xi | < \\xi _ { d }$ , which implies that $\\nabla _ { \\boldsymbol { x } } D ( \\boldsymbol { x } ; \\psi ^ { * } + \\xi \\boldsymbol { v } ) = 0$ for $x \\in s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )$ and $| \\xi | < \\xi _ { d }$ . Thus, we obtain ", + "bbox": [ + 173, + 329, + 825, + 372 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a07c3a8482ee736b654d5b78b767398ae13e3cb218ba12b7492880cf2882b64f.jpg", + "text": "$$\n\\mathbb { E } _ { \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } } [ \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } + \\xi v ) \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) ] = 0\n$$", + "text_format": "latex", + "bbox": [ + 316, + 375, + 679, + 395 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 401, + 200, + 415 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2918b6465467384c47bed217182720336ee90009f7c7157f2822984c697c37c4.jpg", + "text": "$$\n\\int _ { s u p p ( \\mu ^ { * } ) } \\left\\| \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) \\right\\| ^ { 2 } d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ^ { \\prime } = 0\n$$", + "text_format": "latex", + "bbox": [ + 338, + 422, + 661, + 458 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "By Assumption 4, $\\begin{array} { r } { \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] = 0 } \\end{array}$ since $p _ { d } = p _ { \\theta ^ { \\ast } }$ . By adding these equations, we obtain ", + "bbox": [ + 173, + 474, + 825, + 505 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1db21d03fbbfe5903967476b04abe7e3bc2506de90ac37988c65eac94f9f8948.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\dot { \\psi } = \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] } \\\\ { \\displaystyle - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) } 2 \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } + \\xi v ) \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } } \\\\ { \\displaystyle - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) } \\| \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) \\| ^ { 2 } d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ^ { \\prime } } \\\\ { \\displaystyle = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 512, + 745, + 627 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In addition, ", + "bbox": [ + 173, + 636, + 251, + 650 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f02b947c82e979834207e8549915edbc7e86af58f4507735e67988ef442bec30.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\dot { \\theta } = \\frac { \\partial } { \\partial \\theta } \\int _ { \\mathcal X } D ( x ; \\psi ^ { * } + \\xi v ) d p _ { \\theta } \\bigg \\vert _ { \\theta = \\theta ^ { * } } } \\\\ { \\displaystyle = \\int _ { \\mathcal Z } \\nabla _ { \\theta } ^ { T } G ( z ; \\theta ^ { * } ) \\nabla _ { x } D ( G ( z ; \\theta ^ { * } ) ; \\psi ^ { * } + \\xi v ) p _ { l a t e n t } ( z ) d z = 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 660, + 710, + 729 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Therefore, the point $( \\psi ^ { * } + \\xi v , \\theta ^ { * } )$ with $| \\xi | < \\xi _ { d }$ is an equilibrium point. According to Assumption 4, $D ( x ; \\psi ^ { * } + \\xi v )$ is an equilibrium discriminator for $| \\xi | < \\delta _ { d }$ , and thus $D ( x ; \\psi ^ { * } + \\bar { \\xi } v )$ is already an optimal discriminator for $| \\xi | < \\operatorname* { m i n } ( \\xi _ { d } , \\delta _ { d } )$ . ", + "bbox": [ + 174, + 738, + 821, + 782 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Suppose that $w \\ \\in \\ N ( R ^ { T } R )$ . By Assumption 2, $g ( \\theta ^ { * } ) ~ = ~ g ( \\theta ^ { * } + \\nu w ) ~ = ~ 0$ for $| \\nu | < \\nu _ { g }$ , and thus $\\begin{array} { r l } { \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } + \\nu w } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] ~ = ~ 0 } \\end{array}$ for $| \\nu | ~ < ~ \\nu _ { g }$ . Furthermore, Assumption 3 gives $\\mathbb { E } _ { p _ { \\theta ^ { * } + \\nu w } } [ D ( x ; \\psi ^ { * } ) ] = 0$ for a sufficiently close $| \\nu | < \\epsilon _ { g }$ , which implies that $\\begin{array} { r } { \\dot { \\theta } = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ^ { * } ) ] \\bigg \\rvert _ { \\theta = \\theta ^ { * } + \\nu w } = 0 } \\end{array}$ for $| \\nu | < \\epsilon _ { g }$ . Finally, ", + "bbox": [ + 173, + 799, + 826, + 882 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/4309bb51c96b0c9268510e9ec92222fb87d98c6ee8548d64c17cb2b707246508.jpg", + "text": "$$\n\\int _ { s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + s w } ) } 2 \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) \\nabla _ { x } D ( x ; \\psi ^ { * } ) d \\mu _ { i \\psi ^ { * } , \\theta ^ { * } + \\nu w } + \\int _ { s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + s w } ) } \\| \\nabla _ { x } D ( x ; \\psi ^ { * } ) \\| ^ { 2 } d \\mu _ { i \\psi ^ { * } , \\theta ^ { * } + \\nu w } ^ { \\prime }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 890, + 839, + 926 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "since $s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ) \\subset V$ and $\\nabla _ { x } D ( x ; \\psi ^ { * } ) = 0$ on $V$ for a sufficiently small $| \\nu | < \\epsilon _ { \\mu }$ (Assumption 6c). By adding these results, we obtain ", + "bbox": [ + 171, + 102, + 823, + 132 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/76d5b6aa3c75ebd421f3f35d18e3417930c6864d902a0d1b797e9eb7b9c47422.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\dot { \\psi } = \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } + \\nu w } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] } \\\\ { \\displaystyle \\quad - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ) } 2 \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) \\nabla _ { x } D ( x ; \\psi ^ { * } ) d \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } } \\\\ { \\displaystyle \\quad - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ) } \\left\\| \\nabla _ { x } D ( x ; \\psi ^ { * } ) \\right\\| ^ { 2 } d \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ^ { \\prime } } \\\\ { \\displaystyle = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 137, + 714, + 251 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore, the point $( \\psi ^ { * } , \\theta ^ { * } + \\nu w )$ with $| \\nu | < \\operatorname* { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\nu _ { g } , \\delta _ { g } )$ is an equilibrium point, which implies that $p _ { \\theta ^ { * } + \\nu w } = p _ { d }$ according to Assumption 4. ", + "bbox": [ + 171, + 255, + 823, + 285 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "If we consider the projected system $( \\alpha , \\beta ) \\ : = \\ : ( T _ { D } ^ { T } \\psi , T _ { G } ^ { T } \\theta )$ , then the projected dynamic system’s Jacobian at $( T _ { D } ^ { T } \\psi ^ { * } , T _ { G } ^ { T } \\theta ^ { * } )$ is given as follows. ", + "bbox": [ + 174, + 303, + 823, + 335 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ab5cc9bede8049eb9fc625a17955b81b82c31d9ae1628ce9c6a08279a683c068.jpg", + "text": "$$\nJ ^ { \\prime } = \\left[ \\begin{array} { c c } { { - \\rho T _ { D } ^ { T } Q T _ { D } } } & { { - T _ { D } ^ { T } R T _ { G } } } \\\\ { { T _ { G } ^ { T } R ^ { T } T _ { D } } } & { { 0 } } \\end{array} \\right] = \\left[ \\begin{array} { c c } { { - \\rho \\Lambda _ { D } ^ { ( + ) } } } & { { - T _ { D } ^ { T } R T _ { G } } } \\\\ { { T _ { G } ^ { T } R ^ { T } T _ { D } } } & { { 0 } } \\end{array} \\right]\n$$", + "text_format": "latex", + "bbox": [ + 294, + 342, + 702, + 377 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore, we only need to prove that $T _ { D } ^ { T } R T _ { G }$ is of full column rank. Suppose that $u \\in N ( Q ^ { T } ) =$ $N ( Q )$ . According to Assumption 2, $h ( \\psi )$ is locally constant at $\\psi ^ { * }$ along the direction $u$ . Therefore, for a sufficiently small scalar $\\xi$ with $| \\xi | < \\xi _ { u }$ , ", + "bbox": [ + 173, + 385, + 825, + 428 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/e27853ff8e6a35307548bedb3dbff0c2e9f57e804513bf206b04712721e00743.jpg", + "text": "$$\nh ( \\psi ^ { * } + \\xi u ) = h ( \\psi ^ { * } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 411, + 434, + 586, + 450 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the last equality comes from the Assumption 6. This implies that $\\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi u ) = 0$ on $x \\in s u p p ( \\mu ^ { * } )$ for a small value of $| \\xi | < \\epsilon _ { u }$ . By taking directional derivative w.r.t. $\\psi$ along the direction $u$ , we obtain: ", + "bbox": [ + 173, + 457, + 825, + 500 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/46e6de64509dff368771e4fd30ad7bd6b75843574602bedb44e2c0a2bdb55ee9.jpg", + "text": "$$\nu ^ { T } \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) = 0 , x \\in s u p p ( \\mu _ { \\psi ^ { * } + \\xi u , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 307, + 503, + 691, + 523 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "and thus ", + "bbox": [ + 173, + 529, + 232, + 542 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f12156480932f019af6c15a1635e14a7f641e3026b4726e22c8d89fa0e6d3833.jpg", + "text": "$$\nu ^ { T } \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) = u ^ { T } \\nabla _ { x \\psi } D ( x ; \\psi ^ { * } ) = 0 , x \\in s u p p ( p _ { \\theta ^ { + } } ) = s u p p ( p _ { d } )\n$$", + "text_format": "latex", + "bbox": [ + 263, + 540, + 733, + 559 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "according to Assumption $^ \\mathrm { 6 b }$ (the inclusion condition that $s u p p ( p _ { d } ) = s u p p ( p _ { \\theta ^ { * } } ) \\subset s u p p ( \\mu ^ { * } )$ is required). By calculating $u ^ { T } R$ directly, we obtain ", + "bbox": [ + 176, + 563, + 823, + 592 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/736d668fa30bc070ae5aff23b93fc12ed49a8c5f8555af3e9d4c60707f61e42d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\boldsymbol { u } ^ { T } \\boldsymbol { R } = u ^ { T } \\frac { \\partial } { \\partial \\boldsymbol { \\theta } } \\int _ { \\mathcal { X } } \\nabla _ { \\boldsymbol { \\psi } } D ( \\boldsymbol { x } ; \\boldsymbol { \\psi } ^ { * } ) d p _ { \\boldsymbol { \\theta } } \\bigg \\rvert _ { \\boldsymbol { \\theta = \\theta } ^ { = } } } \\\\ & { \\qquad = u ^ { T } \\frac { \\partial } { \\partial \\boldsymbol { \\theta } } \\int _ { \\mathcal { X } } \\nabla _ { \\boldsymbol { \\psi } } D ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ) ; \\boldsymbol { \\psi } ^ { * } ) p _ { l a t e n t } ( \\boldsymbol { z } ) d \\boldsymbol { z } \\bigg \\rvert _ { \\boldsymbol { \\theta = \\theta } ^ { * } } } \\\\ & { \\quad = \\int _ { \\mathcal { X } } u ^ { T } \\nabla _ { \\boldsymbol { x } \\boldsymbol { \\psi } } D ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { * } ) ; \\boldsymbol { \\psi } ^ { * } ) \\nabla _ { \\boldsymbol { \\theta } } G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { * } ) p _ { l a t e n t } ( \\boldsymbol { z } ) d \\boldsymbol { z } = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 597, + 715, + 703 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Thus, we obtain $u \\in N ( R ^ { T } )$ , which implies that $N ( Q ^ { T } ) \\subset N ( R ^ { T } )$ and $C ( R ) \\subset C ( Q )$ . Now, we can check that $R T _ { G }$ is of full column rank since $T _ { G } ^ { T } R ^ { T } R T _ { G } = \\Lambda _ { G } ^ { ( + ) }$ is positive definite. Therefore, ", + "bbox": [ + 174, + 707, + 823, + 742 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6f2616ac722c0754a8514a060e01e37c5471978672ae07a49e954b709ee13d7c.jpg", + "text": "$$\nR T _ { G } w = 0 \\Rightarrow w = 0\n$$", + "text_format": "latex", + "bbox": [ + 426, + 748, + 573, + 763 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We note that the projection matrix on $C ( Q )$ is given by $T _ { D } ( T _ { D } ^ { T } T _ { D } ) ^ { - 1 } T _ { D } ^ { T } = T _ { D } T _ { D } ^ { T }$ . In addition, we know that $C ( R T _ { G } ) \\subset C ( R ) \\subset C ( Q )$ . Therefore, ", + "bbox": [ + 174, + 768, + 823, + 800 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/52492c755402f31d1ef3057171ce82d03f94de7d1b318a0b60ee5be158cc906c.jpg", + "text": "$$\n\\begin{array} { r l } & { T _ { D } ^ { T } R T _ { G } w = 0 } \\\\ & { \\Rightarrow T _ { D } T _ { D } ^ { T } R T _ { G } w = 0 } \\\\ & { \\Rightarrow T _ { D } T _ { D } ^ { T } w ^ { \\prime } = 0 , w ^ { \\prime } = R T _ { G } w \\in C ( R T _ { G } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 806, + 550, + 867 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c7f7e00a5d741a90f9383c383335f1162a67d4574e4d2100316c4460f7140853.jpg", + "text": "$$\n\\begin{array} { l } { { \\Rightarrow w ^ { \\prime } = R T _ { G } w = 0 } } \\\\ { { \\Rightarrow w = 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 886, + 405, + 920 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "which completes the proof that $T _ { D } ^ { T } R T _ { G }$ is a full column rank matrix. ", + "bbox": [ + 173, + 102, + 627, + 119 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Now, we only need to obtain proofs for the trivial cases where either one of $T _ { D }$ or $T _ { G }$ is empty. First, suppose that $T _ { G }$ is empty. Similar to the analysis given above, we can find that the point $( \\psi ^ { * } , \\theta )$ with $| \\theta - \\theta ^ { * } | < \\mathrm { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )$ is an equilibrium point, where $g ( \\theta ^ { * } ) = g ( \\theta )$ for a sufficiently small $\\lvert \\theta - \\theta ^ { * } \\rvert < \\nu$ . We conclude that $p _ { \\theta } = p _ { d }$ for $| \\theta - \\theta ^ { * } | < \\operatorname * { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )$ . Under the generator initialization that is sufficiently close according to $\\theta ^ { * }$ , we can only observe the discriminator update ", + "bbox": [ + 173, + 138, + 825, + 209 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/854fc98497939e024ca6a6b6523d6b1f6d414dfbb440733759a9e9fd04af9bf4.jpg", + "text": "$$\n\\dot { \\psi } = - \\frac { \\rho } { 2 } \\nabla _ { \\psi } \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ]\n$$", + "text_format": "latex", + "bbox": [ + 382, + 215, + 616, + 242 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "since $\\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] = 0$ for any $\\psi$ and $| \\theta - \\theta ^ { * } | < \\mathrm { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )$ . The discriminator update described above is locally stable system near the equilibrium $\\psi = \\psi ^ { * }$ since the Jacobian of the update on $\\psi$ is given as $- \\rho Q$ and the zero eigenvalues can be ignored in a similar manner to the previous step. Therefore, the given system is stable near the equilibrium. ", + "bbox": [ + 173, + 247, + 825, + 306 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Suppose that $T _ { D }$ is empty. Given that $N ( Q ^ { T } ) \\subset N ( R ^ { T } ) .$ , $R = 0$ , then the results are similar to those presented above, but our goal is to show that $( \\psi , \\theta )$ is an equilibrium point, where $( \\psi , \\theta )$ is sufficiently close to the original equilibrium point. We note that $( \\psi ^ { * } , \\theta )$ is also an equilibrium point that satisfies the assumptions. ", + "bbox": [ + 173, + 324, + 825, + 382 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "By Assumption 2, $h ( \\psi ) = h ( \\psi ^ { * } ) = 0$ for $| \\psi - \\psi ^ { * } | < \\xi$ , which implies that $\\nabla _ { x } D ( x ; \\psi ) = 0$ for $x \\in s u p p ( \\mu _ { \\psi , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )$ and $| \\psi - \\psi ^ { * } | < \\xi$ . Thus, we obtain ", + "bbox": [ + 171, + 387, + 823, + 417 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/17dc89005db468985991016b79ee257f8a13b8344c3cffd801396e23132fb758.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { \\mu _ { \\psi , \\theta ^ { * } } } \\big [ \\nabla _ { \\psi x } ^ { T } D ( \\boldsymbol { x } ; \\psi ) \\nabla _ { x } D ( \\boldsymbol { x } ; \\psi ) \\big ] = 0 } \\\\ { \\displaystyle \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu ^ { * } ) } \\big \\| \\nabla _ { x } D \\big \\| ^ { 2 } d \\mu _ { \\psi , \\theta ^ { * } } ^ { \\prime } d \\boldsymbol { x } = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 424, + 624, + 482 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "By Assumption 4, $\\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi } D ( x ; \\psi ) ] = 0$ since $p _ { d } = p _ { \\theta ^ { \\ast } }$ . In addition, ", + "bbox": [ + 169, + 486, + 766, + 503 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d89dfc390709cebbcfe75ef24ebb0dafdd158dbfffd4596ff0be85e3bf0b0adf.jpg", + "text": "$$\n\\dot { \\theta } = \\frac { \\partial } { \\partial \\theta } \\int _ { \\mathcal { X } } D ( x ; \\psi ) d p _ { \\theta } \\bigg | _ { \\theta = \\theta ^ { * } } = \\int _ { \\mathcal { Z } } \\nabla _ { \\theta } ^ { T } G ( z ; \\theta ^ { * } ) \\nabla _ { x } D ( G ( z ; \\theta ^ { * } ) ; \\psi ) p _ { l a t e n t } ( z ) d z = 0\n$$", + "text_format": "latex", + "bbox": [ + 220, + 510, + 776, + 546 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Therefore, the point $( \\psi , \\theta ^ { * } )$ with $| \\psi - \\psi ^ { * } | < \\operatorname* { m i n } ( \\xi , \\delta _ { d } )$ is an equilibrium point. From Assumption 4, $D ( x ; \\psi )$ is an equilibrium discriminator, and thus $D ( x ; \\psi )$ is already an optimal discriminator for $| \\psi - \\psi ^ { * } | < \\operatorname* { m i n } ( \\xi , \\delta _ { d } )$ and $p _ { \\theta }$ coincides with the data distribution $p _ { d }$ for $| \\theta - \\bar { \\theta } ^ { * } | < \\operatorname* { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } )$ , which indicates that every discriminator and generator near $( \\psi ^ { * } , \\theta ^ { * } )$ is an equilibrium point and this completes the proof of the main theorem. □ ", + "bbox": [ + 171, + 551, + 826, + 622 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "APPENDIX D : DETAILED EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 604, + 118 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/c27519b391c821a8a732501033759f5a0bf8d62bd8c66e63574a2babdc1debb6.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 199, + 142, + 799, + 391 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Figure 4: 2D example on 8 Gaussians, swissroll, 25 Gaussians datasets. Images generated with 5 penalty measures: $\\mu _ { G P } , \\mu _ { m i d } , p _ { g } , p _ { d } , \\mu _ { g , a n c }$ . ", + "bbox": [ + 176, + 416, + 823, + 448 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e9add33f13a815a66a7c30fd8fcb4008bda276bf58cd9026c58fd8877cc49d1a.jpg", + "image_caption": [ + "Figure 5: MNIST example. Images generated with $\\mu _ { G P } , \\mu _ { m i d } , p _ { g } , p _ { d } , \\mu _ { g , a n c }$ " + ], + "image_footnote": [], + "bbox": [ + 204, + 102, + 794, + 500 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/2dd27c24c109293e33374a284d06f390455d2abd1929b0a3508768fbd5710adf.jpg", + "image_caption": [ + "Figure 6: CIFAR-10 example. Images generated with WGAN, WGAN-GP, $\\mu _ { G P } , \\mu _ { m i d } , p _ { g } , p _ { d } , \\mu _ { g , a n c }$ under the DCGAN architecture. " + ], + "image_footnote": [], + "bbox": [ + 173, + 77, + 803, + 691 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/d9c4fa273546a3058613783e5118feaf32c9be46e77bdd5be9b4add130cb07b7.jpg", + "image_caption": [ + "Figure 7: CIFAR-10 example. Images generated with WGAN, WGAN-GP, $\\mu _ { G P } , \\mu _ { m i d } , p _ { g } , p _ { d } , \\mu _ { g , a n c }$ under the ResNet architecture. " + ], + "image_footnote": [], + "bbox": [ + 173, + 98, + 826, + 683 + ], + "page_idx": 19 + } +] \ No newline at end of file diff --git a/parse/train/H1ecDoR5Y7/H1ecDoR5Y7_middle.json b/parse/train/H1ecDoR5Y7/H1ecDoR5Y7_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..ca29f13ae5d7b3d3445e7ee5f671d9097b80c029 --- /dev/null +++ b/parse/train/H1ecDoR5Y7/H1ecDoR5Y7_middle.json @@ -0,0 +1,51892 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 492, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 423, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 423, + 98 + ], + "score": 1.0, + "content": "LOCAL STABILITY AND PERFORMANCE OF", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 494, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 323, + 118 + ], + "score": 1.0, + "content": "SIMPLE GRADIENT PENALTY", + "type": "text" + }, + { + "bbox": [ + 324, + 101, + 336, + 118 + ], + "score": 0.76, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 97, + 494, + 118 + ], + "score": 1.0, + "content": "-WASSERSTEIN GAN", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 138, + 244, + 160 + ], + "lines": [ + { + "bbox": [ + 113, + 138, + 201, + 150 + ], + "spans": [ + { + "bbox": [ + 113, + 138, + 201, + 150 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 149, + 245, + 161 + ], + "spans": [ + { + "bbox": [ + 111, + 149, + 245, + 161 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 189, + 333, + 201 + ], + "lines": [ + { + "bbox": [ + 277, + 189, + 335, + 203 + ], + "spans": [ + { + "bbox": [ + 277, + 189, + 335, + 203 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 218, + 468, + 351 + ], + "lines": [ + { + "bbox": [ + 142, + 218, + 469, + 231 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 469, + 231 + ], + "score": 1.0, + "content": "Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 230, + 469, + 243 + ], + "spans": [ + { + "bbox": [ + 141, + 230, + 469, + 243 + ], + "score": 1.0, + "content": "between a data distribution and sample distribution. 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However, WGAN often fails", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 647, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 663 + ], + "score": 1.0, + "content": "with simple examples because the Lipschitz constraint on discriminator is rarely achieved during", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "the optimization process and weight clipping. Thus, mimicking the Lipschitz constraint on the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 670, + 435, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 435, + 685 + ], + "score": 1.0, + "content": "discriminator by using a gradient penalty was proposed by Gulrajani et al. 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Recent studies have proposed", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 241, + 470, + 254 + ], + "spans": [ + { + "bbox": [ + 141, + 241, + 470, + 254 + ], + "score": 1.0, + "content": "stabilizing the training process for the WGAN and implementing the Lipschitz", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 252, + 469, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 469, + 265 + ], + "score": 1.0, + "content": "constraint. 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The measure valued differentiation concept", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 285, + 470, + 297 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 470, + 297 + ], + "score": 1.0, + "content": "is employed to deal with the derivative of the penalty terms, which is helpful for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 296, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 296, + 470, + 308 + ], + "score": 1.0, + "content": "handling abstract singular measures with lower dimensional support. Based on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 306, + 470, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 320 + ], + "score": 1.0, + "content": "this analysis, we claim that penalizing the data manifold or sample manifold is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 318, + 469, + 330 + ], + "spans": [ + { + "bbox": [ + 142, + 318, + 469, + 330 + ], + "score": 1.0, + "content": "the key to regularizing the original WGAN with a gradient penalty. 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(2014). GANs are capable of modeling data with complex structures.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "For example, DCGAN can sample realistic images using a convolutional neural network (CNN)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "structure(Radford et al., 2015). GANs have been implemented in many applications in the field of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "computer vision with good results, such as super-resolution, image translation, and text-to-image", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 466, + 452, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 452, + 479 + ], + "score": 1.0, + "content": "generation(Ledig et al., 2017; Isola et al., 2017; Zhang et al., 2017; Reed et al., 2016).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 412, + 506, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 483, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "However, despite these successes, GANs are affected by training instability and mode collapse prob-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "lems. GANs often fail to converge, which can result in unrealistic fake samples. 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In", + "type": "text" + }, + { + "bbox": [ + 362, + 567, + 370, + 578 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 565, + 398, + 579 + ], + "score": 1.0, + "content": "-GAN,", + "type": "text" + }, + { + "bbox": [ + 399, + 567, + 406, + 578 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "-divergence between the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "target and generator distributions was suggested which generalizes the divergence between two dis-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "tributions(Nowozin et al., 2016). 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(2018) who provided a proof of its stability.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Based on a theoretical analysis of the dynamic system, Nagarajan & Kolter (2017) proved the local", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "exponential stability of the gradient-based optimization dynamics in GANs by treating the simul-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "taneous gradient descent algorithm with a dynamic system approach. These previous studies were", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "useful because they showed that the local behavior of GANs can be explained using dynamic system", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 285, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 285, + 139 + ], + "score": 1.0, + "content": "tools and the related Jacobian’s eigenvalues.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 446, + 156 + ], + "score": 1.0, + "content": "In this study, we aim to prove the convergence property of the simple gradient penalty", + "type": "text" + }, + { + "bbox": [ + 446, + 145, + 453, + 155 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "-Wasserstein", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 152, + 167 + ], + "score": 1.0, + "content": "GAN(SGP", + "type": "text" + }, + { + "bbox": [ + 153, + 155, + 160, + 166 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 154, + 433, + 167 + ], + "score": 1.0, + "content": "-WGAN) dynamic system under general gradient penalty measures", + "type": "text" + }, + { + "bbox": [ + 433, + 156, + 441, + 166 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ". To the best of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "our knowledge, our study is the first theoretical approach to GAN stability analysis which deals", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "with abstract singular penalty measure. In addition, measure valued differentiation(Heidergott &", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "Vazquez-Abad, 2008) is applied to take the derivative on the integral with a parametric measure, ´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 420, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 420, + 211 + ], + "score": 1.0, + "content": "which is helpful for handling an abstract measure and its integral in our proof.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 108, + 215, + 312, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 314, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 314, + 228 + ], + "score": 1.0, + "content": "The main contributions of this study are as follows.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 132, + 236, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 133, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 133, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "• We prove the regularized effect and local stability of the dynamic system for a gen-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "eral penalty measure under suitable assumptions. The assumptions are written as both a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 142, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "tractable strong version and intractable weak version. To prove the main theorem, we also", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 269, + 491, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 491, + 281 + ], + "score": 1.0, + "content": "introduce the measure valued differentiation concept to handle the parametric measure.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 283, + 504, + 296 + ], + "spans": [ + { + "bbox": [ + 140, + 283, + 504, + 296 + ], + "score": 1.0, + "content": "Based on the proof of the stability, we explain the reason for the success of previous penalty", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "measures. We claim that the support of a penalty measure will be strongly related to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "stability, where the weight on the limiting penalty measure might affect the speed of con-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 318, + 183, + 329 + ], + "spans": [ + { + "bbox": [ + 142, + 318, + 183, + 329 + ], + "score": 1.0, + "content": "vergence.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 136, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 136, + 330, + 505, + 345 + ], + "score": 1.0, + "content": "• We experimentally examined the general convergence results by applying two test penalty", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "measures to several examples. The proposed test measures are unintuitive but they still", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 354, + 497, + 366 + ], + "spans": [ + { + "bbox": [ + 142, + 354, + 497, + 366 + ], + "score": 1.0, + "content": "satisfy the assumptions and similar convergence results were obtained in the experiment.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 208, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 209, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 209, + 395 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "First, we introduce our notations and basic measure-theoretic concepts. Second, we define our SGP", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 114, + 428 + ], + "score": 0.76, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 416, + 504, + 429 + ], + "score": 1.0, + "content": "-WGAN optimization problem and treat this problem as a continuous dynamic system. Preliminary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "measure theoretic concepts are required to justify that the dynamic system changes in a sufficiently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 450 + ], + "score": 1.0, + "content": "smooth manner as the parameter changes, so it is possible to use linearization theorem. They are", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "also important for dealing with the parametric measure and its derivative. The problem setting with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "a simple gradient term is also discussed. The squared gradient size and simple gradient penalty term", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "are used to build a differentiable dynamic system and to apply soft regularization as a resolving con-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "straint, respectively. The continuous dynamic system approach, which is a so-called ODE method,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "is used to analyze the GAN optimization problem with the simultaneous gradient descent algorithm,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 504, + 282, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 282, + 516 + ], + "score": 1.0, + "content": "as described by Nagarajan & Kolter (2017).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 529, + 412, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 414, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 414, + 542 + ], + "score": 1.0, + "content": "2.1 NOTATIONS AND PRELIMINARIES REGARDING MEASURE THEORY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 107, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 107, + 549, + 190, + 561 + ], + "score": 0.91, + "content": "D ( x ; 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Throughout this study, we assume that the measures in the sample space are all", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 185, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 185, + 654 + ], + "score": 1.0, + "content": "finite and bounded.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 104, + 657, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 275, + 671 + ], + "score": 1.0, + "content": "Definition 1. 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These previous studies were", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 129 + ], + "score": 1.0, + "content": "useful because they showed that the local behavior of GANs can be explained using dynamic system", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 285, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 285, + 139 + ], + "score": 1.0, + "content": "tools and the related Jacobian’s eigenvalues.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 505, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 446, + 156 + ], + "score": 1.0, + "content": "In this study, we aim to prove the convergence property of the simple gradient penalty", + "type": "text" + }, + { + "bbox": [ + 446, + 145, + 453, + 155 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "-Wasserstein", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 152, + 167 + ], + "score": 1.0, + "content": "GAN(SGP", + "type": "text" + }, + { + "bbox": [ + 153, + 155, + 160, + 166 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 154, + 433, + 167 + ], + "score": 1.0, + "content": "-WGAN) dynamic system under general gradient penalty measures", + "type": "text" + }, + { + "bbox": [ + 433, + 156, + 441, + 166 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ". To the best of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "our knowledge, our study is the first theoretical approach to GAN stability analysis which deals", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "with abstract singular penalty measure. In addition, measure valued differentiation(Heidergott &", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "Vazquez-Abad, 2008) is applied to take the derivative on the integral with a parametric measure, ´", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 420, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 420, + 211 + ], + "score": 1.0, + "content": "which is helpful for handling an abstract measure and its integral in our proof.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 142, + 506, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 215, + 312, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 314, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 314, + 228 + ], + "score": 1.0, + "content": "The main contributions of this study are as follows.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 213, + 314, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 236, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 133, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 133, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "• We prove the regularized effect and local stability of the dynamic system for a gen-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "eral penalty measure under suitable assumptions. The assumptions are written as both a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 142, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "tractable strong version and intractable weak version. To prove the main theorem, we also", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 269, + 491, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 491, + 281 + ], + "score": 1.0, + "content": "introduce the measure valued differentiation concept to handle the parametric measure.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 283, + 504, + 296 + ], + "spans": [ + { + "bbox": [ + 140, + 283, + 504, + 296 + ], + "score": 1.0, + "content": "Based on the proof of the stability, we explain the reason for the success of previous penalty", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "measures. We claim that the support of a penalty measure will be strongly related to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "stability, where the weight on the limiting penalty measure might affect the speed of con-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 318, + 183, + 329 + ], + "spans": [ + { + "bbox": [ + 142, + 318, + 183, + 329 + ], + "score": 1.0, + "content": "vergence.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 136, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 136, + 330, + 505, + 345 + ], + "score": 1.0, + "content": "• We experimentally examined the general convergence results by applying two test penalty", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "measures to several examples. The proposed test measures are unintuitive but they still", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 354, + 497, + 366 + ], + "spans": [ + { + "bbox": [ + 142, + 354, + 497, + 366 + ], + "score": 1.0, + "content": "satisfy the assumptions and similar convergence results were obtained in the experiment.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 133, + 235, + 505, + 366 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 208, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 209, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 209, + 395 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "First, we introduce our notations and basic measure-theoretic concepts. Second, we define our SGP", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 114, + 428 + ], + "score": 0.76, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 416, + 504, + 429 + ], + "score": 1.0, + "content": "-WGAN optimization problem and treat this problem as a continuous dynamic system. Preliminary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "measure theoretic concepts are required to justify that the dynamic system changes in a sufficiently", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 450 + ], + "score": 1.0, + "content": "smooth manner as the parameter changes, so it is possible to use linearization theorem. They are", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "also important for dealing with the parametric measure and its derivative. The problem setting with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "a simple gradient term is also discussed. The squared gradient size and simple gradient penalty term", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "are used to build a differentiable dynamic system and to apply soft regularization as a resolving con-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "straint, respectively. The continuous dynamic system approach, which is a so-called ODE method,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "is used to analyze the GAN optimization problem with the simultaneous gradient descent algorithm,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 504, + 282, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 282, + 516 + ], + "score": 1.0, + "content": "as described by Nagarajan & Kolter (2017).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 405, + 505, + 516 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 529, + 412, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 414, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 414, + 542 + ], + "score": 1.0, + "content": "2.1 NOTATIONS AND PRELIMINARIES REGARDING MEASURE THEORY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 107, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 107, + 549, + 190, + 561 + ], + "score": 0.91, + "content": "D ( x ; 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For a set of finite measures", + "type": "text" + }, + { + "bbox": [ + 275, + 658, + 308, + 670 + ], + "score": 0.94, + "content": "\\{ \\mu _ { i } \\} _ { i \\in \\mathcal { I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 657, + 321, + 671 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 321, + 658, + 351, + 669 + ], + "score": 0.92, + "content": "( \\mathbb { R } ^ { n } , d )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 657, + 450, + 671 + ], + "score": 1.0, + "content": "with euclidean distance", + "type": "text" + }, + { + "bbox": [ + 450, + 658, + 457, + 668 + ], + "score": 0.31, + "content": "d _ { \\mathrm { { z } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 657, + 460, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 461, + 658, + 495, + 670 + ], + "score": 0.91, + "content": "\\{ \\mu _ { i } \\} _ { i \\in \\mathbb { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 668, + 401, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 277, + 682 + ], + "score": 1.0, + "content": "referred to as bounded if there exists some", + "type": "text" + }, + { + "bbox": [ + 278, + 670, + 307, + 679 + ], + "score": 0.88, + "content": "M > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 668, + 375, + 682 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 375, + 669, + 397, + 679 + ], + "score": 0.88, + "content": "i \\in \\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 668, + 401, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 657, + 506, + 682 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 278, + 685, + 333, + 699 + ], + "lines": [ + { + "bbox": [ + 278, + 685, + 333, + 699 + ], + "spans": [ + { + "bbox": [ + 278, + 685, + 333, + 699 + ], + "score": 0.9, + "content": "\\mu _ { i } ( \\mathbb { R } ^ { n } ) \\leq M", + "type": "interline_equation", + "image_path": "1f3e3bf2c0b71234bc7053e9d719032d7c76614e8813c18b3c6d88dfe18599ba.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 278, + 685, + 333, + 699 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 160, + 723 + ], + "score": 1.0, + "content": "For instance,", + "type": "text" + }, + { + "bbox": [ + 161, + 710, + 173, + 720 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 708, + 244, + 723 + ], + "score": 1.0, + "content": "can be set as 1 if", + "type": "text" + }, + { + "bbox": [ + 244, + 710, + 264, + 722 + ], + "score": 0.92, + "content": "\\{ \\mu _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 708, + 379, + 723 + ], + "score": 1.0, + "content": "are probability measures on", + "type": "text" + }, + { + "bbox": [ + 379, + 711, + 393, + 720 + ], + "score": 0.85, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 708, + 505, + 723 + ], + "score": 1.0, + "content": ". Assuming that the penalty", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "measures are bounded, Portmanteau theorem offers the equivalent definition of the weak conver-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 81, + 504, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 466, + 97 + ], + "score": 1.0, + "content": "gence for finite measures. 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If our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "penalty measure is either absolutely continuous or discrete, then it is easy to deal with the integral.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "However, in the case of singular penalty measure, dealing with the integral term is not an easy task.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "Therefore, we introduce the concept of a weak derivative of a probability measure in the follow-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "ing(Heidergott & Vazquez-Abad, 2008). 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If our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "penalty measure is either absolutely continuous or discrete, then it is easy to deal with the integral.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "However, in the case of singular penalty measure, dealing with the integral term is not an easy task.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "Therefore, we introduce the concept of a weak derivative of a probability measure in the follow-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "ing(Heidergott & Vazquez-Abad, 2008). 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The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 437, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 437, + 529 + ], + "score": 1.0, + "content": "product rule for differentiating can also be applied in a similar manner to calculus.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 505, + 505, + 529 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 533, + 417, + 560 + ], + "lines": [ + { + "bbox": [ + 193, + 533, + 417, + 560 + ], + "spans": [ + { + "bbox": [ + 193, + 533, + 417, + 560 + ], + "score": 0.92, + "content": "\\frac { d } { d \\theta } \\int \\phi ( x ; \\theta ) d P _ { \\theta } = \\int \\nabla _ { \\theta } \\phi ( x ; \\theta ) d P _ { \\theta } + \\int \\phi ( x ; \\theta ) d P _ { \\theta } ^ { \\prime }", + "type": "interline_equation", + "image_path": "79a5b8689162a000bbbbc5d53e4cdca2650cd9d1d4cff86ee28804ef8f7b217b.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 193, + 533, + 417, + 546.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 193, + 546.5, + 417, + 560.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 564, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 275, + 578 + ], + "score": 1.0, + "content": "Therefore, for the general finite measure", + "type": "text" + }, + { + "bbox": [ + 275, + 564, + 339, + 577 + ], + "score": 0.93, + "content": "Q _ { \\theta } = M ( \\theta ) P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 563, + 399, + 578 + ], + "score": 1.0, + "content": ", its derivative", + "type": "text" + }, + { + "bbox": [ + 399, + 565, + 413, + 577 + ], + "score": 0.91, + "content": "Q _ { \\theta } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 563, + 506, + 578 + ], + "score": 1.0, + "content": "can be represented as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 136, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 136, + 589 + ], + "score": 1.0, + "content": "below.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 563, + 506, + 589 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 585, + 446, + 600 + ], + "lines": [ + { + "bbox": [ + 165, + 585, + 446, + 600 + ], + "spans": [ + { + "bbox": [ + 165, + 585, + 446, + 600 + ], + "score": 0.88, + "content": "Q _ { \\theta } ^ { \\prime } = M ^ { \\prime } ( \\theta ) P _ { \\theta } + M ( \\theta ) P _ { \\theta } ^ { \\prime } = M ^ { \\prime } ( \\theta ) P _ { \\theta } + c _ { \\theta } M ( \\theta ) P _ { \\theta } ^ { + } - c _ { \\theta } M ( \\theta ) P _ { \\theta } ^ { - }", + "type": "interline_equation", + "image_path": "a55fa1d1447265063e8b6151dc2adfe2dd6f743e19268f0965e13751407c290c.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 165, + 585, + 446, + 600 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 320, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 320, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 320, + 624 + ], + "score": 1.0, + "content": "2.2 PROBLEM SETTING AS A DYNAMIC SYSTEM", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "Previous work of Mescheder et al. 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The WGAN optimization problem with a simple gradient penalty term", + "type": "text" + }, + { + "bbox": [ + 464, + 193, + 501, + 207 + ], + "score": 0.92, + "content": "\\| \\nabla _ { x } D \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 193, + 504, + 208 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 174, + 218 + ], + "score": 1.0, + "content": "penalty measure", + "type": "text" + }, + { + "bbox": [ + 174, + 208, + 181, + 217 + ], + "score": 0.75, + "content": "\\mu _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 206, + 329, + 218 + ], + "score": 1.0, + "content": ", and penalty weight hyperparameter", + "type": "text" + }, + { + "bbox": [ + 329, + 207, + 354, + 218 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "is given as follows, where the penalty", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 314, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 314, + 229 + ], + "score": 1.0, + "content": "term is only introduced to update the discriminator.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 234, + 430, + 277 + ], + "lines": [ + { + "bbox": [ + 181, + 234, + 430, + 277 + ], + "spans": [ + { + "bbox": [ + 181, + 234, + 430, + 277 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\displaystyle \\operatorname* { m a x } _ { \\psi } : \\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] - \\frac { \\rho } { 2 } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\\\ & { \\displaystyle \\operatorname* { m i n } _ { \\theta } : \\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "b912dbb6e2bbe4f8ba417c3fb9e6808043d8e5aaa17dc4700d78e6e7cd0d8d1a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 181, + 234, + 430, + 248.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 181, + 248.33333333333334, + 430, + 262.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 181, + 262.6666666666667, + 430, + 277.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "According to Nagarajan & Kolter (2017) and many other optimization problem studies, the simul-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "taneous gradient descent algorithm for GAN updating can be viewed as an autonomous dynamic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 443, + 322 + ], + "score": 1.0, + "content": "system of discriminator parameters and generator parameters, which we denote as", + "type": "text" + }, + { + "bbox": [ + 444, + 310, + 452, + 321 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 309, + 471, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 310, + 477, + 319 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 309, + 506, + 322 + ], + "score": 1.0, + "content": ". As a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 321, + 324, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 324, + 332 + ], + "score": 1.0, + "content": "result, the related dynamic system is given as follows.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 336, + 415, + 376 + ], + "lines": [ + { + "bbox": [ + 196, + 336, + 415, + 376 + ], + "spans": [ + { + "bbox": [ + 196, + 336, + 415, + 376 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\dot { \\boldsymbol { \\psi } } = \\mathbb { E } _ { \\boldsymbol { p } _ { d } } [ \\nabla _ { \\boldsymbol { \\psi } } D ] - \\mathbb { E } _ { \\boldsymbol { p } _ { \\boldsymbol { \\theta } } } [ \\nabla _ { \\boldsymbol { \\psi } } D ] - \\frac { \\rho } { 2 } \\nabla _ { \\boldsymbol { \\psi } } \\mathbb { E } _ { \\boldsymbol { \\mu } } [ \\nabla _ { \\boldsymbol { x } } ^ { T } D \\nabla _ { \\boldsymbol { x } } D ] } \\\\ & { \\dot { \\boldsymbol { \\theta } } = \\nabla _ { \\boldsymbol { \\theta } } \\mathbb { E } _ { \\boldsymbol { p } _ { \\boldsymbol { \\theta } } } [ D ] } \\end{array}", + "type": "interline_equation", + "image_path": "0644b4f346d433451357f27f989dccf843641a706b9f912012549e316225522f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 196, + 336, + 415, + 349.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 196, + 349.3333333333333, + 415, + 362.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 196, + 362.66666666666663, + 415, + 375.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 389, + 209, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 209, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 209, + 405 + ], + "score": 1.0, + "content": "3 TOY EXAMPLES", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "score": 1.0, + "content": "We investigate two examples considered in previous studies by Mescheder et al. (2018) and Nagara-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "jan & Kolter (2017). We then generalize the results to a finite measure case. The first example is the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 401, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 401, + 449 + ], + "score": 1.0, + "content": "univariate Dirac GAN, which was introduced by Mescheder et al. (2018).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 504, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 419, + 464 + ], + "score": 1.0, + "content": "Definition 5. (Dirac GAN) The Dirac GAN comprises a linear discriminator", + "type": "text" + }, + { + "bbox": [ + 420, + 452, + 480, + 464 + ], + "score": 0.92, + "content": "D ( x ; \\psi ) = \\psi x", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 451, + 505, + 464 + ], + "score": 1.0, + "content": ", data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 325, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 155, + 477 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 463, + 188, + 474 + ], + "score": 0.92, + "content": "p _ { d } = \\delta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 461, + 289, + 477 + ], + "score": 1.0, + "content": ", and sample distribution", + "type": "text" + }, + { + "bbox": [ + 289, + 463, + 321, + 474 + ], + "score": 0.92, + "content": "p _ { \\theta } = \\delta _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 461, + 325, + 477 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 504, + 496 + ], + "score": 1.0, + "content": "The Dirac GAN with a gradient penalty with an arbitrary probability measure is known to be globally", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "score": 1.0, + "content": "convergent(Mescheder et al., 2018). 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Suppose that", + "type": "text" + }, + { + "bbox": [ + 200, + 710, + 281, + 721 + ], + "score": 0.9, + "content": "\\mu _ { \\psi , \\theta } = M ( \\psi , \\bar { \\theta ) \\mu _ { \\psi , \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 710, + 307, + 722 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 308, + 711, + 326, + 721 + ], + "score": 0.87, + "content": "\\bar { \\mu } _ { \\psi , \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "is normalized to the probability measure. Then,", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 487, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 487, + 733 + ], + "score": 0.8, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] = \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ M ( \\psi , \\theta ) \\| \\nabla _ { x } D \\| ^ { 2 } ] = \\int \\| \\nabla _ { x } D \\| ^ { 2 } M ( \\psi , \\theta ) d \\bar { \\mu } _ { \\psi , \\theta } ( x ) = \\int \\| \\nabla _ { x } D \\| ^ { 2 } d \\mu _ { \\psi , \\theta } ( x ) } \\end{array}", + "type": "inline_equation" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 107 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 190 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 278, + 126 + ], + "score": 1.0, + "content": "This advantage of squared gradient term1,", + "type": "text" + }, + { + "bbox": [ + 279, + 110, + 334, + 124 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 111, + 506, + 126 + ], + "score": 1.0, + "content": ", makes the dynamic system differentiable", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 123, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 505, + 136 + ], + "score": 1.0, + "content": "and we define the WGAN problem with the square of the gradient’s norm as a simple gradient", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "penalty. This simple gradient penalty can be treated as soft regularization based on the size of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 304, + 158 + ], + "score": 1.0, + "content": "discriminator’s gradient, especially in case where", + "type": "text" + }, + { + "bbox": [ + 305, + 147, + 312, + 157 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 146, + 506, + 158 + ], + "score": 1.0, + "content": "is the probability measure (Roth et al., 2017). It", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 504, + 169 + ], + "score": 1.0, + "content": "is convenient to determine whether the system is stable by observing the spectrum of the Jacobian", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 207, + 180 + ], + "score": 1.0, + "content": "matrix. In the following,", + "type": "text" + }, + { + "bbox": [ + 208, + 167, + 288, + 179 + ], + "score": 0.92, + "content": "( D ( x ; \\psi ) , p _ { d } , p _ { \\theta } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 167, + 373, + 180 + ], + "score": 1.0, + "content": "is defined as an SGP", + "type": "text" + }, + { + "bbox": [ + 374, + 168, + 381, + 179 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "-WGAN optimization problem", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 178, + 406, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 395, + 192 + ], + "score": 1.0, + "content": "(SGP-form) with a simple gradient penalty term on the penalty measure", + "type": "text" + }, + { + "bbox": [ + 395, + 181, + 402, + 190 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 178, + 406, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 110, + 506, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 104, + 193, + 504, + 208 + ], + "spans": [ + { + "bbox": [ + 104, + 193, + 464, + 208 + ], + "score": 1.0, + "content": "Definition 4. The WGAN optimization problem with a simple gradient penalty term", + "type": "text" + }, + { + "bbox": [ + 464, + 193, + 501, + 207 + ], + "score": 0.92, + "content": "\\| \\nabla _ { x } D \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 193, + 504, + 208 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 174, + 218 + ], + "score": 1.0, + "content": "penalty measure", + "type": "text" + }, + { + "bbox": [ + 174, + 208, + 181, + 217 + ], + "score": 0.75, + "content": "\\mu _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 206, + 329, + 218 + ], + "score": 1.0, + "content": ", and penalty weight hyperparameter", + "type": "text" + }, + { + "bbox": [ + 329, + 207, + 354, + 218 + ], + "score": 0.9, + "content": "\\rho > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "is given as follows, where the penalty", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 314, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 314, + 229 + ], + "score": 1.0, + "content": "term is only introduced to update the discriminator.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 193, + 505, + 229 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 234, + 430, + 277 + ], + "lines": [ + { + "bbox": [ + 181, + 234, + 430, + 277 + ], + "spans": [ + { + "bbox": [ + 181, + 234, + 430, + 277 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\displaystyle \\operatorname* { m a x } _ { \\psi } : \\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] - \\frac { \\rho } { 2 } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\\\ & { \\displaystyle \\operatorname* { m i n } _ { \\theta } : \\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "b912dbb6e2bbe4f8ba417c3fb9e6808043d8e5aaa17dc4700d78e6e7cd0d8d1a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 181, + 234, + 430, + 248.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 181, + 248.33333333333334, + 430, + 262.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 181, + 262.6666666666667, + 430, + 277.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "According to Nagarajan & Kolter (2017) and many other optimization problem studies, the simul-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "taneous gradient descent algorithm for GAN updating can be viewed as an autonomous dynamic", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 443, + 322 + ], + "score": 1.0, + "content": "system of discriminator parameters and generator parameters, which we denote as", + "type": "text" + }, + { + "bbox": [ + 444, + 310, + 452, + 321 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 309, + 471, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 310, + 477, + 319 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 309, + 506, + 322 + ], + "score": 1.0, + "content": ". As a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 321, + 324, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 324, + 332 + ], + "score": 1.0, + "content": "result, the related dynamic system is given as follows.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 287, + 506, + 332 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 336, + 415, + 376 + ], + "lines": [ + { + "bbox": [ + 196, + 336, + 415, + 376 + ], + "spans": [ + { + "bbox": [ + 196, + 336, + 415, + 376 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\dot { \\boldsymbol { \\psi } } = \\mathbb { E } _ { \\boldsymbol { p } _ { d } } [ \\nabla _ { \\boldsymbol { \\psi } } D ] - \\mathbb { E } _ { \\boldsymbol { p } _ { \\boldsymbol { \\theta } } } [ \\nabla _ { \\boldsymbol { \\psi } } D ] - \\frac { \\rho } { 2 } \\nabla _ { \\boldsymbol { \\psi } } \\mathbb { E } _ { \\boldsymbol { \\mu } } [ \\nabla _ { \\boldsymbol { x } } ^ { T } D \\nabla _ { \\boldsymbol { x } } D ] } \\\\ & { \\dot { \\boldsymbol { \\theta } } = \\nabla _ { \\boldsymbol { \\theta } } \\mathbb { E } _ { \\boldsymbol { p } _ { \\boldsymbol { \\theta } } } [ D ] } \\end{array}", + "type": "interline_equation", + "image_path": "0644b4f346d433451357f27f989dccf843641a706b9f912012549e316225522f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 196, + 336, + 415, + 349.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 196, + 349.3333333333333, + 415, + 362.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 196, + 362.66666666666663, + 415, + 375.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 389, + 209, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 209, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 209, + 405 + ], + "score": 1.0, + "content": "3 TOY EXAMPLES", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "score": 1.0, + "content": "We investigate two examples considered in previous studies by Mescheder et al. (2018) and Nagara-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "jan & Kolter (2017). We then generalize the results to a finite measure case. The first example is the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 401, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 401, + 449 + ], + "score": 1.0, + "content": "univariate Dirac GAN, which was introduced by Mescheder et al. (2018).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 413, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 504, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 419, + 464 + ], + "score": 1.0, + "content": "Definition 5. (Dirac GAN) The Dirac GAN comprises a linear discriminator", + "type": "text" + }, + { + "bbox": [ + 420, + 452, + 480, + 464 + ], + "score": 0.92, + "content": "D ( x ; \\psi ) = \\psi x", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 451, + 505, + 464 + ], + "score": 1.0, + "content": ", data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 325, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 155, + 477 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 463, + 188, + 474 + ], + "score": 0.92, + "content": "p _ { d } = \\delta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 461, + 289, + 477 + ], + "score": 1.0, + "content": ", and sample distribution", + "type": "text" + }, + { + "bbox": [ + 289, + 463, + 321, + 474 + ], + "score": 0.92, + "content": "p _ { \\theta } = \\delta _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 461, + 325, + 477 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 451, + 505, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 504, + 496 + ], + "score": 1.0, + "content": "The Dirac GAN with a gradient penalty with an arbitrary probability measure is known to be globally", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 504, + 507 + ], + "score": 1.0, + "content": "convergent(Mescheder et al., 2018). 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Consider the Dirac GAN problem with SGP form", + "type": "text" + }, + { + "bbox": [ + 360, + 521, + 477, + 533 + ], + "score": 0.91, + "content": "\\begin{array} { r } { ( D ( x ; \\psi ) = \\psi x , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { \\psi , \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 518, + 505, + 535 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 361, + 128 + ], + "score": 1.0, + "content": "dynamic system is locally stable near the desired equilibrium", + "type": "text" + }, + { + "bbox": [ + 361, + 116, + 391, + 127 + ], + "score": 0.89, + "content": "( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ", where the spectrum of the", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 397, + 145 + ], + "lines": [ + { + "bbox": [ + 103, + 127, + 395, + 148 + ], + "spans": [ + { + "bbox": [ + 103, + 127, + 156, + 148 + ], + "score": 1.0, + "content": "Jacobian at", + "type": "text" + }, + { + "bbox": [ + 156, + 131, + 186, + 143 + ], + "score": 0.91, + "content": "( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 127, + 233, + 148 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 233, + 127, + 395, + 147 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda = - 2 \\rho \\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] \\pm \\sqrt { 4 \\rho ^ { 2 } \\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] ^ { 2 } - \\frac { 4 } { 9 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 159, + 293, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 295, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 295, + 174 + ], + "score": 1.0, + "content": "4 MAIN CONVERGENCE THEOREM", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "We propose the convergence property of WGAN with a simple gradient penalty on an arbitrary", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 173, + 209 + ], + "score": 1.0, + "content": "penalty measure", + "type": "text" + }, + { + "bbox": [ + 174, + 197, + 182, + 207 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 195, + 267, + 209 + ], + "score": 1.0, + "content": "for a realizable case:", + "type": "text" + }, + { + "bbox": [ + 268, + 196, + 296, + 206 + ], + "score": 0.9, + "content": "\\theta = \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 195, + 318, + 209 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 318, + 197, + 355, + 207 + ], + "score": 0.89, + "content": "p _ { d } = p _ { \\theta ^ { \\ast } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "exists. 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A more rigorous analysis is given in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 228, + 165, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 165, + 241 + ], + "score": 1.0, + "content": "the Appendix.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 253, + 194, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 195, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 195, + 266 + ], + "score": 1.0, + "content": "4.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "The first assumption is made regarding the equilibrium condition for GANs, where we state the ideal", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "conditions for the discriminator parameter and generator parameter. 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For simplicity, we denote", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 344, + 264, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 193, + 357 + ], + "score": 0.92, + "content": "\\cup _ { x ^ { \\prime } \\in s u p p ( p _ { d } ) } B _ { \\epsilon _ { x ^ { \\prime } } } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 344, + 206, + 358 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 206, + 345, + 260, + 357 + ], + "score": 0.91, + "content": "B ( s u p p ( p _ { d } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 344, + 264, + 358 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "The second assumption ensures that the higher order terms cannot affect the stability of the SGP", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 114, + 387 + ], + "score": 0.74, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "-WGAN. In the Appendix, we consider the case where the WGAN fails to converge when As-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "sumption 2 is not satisfied. Compared with the previous study by Nagarajan & Kolter (2017), the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 397, + 364, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 364, + 409 + ], + "score": 1.0, + "content": "conditions for the discriminator parameter are slightly modified.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 412, + 169, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 170, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 170, + 426 + ], + "score": 1.0, + "content": "Assumption 2.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 428, + 476, + 444 + ], + "lines": [ + { + "bbox": [ + 135, + 428, + 476, + 444 + ], + "spans": [ + { + "bbox": [ + 135, + 428, + 476, + 444 + ], + "score": 0.88, + "content": "\\begin{array} { r } { g ( \\theta ) = \\| \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] - \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] \\| ^ { 2 } , h ( \\psi ) = \\mathbb { E } _ { \\mu _ { \\psi , \\theta ^ { * } } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\end{array}", + "type": "interline_equation", + "image_path": "57cde38dc1d5fe809648c3b6e3c15199978983fa1d367eb6377e6bd493e47cd9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 135, + 428, + 476, + 444 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 362, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 358, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 358, + 460 + ], + "score": 1.0, + "content": "are locally constant along the nullspace of the Hessian matrix.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 504, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "The third assumption allows us to extend our results to discrete probability distribution cases, as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 479, + 259, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 259, + 491 + ], + "score": 1.0, + "content": "described by Mescheder et al. 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A bad equilibrium does not exist near the desired equilibrium point. Thus,", + "type": "text" + }, + { + "bbox": [ + 470, + 541, + 504, + 552 + ], + "score": 0.89, + "content": "( \\psi ^ { * } , \\theta ^ { * } )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 270, + 566 + ], + "score": 1.0, + "content": "is an isolated equilibrium or there exist", + "type": "text" + }, + { + "bbox": [ + 270, + 552, + 314, + 564 + ], + "score": 0.93, + "content": "\\delta _ { d } , \\delta _ { g } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 551, + 457, + 566 + ], + "score": 1.0, + "content": "such that all equilibrium points in", + "type": "text" + }, + { + "bbox": [ + 457, + 552, + 505, + 564 + ], + "score": 0.9, + "content": "B _ { \\delta _ { d } } ( \\psi ^ { * } ) \\times", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 562, + 262, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 141, + 576 + ], + "score": 0.93, + "content": "B _ { \\delta _ { g } } ( \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 562, + 262, + 576 + ], + "score": 1.0, + "content": "satisfy the other assumptions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "The last assumption is related to the necessary conditions for the penalty measure. A calculation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "of the gradient penalty based on samples from the data manifold and generator manifold or the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "interpolation of both was introduced in recent studies (Gulrajani et al., 2017; Roth et al., 2017;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 447, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 447, + 629 + ], + "score": 1.0, + "content": "Mescheder et al., 2018). First, we propose strong conditions for the penalty measure.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 108, + 631, + 407, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 408, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 280, + 645 + ], + "score": 1.0, + "content": "Assumption 5. The finite penalty measure", + "type": "text" + }, + { + "bbox": [ + 280, + 633, + 311, + 642 + ], + "score": 0.9, + "content": "\\mu = \\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 629, + 408, + 645 + ], + "score": 1.0, + "content": "satisfies the followings:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 130, + 650, + 436, + 703 + ], + "lines": [ + { + "bbox": [ + 130, + 650, + 437, + 665 + ], + "spans": [ + { + "bbox": [ + 130, + 650, + 142, + 665 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 142, + 652, + 208, + 663 + ], + "score": 0.89, + "content": "\\mu _ { \\theta } \\to \\mu _ { \\theta ^ { * } } = \\mu ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 650, + 227, + 665 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 227, + 653, + 238, + 663 + ], + "score": 0.78, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 650, + 425, + 665 + ], + "score": 1.0, + "content": "is independent of the discriminator parameter", + "type": "text" + }, + { + "bbox": [ + 426, + 652, + 433, + 663 + ], + "score": 0.82, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 650, + 437, + 665 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 130, + 668, + 402, + 703 + ], + "spans": [ + { + "bbox": [ + 130, + 668, + 402, + 703 + ], + "score": 0.63, + "content": "\\begin{array} { r l } & { b \\ s u p p ( p _ { d } ) \\subset s u p p ( \\mu ^ { * } ) } \\\\ & { c \\ \\exists \\epsilon _ { \\mu } > 0 \\ s u c h \\ t h a t \\ s u p p ( \\mu _ { \\theta } ) \\subset B ( s u p p ( p _ { d } ) ) f o r \\vert \\theta - \\theta ^ { * } \\vert < \\epsilon _ { \\mu } . } \\end{array}", + "type": "inline_equation", + "image_path": "3481b92d85c73f18d26978c32171e1709759867abb05ec1ce246126d4e9d4fac.jpg" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 408, + 722 + ], + "score": 1.0, + "content": "The assumption given above means that the support of the penalty measure", + "type": "text" + }, + { + "bbox": [ + 409, + 712, + 421, + 721 + ], + "score": 0.86, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "should approach the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 218, + 732 + ], + "score": 1.0, + "content": "data manifolds smoothly as", + "type": "text" + }, + { + "bbox": [ + 219, + 721, + 249, + 730 + ], + "score": 0.91, + "content": "\\theta \\to \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 721, + 505, + 732 + ], + "score": 1.0, + "content": ". 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Consider the toy example", + "type": "text" + }, + { + "bbox": [ + 278, + 82, + 474, + 95 + ], + "score": 0.89, + "content": "\\begin{array} { r c l } { ( D ( x ; \\psi ) } & { = } & { \\psi x ^ { 2 } , U ( - 1 , 1 ) , U ( - | \\theta | , | \\theta | ) , \\mu _ { \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 160, + 106 + ], + "score": 0.91, + "content": "U ( 0 , 0 ) = \\delta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 92, + 342, + 107 + ], + "score": 1.0, + "content": "and the ideal equilibrium points are given by", + "type": "text" + }, + { + "bbox": [ + 342, + 94, + 419, + 105 + ], + "score": 0.88, + "content": "( \\psi ^ { * } , \\theta ^ { * } ) = ( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 92, + 505, + 107 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 361, + 128 + ], + "score": 1.0, + "content": "dynamic system is locally stable near the desired equilibrium", + "type": "text" + }, + { + "bbox": [ + 361, + 116, + 391, + 127 + ], + "score": 0.89, + "content": "( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ", where the spectrum of the", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 397, + 145 + ], + "lines": [ + { + "bbox": [ + 103, + 127, + 395, + 148 + ], + "spans": [ + { + "bbox": [ + 103, + 127, + 156, + 148 + ], + "score": 1.0, + "content": "Jacobian at", + "type": "text" + }, + { + "bbox": [ + 156, + 131, + 186, + 143 + ], + "score": 0.91, + "content": "( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 127, + 233, + 148 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 233, + 127, + 395, + 147 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda = - 2 \\rho \\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] \\pm \\sqrt { 4 \\rho ^ { 2 } \\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] ^ { 2 } - \\frac { 4 } { 9 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 103, + 127, + 395, + 148 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 159, + 293, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 295, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 295, + 174 + ], + "score": 1.0, + "content": "4 MAIN CONVERGENCE THEOREM", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 198 + ], + "score": 1.0, + "content": "We propose the convergence property of WGAN with a simple gradient penalty on an arbitrary", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 173, + 209 + ], + "score": 1.0, + "content": "penalty measure", + "type": "text" + }, + { + "bbox": [ + 174, + 197, + 182, + 207 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 195, + 267, + 209 + ], + "score": 1.0, + "content": "for a realizable case:", + "type": "text" + }, + { + "bbox": [ + 268, + 196, + 296, + 206 + ], + "score": 0.9, + "content": "\\theta = \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 195, + 318, + 209 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 318, + 197, + 355, + 207 + ], + "score": 0.89, + "content": "p _ { d } = p _ { \\theta ^ { \\ast } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "exists. In subsection 4.1, we provide", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "the necessary assumptions, which comprise our main convergence theorem. In subsection 4.2, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "give the main convergence theorem with a sketch of the proof. A more rigorous analysis is given in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 228, + 165, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 165, + 241 + ], + "score": 1.0, + "content": "the Appendix.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 184, + 506, + 241 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 253, + 194, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 195, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 195, + 266 + ], + "score": 1.0, + "content": "4.1 ASSUMPTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "The first assumption is made regarding the equilibrium condition for GANs, where we state the ideal", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 297 + ], + "score": 1.0, + "content": "conditions for the discriminator parameter and generator parameter. As the parameters converge to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 286, + 308 + ], + "score": 1.0, + "content": "the ideal equilibrium, the sample distribution", + "type": "text" + }, + { + "bbox": [ + 286, + 296, + 303, + 307 + ], + "score": 0.83, + "content": "\\left( p _ { \\theta } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 295, + 455, + 308 + ], + "score": 1.0, + "content": "converges to the real data distribution", + "type": "text" + }, + { + "bbox": [ + 455, + 296, + 472, + 307 + ], + "score": 0.85, + "content": "\\left( p _ { d } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 307, + 394, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 394, + 319 + ], + "score": 1.0, + "content": "discriminator cannot distinguish the generated sample and the real data.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 272, + 506, + 319 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 175, + 334 + ], + "score": 1.0, + "content": "Assumption 1.", + "type": "text" + }, + { + "bbox": [ + 176, + 322, + 220, + 333 + ], + "score": 0.85, + "content": "p _ { \\theta } p _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 319, + 236, + 334 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 236, + 321, + 275, + 331 + ], + "score": 0.88, + "content": "\\theta ~ \\to ~ \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 319, + 298, + 334 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 321, + 363, + 333 + ], + "score": 0.9, + "content": "D ( x ; \\psi ^ { * } ) ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 319, + 402, + 334 + ], + "score": 1.0, + "content": "on supp", + "type": "text" + }, + { + "bbox": [ + 402, + 321, + 420, + 333 + ], + "score": 0.51, + "content": "\\left( p _ { d } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 319, + 505, + 334 + ], + "score": 1.0, + "content": "and its small open", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 330, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 187, + 347 + ], + "score": 1.0, + "content": "neighborhood, i.e.,", + "type": "text" + }, + { + "bbox": [ + 187, + 333, + 296, + 345 + ], + "score": 0.89, + "content": "x \\in \\cup _ { x ^ { \\prime } \\in s u p p ( p _ { d } ) } B _ { \\epsilon _ { x ^ { \\prime } } } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 330, + 330, + 347 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 331, + 333, + 392, + 344 + ], + "score": 0.91, + "content": "D ( x ; \\psi ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 330, + 506, + 347 + ], + "score": 1.0, + "content": ". For simplicity, we denote", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 344, + 264, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 193, + 357 + ], + "score": 0.92, + "content": "\\cup _ { x ^ { \\prime } \\in s u p p ( p _ { d } ) } B _ { \\epsilon _ { x ^ { \\prime } } } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 344, + 206, + 358 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 206, + 345, + 260, + 357 + ], + "score": 0.91, + "content": "B ( s u p p ( p _ { d } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 344, + 264, + 358 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 319, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "The second assumption ensures that the higher order terms cannot affect the stability of the SGP", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 114, + 387 + ], + "score": 0.74, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "-WGAN. In the Appendix, we consider the case where the WGAN fails to converge when As-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "sumption 2 is not satisfied. Compared with the previous study by Nagarajan & Kolter (2017), the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 397, + 364, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 364, + 409 + ], + "score": 1.0, + "content": "conditions for the discriminator parameter are slightly modified.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 365, + 505, + 409 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 412, + 169, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 170, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 170, + 426 + ], + "score": 1.0, + "content": "Assumption 2.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 428, + 476, + 444 + ], + "lines": [ + { + "bbox": [ + 135, + 428, + 476, + 444 + ], + "spans": [ + { + "bbox": [ + 135, + 428, + 476, + 444 + ], + "score": 0.88, + "content": "\\begin{array} { r } { g ( \\theta ) = \\| \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] - \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] \\| ^ { 2 } , h ( \\psi ) = \\mathbb { E } _ { \\mu _ { \\psi , \\theta ^ { * } } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\end{array}", + "type": "interline_equation", + "image_path": "57cde38dc1d5fe809648c3b6e3c15199978983fa1d367eb6377e6bd493e47cd9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 135, + 428, + 476, + 444 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 362, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 358, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 358, + 460 + ], + "score": 1.0, + "content": "are locally constant along the nullspace of the Hessian matrix.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 447, + 358, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 504, + 491 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "The third assumption allows us to extend our results to discrete probability distribution cases, as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 479, + 259, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 259, + 491 + ], + "score": 1.0, + "content": "described by Mescheder et al. (2018).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 466, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 493, + 403, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 403, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 172, + 509 + ], + "score": 1.0, + "content": "Assumption 3.", + "type": "text" + }, + { + "bbox": [ + 172, + 493, + 206, + 506 + ], + "score": 0.91, + "content": "\\exists \\epsilon _ { g } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 491, + 246, + 509 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 246, + 493, + 303, + 506 + ], + "score": 0.87, + "content": "D ( x ; \\psi ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 491, + 316, + 509 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 317, + 494, + 401, + 507 + ], + "score": 0.62, + "content": "\\cup _ { | \\theta - \\theta ^ { * } | < \\epsilon _ { g } } s u p p ( p _ { \\theta } ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 491, + 403, + 509 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 491, + 403, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 467, + 530 + ], + "score": 1.0, + "content": "The fourth assumption indicates that there are no other “bad” equilibrium points near", + "type": "text" + }, + { + "bbox": [ + 468, + 515, + 501, + 528 + ], + "score": 0.91, + "content": "( \\psi ^ { * } , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 514, + 506, + 530 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 525, + 410, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 410, + 540 + ], + "score": 1.0, + "content": "which justifies the projection along the axis perpendicular to the null space.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 514, + 506, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 470, + 555 + ], + "score": 1.0, + "content": "Assumption 4. A bad equilibrium does not exist near the desired equilibrium point. Thus,", + "type": "text" + }, + { + "bbox": [ + 470, + 541, + 504, + 552 + ], + "score": 0.89, + "content": "( \\psi ^ { * } , \\theta ^ { * } )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 270, + 566 + ], + "score": 1.0, + "content": "is an isolated equilibrium or there exist", + "type": "text" + }, + { + "bbox": [ + 270, + 552, + 314, + 564 + ], + "score": 0.93, + "content": "\\delta _ { d } , \\delta _ { g } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 551, + 457, + 566 + ], + "score": 1.0, + "content": "such that all equilibrium points in", + "type": "text" + }, + { + "bbox": [ + 457, + 552, + 505, + 564 + ], + "score": 0.9, + "content": "B _ { \\delta _ { d } } ( \\psi ^ { * } ) \\times", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 562, + 262, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 141, + 576 + ], + "score": 0.93, + "content": "B _ { \\delta _ { g } } ( \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 562, + 262, + 576 + ], + "score": 1.0, + "content": "satisfy the other assumptions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 540, + 505, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "The last assumption is related to the necessary conditions for the penalty measure. A calculation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "of the gradient penalty based on samples from the data manifold and generator manifold or the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "interpolation of both was introduced in recent studies (Gulrajani et al., 2017; Roth et al., 2017;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 447, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 447, + 629 + ], + "score": 1.0, + "content": "Mescheder et al., 2018). First, we propose strong conditions for the penalty measure.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 583, + 505, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 631, + 407, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 408, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 280, + 645 + ], + "score": 1.0, + "content": "Assumption 5. The finite penalty measure", + "type": "text" + }, + { + "bbox": [ + 280, + 633, + 311, + 642 + ], + "score": 0.9, + "content": "\\mu = \\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 629, + 408, + 645 + ], + "score": 1.0, + "content": "satisfies the followings:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 629, + 408, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 650, + 436, + 703 + ], + "lines": [ + { + "bbox": [ + 130, + 650, + 437, + 665 + ], + "spans": [ + { + "bbox": [ + 130, + 650, + 142, + 665 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 142, + 652, + 208, + 663 + ], + "score": 0.89, + "content": "\\mu _ { \\theta } \\to \\mu _ { \\theta ^ { * } } = \\mu ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 650, + 227, + 665 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 227, + 653, + 238, + 663 + ], + "score": 0.78, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 650, + 425, + 665 + ], + "score": 1.0, + "content": "is independent of the discriminator parameter", + "type": "text" + }, + { + "bbox": [ + 426, + 652, + 433, + 663 + ], + "score": 0.82, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 650, + 437, + 665 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 130, + 668, + 402, + 703 + ], + "spans": [ + { + "bbox": [ + 130, + 668, + 402, + 703 + ], + "score": 0.63, + "content": "\\begin{array} { r l } & { b \\ s u p p ( p _ { d } ) \\subset s u p p ( \\mu ^ { * } ) } \\\\ & { c \\ \\exists \\epsilon _ { \\mu } > 0 \\ s u c h \\ t h a t \\ s u p p ( \\mu _ { \\theta } ) \\subset B ( s u p p ( p _ { d } ) ) f o r \\vert \\theta - \\theta ^ { * } \\vert < \\epsilon _ { \\mu } . } \\end{array}", + "type": "inline_equation", + "image_path": "3481b92d85c73f18d26978c32171e1709759867abb05ec1ce246126d4e9d4fac.jpg" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 130, + 650, + 437, + 703 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 408, + 722 + ], + "score": 1.0, + "content": "The assumption given above means that the support of the penalty measure", + "type": "text" + }, + { + "bbox": [ + 409, + 712, + 421, + 721 + ], + "score": 0.86, + "content": "\\mu _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "should approach the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 218, + 732 + ], + "score": 1.0, + "content": "data manifolds smoothly as", + "type": "text" + }, + { + "bbox": [ + 219, + 721, + 249, + 730 + ], + "score": 0.91, + "content": "\\theta \\to \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 721, + 505, + 732 + ], + "score": 1.0, + "content": ". However, the penalty measure from WGAN-GP with a simple", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "gradient penalty still reaches equilibrium without satisfying Assumption 5c. Therefore, we suggest", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 387, + 107 + ], + "score": 1.0, + "content": "Assumption 6, which is a weak version of Assumption 5. Assumption", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 388, + 93, + 402, + 104 + ], + "score": 0.89, + "content": "6 \\mathrm { a } ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 403, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "is technically required to", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 405, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 240, + 120 + ], + "score": 1.0, + "content": "take the derivative of the integral", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 240, + 105, + 330, + 119 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 331, + 105, + 393, + 120 + ], + "score": 1.0, + "content": "with respect to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 393, + 106, + 401, + 118 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 401, + 105, + 405, + 120 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "gradient penalty still reaches equilibrium without satisfying Assumption 5c. Therefore, we suggest", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 387, + 107 + ], + "score": 1.0, + "content": "Assumption 6, which is a weak version of Assumption 5. Assumption", + "type": "text" + }, + { + "bbox": [ + 388, + 93, + 402, + 104 + ], + "score": 0.89, + "content": "6 \\mathrm { a } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "is technically required to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 405, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 240, + 120 + ], + "score": 1.0, + "content": "take the derivative of the integral", + "type": "text" + }, + { + "bbox": [ + 240, + 105, + 330, + 119 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 105, + 393, + 120 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 393, + 106, + 401, + 118 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 105, + 405, + 120 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 119, + 505, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 413, + 133 + ], + "score": 1.0, + "content": "Assumption 6. (Weak version of Assumption 5) The finite penalty measure", + "type": "text" + }, + { + "bbox": [ + 414, + 122, + 454, + 132 + ], + "score": 0.9, + "content": "\\mu = \\mu _ { \\psi , \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 118, + 505, + 133 + ], + "score": 1.0, + "content": "satisfies the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 128, + 150, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 150, + 145 + ], + "score": 1.0, + "content": "following.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 131, + 149, + 505, + 184 + ], + "lines": [ + { + "bbox": [ + 129, + 148, + 503, + 165 + ], + "spans": [ + { + "bbox": [ + 129, + 148, + 142, + 165 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 142, + 151, + 231, + 162 + ], + "score": 0.89, + "content": "\\mu _ { \\psi , \\theta } \\mu _ { \\psi ^ { * } , \\theta ^ { * } } = \\mu ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 148, + 284, + 165 + ], + "score": 1.0, + "content": ", where supp", + "type": "text" + }, + { + "bbox": [ + 285, + 150, + 311, + 162 + ], + "score": 0.73, + "content": "( \\mu _ { \\psi , \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 148, + 381, + 165 + ], + "score": 1.0, + "content": "only depends on", + "type": "text" + }, + { + "bbox": [ + 382, + 150, + 388, + 160 + ], + "score": 0.6, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 148, + 484, + 165 + ], + "score": 1.0, + "content": ". Near the equilibrium,", + "type": "text" + }, + { + "bbox": [ + 484, + 153, + 503, + 162 + ], + "score": 0.85, + "content": "\\mu _ { \\psi , \\theta }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 141, + 160, + 351, + 173 + ], + "score": 1.0, + "content": "can be weakly differentiated twice with respect to", + "type": "text" + }, + { + "bbox": [ + 351, + 162, + 358, + 172 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 160, + 455, + 173 + ], + "score": 1.0, + "content": ". In addition, its mass", + "type": "text" + }, + { + "bbox": [ + 455, + 161, + 505, + 173 + ], + "score": 0.92, + "content": "M ( \\psi , \\theta ) \\stackrel { \\cdot } { = }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 170, + 474, + 186 + ], + "spans": [ + { + "bbox": [ + 142, + 171, + 180, + 185 + ], + "score": 0.91, + "content": "\\int 1 d \\mu _ { \\psi , \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 170, + 324, + 186 + ], + "score": 1.0, + "content": "is a twice-differentiable function of", + "type": "text" + }, + { + "bbox": [ + 324, + 172, + 332, + 183 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 170, + 474, + 186 + ], + "score": 1.0, + "content": "and bounded near the equilibrium.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 131, + 190, + 401, + 205 + ], + "lines": [ + { + "bbox": [ + 130, + 188, + 399, + 208 + ], + "spans": [ + { + "bbox": [ + 130, + 188, + 142, + 208 + ], + "score": 1.0, + "content": "b", + "type": "text" + }, + { + "bbox": [ + 142, + 190, + 221, + 205 + ], + "score": 0.92, + "content": "E _ { \\mu ^ { * } } [ \\nabla _ { \\psi x } D \\nabla _ { \\psi x } ^ { T } D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 188, + 310, + 208 + ], + "score": 1.0, + "content": "is positive definite or", + "type": "text" + }, + { + "bbox": [ + 310, + 191, + 399, + 204 + ], + "score": 0.79, + "content": "s u p p ( p _ { d } ) \\subset s u p p ( \\mu ^ { * } ) .", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 130, + 210, + 492, + 224 + ], + "lines": [ + { + "bbox": [ + 131, + 210, + 493, + 226 + ], + "spans": [ + { + "bbox": [ + 131, + 210, + 137, + 226 + ], + "score": 1.0, + "content": "c", + "type": "text" + }, + { + "bbox": [ + 137, + 212, + 177, + 224 + ], + "score": 0.8, + "content": "\\ : \\exists \\epsilon _ { \\mu } > 0 \\ :", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 210, + 220, + 226 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 221, + 211, + 278, + 224 + ], + "score": 0.53, + "content": "\\operatorname { s u p p } ( \\mu _ { \\theta } ) \\subset V", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 210, + 293, + 226 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 293, + 212, + 349, + 224 + ], + "score": 0.43, + "content": "| \\theta - \\theta ^ { * } | < \\epsilon _ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 210, + 380, + 226 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 380, + 211, + 490, + 224 + ], + "score": 0.89, + "content": "V = \\{ x | \\nabla _ { x } D ( x ; \\psi ^ { * } ) = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 210, + 493, + 226 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "The assumption above implies the following situations; The penalty measure’s support approaches", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 381, + 255 + ], + "score": 1.0, + "content": "to data manifold and its weight changes smoothly with respect to", + "type": "text" + }, + { + "bbox": [ + 381, + 243, + 389, + 253 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 241, + 410, + 255 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 410, + 243, + 416, + 252 + ], + "score": 0.76, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 241, + 505, + 255 + ], + "score": 1.0, + "content": ". At the equilibrium,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 254, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 505, + 265 + ], + "score": 1.0, + "content": "penalty measure’s support contains data manifold. Also, ideal discriminator will remain flat on the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 161, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 161, + 277 + ], + "score": 1.0, + "content": "penalty area.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "In summary, the gradient penalty regularization term with any penalty measure where the support", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 154, + 305 + ], + "score": 1.0, + "content": "approaches", + "type": "text" + }, + { + "bbox": [ + 154, + 292, + 208, + 304 + ], + "score": 0.92, + "content": "B ( s u p p ( p _ { d } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "in a smooth manner works well and this main result can explain the regu-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 499, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 368, + 316 + ], + "score": 1.0, + "content": "larization effect of previously proposed penalty measures such as", + "type": "text" + }, + { + "bbox": [ + 368, + 304, + 415, + 315 + ], + "score": 0.27, + "content": "\\mu _ { G P } , p _ { d } , p _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 302, + 499, + 316 + ], + "score": 1.0, + "content": ", and their mixtures.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 327, + 271, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 272, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 272, + 339 + ], + "score": 1.0, + "content": "4.2 MAIN CONVERGENCE THEOREM", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "According to the modified assumptions given above, we prove that the related dynamic system is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "locally stable near the equilibrium. The tools used for analyzing stability are mainly based on those", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "described by Nagarajan & Kolter (2017). Our main contributions comprise proposing the necessary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 504, + 394 + ], + "score": 1.0, + "content": "conditions for the penalty measure and proving the local stability for all penalty measures that satisfy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 167, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 167, + 404 + ], + "score": 1.0, + "content": "Assumption 6.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 251, + 418 + ], + "score": 1.0, + "content": "Theorem 1. Suppose that our SGP", + "type": "text" + }, + { + "bbox": [ + 251, + 406, + 258, + 416 + ], + "score": 0.74, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 404, + 379, + 418 + ], + "score": 1.0, + "content": "-WGAN optimization problem", + "type": "text" + }, + { + "bbox": [ + 380, + 405, + 435, + 417 + ], + "score": 0.92, + "content": "( D , p _ { d } , p _ { \\theta } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "with equilibrium", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 131, + 429 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 131, + 416, + 164, + 428 + ], + "score": 0.94, + "content": "( \\psi ^ { * } , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "satisfies the assumptions given above. Then, the related dynamic system is locally", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 427, + 209, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 209, + 439 + ], + "score": 1.0, + "content": "stable at the equilibrium.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "A detailed proof of the main convergence theorem is given in the Appendix. A sketch of the proof is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "given in three steps. First, the undesired terms in the Jacobian matrix of the system at the equilibrium", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 468, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 395, + 489 + ], + "score": 1.0, + "content": "are cancelled out. Next, the Jacobian matrix at equilibrium is given by", + "type": "text" + }, + { + "bbox": [ + 395, + 468, + 452, + 494 + ], + "score": 0.93, + "content": "\\left[ \\begin{array} { c c } { \\therefore } & { \\mathbf { 0 } } \\\\ { R ^ { T } } & { 0 } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 473, + 483, + 489 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 483, + 474, + 506, + 486 + ], + "score": 0.84, + "content": "Q =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 490, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 184, + 507 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mu ^ { * } } [ \\nabla _ { \\psi x } D \\nabla _ { \\psi x } ^ { T } D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 490, + 204, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 493, + 307, + 506 + ], + "score": 0.94, + "content": "R = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ] | _ { \\theta = \\theta ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 490, + 477, + 510 + ], + "score": 1.0, + "content": ". The system is locally stable when both", + "type": "text" + }, + { + "bbox": [ + 477, + 494, + 486, + 505 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 490, + 507, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 505, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 129, + 517 + ], + "score": 0.88, + "content": "R ^ { T } R", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 505, + 506, + 521 + ], + "score": 1.0, + "content": "are positive definite. We can complete the proof by dealing with zero eigenvalues by showing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 497, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 124, + 532 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 517, + 200, + 530 + ], + "score": 0.93, + "content": "{ \\cal N } ( Q ^ { \\hat { T } } ) \\subset { \\cal N } ( R ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 517, + 497, + 532 + ], + "score": 1.0, + "content": "and the projected system’s stability implies the original system’s stability.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 548 + ], + "score": 1.0, + "content": "Our analysis mainly focuses on WGAN, which is the simplest case of general GAN minimax opti-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 546, + 145, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 145, + 558 + ], + "score": 1.0, + "content": "mization", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 556, + 448, + 599 + ], + "lines": [ + { + "bbox": [ + 163, + 556, + 448, + 599 + ], + "spans": [ + { + "bbox": [ + 163, + 556, + 448, + 599 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\underset { \\psi } { \\operatorname* { m a x } } : \\mathbb { E } _ { p _ { d } } [ f ( D ( x ; \\psi ) ) ] + \\mathbb { E } _ { p _ { \\theta } } [ f ( - D ( x ; \\psi ) ) ] - \\frac { \\rho } { 2 } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ] } \\\\ & { \\underset { \\theta } { \\operatorname* { m i n } } : \\mathbb { E } _ { p _ { d } } [ f ( D ( x ; \\psi ) ) ] + \\mathbb { E } _ { p _ { \\theta } } [ f ( - D ( x ; \\psi ) ) ] } \\end{array}", + "type": "interline_equation", + "image_path": "c9cfe94c0eec51cce17d86c66b5deff479567e4073bef83271dac237aa8ce112.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 163, + 556, + 448, + 570.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 163, + 570.3333333333334, + 448, + 584.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 163, + 584.6666666666667, + 448, + 599.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 502, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 128, + 613 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 600, + 172, + 612 + ], + "score": 0.92, + "content": "f ( x ) = x", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 599, + 475, + 613 + ], + "score": 1.0, + "content": ". Similar approach is still valid for general GANs with concave function", + "type": "text" + }, + { + "bbox": [ + 475, + 601, + 483, + 612 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 610, + 214, + 625 + ], + "spans": [ + { + "bbox": [ + 107, + 612, + 151, + 623 + ], + "score": 0.92, + "content": "f ^ { \\prime \\prime } ( x ) < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 610, + 168, + 625 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 611, + 209, + 623 + ], + "score": 0.93, + "content": "f ^ { \\prime } ( 0 ) \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 610, + 214, + 625 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 108, + 638, + 257, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 258, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 258, + 652 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 662, + 504, + 685 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 675 + ], + "score": 1.0, + "content": "We claim that every penalty measure that satisfies the assumptions can regularize the WGAN and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 506, + 687 + ], + "score": 1.0, + "content": "generate similar results to the recently proposed gradient penalty methods. Several penalty measures", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 691, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 689, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 117, + 689, + 505, + 704 + ], + "score": 1.0, + "content": "2This condition is technically required to handle the derivative of the measure in a convenient manner using", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 702, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 712 + ], + "score": 1.0, + "content": "the weak formulation. Even if the measure is not differentiable, it may possible to differentiate the integral. For", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 140, + 723 + ], + "score": 1.0, + "content": "instance,", + "type": "text" + }, + { + "bbox": [ + 141, + 712, + 151, + 722 + ], + "score": 0.85, + "content": "\\delta _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "is continuous but it does not have its weak derivative. 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At the equilibrium,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 254, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 505, + 265 + ], + "score": 1.0, + "content": "penalty measure’s support contains data manifold. Also, ideal discriminator will remain flat on the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 161, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 161, + 277 + ], + "score": 1.0, + "content": "penalty area.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 230, + 505, + 277 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "In summary, the gradient penalty regularization term with any penalty measure where the support", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 154, + 305 + ], + "score": 1.0, + "content": "approaches", + "type": "text" + }, + { + "bbox": [ + 154, + 292, + 208, + 304 + ], + "score": 0.92, + "content": "B ( s u p p ( p _ { d } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 292, + 505, + 305 + ], + "score": 1.0, + "content": "in a smooth manner works well and this main result can explain the regu-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 499, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 368, + 316 + ], + "score": 1.0, + "content": "larization effect of previously proposed penalty measures such as", + "type": "text" + }, + { + "bbox": [ + 368, + 304, + 415, + 315 + ], + "score": 0.27, + "content": "\\mu _ { G P } , p _ { d } , p _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 302, + 499, + 316 + ], + "score": 1.0, + "content": ", and their mixtures.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 280, + 505, + 316 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 327, + 271, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 272, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 272, + 339 + ], + "score": 1.0, + "content": "4.2 MAIN CONVERGENCE THEOREM", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 361 + ], + "score": 1.0, + "content": "According to the modified assumptions given above, we prove that the related dynamic system is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "locally stable near the equilibrium. The tools used for analyzing stability are mainly based on those", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "described by Nagarajan & Kolter (2017). Our main contributions comprise proposing the necessary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 504, + 394 + ], + "score": 1.0, + "content": "conditions for the penalty measure and proving the local stability for all penalty measures that satisfy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 167, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 167, + 404 + ], + "score": 1.0, + "content": "Assumption 6.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 347, + 505, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 251, + 418 + ], + "score": 1.0, + "content": "Theorem 1. Suppose that our SGP", + "type": "text" + }, + { + "bbox": [ + 251, + 406, + 258, + 416 + ], + "score": 0.74, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 404, + 379, + 418 + ], + "score": 1.0, + "content": "-WGAN optimization problem", + "type": "text" + }, + { + "bbox": [ + 380, + 405, + 435, + 417 + ], + "score": 0.92, + "content": "( D , p _ { d } , p _ { \\theta } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 404, + 505, + 418 + ], + "score": 1.0, + "content": "with equilibrium", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 131, + 429 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 131, + 416, + 164, + 428 + ], + "score": 0.94, + "content": "( \\psi ^ { * } , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "satisfies the assumptions given above. Then, the related dynamic system is locally", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 427, + 209, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 209, + 439 + ], + "score": 1.0, + "content": "stable at the equilibrium.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 404, + 505, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "A detailed proof of the main convergence theorem is given in the Appendix. A sketch of the proof is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "given in three steps. First, the undesired terms in the Jacobian matrix of the system at the equilibrium", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 468, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 395, + 489 + ], + "score": 1.0, + "content": "are cancelled out. Next, the Jacobian matrix at equilibrium is given by", + "type": "text" + }, + { + "bbox": [ + 395, + 468, + 452, + 494 + ], + "score": 0.93, + "content": "\\left[ \\begin{array} { c c } { \\therefore } & { \\mathbf { 0 } } \\\\ { R ^ { T } } & { 0 } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 473, + 483, + 489 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 483, + 474, + 506, + 486 + ], + "score": 0.84, + "content": "Q =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 490, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 184, + 507 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { \\mu ^ { * } } [ \\nabla _ { \\psi x } D \\nabla _ { \\psi x } ^ { T } D ]", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 490, + 204, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 204, + 493, + 307, + 506 + ], + "score": 0.94, + "content": "R = \\nabla _ { \\theta } \\mathbb { E } _ { p _ { \\theta } } [ \\nabla _ { \\psi } D ] | _ { \\theta = \\theta ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 490, + 477, + 510 + ], + "score": 1.0, + "content": ". The system is locally stable when both", + "type": "text" + }, + { + "bbox": [ + 477, + 494, + 486, + 505 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 490, + 507, + 510 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 505, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 107, + 506, + 129, + 517 + ], + "score": 0.88, + "content": "R ^ { T } R", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 505, + 506, + 521 + ], + "score": 1.0, + "content": "are positive definite. 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PenaltyPenalty termPenalty measure, sampling method
WGAN WGAN-GPNone(Weight Clipping) Eμ[(IVxD|-1)2]None x=axd+(1-α)xg
Eμ[VxD|2]
Pg PdEμ[VxDii2]x=xg x=xd
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μmidEμ[VxDii2]x= 0.5xd+0.5xg
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PenaltyPenalty termPenalty measure, sampling method
WGAN WGAN-GPNone(Weight Clipping) Eμ[(IVxD|-1)2]None x=axd+(1-α)xg
Eμ[VxD|2]
Pg PdEμ[VxDii2]x=xg x=xd
μGPEμVDii2]x=axd+(1-α)xg
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μg,ancEμVDii2]x=αA+(1-α)xg
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PenaltyDCGAN Inception FIDResNet InceptionFID
WGAN 35.64 ± 0.09 48.7==
WGAN-GP6.48 ± 0.10 35.07.82 ± 0.0918.1
Pg6.46 ± 0.09 38.07.63 ± 0.1020.9
pd6.33 ± 0.0738.9 7.63 ± 0.0920.3
μGP6.40 ±0.0835.4 7.60 ± 0.0918.3
μmid6.60 ± 0.0733.9 7.86 ± 0.0716.4
μg,anc6.45 ± 0.0733.7 7.36 ± 0.0922.4
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PenaltyDCGAN Inception FIDResNet InceptionFID
WGAN 35.64 ± 0.09 48.7==
WGAN-GP6.48 ± 0.10 35.07.82 ± 0.0918.1
Pg6.46 ± 0.09 38.07.63 ± 0.1020.9
pd6.33 ± 0.0738.9 7.63 ± 0.0920.3
μGP6.40 ±0.0835.4 7.60 ± 0.0918.3
μmid6.60 ± 0.0733.9 7.86 ± 0.0716.4
μg,anc6.45 ± 0.0733.7 7.36 ± 0.0922.4
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Stackgan: Text to photo-realistic image synthesis with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 115, + 133, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 115, + 133, + 505, + 148 + ], + "score": 1.0, + "content": "stacked generative adversarial networks. 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The related dynamic system of", + "type": "text" + }, + { + "bbox": [ + 315, + 106, + 432, + 119 + ], + "score": 0.91, + "content": "\\begin{array} { r } { ( D ( x ; \\psi ) = \\psi x , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { \\psi , \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 105, + 505, + 120 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 115, + 142, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 142, + 131 + ], + "score": 1.0, + "content": "follows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 132, + 360, + 171 + ], + "lines": [ + { + "bbox": [ + 250, + 132, + 360, + 171 + ], + "spans": [ + { + "bbox": [ + 250, + 132, + 360, + 171 + ], + "score": 0.92, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\psi } = - \\theta - \\frac { \\rho } { 2 } \\nabla \\psi \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\psi ^ { 2 } ] } \\\\ { \\displaystyle \\dot { \\theta } = \\psi } \\end{array}", + "type": "interline_equation", + "image_path": "73bdd30368acc5ebfd92175ccbcc4e3aa653d0a11d2e143c01ca06dae90d2a83.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 132, + 360, + 151.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 250, + 151.5, + 360, + 171.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 175, + 375, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 375, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 283, + 189 + ], + "score": 1.0, + "content": "First, the only equilibrium point is given by", + "type": "text" + }, + { + "bbox": [ + 283, + 175, + 352, + 188 + ], + "score": 0.91, + "content": "( \\psi ^ { * } , \\theta ^ { * } ) = ( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 174, + 375, + 189 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 193, + 388, + 223 + ], + "lines": [ + { + "bbox": [ + 221, + 193, + 388, + 223 + ], + "spans": [ + { + "bbox": [ + 221, + 193, + 388, + 223 + ], + "score": 0.92, + "content": "\\begin{array} { l } { 0 = - \\theta - 2 \\psi M ( \\psi , \\theta ) - \\psi ^ { 2 } \\nabla _ { \\psi } M ( \\psi , \\theta ) } \\\\ { \\quad 0 = \\psi } \\end{array}", + "type": "interline_equation", + "image_path": "11feeb6b3820e36c142d1e8b6aef9276fb59080ccfe9959982cc90756eaaf513.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 221, + 193, + 388, + 223 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 398, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 399, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 399, + 242 + ], + "score": 1.0, + "content": "The corresponding Jacobian matrix for the dynamic system is written as:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 244, + 337, + 273 + ], + "lines": [ + { + "bbox": [ + 274, + 244, + 337, + 273 + ], + "spans": [ + { + "bbox": [ + 274, + 244, + 337, + 273 + ], + "score": 0.94, + "content": "J = \\left[ { \\begin{array} { r r } { Z } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} } \\right]", + "type": "interline_equation", + "image_path": "a595bbd7f831617f8cd58599e2e1aff778bc300bdd3058a707ea5231d7ddabba.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 274, + 244, + 337, + 273 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 133, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 135, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 135, + 290 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 286, + 373, + 316 + ], + "lines": [ + { + "bbox": [ + 238, + 286, + 373, + 316 + ], + "spans": [ + { + "bbox": [ + 238, + 286, + 373, + 316 + ], + "score": 0.94, + "content": "Z = - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\psi ^ { 2 } ] \\bigg | _ { \\psi = 0 , \\theta = 0 }", + "type": "interline_equation", + "image_path": "8ca3fa140aa723f934b3ad2afb4750c4a94f366b1468f44a121b9153fa19e56d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 238, + 286, + 373, + 316 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 374, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 375, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 175, + 331 + ], + "score": 0.91, + "content": "\\nabla _ { \\psi } D ( x ; \\psi ) = \\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 318, + 256, + 333 + ], + "score": 1.0, + "content": "does not depend on", + "type": "text" + }, + { + "bbox": [ + 256, + 322, + 263, + 329 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 318, + 375, + 333 + ], + "score": 1.0, + "content": ", so this can be rewritten as:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 337, + 489, + 373 + ], + "lines": [ + { + "bbox": [ + 122, + 337, + 489, + 373 + ], + "spans": [ + { + "bbox": [ + 122, + 337, + 489, + 373 + ], + "score": 0.83, + "content": "Z = - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } ( \\psi ^ { 2 } \\mathbb { E } _ { \\mu _ { \\psi } , \\theta } [ 1 ] ) = - \\frac { \\rho } { 2 } ( 2 M ( \\psi , \\theta ) + 4 \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) + \\psi ^ { 2 } M _ { \\psi \\psi } ( \\psi , \\theta ) ) \\bigg | _ { \\psi = 0 , \\theta = 0 }", + "type": "interline_equation", + "image_path": "fbaf1589dc439a233d0447e16c827632565803a15d7716a00ca39b2bd8ab939e.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 122, + 337, + 489, + 349.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 122, + 349.0, + 489, + 361.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 122, + 361.0, + 489, + 373.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 387, + 504, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 161, + 401 + ], + "score": 1.0, + "content": "Therefore, if", + "type": "text" + }, + { + "bbox": [ + 162, + 387, + 217, + 399 + ], + "score": 0.92, + "content": "M ( 0 , 0 ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 385, + 506, + 401 + ], + "score": 1.0, + "content": ", then the given system is locally stable because the eigenvalues of its", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 290, + 411 + ], + "score": 1.0, + "content": "linearized system have negative real parts. If", + "type": "text" + }, + { + "bbox": [ + 291, + 397, + 344, + 410 + ], + "score": 0.93, + "content": "M ( 0 , 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 397, + 506, + 411 + ], + "score": 1.0, + "content": ", then the stability of the system cannot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 409, + 502, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 502, + 421 + ], + "score": 1.0, + "content": "be proved by the linearization theorem. In this case, we consider the following Lyapunov function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 425, + 367, + 440 + ], + "lines": [ + { + "bbox": [ + 243, + 425, + 367, + 440 + ], + "spans": [ + { + "bbox": [ + 243, + 425, + 367, + 440 + ], + "score": 0.92, + "content": "L ( \\psi ( t ) , \\theta ( t ) ) = \\psi ( t ) ^ { 2 } + \\theta ( t ) ^ { 2 }", + "type": "interline_equation", + "image_path": "c80231cde598e57e89c34285fc160a222055cae6b66ba8d6e44bdd43023c446e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 243, + 425, + 367, + 440 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 250, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 250, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 200, + 459 + ], + "score": 1.0, + "content": "By differentiating with", + "type": "text" + }, + { + "bbox": [ + 200, + 447, + 205, + 455 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 444, + 250, + 459 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 462, + 475, + 496 + ], + "lines": [ + { + "bbox": [ + 136, + 462, + 475, + 496 + ], + "spans": [ + { + "bbox": [ + 136, + 462, + 475, + 496 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\dot { L } = 2 ( \\psi \\psi ^ { \\prime } + \\theta \\theta ^ { \\prime } ) = - \\rho \\psi \\nabla _ { \\psi } ( \\psi ^ { 2 } M ( \\psi , \\theta ) ) = - \\rho \\psi ( 2 \\psi M ( \\psi , \\theta ) + \\psi ^ { 2 } \\nabla _ { \\psi } M ( \\psi , \\theta ) ) } \\\\ & { \\quad = - \\rho \\psi ^ { 2 } ( 2 M ( \\psi , \\theta ) + \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) ) \\leq 0 } \\end{array}", + "type": "interline_equation", + "image_path": "a07979e4d63cf803a5e69d8098d7bec2cbff2467fae906f1e2dccac861941cb3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 136, + 462, + 475, + 473.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 136, + 473.3333333333333, + 475, + 484.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 136, + 484.66666666666663, + 475, + 495.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 506, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 141, + 516 + ], + "score": 1.0, + "content": "Clearly,", + "type": "text" + }, + { + "bbox": [ + 141, + 502, + 193, + 514 + ], + "score": 0.91, + "content": "L ( \\psi , \\theta ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 501, + 299, + 516 + ], + "score": 1.0, + "content": "and the equality holds iff", + "type": "text" + }, + { + "bbox": [ + 299, + 502, + 348, + 514 + ], + "score": 0.92, + "content": "\\psi = \\theta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 501, + 403, + 516 + ], + "score": 1.0, + "content": ". In addition,", + "type": "text" + }, + { + "bbox": [ + 403, + 501, + 431, + 513 + ], + "score": 0.91, + "content": "\\dot { L } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 501, + 456, + 516 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 456, + 502, + 505, + 514 + ], + "score": 0.9, + "content": "M ( \\psi , \\theta ) \\geq", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 133, + 527 + ], + "score": 1.0, + "content": "0 and", + "type": "text" + }, + { + "bbox": [ + 134, + 514, + 214, + 525 + ], + "score": 0.88, + "content": "\\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 513, + 441, + 527 + ], + "score": 1.0, + "content": "from the assumption. Furthermore, it is clear that if", + "type": "text" + }, + { + "bbox": [ + 442, + 514, + 505, + 525 + ], + "score": 0.89, + "content": "( \\psi ( 0 ) , \\theta ( 0 ) ) \\ \\in \\qquad ", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 107, + 524, + 149, + 536 + ], + "score": 0.92, + "content": "B _ { \\eta } ( ( 0 , 0 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 523, + 173, + 538 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 174, + 524, + 280, + 536 + ], + "score": 0.89, + "content": "( \\psi ( \\tau ) , \\theta ( \\tau ) ) \\in B _ { \\eta } ( ( 0 , 0 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 523, + 308, + 538 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 308, + 525, + 334, + 535 + ], + "score": 0.89, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 523, + 506, + 538 + ], + "score": 1.0, + "content": "because the Lyapunov function (square of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 249, + 550 + ], + "score": 1.0, + "content": "the distance between the origin and", + "type": "text" + }, + { + "bbox": [ + 249, + 536, + 302, + 547 + ], + "score": 0.91, + "content": "( \\psi ( \\tau ) , \\theta ( \\tau ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 534, + 385, + 550 + ], + "score": 1.0, + "content": "always decreases as", + "type": "text" + }, + { + "bbox": [ + 386, + 537, + 418, + 546 + ], + "score": 0.88, + "content": "\\tau \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 534, + 506, + 550 + ], + "score": 1.0, + "content": ". Therefore, the given", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 546, + 353, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 353, + 561 + ], + "score": 1.0, + "content": "system is stable according to the Lyapunov stability theorem.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 224, + 576 + ], + "score": 1.0, + "content": "Again, we can check that if", + "type": "text" + }, + { + "bbox": [ + 225, + 565, + 244, + 575 + ], + "score": 0.87, + "content": "\\mu _ { \\psi , \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "is a probability measure, then the system is globally stable, as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 442, + 587 + ], + "score": 1.0, + "content": "shown by Mescheder et al. (2018). The basin of attraction is given by the whole", + "type": "text" + }, + { + "bbox": [ + 442, + 574, + 455, + 585 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "plane since", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 160, + 600 + ], + "score": 0.92, + "content": "M ( \\psi , \\theta ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 586, + 176, + 600 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 176, + 586, + 349, + 600 + ], + "score": 0.93, + "content": "\\dot { L } = - \\rho \\psi ^ { 2 } ( 2 M + \\psi \\nabla _ { \\psi } M ) = - 2 \\rho \\psi ^ { 2 } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 586, + 389, + 600 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 390, + 586, + 438, + 600 + ], + "score": 0.92, + "content": "( \\psi , \\theta ) \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 586, + 443, + 600 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 504, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 302, + 626 + ], + "score": 1.0, + "content": "Proof of Lemma 2. From the general setup of the", + "type": "text" + }, + { + "bbox": [ + 302, + 613, + 329, + 624 + ], + "score": 0.4, + "content": "\\operatorname { S G P } \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 613, + 504, + 626 + ], + "score": 1.0, + "content": "-WGAN optimization problem, the dynamic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 624, + 440, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 440, + 637 + ], + "score": 1.0, + "content": "system corresponding to the simple-GAN in Definition 6 can be written as follows.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 641, + 361, + 693 + ], + "lines": [ + { + "bbox": [ + 250, + 641, + 361, + 693 + ], + "spans": [ + { + "bbox": [ + 250, + 641, + 361, + 693 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\dot { \\psi } = \\frac { 1 } { 3 } - \\frac { \\theta ^ { 2 } } { 3 } - 4 \\rho \\psi \\mathbb { E } _ { \\mu } [ x ^ { 2 } ] } } } \\\\ { { \\displaystyle { \\dot { \\theta } = \\frac { 2 \\psi \\theta } { 3 } } } } \\end{array}", + "type": "interline_equation", + "image_path": "4e998f6091eba7f6033f5fd09344d95a68f943a8409f4336df57850b4d236018.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 641, + 361, + 667.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 250, + 667.0, + 361, + 693.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 505, + 736 + ], + "lines": [ + { + "bbox": [ + 104, + 695, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 695, + 146, + 711 + ], + "score": 1.0, + "content": "If we let", + "type": "text" + }, + { + "bbox": [ + 147, + 697, + 210, + 710 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] = A ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 695, + 399, + 711 + ], + "score": 1.0, + "content": ", then the Jacobian matrix at the equilibrium", + "type": "text" + }, + { + "bbox": [ + 400, + 698, + 430, + 710 + ], + "score": 0.92, + "content": "( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 695, + 483, + 711 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 483, + 698, + 505, + 709 + ], + "score": 0.84, + "content": "{ \\boldsymbol { J } } =", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 709, + 410, + 736 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 172, + 736 + ], + "score": 0.92, + "content": "\\left[ { \\begin{array} { c c } { - 4 \\rho A ^ { 2 } } & { { \\mp } { \\frac { 2 } { 3 } } } \\\\ { \\pm { \\frac { 2 } { 3 } } } & { 0 } \\end{array} } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 713, + 379, + 731 + ], + "score": 1.0, + "content": "Therefore, the given system is locally stable when", + "type": "text" + }, + { + "bbox": [ + 379, + 717, + 405, + 728 + ], + "score": 0.9, + "content": "A \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 713, + 410, + 731 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 761 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "11", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 421, + 95 + ], + "lines": [ + { + "bbox": [ + 107, + 81, + 421, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 421, + 95 + ], + "score": 1.0, + "content": "APPENDIX A : PROOF OF LEMMAS BASED ON TOY EXAMPLES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 315, + 120 + ], + "score": 1.0, + "content": "Proof of Lemma 1. The related dynamic system of", + "type": "text" + }, + { + "bbox": [ + 315, + 106, + 432, + 119 + ], + "score": 0.91, + "content": "\\begin{array} { r } { ( D ( x ; \\psi ) = \\psi x , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { \\psi , \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 105, + 505, + 120 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 115, + 142, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 142, + 131 + ], + "score": 1.0, + "content": "follows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 105, + 505, + 131 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 132, + 360, + 171 + ], + "lines": [ + { + "bbox": [ + 250, + 132, + 360, + 171 + ], + "spans": [ + { + "bbox": [ + 250, + 132, + 360, + 171 + ], + "score": 0.92, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\psi } = - \\theta - \\frac { \\rho } { 2 } \\nabla \\psi \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\psi ^ { 2 } ] } \\\\ { \\displaystyle \\dot { \\theta } = \\psi } \\end{array}", + "type": "interline_equation", + "image_path": "73bdd30368acc5ebfd92175ccbcc4e3aa653d0a11d2e143c01ca06dae90d2a83.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 132, + 360, + 151.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 250, + 151.5, + 360, + 171.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 175, + 375, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 375, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 283, + 189 + ], + "score": 1.0, + "content": "First, the only equilibrium point is given by", + "type": "text" + }, + { + "bbox": [ + 283, + 175, + 352, + 188 + ], + "score": 0.91, + "content": "( \\psi ^ { * } , \\theta ^ { * } ) = ( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 174, + 375, + 189 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 174, + 375, + 189 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 193, + 388, + 223 + ], + "lines": [ + { + "bbox": [ + 221, + 193, + 388, + 223 + ], + "spans": [ + { + "bbox": [ + 221, + 193, + 388, + 223 + ], + "score": 0.92, + "content": "\\begin{array} { l } { 0 = - \\theta - 2 \\psi M ( \\psi , \\theta ) - \\psi ^ { 2 } \\nabla _ { \\psi } M ( \\psi , \\theta ) } \\\\ { \\quad 0 = \\psi } \\end{array}", + "type": "interline_equation", + "image_path": "11feeb6b3820e36c142d1e8b6aef9276fb59080ccfe9959982cc90756eaaf513.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 221, + 193, + 388, + 223 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 398, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 399, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 399, + 242 + ], + "score": 1.0, + "content": "The corresponding Jacobian matrix for the dynamic system is written as:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 227, + 399, + 242 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 244, + 337, + 273 + ], + "lines": [ + { + "bbox": [ + 274, + 244, + 337, + 273 + ], + "spans": [ + { + "bbox": [ + 274, + 244, + 337, + 273 + ], + "score": 0.94, + "content": "J = \\left[ { \\begin{array} { r r } { Z } & { - 1 } \\\\ { 1 } & { 0 } \\end{array} } \\right]", + "type": "interline_equation", + "image_path": "a595bbd7f831617f8cd58599e2e1aff778bc300bdd3058a707ea5231d7ddabba.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 274, + 244, + 337, + 273 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 133, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 135, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 135, + 290 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 276, + 135, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 286, + 373, + 316 + ], + "lines": [ + { + "bbox": [ + 238, + 286, + 373, + 316 + ], + "spans": [ + { + "bbox": [ + 238, + 286, + 373, + 316 + ], + "score": 0.94, + "content": "Z = - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\psi ^ { 2 } ] \\bigg | _ { \\psi = 0 , \\theta = 0 }", + "type": "interline_equation", + "image_path": "8ca3fa140aa723f934b3ad2afb4750c4a94f366b1468f44a121b9153fa19e56d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 238, + 286, + 373, + 316 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 374, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 375, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 175, + 331 + ], + "score": 0.91, + "content": "\\nabla _ { \\psi } D ( x ; \\psi ) = \\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 318, + 256, + 333 + ], + "score": 1.0, + "content": "does not depend on", + "type": "text" + }, + { + "bbox": [ + 256, + 322, + 263, + 329 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 318, + 375, + 333 + ], + "score": 1.0, + "content": ", so this can be rewritten as:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 318, + 375, + 333 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 337, + 489, + 373 + ], + "lines": [ + { + "bbox": [ + 122, + 337, + 489, + 373 + ], + "spans": [ + { + "bbox": [ + 122, + 337, + 489, + 373 + ], + "score": 0.83, + "content": "Z = - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } ( \\psi ^ { 2 } \\mathbb { E } _ { \\mu _ { \\psi } , \\theta } [ 1 ] ) = - \\frac { \\rho } { 2 } ( 2 M ( \\psi , \\theta ) + 4 \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) + \\psi ^ { 2 } M _ { \\psi \\psi } ( \\psi , \\theta ) ) \\bigg | _ { \\psi = 0 , \\theta = 0 }", + "type": "interline_equation", + "image_path": "fbaf1589dc439a233d0447e16c827632565803a15d7716a00ca39b2bd8ab939e.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 122, + 337, + 489, + 349.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 122, + 349.0, + 489, + 361.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 122, + 361.0, + 489, + 373.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 387, + 504, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 161, + 401 + ], + "score": 1.0, + "content": "Therefore, if", + "type": "text" + }, + { + "bbox": [ + 162, + 387, + 217, + 399 + ], + "score": 0.92, + "content": "M ( 0 , 0 ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 385, + 506, + 401 + ], + "score": 1.0, + "content": ", then the given system is locally stable because the eigenvalues of its", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 290, + 411 + ], + "score": 1.0, + "content": "linearized system have negative real parts. If", + "type": "text" + }, + { + "bbox": [ + 291, + 397, + 344, + 410 + ], + "score": 0.93, + "content": "M ( 0 , 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 397, + 506, + 411 + ], + "score": 1.0, + "content": ", then the stability of the system cannot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 409, + 502, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 502, + 421 + ], + "score": 1.0, + "content": "be proved by the linearization theorem. In this case, we consider the following Lyapunov function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 385, + 506, + 421 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 425, + 367, + 440 + ], + "lines": [ + { + "bbox": [ + 243, + 425, + 367, + 440 + ], + "spans": [ + { + "bbox": [ + 243, + 425, + 367, + 440 + ], + "score": 0.92, + "content": "L ( \\psi ( t ) , \\theta ( t ) ) = \\psi ( t ) ^ { 2 } + \\theta ( t ) ^ { 2 }", + "type": "interline_equation", + "image_path": "c80231cde598e57e89c34285fc160a222055cae6b66ba8d6e44bdd43023c446e.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 243, + 425, + 367, + 440 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 250, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 250, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 200, + 459 + ], + "score": 1.0, + "content": "By differentiating with", + "type": "text" + }, + { + "bbox": [ + 200, + 447, + 205, + 455 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 444, + 250, + 459 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 444, + 250, + 459 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 462, + 475, + 496 + ], + "lines": [ + { + "bbox": [ + 136, + 462, + 475, + 496 + ], + "spans": [ + { + "bbox": [ + 136, + 462, + 475, + 496 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\dot { L } = 2 ( \\psi \\psi ^ { \\prime } + \\theta \\theta ^ { \\prime } ) = - \\rho \\psi \\nabla _ { \\psi } ( \\psi ^ { 2 } M ( \\psi , \\theta ) ) = - \\rho \\psi ( 2 \\psi M ( \\psi , \\theta ) + \\psi ^ { 2 } \\nabla _ { \\psi } M ( \\psi , \\theta ) ) } \\\\ & { \\quad = - \\rho \\psi ^ { 2 } ( 2 M ( \\psi , \\theta ) + \\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) ) \\leq 0 } \\end{array}", + "type": "interline_equation", + "image_path": "a07979e4d63cf803a5e69d8098d7bec2cbff2467fae906f1e2dccac861941cb3.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 136, + 462, + 475, + 473.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 136, + 473.3333333333333, + 475, + 484.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 136, + 484.66666666666663, + 475, + 495.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 506, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 141, + 516 + ], + "score": 1.0, + "content": "Clearly,", + "type": "text" + }, + { + "bbox": [ + 141, + 502, + 193, + 514 + ], + "score": 0.91, + "content": "L ( \\psi , \\theta ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 501, + 299, + 516 + ], + "score": 1.0, + "content": "and the equality holds iff", + "type": "text" + }, + { + "bbox": [ + 299, + 502, + 348, + 514 + ], + "score": 0.92, + "content": "\\psi = \\theta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 501, + 403, + 516 + ], + "score": 1.0, + "content": ". In addition,", + "type": "text" + }, + { + "bbox": [ + 403, + 501, + 431, + 513 + ], + "score": 0.91, + "content": "\\dot { L } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 501, + 456, + 516 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 456, + 502, + 505, + 514 + ], + "score": 0.9, + "content": "M ( \\psi , \\theta ) \\geq", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 133, + 527 + ], + "score": 1.0, + "content": "0 and", + "type": "text" + }, + { + "bbox": [ + 134, + 514, + 214, + 525 + ], + "score": 0.88, + "content": "\\psi \\nabla _ { \\psi } M ( \\psi , \\theta ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 513, + 441, + 527 + ], + "score": 1.0, + "content": "from the assumption. Furthermore, it is clear that if", + "type": "text" + }, + { + "bbox": [ + 442, + 514, + 505, + 525 + ], + "score": 0.89, + "content": "( \\psi ( 0 ) , \\theta ( 0 ) ) \\ \\in \\qquad ", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 523, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 107, + 524, + 149, + 536 + ], + "score": 0.92, + "content": "B _ { \\eta } ( ( 0 , 0 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 523, + 173, + 538 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 174, + 524, + 280, + 536 + ], + "score": 0.89, + "content": "( \\psi ( \\tau ) , \\theta ( \\tau ) ) \\in B _ { \\eta } ( ( 0 , 0 ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 523, + 308, + 538 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 308, + 525, + 334, + 535 + ], + "score": 0.89, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 523, + 506, + 538 + ], + "score": 1.0, + "content": "because the Lyapunov function (square of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 249, + 550 + ], + "score": 1.0, + "content": "the distance between the origin and", + "type": "text" + }, + { + "bbox": [ + 249, + 536, + 302, + 547 + ], + "score": 0.91, + "content": "( \\psi ( \\tau ) , \\theta ( \\tau ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 534, + 385, + 550 + ], + "score": 1.0, + "content": "always decreases as", + "type": "text" + }, + { + "bbox": [ + 386, + 537, + 418, + 546 + ], + "score": 0.88, + "content": "\\tau \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 534, + 506, + 550 + ], + "score": 1.0, + "content": ". Therefore, the given", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 546, + 353, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 353, + 561 + ], + "score": 1.0, + "content": "system is stable according to the Lyapunov stability theorem.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 501, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 562, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 224, + 576 + ], + "score": 1.0, + "content": "Again, we can check that if", + "type": "text" + }, + { + "bbox": [ + 225, + 565, + 244, + 575 + ], + "score": 0.87, + "content": "\\mu _ { \\psi , \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "is a probability measure, then the system is globally stable, as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 442, + 587 + ], + "score": 1.0, + "content": "shown by Mescheder et al. (2018). The basin of attraction is given by the whole", + "type": "text" + }, + { + "bbox": [ + 442, + 574, + 455, + 585 + ], + "score": 0.86, + "content": "\\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "plane since", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 160, + 600 + ], + "score": 0.92, + "content": "M ( \\psi , \\theta ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 586, + 176, + 600 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 176, + 586, + 349, + 600 + ], + "score": 0.93, + "content": "\\dot { L } = - \\rho \\psi ^ { 2 } ( 2 M + \\psi \\nabla _ { \\psi } M ) = - 2 \\rho \\psi ^ { 2 } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 586, + 389, + 600 + ], + "score": 1.0, + "content": "for every", + "type": "text" + }, + { + "bbox": [ + 390, + 586, + 438, + 600 + ], + "score": 0.92, + "content": "( \\psi , \\theta ) \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 586, + 443, + 600 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 563, + 505, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 504, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 302, + 626 + ], + "score": 1.0, + "content": "Proof of Lemma 2. From the general setup of the", + "type": "text" + }, + { + "bbox": [ + 302, + 613, + 329, + 624 + ], + "score": 0.4, + "content": "\\operatorname { S G P } \\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 613, + 504, + 626 + ], + "score": 1.0, + "content": "-WGAN optimization problem, the dynamic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 624, + 440, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 440, + 637 + ], + "score": 1.0, + "content": "system corresponding to the simple-GAN in Definition 6 can be written as follows.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 613, + 504, + 637 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 641, + 361, + 693 + ], + "lines": [ + { + "bbox": [ + 250, + 641, + 361, + 693 + ], + "spans": [ + { + "bbox": [ + 250, + 641, + 361, + 693 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\dot { \\psi } = \\frac { 1 } { 3 } - \\frac { \\theta ^ { 2 } } { 3 } - 4 \\rho \\psi \\mathbb { E } _ { \\mu } [ x ^ { 2 } ] } } } \\\\ { { \\displaystyle { \\dot { \\theta } = \\frac { 2 \\psi \\theta } { 3 } } } } \\end{array}", + "type": "interline_equation", + "image_path": "4e998f6091eba7f6033f5fd09344d95a68f943a8409f4336df57850b4d236018.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 641, + 361, + 667.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 250, + 667.0, + 361, + 693.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 505, + 736 + ], + "lines": [ + { + "bbox": [ + 104, + 695, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 695, + 146, + 711 + ], + "score": 1.0, + "content": "If we let", + "type": "text" + }, + { + "bbox": [ + 147, + 697, + 210, + 710 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { \\mu ^ { * } } [ x ^ { 2 } ] = A ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 695, + 399, + 711 + ], + "score": 1.0, + "content": ", then the Jacobian matrix at the equilibrium", + "type": "text" + }, + { + "bbox": [ + 400, + 698, + 430, + 710 + ], + "score": 0.92, + "content": "( 0 , \\pm 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 695, + 483, + 711 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 483, + 698, + 505, + 709 + ], + "score": 0.84, + "content": "{ \\boldsymbol { J } } =", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 709, + 410, + 736 + ], + "spans": [ + { + "bbox": [ + 107, + 709, + 172, + 736 + ], + "score": 0.92, + "content": "\\left[ { \\begin{array} { c c } { - 4 \\rho A ^ { 2 } } & { { \\mp } { \\frac { 2 } { 3 } } } \\\\ { \\pm { \\frac { 2 } { 3 } } } & { 0 } \\end{array} } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 713, + 379, + 731 + ], + "score": 1.0, + "content": "Therefore, the given system is locally stable when", + "type": "text" + }, + { + "bbox": [ + 379, + 717, + 405, + 728 + ], + "score": 0.9, + "content": "A \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 713, + 410, + 731 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 695, + 505, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 439, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 440, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 440, + 95 + ], + "score": 1.0, + "content": "APPENDIX B : PROOF OF LEMMA RELATED WITH ASSUMPTION 2", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 104, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 322, + 119 + ], + "score": 1.0, + "content": "Lemma 3. Consider the Dirac-GAN setup and SGP", + "type": "text" + }, + { + "bbox": [ + 322, + 108, + 329, + 118 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "-WGAN optimization system with a slightly", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 115, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 235, + 131 + ], + "score": 1.0, + "content": "changed discriminator function", + "type": "text" + }, + { + "bbox": [ + 236, + 117, + 304, + 129 + ], + "score": 0.92, + "content": "D _ { 2 } ( x ; \\psi ) \\stackrel { \\textstyle - } { = } \\psi x ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 115, + 358, + 131 + ], + "score": 1.0, + "content": ". 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Consider the Dirac-GAN setup and SGP", + "type": "text" + }, + { + "bbox": [ + 322, + 108, + 329, + 118 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "-WGAN optimization system with a slightly", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 115, + 506, + 131 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 235, + 131 + ], + "score": 1.0, + "content": "changed discriminator function", + "type": "text" + }, + { + "bbox": [ + 236, + 117, + 304, + 129 + ], + "score": 0.92, + "content": "D _ { 2 } ( x ; \\psi ) \\stackrel { \\textstyle - } { = } \\psi x ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 115, + 358, + 131 + ], + "score": 1.0, + "content": ". The system", + "type": "text" + }, + { + "bbox": [ + 358, + 117, + 429, + 129 + ], + "score": 0.9, + "content": "( D _ { 2 } , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { G P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 115, + 506, + 131 + ], + "score": 1.0, + "content": "does not converge", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 117, + 141 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 128, + 140, + 140 + ], + "score": 0.9, + "content": "( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 127, + 212, + 141 + ], + "score": 1.0, + "content": "but for any point", + "type": "text" + }, + { + "bbox": [ + 212, + 128, + 235, + 140 + ], + "score": 0.91, + "content": "( a , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 127, + 256, + 141 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 257, + 129, + 281, + 139 + ], + "score": 0.9, + "content": "a < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 127, + 476, + 141 + ], + "score": 1.0, + "content": ", the system has equilibrium points on the whole", + "type": "text" + }, + { + "bbox": [ + 477, + 129, + 484, + 140 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 127, + 505, + 141 + ], + "score": 1.0, + "content": "-axis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 224, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 224, + 151 + ], + "score": 1.0, + "content": "and it violates Assumption 2.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 105, + 506, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 504, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 240, + 176 + ], + "score": 1.0, + "content": "Proof of Lemma 3. For the SGP", + "type": "text" + }, + { + "bbox": [ + 241, + 164, + 248, + 175 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 162, + 376, + 176 + ], + "score": 1.0, + "content": "-WGAN optimization problem", + "type": "text" + }, + { + "bbox": [ + 376, + 163, + 447, + 175 + ], + "score": 0.91, + "content": "( D _ { 2 } , \\delta _ { 0 } , \\delta _ { \\theta } , \\mu _ { G P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 162, + 505, + 176 + ], + "score": 1.0, + "content": ", the dynamic", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 241, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 241, + 187 + ], + "score": 1.0, + "content": "system can be written as follows.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 162, + 505, + 187 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 190, + 346, + 231 + ], + "lines": [ + { + "bbox": [ + 264, + 190, + 346, + 231 + ], + "spans": [ + { + "bbox": [ + 264, + 190, + 346, + 231 + ], + "score": 0.93, + "content": "\\begin{array} { l } { { \\dot { \\psi } = - 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Every solution curve that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 538, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 188, + 554 + ], + "score": 1.0, + "content": "passes the nullcline", + "type": "text" + }, + { + "bbox": [ + 188, + 540, + 217, + 551 + ], + "score": 0.91, + "content": "\\psi = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 539, + 235, + 554 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 235, + 538, + 262, + 550 + ], + "score": 0.91, + "content": "\\dot { \\theta } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 539, + 338, + 554 + ], + "score": 1.0, + "content": ". For the nullcline", + "type": "text" + }, + { + "bbox": [ + 339, + 538, + 408, + 554 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\psi = - \\frac { 3 } { 4 \\rho } = - 2 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 539, + 477, + 554 + ], + "score": 1.0, + "content": ", no updating on", + "type": "text" + }, + { + "bbox": [ + 477, + 540, + 485, + 551 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 539, + 505, + 554 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 169, + 565 + ], + "score": 1.0, + "content": "occur and only", + "type": "text" + }, + { + "bbox": [ + 170, + 552, + 177, + 563 + ], + "score": 0.74, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "will be updated. Given that the solution curves do not intersect with each other,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 564, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 303, + 594 + ], + "score": 1.0, + "content": "every solution curve is exactly one of the followinstays in area A. (2) Solution curve converges to", + "type": "text" + }, + { + "bbox": [ + 303, + 573, + 379, + 588 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\bar { ( \\psi , \\theta ) } \\overset { * } { = } ( - \\frac { 3 } { 4 \\rho } , 0 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 564, + 459, + 594 + ], + "score": 1.0, + "content": "trivial cases; (1) Solalong the nullcline", + "type": "text" + }, + { + "bbox": [ + 460, + 573, + 501, + 588 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\psi = - \\frac { 3 } { 4 \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 564, + 505, + 594 + ], + "score": 1.0, + "content": "e.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 587, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 477, + 600 + ], + "score": 1.0, + "content": "(3) Solution curve stays in area B. (4) Solution curve starts from area C, crosses the nullcline", + "type": "text" + }, + { + "bbox": [ + 478, + 588, + 504, + 598 + ], + "score": 0.88, + "content": "\\psi = 0", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 241, + 611 + ], + "score": 1.0, + "content": "perpendicularly, and converges to", + "type": "text" + }, + { + "bbox": [ + 241, + 598, + 263, + 610 + ], + "score": 0.92, + "content": "( b , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 597, + 284, + 611 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 284, + 598, + 308, + 609 + ], + "score": 0.89, + "content": "b < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 597, + 478, + 611 + ], + "score": 1.0, + "content": ". Therefore, no solution curve converges to", + "type": "text" + }, + { + "bbox": [ + 479, + 599, + 501, + 610 + ], + "score": 0.91, + "content": "( 0 , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 597, + 505, + 611 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + } + ], + "index": 15.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 80, + 428, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 429, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 429, + 96 + ], + "score": 1.0, + "content": "APPENDIX C : PROOF OF THE MAIN CONVERGENCE THEOREM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 501, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 498, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 280, + 123 + ], + "score": 1.0, + "content": "Proof. Let us consider the Jacobian matrix", + "type": "text" + }, + { + "bbox": [ + 280, + 103, + 365, + 130 + ], + "score": 0.93, + "content": "J = \\left[ \\begin{array} { l l } { K _ { D D } } & { K _ { D G } } \\\\ { K _ { G D } } & { K _ { G G } } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 101, + 459, + 132 + ], + "score": 1.0, + "content": "at the first equilibrium", + "type": "text" + }, + { + "bbox": [ + 459, + 110, + 498, + 123 + ], + "score": 0.89, + "content": "( \\psi ^ { * } , \\theta ^ { * } ) ^ { 4 }", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 133, + 504, + 161 + ], + "lines": [ + { + "bbox": [ + 111, + 133, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 111, + 133, + 504, + 161 + ], + "score": 0.9, + "content": "\\begin{array} { r l } { J = \\left[ \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi \\psi } D ] - \\mathbb { E } _ { p _ { \\theta ^ { \\star } } } [ \\nabla _ { \\psi \\psi } D ] - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] } & { - \\nabla _ { \\theta \\psi } \\mathbb { E } _ { p _ { \\theta } } [ D ] - \\frac { \\rho } { 2 } \\nabla _ { \\theta \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] \\right] } \\\\ & { \\qquad \\nabla _ { \\psi \\theta } \\mathbb { E } _ { p _ { \\theta } } [ D ] ^ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "e7c9c27b2a41ee6f9e13ea7a81eab9b1746accc232b6bf11ad215c5545de1b58.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 133, + 504, + 161 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 237, + 185 + ], + "score": 1.0, + "content": "First, Assumption 1 implies that", + "type": "text" + }, + { + "bbox": [ + 238, + 171, + 372, + 183 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi \\psi } D ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi \\psi } D ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 170, + 396, + 185 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 397, + 173, + 433, + 182 + ], + "score": 0.89, + "content": "p _ { \\theta } p _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 170, + 444, + 185 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 445, + 172, + 476, + 181 + ], + "score": 0.9, + "content": "\\theta \\to \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 170, + 506, + 185 + ], + "score": 1.0, + "content": ". 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Let us consider the Jacobian matrix", + "type": "text" + }, + { + "bbox": [ + 280, + 103, + 365, + 130 + ], + "score": 0.93, + "content": "J = \\left[ \\begin{array} { l l } { K _ { D D } } & { K _ { D G } } \\\\ { K _ { G D } } & { K _ { G G } } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 101, + 459, + 132 + ], + "score": 1.0, + "content": "at the first equilibrium", + "type": "text" + }, + { + "bbox": [ + 459, + 110, + 498, + 123 + ], + "score": 0.89, + "content": "( \\psi ^ { * } , \\theta ^ { * } ) ^ { 4 }", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 101, + 498, + 132 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 133, + 504, + 161 + ], + "lines": [ + { + "bbox": [ + 111, + 133, + 504, + 161 + ], + "spans": [ + { + "bbox": [ + 111, + 133, + 504, + 161 + ], + "score": 0.9, + "content": "\\begin{array} { r l } { J = \\left[ \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi \\psi } D ] - \\mathbb { E } _ { p _ { \\theta ^ { \\star } } } [ \\nabla _ { \\psi \\psi } D ] - \\frac { \\rho } { 2 } \\nabla _ { \\psi \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] } & { - \\nabla _ { \\theta \\psi } \\mathbb { E } _ { p _ { \\theta } } [ D ] - \\frac { \\rho } { 2 } \\nabla _ { \\theta \\psi } \\mathbb { E } _ { \\mu } [ \\| \\nabla _ { x } D \\| ^ { 2 } ] \\right] } \\\\ & { \\qquad \\nabla _ { \\psi \\theta } \\mathbb { E } _ { p _ { \\theta } } [ D ] ^ { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "e7c9c27b2a41ee6f9e13ea7a81eab9b1746accc232b6bf11ad215c5545de1b58.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 133, + 504, + 161 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 169, + 504, + 194 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 237, + 185 + ], + "score": 1.0, + "content": "First, Assumption 1 implies that", + "type": "text" + }, + { + "bbox": [ + 238, + 171, + 372, + 183 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi \\psi } D ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi \\psi } D ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 170, + 396, + 185 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 397, + 173, + 433, + 182 + ], + "score": 0.89, + "content": "p _ { \\theta } p _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 170, + 444, + 185 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 445, + 172, + 476, + 181 + ], + "score": 0.9, + "content": "\\theta \\to \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 170, + 506, + 185 + ], + "score": 1.0, + "content": ". 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If the system does not update at the equilibrium point", + "type": "text" + }, + { + "bbox": [ + 358, + 213, + 392, + 224 + ], + "score": 0.93, + "content": "( \\psi ^ { * } , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "and its small neighborhood", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 222, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 107, + 223, + 191, + 235 + ], + "score": 0.9, + "content": "\\left( \\psi ^ { * } + \\xi v , \\theta ^ { * } + \\nu w \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 222, + 271, + 237 + ], + "score": 1.0, + "content": "is perturbed along", + "type": "text" + }, + { + "bbox": [ + 271, + 223, + 297, + 235 + ], + "score": 0.92, + "content": "N ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 222, + 317, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 223, + 357, + 235 + ], + "score": 0.93, + "content": "N ( \\bar { R } ^ { T } R )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 222, + 506, + 237 + ], + "score": 1.0, + "content": ", then it is reasonable to project the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 280, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 192, + 248 + ], + "score": 1.0, + "content": "system orthogonal to", + "type": "text" + }, + { + "bbox": [ + 193, + 235, + 218, + 246 + ], + "score": 0.92, + "content": "N ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 234, + 236, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 234, + 276, + 246 + ], + "score": 0.93, + "content": "N ( R ^ { T } { \\bar { R } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 234, + 280, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 261, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 194, + 275 + ], + "score": 1.0, + "content": "First, we assume that", + "type": "text" + }, + { + "bbox": [ + 194, + 262, + 237, + 273 + ], + "score": 0.91, + "content": "v \\in N ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 261, + 318, + 275 + ], + "score": 1.0, + "content": ". By Assumption 2,", + "type": "text" + }, + { + "bbox": [ + 318, + 262, + 424, + 274 + ], + "score": 0.89, + "content": "h ( \\psi ^ { * } + \\xi v ) = h ( \\psi ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 261, + 439, + 275 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 440, + 262, + 474, + 274 + ], + "score": 0.9, + "content": "| \\xi | < \\xi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 261, + 505, + 275 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 271, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 156, + 287 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 157, + 273, + 250, + 285 + ], + "score": 0.91, + "content": "\\nabla _ { \\boldsymbol { x } } D ( \\boldsymbol { x } ; \\psi ^ { * } + \\xi \\boldsymbol { v } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 271, + 266, + 287 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 266, + 274, + 407, + 286 + ], + "score": 0.89, + "content": "x \\in s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 271, + 426, + 287 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 273, + 461, + 285 + ], + "score": 0.92, + "content": "| \\xi | < \\xi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 271, + 506, + 287 + ], + "score": 1.0, + "content": ". Thus, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 135, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 135, + 297 + ], + "score": 1.0, + "content": "obtain", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 297, + 416, + 313 + ], + "lines": [ + { + "bbox": [ + 194, + 297, + 416, + 313 + ], + "spans": [ + { + "bbox": [ + 194, + 297, + 416, + 313 + ], + "score": 0.84, + "content": "\\mathbb { E } _ { \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } } [ \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } + \\xi v ) \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) ] = 0", + "type": "interline_equation", + "image_path": "a07c3a8482ee736b654d5b78b767398ae13e3cb218ba12b7492880cf2882b64f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 194, + 297, + 416, + 313 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 123, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 123, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 123, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 335, + 405, + 363 + ], + "lines": [ + { + "bbox": [ + 207, + 335, + 405, + 363 + ], + "spans": [ + { + "bbox": [ + 207, + 335, + 405, + 363 + ], + "score": 0.89, + "content": "\\int _ { s u p p ( \\mu ^ { * } ) } \\left\\| \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) \\right\\| ^ { 2 } d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ^ { \\prime } = 0", + "type": "interline_equation", + "image_path": "2918b6465467384c47bed217182720336ee90009f7c7157f2822984c697c37c4.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 207, + 335, + 405, + 363 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 104, + 375, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 183, + 392 + ], + "score": 1.0, + "content": "By Assumption 4,", + "type": "text" + }, + { + "bbox": [ + 184, + 377, + 417, + 390 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 375, + 443, + 392 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 443, + 379, + 483, + 389 + ], + "score": 0.88, + "content": "p _ { d } = p _ { \\theta ^ { \\ast } }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 375, + 505, + 392 + ], + "score": 1.0, + "content": ". By", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 388, + 244, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 244, + 401 + ], + "score": 1.0, + "content": "adding these equations, we obtain", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 406, + 456, + 497 + ], + "lines": [ + { + "bbox": [ + 154, + 406, + 456, + 497 + ], + "spans": [ + { + "bbox": [ + 154, + 406, + 456, + 497 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\psi } = \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] } \\\\ { \\displaystyle - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) } 2 \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } + \\xi v ) \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } } \\\\ { \\displaystyle - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) } \\| \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) \\| ^ { 2 } d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ^ { \\prime } } \\\\ { \\displaystyle = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "1db21d03fbbfe5903967476b04abe7e3bc2506de90ac37988c65eac94f9f8948.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 154, + 406, + 456, + 436.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 154, + 436.3333333333333, + 456, + 466.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 154, + 466.66666666666663, + 456, + 496.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 504, + 154, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 156, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 156, + 517 + ], + "score": 1.0, + "content": "In addition,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 523, + 435, + 578 + ], + "lines": [ + { + "bbox": [ + 176, + 523, + 435, + 578 + ], + "spans": [ + { + "bbox": [ + 176, + 523, + 435, + 578 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\theta } = \\frac { \\partial } { \\partial \\theta } \\int _ { \\mathcal X } D ( x ; \\psi ^ { * } + \\xi v ) d p _ { \\theta } \\bigg \\vert _ { \\theta = \\theta ^ { * } } } \\\\ { \\displaystyle = \\int _ { \\mathcal Z } \\nabla _ { \\theta } ^ { T } G ( z ; \\theta ^ { * } ) \\nabla _ { x } D ( G ( z ; \\theta ^ { * } ) ; \\psi ^ { * } + \\xi v ) p _ { l a t e n t } ( z ) d z = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "f02b947c82e979834207e8549915edbc7e86af58f4507735e67988ef442bec30.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 176, + 523, + 435, + 541.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 176, + 541.3333333333334, + 435, + 559.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 176, + 559.6666666666667, + 435, + 578.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 503, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 189, + 599 + ], + "score": 1.0, + "content": "Therefore, the point", + "type": "text" + }, + { + "bbox": [ + 189, + 585, + 245, + 597 + ], + "score": 0.92, + "content": "( \\psi ^ { * } + \\xi v , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 583, + 267, + 599 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 267, + 585, + 301, + 597 + ], + "score": 0.92, + "content": "| \\xi | < \\xi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 583, + 505, + 599 + ], + "score": 1.0, + "content": "is an equilibrium point. According to Assumption", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 117, + 609 + ], + "score": 1.0, + "content": "4,", + "type": "text" + }, + { + "bbox": [ + 117, + 596, + 177, + 608 + ], + "score": 0.9, + "content": "D ( x ; \\psi ^ { * } + \\xi v )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 596, + 317, + 609 + ], + "score": 1.0, + "content": "is an equilibrium discriminator for", + "type": "text" + }, + { + "bbox": [ + 318, + 596, + 351, + 608 + ], + "score": 0.93, + "content": "| \\xi | < \\delta _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 596, + 391, + 609 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 391, + 596, + 451, + 608 + ], + "score": 0.94, + "content": "D ( x ; \\psi ^ { * } + \\bar { \\xi } v )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "is already an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 606, + 285, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 209, + 621 + ], + "score": 1.0, + "content": "optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 209, + 607, + 281, + 619 + ], + "score": 0.93, + "content": "| \\xi | < \\operatorname* { m i n } ( \\xi _ { d } , \\delta _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 606, + 285, + 621 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 104, + 630, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 164, + 651 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 164, + 634, + 230, + 646 + ], + "score": 0.91, + "content": "w \\ \\in \\ N ( R ^ { T } R )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 630, + 321, + 651 + ], + "score": 1.0, + "content": ". 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By", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 388, + 244, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 244, + 401 + ], + "score": 1.0, + "content": "adding these equations, we obtain", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 375, + 505, + 401 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 406, + 456, + 497 + ], + "lines": [ + { + "bbox": [ + 154, + 406, + 456, + 497 + ], + "spans": [ + { + "bbox": [ + 154, + 406, + 456, + 497 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\psi } = \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } + \\xi v ) ] } \\\\ { \\displaystyle - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) } 2 \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } + \\xi v ) \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } } \\\\ { \\displaystyle - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ) } \\| \\nabla _ { x } D ( x ; \\psi ^ { * } + \\xi v ) \\| ^ { 2 } d \\mu _ { \\psi ^ { * } + \\xi v , \\theta ^ { * } } ^ { \\prime } } \\\\ { \\displaystyle = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "1db21d03fbbfe5903967476b04abe7e3bc2506de90ac37988c65eac94f9f8948.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 154, + 406, + 456, + 436.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 154, + 436.3333333333333, + 456, + 466.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 154, + 466.66666666666663, + 456, + 496.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 504, + 154, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 156, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 156, + 517 + ], + "score": 1.0, + "content": "In addition,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 502, + 156, + 517 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 523, + 435, + 578 + ], + "lines": [ + { + "bbox": [ + 176, + 523, + 435, + 578 + ], + "spans": [ + { + "bbox": [ + 176, + 523, + 435, + 578 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\theta } = \\frac { \\partial } { \\partial \\theta } \\int _ { \\mathcal X } D ( x ; \\psi ^ { * } + \\xi v ) d p _ { \\theta } \\bigg \\vert _ { \\theta = \\theta ^ { * } } } \\\\ { \\displaystyle = \\int _ { \\mathcal Z } \\nabla _ { \\theta } ^ { T } G ( z ; \\theta ^ { * } ) \\nabla _ { x } D ( G ( z ; \\theta ^ { * } ) ; \\psi ^ { * } + \\xi v ) p _ { l a t e n t } ( z ) d z = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "f02b947c82e979834207e8549915edbc7e86af58f4507735e67988ef442bec30.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 176, + 523, + 435, + 541.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 176, + 541.3333333333334, + 435, + 559.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 176, + 559.6666666666667, + 435, + 578.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 503, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 189, + 599 + ], + "score": 1.0, + "content": "Therefore, the point", + "type": "text" + }, + { + "bbox": [ + 189, + 585, + 245, + 597 + ], + "score": 0.92, + "content": "( \\psi ^ { * } + \\xi v , \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 583, + 267, + 599 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 267, + 585, + 301, + 597 + ], + "score": 0.92, + "content": "| \\xi | < \\xi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 583, + 505, + 599 + ], + "score": 1.0, + "content": "is an equilibrium point. According to Assumption", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 117, + 609 + ], + "score": 1.0, + "content": "4,", + "type": "text" + }, + { + "bbox": [ + 117, + 596, + 177, + 608 + ], + "score": 0.9, + "content": "D ( x ; \\psi ^ { * } + \\xi v )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 596, + 317, + 609 + ], + "score": 1.0, + "content": "is an equilibrium discriminator for", + "type": "text" + }, + { + "bbox": [ + 318, + 596, + 351, + 608 + ], + "score": 0.93, + "content": "| \\xi | < \\delta _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 596, + 391, + 609 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 391, + 596, + 451, + 608 + ], + "score": 0.94, + "content": "D ( x ; \\psi ^ { * } + \\bar { \\xi } v )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "is already an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 606, + 285, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 209, + 621 + ], + "score": 1.0, + "content": "optimal discriminator for", + "type": "text" + }, + { + "bbox": [ + 209, + 607, + 281, + 619 + ], + "score": 0.93, + "content": "| \\xi | < \\operatorname* { m i n } ( \\xi _ { d } , \\delta _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 606, + 285, + 621 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 583, + 505, + 621 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 104, + 630, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 164, + 651 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 164, + 634, + 230, + 646 + ], + "score": 0.91, + "content": "w \\ \\in \\ N ( R ^ { T } R )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 630, + 321, + 651 + ], + "score": 1.0, + "content": ". 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By adding these results, we obtain", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 109, + 437, + 199 + ], + "lines": [ + { + "bbox": [ + 174, + 109, + 437, + 199 + ], + "spans": [ + { + "bbox": [ + 174, + 109, + 437, + 199 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\dot { \\psi } = \\mathbb { E } _ { p _ { d } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] - \\mathbb { E } _ { p _ { \\theta ^ { * } + \\nu w } } [ \\nabla _ { \\psi } D ( x ; \\psi ^ { * } ) ] } \\\\ { \\displaystyle \\quad - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ) } 2 \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) \\nabla _ { x } D ( x ; \\psi ^ { * } ) d \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } } \\\\ { \\displaystyle \\quad - \\frac { \\rho } { 2 } \\int _ { s u p p ( \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ) } \\left\\| \\nabla _ { x } D ( x ; \\psi ^ { * } ) \\right\\| ^ { 2 } d \\mu _ { \\psi ^ { * } , \\theta ^ { * } + \\nu w } ^ { \\prime } } \\\\ { \\displaystyle = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "76d5b6aa3c75ebd421f3f35d18e3417930c6864d902a0d1b797e9eb7b9c47422.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 174, + 109, + 437, + 139.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 174, + 139.0, + 437, + 169.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 174, + 169.0, + 437, + 199.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 202, + 504, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 201, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 104, + 201, + 191, + 218 + ], + "score": 1.0, + "content": "Therefore, the point", + "type": "text" + }, + { + "bbox": [ + 192, + 203, + 252, + 215 + ], + "score": 0.91, + "content": "( \\psi ^ { * } , \\theta ^ { * } + \\nu w )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 201, + 275, + 218 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 275, + 203, + 377, + 216 + ], + "score": 0.92, + "content": "| \\nu | < \\operatorname* { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\nu _ { g } , \\delta _ { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 201, + 506, + 218 + ], + "score": 1.0, + "content": "is an equilibrium point, which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 213, + 325, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 155, + 228 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 156, + 216, + 210, + 226 + ], + "score": 0.91, + "content": "p _ { \\theta ^ { * } + \\nu w } = p _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 213, + 325, + 228 + ], + "score": 1.0, + "content": "according to Assumption 4.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 256, + 256 + ], + "score": 1.0, + "content": "If we consider the projected system", + "type": "text" + }, + { + "bbox": [ + 256, + 241, + 349, + 254 + ], + "score": 0.93, + "content": "( \\alpha , \\beta ) \\ : = \\ : ( T _ { D } ^ { T } \\psi , T _ { G } ^ { T } \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 240, + 506, + 256 + ], + "score": 1.0, + "content": ", then the projected dynamic system’s", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 253, + 295, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 154, + 267 + ], + "score": 1.0, + "content": "Jacobian at", + "type": "text" + }, + { + "bbox": [ + 154, + 253, + 214, + 266 + ], + "score": 0.94, + "content": "( T _ { D } ^ { T } \\psi ^ { * } , T _ { G } ^ { T } \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 253, + 295, + 267 + ], + "score": 1.0, + "content": "is given as follows.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 271, + 430, + 299 + ], + "lines": [ + { + "bbox": [ + 180, + 271, + 430, + 299 + ], + "spans": [ + { + "bbox": [ + 180, + 271, + 430, + 299 + ], + "score": 0.94, + "content": "J ^ { \\prime } = \\left[ \\begin{array} { c c } { { - \\rho T _ { D } ^ { T } Q T _ { D } } } & { { - T _ { D } ^ { T } R T _ { G } } } \\\\ { { T _ { G } ^ { T } R ^ { T } T _ { D } } } & { { 0 } } \\end{array} \\right] = \\left[ \\begin{array} { c c } { { - \\rho \\Lambda _ { D } ^ { ( + ) } } } & { { - T _ { D } ^ { T } R T _ { G } } } \\\\ { { T _ { G } ^ { T } R ^ { T } T _ { D } } } & { { 0 } } \\end{array} \\right]", + "type": "interline_equation", + "image_path": "ab5cc9bede8049eb9fc625a17955b81b82c31d9ae1628ce9c6a08279a683c068.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 180, + 271, + 430, + 299 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 305, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 259, + 319 + ], + "score": 1.0, + "content": "Therefore, we only need to prove that", + "type": "text" + }, + { + "bbox": [ + 260, + 305, + 294, + 317 + ], + "score": 0.92, + "content": "T _ { D } ^ { T } R T _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 304, + 443, + 319 + ], + "score": 1.0, + "content": "is of full column rank. Suppose that", + "type": "text" + }, + { + "bbox": [ + 443, + 304, + 506, + 317 + ], + "score": 0.91, + "content": "u \\in N ( Q ^ { T } ) =", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 107, + 317, + 132, + 329 + ], + "score": 0.91, + "content": "N ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 315, + 252, + 330 + ], + "score": 1.0, + "content": ". 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By calculating", + "type": "text" + }, + { + "bbox": [ + 209, + 456, + 230, + 467 + ], + "score": 0.9, + "content": "u ^ { T } R", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 456, + 308, + 470 + ], + "score": 1.0, + "content": "directly, we obtain", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 473, + 438, + 557 + ], + "lines": [ + { + "bbox": [ + 173, + 473, + 438, + 557 + ], + "spans": [ + { + "bbox": [ + 173, + 473, + 438, + 557 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\boldsymbol { u } ^ { T } \\boldsymbol { R } = u ^ { T } \\frac { \\partial } { \\partial \\boldsymbol { \\theta } } \\int _ { \\mathcal { X } } \\nabla _ { \\boldsymbol { \\psi } } D ( \\boldsymbol { x } ; \\boldsymbol { \\psi } ^ { * } ) d p _ { \\boldsymbol { \\theta } } \\bigg \\rvert _ { \\boldsymbol { \\theta = \\theta } ^ { = } } } \\\\ & { \\qquad = u ^ { T } \\frac { \\partial } { \\partial \\boldsymbol { \\theta } } \\int _ { \\mathcal { X } } \\nabla _ { \\boldsymbol { \\psi } } D ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ) ; \\boldsymbol { \\psi } ^ { * } ) p _ { l a t e n t } ( \\boldsymbol { z } ) d \\boldsymbol { z } \\bigg \\rvert _ { \\boldsymbol { \\theta = \\theta } ^ { * } } } \\\\ & { \\quad = \\int _ { \\mathcal { X } } u ^ { T } \\nabla _ { \\boldsymbol { x } \\boldsymbol { \\psi } } D ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { * } ) ; \\boldsymbol { \\psi } ^ { * } ) \\nabla _ { \\boldsymbol { \\theta } } G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { * } ) p _ { l a t e n t } ( \\boldsymbol { z } ) d \\boldsymbol { z } = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "736d668fa30bc070ae5aff23b93fc12ed49a8c5f8555af3e9d4c60707f61e42d.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 173, + 473, + 438, + 501.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 173, + 501.0, + 438, + 529.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 173, + 529.0, + 438, + 557.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 173, + 575 + ], + "score": 1.0, + "content": "Thus, we obtain", + "type": "text" + }, + { + "bbox": [ + 174, + 561, + 224, + 574 + ], + "score": 0.93, + "content": "u \\in N ( R ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 560, + 305, + 575 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 305, + 561, + 382, + 574 + ], + "score": 0.91, + "content": "N ( Q ^ { T } ) \\subset N ( R ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 560, + 400, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 562, + 462, + 574 + ], + "score": 0.93, + "content": "C ( R ) \\subset C ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 560, + 506, + 575 + ], + "score": 1.0, + "content": ". Now, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 572, + 507, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 165, + 590 + ], + "score": 1.0, + "content": "can check that", + "type": "text" + }, + { + "bbox": [ + 166, + 576, + 187, + 587 + ], + "score": 0.91, + "content": "R T _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 572, + 299, + 590 + ], + "score": 1.0, + "content": "is of full column rank since", + "type": "text" + }, + { + "bbox": [ + 300, + 574, + 381, + 588 + ], + "score": 0.92, + "content": "T _ { G } ^ { T } R ^ { T } R T _ { G } = \\Lambda _ { G } ^ { ( + ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 572, + 507, + 590 + ], + "score": 1.0, + "content": "is positive definite. Therefore,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 593, + 351, + 605 + ], + "lines": [ + { + "bbox": [ + 261, + 593, + 351, + 605 + ], + "spans": [ + { + "bbox": [ + 261, + 593, + 351, + 605 + ], + "score": 0.89, + "content": "R T _ { G } w = 0 \\Rightarrow w = 0", + "type": "interline_equation", + "image_path": "6f2616ac722c0754a8514a060e01e37c5471978672ae07a49e954b709ee13d7c.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 261, + 593, + 351, + 605 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 261, + 624 + ], + "score": 1.0, + "content": "We note that the projection matrix on", + "type": "text" + }, + { + "bbox": [ + 262, + 610, + 286, + 622 + ], + "score": 0.91, + "content": "C ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 608, + 335, + 624 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 335, + 609, + 450, + 623 + ], + "score": 0.93, + "content": "T _ { D } ( T _ { D } ^ { T } T _ { D } ) ^ { - 1 } T _ { D } ^ { T } = T _ { D } T _ { D } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 608, + 506, + 624 + ], + "score": 1.0, + "content": ". 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Suppose that", + "type": "text" + }, + { + "bbox": [ + 443, + 304, + 506, + 317 + ], + "score": 0.91, + "content": "u \\in N ( Q ^ { T } ) =", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 107, + 317, + 132, + 329 + ], + "score": 0.91, + "content": "N ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 315, + 252, + 330 + ], + "score": 1.0, + "content": ". According to Assumption 2,", + "type": "text" + }, + { + "bbox": [ + 252, + 317, + 273, + 328 + ], + "score": 0.88, + "content": "h ( \\psi )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 315, + 358, + 330 + ], + "score": 1.0, + "content": "is locally constant at", + "type": "text" + }, + { + "bbox": [ + 359, + 317, + 371, + 328 + ], + "score": 0.89, + "content": "\\psi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 315, + 450, + 330 + ], + "score": 1.0, + "content": "along the direction", + "type": "text" + }, + { + "bbox": [ + 450, + 319, + 456, + 326 + ], + "score": 0.77, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 315, + 506, + 330 + ], + "score": 1.0, + "content": ". 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By taking directional derivative w.r.t.", + "type": "text" + }, + { + "bbox": [ + 456, + 375, + 464, + 385 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "along the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 383, + 199, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 144, + 396 + ], + "score": 1.0, + "content": "direction", + "type": "text" + }, + { + "bbox": [ + 145, + 387, + 151, + 394 + ], + "score": 0.79, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 383, + 199, + 396 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 361, + 505, + 396 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 188, + 399, + 423, + 415 + ], + "lines": [ + { + "bbox": [ + 188, + 399, + 423, + 415 + ], + "spans": [ + { + "bbox": [ + 188, + 399, + 423, + 415 + ], + "score": 0.88, + "content": "u ^ { T } \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) = 0 , x \\in s u p p ( \\mu _ { \\psi ^ { * } + \\xi u , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )", + "type": "interline_equation", + "image_path": "46e6de64509dff368771e4fd30ad7bd6b75843574602bedb44e2c0a2bdb55ee9.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 188, + 399, + 423, + 415 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 142, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 144, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 144, + 431 + ], + "score": 1.0, + "content": "and thus", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 418, + 144, + 431 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 428, + 449, + 443 + ], + "lines": [ + { + "bbox": [ + 161, + 428, + 449, + 443 + ], + "spans": [ + { + "bbox": [ + 161, + 428, + 449, + 443 + ], + "score": 0.88, + "content": "u ^ { T } \\nabla _ { \\psi x } ^ { T } D ( x ; \\psi ^ { * } ) = u ^ { T } \\nabla _ { x \\psi } D ( x ; \\psi ^ { * } ) = 0 , x \\in s u p p ( p _ { \\theta ^ { + } } ) = s u p p ( p _ { d } )", + "type": "interline_equation", + "image_path": "f12156480932f019af6c15a1635e14a7f641e3026b4726e22c8d89fa0e6d3833.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 161, + 428, + 449, + 443 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 446, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 211, + 459 + ], + "score": 1.0, + "content": "according to Assumption", + "type": "text" + }, + { + "bbox": [ + 211, + 447, + 223, + 456 + ], + "score": 0.68, + "content": "^ \\mathrm { 6 b }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 445, + 342, + 459 + ], + "score": 1.0, + "content": "(the inclusion condition that", + "type": "text" + }, + { + "bbox": [ + 342, + 446, + 494, + 459 + ], + "score": 0.91, + "content": "s u p p ( p _ { d } ) = s u p p ( p _ { \\theta ^ { * } } ) \\subset s u p p ( \\mu ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 308, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 209, + 470 + ], + "score": 1.0, + "content": "required). By calculating", + "type": "text" + }, + { + "bbox": [ + 209, + 456, + 230, + 467 + ], + "score": 0.9, + "content": "u ^ { T } R", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 456, + 308, + 470 + ], + "score": 1.0, + "content": "directly, we obtain", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 445, + 506, + 470 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 473, + 438, + 557 + ], + "lines": [ + { + "bbox": [ + 173, + 473, + 438, + 557 + ], + "spans": [ + { + "bbox": [ + 173, + 473, + 438, + 557 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\boldsymbol { u } ^ { T } \\boldsymbol { R } = u ^ { T } \\frac { \\partial } { \\partial \\boldsymbol { \\theta } } \\int _ { \\mathcal { X } } \\nabla _ { \\boldsymbol { \\psi } } D ( \\boldsymbol { x } ; \\boldsymbol { \\psi } ^ { * } ) d p _ { \\boldsymbol { \\theta } } \\bigg \\rvert _ { \\boldsymbol { \\theta = \\theta } ^ { = } } } \\\\ & { \\qquad = u ^ { T } \\frac { \\partial } { \\partial \\boldsymbol { \\theta } } \\int _ { \\mathcal { X } } \\nabla _ { \\boldsymbol { \\psi } } D ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ) ; \\boldsymbol { \\psi } ^ { * } ) p _ { l a t e n t } ( \\boldsymbol { z } ) d \\boldsymbol { z } \\bigg \\rvert _ { \\boldsymbol { \\theta = \\theta } ^ { * } } } \\\\ & { \\quad = \\int _ { \\mathcal { X } } u ^ { T } \\nabla _ { \\boldsymbol { x } \\boldsymbol { \\psi } } D ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { * } ) ; \\boldsymbol { \\psi } ^ { * } ) \\nabla _ { \\boldsymbol { \\theta } } G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { * } ) p _ { l a t e n t } ( \\boldsymbol { z } ) d \\boldsymbol { z } = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "736d668fa30bc070ae5aff23b93fc12ed49a8c5f8555af3e9d4c60707f61e42d.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 173, + 473, + 438, + 501.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 173, + 501.0, + 438, + 529.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 173, + 529.0, + 438, + 557.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 173, + 575 + ], + "score": 1.0, + "content": "Thus, we obtain", + "type": "text" + }, + { + "bbox": [ + 174, + 561, + 224, + 574 + ], + "score": 0.93, + "content": "u \\in N ( R ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 560, + 305, + 575 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 305, + 561, + 382, + 574 + ], + "score": 0.91, + "content": "N ( Q ^ { T } ) \\subset N ( R ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 560, + 400, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 401, + 562, + 462, + 574 + ], + "score": 0.93, + "content": "C ( R ) \\subset C ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 560, + 506, + 575 + ], + "score": 1.0, + "content": ". Now, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 572, + 507, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 165, + 590 + ], + "score": 1.0, + "content": "can check that", + "type": "text" + }, + { + "bbox": [ + 166, + 576, + 187, + 587 + ], + "score": 0.91, + "content": "R T _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 572, + 299, + 590 + ], + "score": 1.0, + "content": "is of full column rank since", + "type": "text" + }, + { + "bbox": [ + 300, + 574, + 381, + 588 + ], + "score": 0.92, + "content": "T _ { G } ^ { T } R ^ { T } R T _ { G } = \\Lambda _ { G } ^ { ( + ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 572, + 507, + 590 + ], + "score": 1.0, + "content": "is positive definite. Therefore,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 560, + 507, + 590 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 593, + 351, + 605 + ], + "lines": [ + { + "bbox": [ + 261, + 593, + 351, + 605 + ], + "spans": [ + { + "bbox": [ + 261, + 593, + 351, + 605 + ], + "score": 0.89, + "content": "R T _ { G } w = 0 \\Rightarrow w = 0", + "type": "interline_equation", + "image_path": "6f2616ac722c0754a8514a060e01e37c5471978672ae07a49e954b709ee13d7c.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 261, + 593, + 351, + 605 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 504, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 261, + 624 + ], + "score": 1.0, + "content": "We note that the projection matrix on", + "type": "text" + }, + { + "bbox": [ + 262, + 610, + 286, + 622 + ], + "score": 0.91, + "content": "C ( Q )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 608, + 335, + 624 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 335, + 609, + 450, + 623 + ], + "score": 0.93, + "content": "T _ { D } ( T _ { D } ^ { T } T _ { D } ) ^ { - 1 } T _ { D } ^ { T } = T _ { D } T _ { D } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 608, + 506, + 624 + ], + "score": 1.0, + "content": ". 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First,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 120, + 504, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 160, + 134 + ], + "score": 1.0, + "content": "suppose that", + "type": "text" + }, + { + "bbox": [ + 160, + 122, + 174, + 132 + ], + "score": 0.88, + "content": "T _ { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 120, + 474, + 134 + ], + "score": 1.0, + "content": "is empty. Similar to the analysis given above, we can find that the point", + "type": "text" + }, + { + "bbox": [ + 475, + 122, + 504, + 133 + ], + "score": 0.91, + "content": "( \\psi ^ { * } , \\theta )", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 127, + 147 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 132, + 248, + 145 + ], + "score": 0.92, + "content": "| \\theta - \\theta ^ { * } | < \\mathrm { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 130, + 376, + 147 + ], + "score": 1.0, + "content": "is an equilibrium point, where", + "type": "text" + }, + { + "bbox": [ + 376, + 132, + 433, + 144 + ], + "score": 0.92, + "content": "g ( \\theta ^ { * } ) = g ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 130, + 505, + 147 + ], + "score": 1.0, + "content": "for a sufficiently", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 141, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 131, + 157 + ], + "score": 1.0, + "content": "small", + "type": "text" + }, + { + "bbox": [ + 131, + 144, + 181, + 155 + ], + "score": 0.92, + "content": "\\lvert \\theta - \\theta ^ { * } \\rvert < \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 141, + 255, + 157 + ], + "score": 1.0, + "content": ". We conclude that", + "type": "text" + }, + { + "bbox": [ + 256, + 145, + 289, + 155 + ], + "score": 0.89, + "content": "p _ { \\theta } = p _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 141, + 304, + 157 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 305, + 144, + 419, + 156 + ], + "score": 0.93, + "content": "| \\theta - \\theta ^ { * } | < \\operatorname * { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 141, + 506, + 157 + ], + "score": 1.0, + "content": ". Under the generator", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 308, + 167 + ], + "score": 1.0, + "content": "initialization that is sufficiently close according to", + "type": "text" + }, + { + "bbox": [ + 308, + 155, + 318, + 164 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 153, + 505, + 167 + ], + "score": 1.0, + "content": ", we can only observe the discriminator update", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 171, + 377, + 192 + ], + "lines": [ + { + "bbox": [ + 234, + 171, + 377, + 192 + ], + "spans": [ + { + "bbox": [ + 234, + 171, + 377, + 192 + ], + "score": 0.93, + "content": "\\dot { \\psi } = - \\frac { \\rho } { 2 } \\nabla _ { \\psi } \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ]", + "type": "interline_equation", + "image_path": "854fc98497939e024ca6a6b6523d6b1f6d414dfbb440733759a9e9fd04af9bf4.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 234, + 171, + 377, + 192 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 131, + 210 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 131, + 197, + 281, + 210 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 197, + 319, + 210 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 320, + 198, + 328, + 209 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 197, + 349, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 198, + 477, + 210 + ], + "score": 0.93, + "content": "| \\theta - \\theta ^ { * } | < \\mathrm { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 197, + 506, + 210 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 434, + 220 + ], + "score": 1.0, + "content": "discriminator update described above is locally stable system near the equilibrium", + "type": "text" + }, + { + "bbox": [ + 434, + 210, + 466, + 220 + ], + "score": 0.89, + "content": "\\psi = \\psi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "since the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 213, + 233 + ], + "score": 1.0, + "content": "Jacobian of the update on", + "type": "text" + }, + { + "bbox": [ + 213, + 220, + 221, + 231 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 219, + 268, + 233 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + }, + { + "bbox": [ + 269, + 220, + 291, + 231 + ], + "score": 0.91, + "content": "- \\rho Q", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "and the zero eigenvalues can be ignored in a similar", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 455, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 455, + 243 + ], + "score": 1.0, + "content": "manner to the previous step. Therefore, the given system is stable near the equilibrium.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 161, + 272 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 162, + 259, + 176, + 270 + ], + "score": 0.89, + "content": "T _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 257, + 265, + 272 + ], + "score": 1.0, + "content": "is empty. Given that", + "type": "text" + }, + { + "bbox": [ + 266, + 258, + 345, + 270 + ], + "score": 0.89, + "content": "N ( Q ^ { T } ) \\subset N ( R ^ { T } ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 257, + 348, + 272 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 349, + 258, + 379, + 269 + ], + "score": 0.83, + "content": "R = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 257, + 506, + 272 + ], + "score": 1.0, + "content": ", then the results are similar to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 316, + 281 + ], + "score": 1.0, + "content": "those presented above, but our goal is to show that", + "type": "text" + }, + { + "bbox": [ + 317, + 270, + 341, + 281 + ], + "score": 0.91, + "content": "( \\psi , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 270, + 469, + 281 + ], + "score": 1.0, + "content": "is an equilibrium point, where", + "type": "text" + }, + { + "bbox": [ + 469, + 269, + 494, + 281 + ], + "score": 0.92, + "content": "( \\psi , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 363, + 294 + ], + "score": 1.0, + "content": "sufficiently close to the original equilibrium point. We note that", + "type": "text" + }, + { + "bbox": [ + 363, + 281, + 392, + 293 + ], + "score": 0.92, + "content": "( \\psi ^ { * } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "is also an equilibrium point", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 227, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 227, + 304 + ], + "score": 1.0, + "content": "that satisfies the assumptions.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 105, + 307, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 183, + 321 + ], + "score": 1.0, + "content": "By Assumption 2,", + "type": "text" + }, + { + "bbox": [ + 183, + 308, + 265, + 320 + ], + "score": 0.92, + "content": "h ( \\psi ) = h ( \\psi ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 307, + 281, + 321 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 282, + 308, + 339, + 320 + ], + "score": 0.92, + "content": "| \\psi - \\psi ^ { * } | < \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 307, + 421, + 321 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 422, + 308, + 489, + 320 + ], + "score": 0.93, + "content": "\\nabla _ { x } D ( x ; \\psi ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 319, + 373, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 320, + 228, + 332 + ], + "score": 0.92, + "content": "x \\in s u p p ( \\mu _ { \\psi , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 319, + 246, + 332 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 247, + 320, + 302, + 331 + ], + "score": 0.92, + "content": "| \\psi - \\psi ^ { * } | < \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 319, + 373, + 332 + ], + "score": 1.0, + "content": ". 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We conclude that", + "type": "text" + }, + { + "bbox": [ + 256, + 145, + 289, + 155 + ], + "score": 0.89, + "content": "p _ { \\theta } = p _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 141, + 304, + 157 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 305, + 144, + 419, + 156 + ], + "score": 0.93, + "content": "| \\theta - \\theta ^ { * } | < \\operatorname * { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 141, + 506, + 157 + ], + "score": 1.0, + "content": ". Under the generator", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 308, + 167 + ], + "score": 1.0, + "content": "initialization that is sufficiently close according to", + "type": "text" + }, + { + "bbox": [ + 308, + 155, + 318, + 164 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 153, + 505, + 167 + ], + "score": 1.0, + "content": ", we can only observe the discriminator update", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 110, + 506, + 167 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 171, + 377, + 192 + ], + "lines": [ + { + "bbox": [ + 234, + 171, + 377, + 192 + ], + "spans": [ + { + "bbox": [ + 234, + 171, + 377, + 192 + ], + "score": 0.93, + "content": "\\dot { \\psi } = - \\frac { \\rho } { 2 } \\nabla _ { \\psi } \\mathbb { E } _ { \\mu _ { \\psi , \\theta } } [ \\| \\nabla _ { x } D ( x ; \\psi ) \\| ^ { 2 } ]", + "type": "interline_equation", + "image_path": "854fc98497939e024ca6a6b6523d6b1f6d414dfbb440733759a9e9fd04af9bf4.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 234, + 171, + 377, + 192 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 131, + 210 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 131, + 197, + 281, + 210 + ], + "score": 0.92, + "content": "\\mathbb { E } _ { p _ { d } } [ D ( x ; \\psi ) ] - \\mathbb { E } _ { p _ { \\theta } } [ D ( x ; \\psi ) ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 197, + 319, + 210 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 320, + 198, + 328, + 209 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 197, + 349, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 198, + 477, + 210 + ], + "score": 0.93, + "content": "| \\theta - \\theta ^ { * } | < \\mathrm { m i n } ( \\epsilon _ { \\mu } , \\epsilon _ { g } , \\delta _ { g } , \\nu )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 197, + 506, + 210 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 434, + 220 + ], + "score": 1.0, + "content": "discriminator update described above is locally stable system near the equilibrium", + "type": "text" + }, + { + "bbox": [ + 434, + 210, + 466, + 220 + ], + "score": 0.89, + "content": "\\psi = \\psi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "since the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 213, + 233 + ], + "score": 1.0, + "content": "Jacobian of the update on", + "type": "text" + }, + { + "bbox": [ + 213, + 220, + 221, + 231 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 219, + 268, + 233 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + }, + { + "bbox": [ + 269, + 220, + 291, + 231 + ], + "score": 0.91, + "content": "- \\rho Q", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "and the zero eigenvalues can be ignored in a similar", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 231, + 455, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 455, + 243 + ], + "score": 1.0, + "content": "manner to the previous step. 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Given that", + "type": "text" + }, + { + "bbox": [ + 266, + 258, + 345, + 270 + ], + "score": 0.89, + "content": "N ( Q ^ { T } ) \\subset N ( R ^ { T } ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 257, + 348, + 272 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 349, + 258, + 379, + 269 + ], + "score": 0.83, + "content": "R = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 257, + 506, + 272 + ], + "score": 1.0, + "content": ", then the results are similar to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 316, + 281 + ], + "score": 1.0, + "content": "those presented above, but our goal is to show that", + "type": "text" + }, + { + "bbox": [ + 317, + 270, + 341, + 281 + ], + "score": 0.91, + "content": "( \\psi , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 270, + 469, + 281 + ], + "score": 1.0, + "content": "is an equilibrium point, where", + "type": "text" + }, + { + "bbox": [ + 469, + 269, + 494, + 281 + ], + "score": 0.92, + "content": "( \\psi , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 363, + 294 + ], + "score": 1.0, + "content": "sufficiently close to the original equilibrium point. We note that", + "type": "text" + }, + { + "bbox": [ + 363, + 281, + 392, + 293 + ], + "score": 0.92, + "content": "( \\psi ^ { * } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "is also an equilibrium point", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 227, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 227, + 304 + ], + "score": 1.0, + "content": "that satisfies the assumptions.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 257, + 506, + 304 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 307, + 504, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 183, + 321 + ], + "score": 1.0, + "content": "By Assumption 2,", + "type": "text" + }, + { + "bbox": [ + 183, + 308, + 265, + 320 + ], + "score": 0.92, + "content": "h ( \\psi ) = h ( \\psi ^ { * } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 307, + 281, + 321 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 282, + 308, + 339, + 320 + ], + "score": 0.92, + "content": "| \\psi - \\psi ^ { * } | < \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 307, + 421, + 321 + ], + "score": 1.0, + "content": ", which implies that", + "type": "text" + }, + { + "bbox": [ + 422, + 308, + 489, + 320 + ], + "score": 0.93, + "content": "\\nabla _ { x } D ( x ; \\psi ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 319, + 373, + 332 + ], + "spans": [ + { + "bbox": [ + 107, + 320, + 228, + 332 + ], + "score": 0.92, + "content": "x \\in s u p p ( \\mu _ { \\psi , \\theta ^ { * } } ) = s u p p ( \\mu ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 319, + 246, + 332 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 247, + 320, + 302, + 331 + ], + "score": 0.92, + "content": "| \\psi - \\psi ^ { * } | < \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 319, + 373, + 332 + ], + "score": 1.0, + "content": ". 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PenaltyPenalty termPenalty measure, sampling method
WGAN WGAN-GPNone(Weight Clipping) Eμ[(IVxD|-1)2]None x=axd+(1-α)xg
Eμ[VxD|2]
Pg PdEμ[VxDii2]x=xg x=xd
μGPEμVDii2]x=axd+(1-α)xg
μmidEμ[VxDii2]x= 0.5xd+0.5xg
μg,ancEμVDii2]x=αA+(1-α)xg
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PenaltyDCGAN Inception FIDResNet InceptionFID
WGAN 35.64 ± 0.09 48.7==
WGAN-GP6.48 ± 0.10 35.07.82 ± 0.0918.1
Pg6.46 ± 0.09 38.07.63 ± 0.1020.9
pd6.33 ± 0.0738.9 7.63 ± 0.0920.3
μGP6.40 ±0.0835.4 7.60 ± 0.0918.3
μmid6.60 ± 0.0733.9 7.86 ± 0.0716.4
μg,anc6.45 ± 0.0733.7 7.36 ± 0.0922.4
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"page_info": { + "page_no": 19, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/HyAddcLge/HyAddcLge.md b/parse/train/HyAddcLge/HyAddcLge.md new file mode 100644 index 0000000000000000000000000000000000000000..e676b01211d10ee28422da90b0f89e291cc0f159 --- /dev/null +++ b/parse/train/HyAddcLge/HyAddcLge.md @@ -0,0 +1,265 @@ +# REVISITING DISTRIBUTED SYNCHRONOUS SGD + +Jianmin Chen∗, Xinghao Pan∗†, Rajat Monga, Samy Bengio +Google Brain +Mountain View, CA, USA +{jmchen,xinghao,rajatmonga,bengio}@google.com + +Rafal Jozefowicz OpenAI San Francisco, CA, USA rafal@openai.com + +# ABSTRACT + +Distributed training of deep learning models on large-scale training data is typically conducted with asynchronous stochastic optimization to maximize the rate of updates, at the cost of additional noise introduced from asynchrony. In contrast, the synchronous approach is often thought to be impractical due to idle time wasted on waiting for straggling workers. We revisit these conventional beliefs in this paper, and examine the weaknesses of both approaches. We demonstrate that a third approach, synchronous optimization with backup workers, can avoid asynchronous noise while mitigating for the worst stragglers. Our approach is empirically validated and shown to converge faster and to better test accuracies. + +# 1 INTRODUCTION + +The recent success of deep learning approaches for domains like speech recognition (Hinton et al., 2012) and computer vision (Ioffe & Szegedy, 2015) stems from many algorithmic improvements but also from the fact that the size of available training data has grown significantly over the years, together with the computing power, in terms of both CPUs and GPUs. While a single GPU often provides algorithmic simplicity and speed up to a given scale of data and model, there exist an operating point where a distributed implementation of training algorithms for deep architectures becomes necessary. + +Currently, popular distributed training algorithms include mini-batch versions of stochastic gradient descent (SGD) and other stochastic optimization algorithms such as AdaGrad (Duchi et al., 2011), RMSProp (Tieleman & Hinton, 2012), and ADAM (Kingma & Ba, 2014). Unfortunately, bulksynchronous implementations of stochastic optimization are often slow in practice due to the need to wait for the slowest machine in each synchronous batch. To circumvent this problem, practitioners have resorted to asynchronous approaches which emphasize speed by using potentially stale information for computation. While asynchronous training have proven to be faster than their synchronous counterparts, they often result in convergence to poorer results. + +In this paper1, we revisit synchronous learning, and propose a method for mitigating stragglers in synchronous stochastic optimization. Specifically, we synchronously compute a mini-batch gradient with only a subset of worker machines, thus alleviating the straggler effect while avoiding any staleness in our gradients. The primary contributions of our paper are: + +• Illustration of how gradient staleness in asynchronous training negatively impacts test accuracy and is exacerbated by deep models. +Measurement of machine response times for synchronous stochastic optimization in a large deployment of 100 GPUs, showing how stragglers in the tail end affect convergence speed. Proposal of synchronous stochastic optimization with backup workers to mitigate straggler effects without gradient staleness. +• Establishing the need to measure both speed of convergence and test accuracy of optimum for empirical validation. + +• Empirical demonstration that our proposed synchronous training method outperforms asynchronous training by converging faster and to better test accuracies. + +The remainder of this paper is organized as follows. We briefly present preliminaries and notation in Section 1.1. Section 2 describes asynchronous stochastic optimization and presents experimental evidence of gradient staleness in deep neural network models. We present our approach in Section 3, and exhibit straggler effects that motivate the approach. We then empirically evaluate our approach in Sections 4. Related work is discussed in Section 5, and we conclude in Section 6. + +# 1.1 PRELIMINARIES AND NOTATION + +Given a dataset $\mathcal { X } = \{ x _ { i } : i = 1 , \ldots , | \mathcal { X } | \}$ , our goal is to learn the parameters $\theta$ of a model with respect to an empirical loss function $f$ , defined as $\begin{array} { r } { f ( \theta ) \stackrel { \Delta } { = } \frac { 1 } { | \mathcal { X } | } \sum _ { i = 1 } ^ { | \mathcal { X } | } F ( x _ { i } ; \theta ) } \end{array}$ , where $F ( x _ { i } ; \theta )$ is the loss with respect to a datapoint $x _ { i }$ and the model $\theta$ . + +A first-order stochastic optimization algorithm achieves this by iteratively updating $\theta$ using a stochastic gradient $G \overset { \Delta } { = } \nabla F ( x _ { i } ; \theta )$ computed at a randomly sampled $x _ { i }$ , producing a sequence of models $\theta ^ { ( 0 ) } , \theta ^ { ( 1 ) } , \ldots$ . Stochastic optimization algorithms differ in their update equations. For example, the update of SGD is $\bar { \theta ^ { ( t + 1 ) } } \bar { \bf \Phi } = \theta ^ { ( t ) } - \gamma _ { t } \bar { G } ^ { ( t ) } = \theta ^ { ( t ) } - \gamma _ { t } \nabla F ( x _ { i } ; \bar { \theta } ^ { ( t ) } )$ , where $\gamma _ { t }$ is the learning rate or step size at iteration $t$ . A mini-batch version of the stochastic optimization algorithm computes the stochastic gradient over mini-batch of size $B$ instead of a single datapoint, i.e., $\begin{array} { r } { G \stackrel { \Delta } { = } \frac { 1 } { B } \sum _ { i = 1 } ^ { B } \nabla F ( \widetilde { x } _ { i } ; \theta ^ { ( t ) } ) } \end{array}$ , where $\widetilde { x } _ { i }$ ’s are randomly sampled from $\mathcal { X }$ . We will often evaluate performance on an exponential moving average $\bar { \theta } ^ { ( t ) } = \alpha \bar { \theta } ^ { ( t - 1 ) } + ( 1 - \alpha ) \theta ^ { ( t ) }$ with decay rate $\alpha$ . + +Our interest is in distributed stochastic optimization using $N$ worker machines in charge of computing stochastic gradients that are sent to $M$ parameter servers. Each parameter server $j$ is responsible for storing a subset $\theta [ j ]$ of the model, and performing updates on $\theta [ j ]$ . In the synchronous setting, we will also introduce additional $b$ backup workers for straggler mitigation. + +# 2 ASYNCHRONOUS STOCHASTIC OPTIMIZATION + +An approach for a distributed stochastic gradient descent algorithm was presented in Dean et al. (2012), consisting of two main ingredients. First, the parameters of the model are distributed on multiple servers, depending on the architecture. This set of servers are called the parameter servers. Second, there can be multiple workers processing data in parallel and communicating with the parameter servers. Each worker processes a mini-batch of data independently of the others, as follows: + +• The worker fetches from the parameter servers the most up-to-date parameters of the model needed to process the current mini-batch; +• It then computes gradients of the loss with respect to these parameters; +• Finally, these gradients are sent back to the parameter servers, which then updates the model accordingly. + +Since each worker communicates with the parameter servers independently of the others, this is called Asynchronous Stochastic Gradient Descent (Async-SGD), or more generally, Asynchronous Stochastic Optimization (Async-Opt). A similar approach was later proposed by Chilimbi et al. (2014). Async-Opt is presented in Algorithms 1 and 2. + +In practice, the updates of Async-Opt are different than those of serially running the stochastic optimization algorithm for two reasons. Firstly, the read operation (Algo 1 Line 2) on a worker may be interleaved with updates by other workers to different parameter servers, so the resultant $\widehat { \theta } _ { k }$ may not be consistent with any parameter incarnation $\boldsymbol { \theta } ^ { ( t ) }$ . Secondly, model updates may have occurred while a worker is computing its stochastic gradient; hence, the resultant gradients are typically computed with respect to outdated parameters. We refer to these as stale gradients, and its staleness as the number of updates that have occurred between its corresponding read and update operations. + +Understanding the theoretical impact of staleness is difficult work and the topic of many recent papers, e.g. Recht et al. (2011); Duchi et al. (2013); Leblond et al. (2016); Reddi et al. (2015); + +# Algorithm 1: Async-SGD worker $k$ + +Input: Dataset $\mathcal { X }$ +Input: $B$ mini-batch size +1 while True do +2 Read $\widehat { \theta } _ { k } = ( \theta [ 0 ] , \ldots , \theta [ M ] )$ from PSs. +3 $G _ { k } ^ { ( t ) } : = 0$ . +4 for $i = 1 , \ldots , B$ do +5 Sample datapoint $\widetilde { x } _ { i }$ from $\mathcal { X }$ . +6 $\begin{array} { r } { G _ { k } ^ { ( t ) } G _ { k } ^ { ( t ) } + \frac { 1 } { B } \nabla F ( \widetilde { x } _ { i } ; \widehat { \theta } _ { k } ) . } \end{array}$ +7 end +8 Send $G _ { k } ^ { ( t ) }$ to parameter servers. +9 end + +# Algorithm 2: Async-SGD Parameter Server j + +Input: $\gamma _ { 0 } , \gamma _ { 1 } , \ldots$ learning rates. Input: $\alpha$ decay rate. Input: $\theta ^ { ( 0 ) }$ model initialization. 1 for $t = 0 , 1 , \ldots$ do 2 Wait for gradient $G$ from any worker. 3 $\theta ^ { ( t + 1 ) } [ j ] \theta ^ { ( t ) } [ j ] - \gamma _ { t } G [ j ]$ . 4 $\bar { \theta } ^ { ( t ) } [ j ] = \alpha \bar { \theta } ^ { ( t - 1 ) } [ j ] + ( 1 - \alpha ) \theta ^ { ( t ) } [ j ] ,$ . 5 end + +![](images/869a74e931a0a7ff1e7216146dd0c69e43230ff7820846197c9038f2307065dd.jpg) +Figure 1: Gradient staleness dependence on model layer. Gradients are computed in a bottom-up forward propagation step followed by a top-down back propagation step. Parameters are read from servers in the forward prop, but gradients are sent to servers during the back prop. Thus, gradients of lower layers are more stale than top layers. + +![](images/067a6882cd5a5473a6949076399e8dc649d5d54797b44d7ec5ef733add6828af.jpg) +Figure 2: Degradation of test classification error with increasing average gradient staleness in MNIST CNN model. + +De Sa et al. (2015); Mania et al. (2015), most of which focus on individual algorithms, under strong assumptions that may not hold up in practice. This is further complicated by deep models with multiple layers, since the times at which model parameters are read and which gradients are computed and sent are dependent on the depth of the layers (Figure 1). To better understand this dependence in real models, we collected staleness statistics on a Async-Opt run with 40 workers on a 18-layer Inception model (Szegedy et al., 2016) trained on the ImageNet Challenge dataset (Russakovsky et al., 2015), as shown in Table 1. + +Despite the abovementioned problems, Async-Opt has been shown to be scale well up to a few dozens of workers for some models. However, at larger scales, increasing the number of machines (and thus staleness of gradients) can result in poorer trained models. + +
LayerMinMeanMedianMaxStd DevCount
18414.5413.94293.8310908
12511.3511.3233.0944478
11819.819.59343.65187
02438.9738.43615.43178
+ +Table 1: Staleness of gradients in a 18-layer Inception model. Gradients were collected in a run of asynchronous training using 40 machines. Staleness of a gradient is measured as the number of updates that have occurred between its corresponding read and update operations. The staleness of gradients increases from a mean of ${ \sim } 1 4 . 5$ in the top layer (Layer 18) to ${ \sim } 3 9 . 0 $ in the bottom layer (Layer 0). + +# 2.1 IMPACT OF STALENESS ON TEST ACCURACY + +We explore how increased staleness contributes to training of poorer models. In order to mimic the setting on a smaller scale, we trained a state-of-the-art MNIST CNN model but simulated staleness by using old gradients for the parameter updates. Details of the model and training are provided in Appendix A.1. + +The best final classification error on a test set was $0 . 3 6 \%$ , which increases to $0 . 4 7 \%$ with average gradient staleness of 20 steps, and up to $0 . 7 9 \%$ with 50 steps (see Figure 2). + +Once the average simulated staleness was chosen to be more than 15 steps, the results started to significantly deteriorate and the training itself became much less stable. We had to employ following tricks to prevent the results from blowing up: + +• Slowly increase the staleness over the first 3 epochs of training. This mimics increasing the number of asynchronous workers and is also very important in practice for some of the models we experimented with (e.g. large word-level language models). The trick was not relevant with a simulated staleness less than 15 but became crucial for larger values. Use lower initial learning rates when staleness is at least 20, which reduces a frequency of explosions (train error goes to $90 \%$ ). This observation is similar to what we found in other experiments - we were able to use much larger learning rates with synchronous training and the results were also more stable. +Even with above tricks the divergence occurs occasionally and we found that restarting training from random weights can lead to more successful runs. The best results were then chosen based on validation set performance. + +# 3 REVISTING SYNCHRONOUS STOCHASTIC OPTIMIZATION + +Both Dean et al. (2012) and Chilimbi et al. (2014) use versions of Async-SGD where the main potential problem is that each worker computes gradients over a potentially old version of the model. In order to remove this discrepancy, we propose here to reconsider a synchronous version of distributed stochastic gradient descent (Sync-SGD), or more generally, Synchronous Stochastic Optimization (Sync-Opt), where the parameter servers wait for all workers to send their gradients, aggregate them, and send the updated parameters to all workers afterward. This ensures that the actual algorithm is a true mini-batch stochastic gradient descent, with an effective batch size equal to the sum of all the mini-batch sizes of the workers. + +While this approach solves the staleness problem, it also introduces the potential problem that the actual update time now depends on the slowest worker. Although workers have equivalent computation and network communication workload, slow stragglers may result from failing hardware, or contention on shared underlying hardware resources in data centers, or even due to preemption by other jobs. + +To alleviate the straggler problem, we introduce backup workers (Dean & Barroso, 2013) as follows: instead of having only $N$ workers, we add $b$ extra workers, but as soon as the parameter servers receive gradients from any $N$ workers, they stop waiting and update their parameters using the $N$ gradients. The slowest $b$ workers’ gradients will be dropped when they arrive. Our method is presented in Algorithms 3, 4. + +# Algorithm 3: Sync-SGD worker $k$ , where $k =$ $1 , \ldots , N + b$ + +Input: Dataset $\mathcal { X }$ Input: $B$ mini-batch size 1 for $t = 0 , 1 , \ldots$ do 2 Wait to read ${ \boldsymbol { \theta } } ^ { ( t ) } = ( { \boldsymbol { \theta } } ^ { ( t ) } [ 0 ] , \dots , { \boldsymbol { \theta } } ^ { ( t ) } [ M ] )$ from parameter servers. 3 $G _ { k } ^ { ( t ) } : = 0$ 4 for $i = 1 , \ldots , B$ do 5 Sample datapoint $\widetilde { x } _ { k , i }$ from $\mathcal { X }$ . 6 $\begin{array} { r } { G _ { k } ^ { ( t ) } G _ { k } ^ { ( t ) } + \frac { 1 } { B } \nabla F ( \widetilde { x } _ { k , i } ; \theta ^ { ( t ) } ) } \end{array}$ . 7 end 8 Send $( G _ { k } ^ { ( t ) } , t )$ to parameter servers. 9 end + +# Algorithm 4: Sync-SGD Parameter Server $j$ + +Input: $\gamma _ { 0 } , \gamma _ { 1 } , \ldots$ learning rates. Input: $\alpha$ decay rate. Input: $N$ number of mini-batches to aggregate. Input: $\theta ^ { ( 0 ) }$ model initialization. for $t = 0 , 1 , \ldots$ do 2 $\mathcal { G } = \{ \}$ 3 while $| \mathcal { G } | < N$ do 4 Wait for $( G , t ^ { \prime } )$ from any worker. 5 if $t ^ { \prime } = = t$ then ${ \mathcal { G } } { \mathcal { G } } \cup \{ G \}$ . 6 else Drop gradient $G$ . 7 end 8 $\begin{array} { r } { \theta ^ { ( t + 1 ) } [ j ] \theta ^ { ( t ) } [ j ] - \frac { \gamma _ { t } } { N } \sum _ { G \in \mathcal { G } } G [ j ] . } \\ { \bar { \theta } ^ { ( t ) } [ j ] = \alpha \bar { \theta } ^ { ( t - 1 ) } [ j ] + ( 1 - \alpha ) \theta ^ { ( t ) } [ j ] . } \end{array}$ 9 10 end + +# 3.1 STRAGGLER EFFECTS + +The use of backup workers is motivated by the need to mitigate slow stragglers while maximizing computation. We investigate the effect of stragglers on Sync-Opt model training here. + +We ran Sync-Opt with $N = 1 0 0$ workers, $b = 0$ backups, and 19 parameter servers on the Inception model. Using one variable as a proxy, we collected for each iteration both the start time of the iteration and the time when the $k$ th gradient of that variable arrived at the parameter server. These times are presented in Figure 3 for $k = 1$ , 50, 90, 97, 98, 99, 100. Note that $80 \%$ of the 98th gradient arrives in under 2s, whereas only $30 \%$ of the final gradient do. Furthermore, the time to collect the final few gradients grows exponentially, resulting in wasted idle resources and time expended to wait for the slowest gradients. This exponential increase is also seen in Figure 4. + +![](images/f142cfb5431917c4256dc4d3a33158daba7d846ffc4487f811dfdda3fff7a939.jpg) +Figure 3: CDF of time taken to aggregate gradients from $N$ machines. For clarity, we only show times of $\leq 6 \mathrm { s }$ ; the maximum observed time is 310s. + +![](images/1faa87f637ff8939e5bbdb8a66acd6d420b49338648b740f8e467c1caef2ee75.jpg) +Figure 4: Mean and median times, across all iterations, to collect $k$ gradients on $N = 1 0 0$ workers and $b = 0$ backups. Most mean times fall between 1.4s and 1.8s, except of final few gradients. + +Thus, one might choose to drop slow stragglers to decrease the iteration time. However, using fewer machines implies a smaller effective mini-batch size and thus greater gradient variance, which in turn could require more iterations for convergence. We examine this relationship by running Sync-Opt2 with $N = 5 0$ , 70, 80, 90, 100 and $b = 6$ , and note the number of iterations required for convergence in Figure 5. Additional details of this training are provided in Appendix A.2. As $N$ is doubled from 50 to 100, the number of iterations to converge nearly halves from $1 3 7 . 5 e 3$ to $7 6 . 2 e 3$ . + +![](images/808b4545e00ca62fea4a8ae6f8ace30f28b4dffb9d1dabc8f1fdf079679549b0.jpg) +Figure 5: Number of iterations to converge when aggregating gradient from $N$ machines. + +![](images/9b0b18171010653451f924f726f76139bfecb911a4efdc1c7e6e56ca49a25b7b.jpg) +Figure 6: Estimated time to converge when aggregating gradients from $N$ machines on a $N + b = \bar { 1 } 0 0$ machine configuration. Convergence is fastest when choosing $N = 9 6$ , $b = 4$ . + +Hence, there is a trade-off between dropping more stragglers to reduce iteration time, and waiting for more gradients to improve the gradient quality. Consider a hypothetical setting where we have $N + b = 1 0 0$ machines, and we wish to choose the best configuration of $N$ and $b$ to minimize running time to convergence3. For each configuration, we can estimate the iterations required from Figure 5 (linearly interpolating for values of $N$ for which we did not collect data). We can multiply this with the mean iteration times (Figure 4) to obtain the running time required to converge for each setting of $N$ and $b$ . These results are shown in Figure 6, indicating that $N = 9 6$ , $b = 4$ converges fastest. Therefore, this motivates our choice to use a few backup workers for mitigating stragglers. + +# 4 EXPERIMENTS + +In this section, we present our empirical comparisons of synchronous and asynchronous distributed stochastic optimization algorithms as applied to models such as Inception and PixelCNN. All experiments in this paper are using the TensorFlow system (Abadi et al., 2015). + +# 4.1 METRICS OF COMPARISON: FASTER CONVERGENCE, BETTER OPTIMUM + +We are interested in two metrics of comparison for our empirical validation: (1) test error or accuracy, and (2) speed of convergence3. We point out that for non-convex deep learning models, it is possible to converge faster to a poorer local optimum. Here we show a simple example with Inception using different learning rates. + +Table 2: Test accuracies at convergence and number of epochs to converge for different initial learning rates $\gamma _ { 0 }$ . Low initial learning rates result in faster convergence to poorer local optimum. + +
Initial rate 20Test precision at convergenceEpochs to converge
1.125 2.2577.29% 77.75%52628 65811
4.5
9.078.15%76209
78.17%77235
+ +![](images/cf7dd8a2918112ad1401df8582e7cb61ac45e2309e2c05a587c225cae4d880a5.jpg) +Figure 7: Convergence of Sync-Opt on Inception model using $N =$ 100 workers and $b = 6$ backups, with varying initial learning rates $\gamma _ { 0 }$ . To reach a lower $\epsilon$ test precision, small $\gamma _ { 0 }$ ’s require fewer epochs than large $\gamma _ { 0 }$ ’s. However, small $\gamma _ { 0 }$ ’s either fail to attain high $\epsilon$ precision, or take more epochs than higher $\gamma _ { 0 }$ ’s. + +We ran Sync-Opt on Inception with $N = 1 0 0$ and $b = 6$ , but varied the initial learning rate $\gamma _ { 0 }$ between 1.125 and 9.0. (Learning rates are exponentially decreased with iterations.) Table 2 shows that smaller $\gamma _ { 0 }$ converge faster, but to poorer test precisions. Focusing on speed on an early phase of training could lead to misleading conclusions if we fail to account for eventual convergence. For example, Figure 3b shows that using $\gamma _ { 0 } = 1 . 1 2 5$ reaches $\epsilon = 7 5 \%$ precision $1 . 5 \times$ faster than $\gamma _ { 0 } = 4 . 5$ , but is slower for $\epsilon = 7 7 . 7 5 \%$ , and fails to reach higher precisions. + +# 4.2 INCEPTION + +We conducted experiments on the Inception model (Szegedy et al., 2016) trained on ImageNet Challenge dataset (Russakovsky et al., 2015), where the task is to classify images out of 1000 categories. We used several configurations, varying $N + b$ from 53 to 212 workers. Additional details of the training are provided in Appendix A.3. An epoch is a synchronous iteration for Sync-Opt, or a full pass of $N$ updates for Async-Opt, which represent similar amounts of computation. Results of this experiment are presented in Figure 8. + +Figure 8b shows that Sync-Opt outperforms Async-Opt in test precision: Sync-Opt attains ${ \sim } 0 . 5 \%$ better test precision than Async-Opt for comparable $N + b$ workers. Furthermore, Sync-Opt converges 6h and 18h faster than Async-Opt for 106 and 212 workers respectively, and is 3h slower when 53 workers are used, as seen in Figure 8d. This difference in speed is largely due to the fewer epochs (Figure 8c) needed by Sync-Opt, but comparable or better epoch time (Figure 8e). + +![](images/4a2795f1f53e652a98aa43d466d16efa8ffdf2c2db7dc3f06c9317fb0d07f46a.jpg) +Figure 8: Convergence of Sync-Opt and Async-Opt on Inception model using varying number of machines. Sync-Opt with backup workers converge faster, with fewer epochs, to higher test accuracies. + +# 4.3 PIXELCNN EXPERIMENTS + +The second model we experimented on is PixelCNN (Oord et al., 2016), a conditional image generation deep neural network, which we train on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset. Configurations of $N + b = 1 , 8 , 1 6$ workers were used; for Sync-Opt, we always used $b = 1$ backup worker. Additional details are provided in Appendix A.4. + +![](images/c0603f807e6115ff2ee2ef1f99ab3da81f0e47d7e4e735a38c20bae4c9c188e1.jpg) +Figure 9: Convergence of synchronous and asynchronous training on PixelCNN model. Sync-Opt achieves lower negative log likelihood in less time than Async-Opt. + +Convergence of the test negative log likelihood (NLL) on PixelCNN is shown in Figure 9a, where lower is better. Observe that Sync-Opt obtains lower NLL than Async-Opt; in fact, Async-Opt is even outperformed by serial RMSProp with $N = 1$ worker, with degrading performance as $N$ increases from 8 to 16. Figure 9b further shows the time taken to reach $\epsilon$ test NLL. Sync-Opt reduces the time to reach $\epsilon = 2 . 1 4 5$ from 247h to $5 8 . 3 \mathrm { h }$ ; this NLL is not even achieved by Async-Opt. + +# 5 RELATED WORK + +Multicore and distributed optimization algorithms have received much attention in recent years. Asynchronous algorithms include Recht et al. (2011); Duchi et al. (2013); Zhang et al. (2015a); Reddi et al. (2015); Leblond et al. (2016). Implementations of asynchronous optimization include Xing et al. (2015); Li et al. (2014); Chilimbi et al. (2014). Attempts have also been made in Zinkevich et al. (2010) and Zhang & Jordan (2015) to algorithmically improve synchronous SGD. + +An alternative solution, “softsync”, was presented in Zhang et al. (2015b), which proposed batching gradients from multiple machines before performing an asynchronous SGD update, thereby reducing the effective staleness of gradients. Similar to our proposal, softsync avoids stragglers by not forcing updates to wait for the slowest worker. However, softsync allows the use of stale gradients but we do not. The two solutions provide different explorations of the trade-off between high accuracy (by minimizing staleness) and fast throughput (by avoiding stragglers). + +Watcharapichat et al. (2016) introduces a distributed deep learning system without parameter servers, by having workers interleave gradient computation and communication in a round-robin pattern. Like Async-Opt, this approach suffers from staleness. We also note that in principle, workers in Sync-Opt can double as parameter servers and execute the update operations and avoid the need to partition hardware resources between workers and servers. + +Das et al. (2016) analyzes distributed stochastic optimization and optimizes the system by solving detailed system balance equations. We believe this approach is complimentary to our work, and could potentially be applied to guide the choice of systems configurations for Sync-Opt. + +Keskar et al. (2016) suggests that large batch sizes for synchronous stochastic optimization leads to poorer generalization. Our effective batch size increases linearly with the number of workers $N$ . However, we did not observe this effect in our experiments; we believe we are not yet in the large batch size regime examined by Keskar et al. (2016). + +# 6 CONCLUSION AND FUTURE WORK + +Distributed training strategies for deep learning architectures will become ever more important as the size of datasets increases. In this work, we have shown how both synchronous and asynchronous distributed stochastic optimization suffer from their respective weaknesses of stragglers and staleness. This has motivated our development of synchronous stochastic optimization with backup workers, which we show to be a viable and scalable strategy. + +We are currently experimenting with different kinds of datasets, including word-level language models where parts of the model (the embedding layers) are often very sparse, which involves very different communication constraints. We are also working on further improving the performance of synchronous training like combining gradients from multiple workers sharing the same machine before sending them to the parameter servers to reduce the communication overhead. An alternative of using time-outs instead of backup workers is also being explored. + +# REFERENCES + +Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mane, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit ´ Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, ´ Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL http://tensorflow.org/. + +Jianmin Chen, Rajat Monga, Samy Bengio, and Rafal Jozefowicz. Revisiting distributed synchronous sgd. arXiv preprint arXiv:1604.00981, 2016. + +T. Chilimbi, Y. Suzue, J. Apacible, and K. Kalyanaraman. Project adam: Building an efficient and scalable deep learning training system. In Proceedings of the 11th USENIX Symposium on Operating Systems Design and Implementation, 2014. + +Dipankar Das, Sasikanth Avancha, Dheevatsa Mudigere, Karthikeyan Vaidynathan, Srinivas Sridharan, Dhiraj Kalamkar, Bharat Kaul, and Pradeep Dubey. Distributed deep learning using synchronous stochastic gradient descent. arXiv preprint arXiv:1602.06709, 2016. + +Christopher M De Sa, Ce Zhang, Kunle Olukotun, and Christopher Re. Taming the wild: A unified analysis of ´ hogwild-style algorithms. In Advances in Neural Information Processing Systems, pp. 2674–2682, 2015. + +J. Dean, G. S. Corrado, R. Monga, K. Chen, M. Devin, Q. V. Le, M. Z. Mao, M. A. Ranzato, A. Senior, P. Tucker, K. Yang, and A. Y. Ng. Large scale distributed deep networks. In Advances in Neural Information Processing Systems, NIPS, 2012. + +Jeffrey Dean and Luiz Andr Barroso. The tail at scale. Communications of the ACM, 56:74–80, 2013. URL http://cacm.acm.org/magazines/2013/2/160173-the-tail-at-scale/fulltext. + +John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12(Jul):2121–2159, 2011. + +John Duchi, Michael I Jordan, and Brendan McMahan. Estimation, optimization, and parallelism when data is sparse. In Advances in Neural Information Processing Systems, pp. 2832–2840, 2013. + +G. Hinton, L. Deng, D. Yu, G. Dahl, A. Mohamed, N. Jaitly, A. Senior, V. Vanhoucke, P. Nguyen, T. N. Sainath, and B. Kingsbury. Deep neural networks for acoustic modeling in speech recognition. 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In Advances in neural information processing systems, pp. 2595–2603, 2010. + +# A DETAILS OF MODELS AND TRAINING + +# A.1 MNIST CNN, SECTION 2.1 + +The model used in our experiments is a 4-layer CNN that have $3 { \tt X } 3$ filters with max-pooling and weight normalization in every layer. We trained the model with SGD for 25 epochs and evaluated performance on the exponential moving average $\bar { \theta }$ using a decay rate of $\alpha = 0 . 9 9 9 9$ . Initial learning rate was set to be 0.1 and linearly annealed to 0 in the last 10 epochs. We also used small image rotations and zooms as a data augmentation scheme. + +# A.2 INCEPTION, SECTION 3.1 + +For our straggler experiments, we trained the Inception (Szegedy et al., 2016) model on the ImageNet Challenge dataset (Russakovsky et al., 2015). 10 parameter servers were used, and each worker was equipped with a k40 GPU. + +The underlying optimizer was RMSProp with momentum, with decay of 0.9 and momentum of 0.9. Mini-batch size $B = 3 2$ was used. Initial learning rates $\gamma _ { 0 }$ were set at $0 . 0 4 5 N$ , which we found to provide good test precisions for Inception. Learning rates were also exponentially decreased with decay rate $\beta = 0 . 9 4$ as $\gamma _ { 0 } \beta ^ { t N / ( 2 T ) }$ , where $T = | { \mathcal { X } } | / B$ is the number of mini-batches in the dataset. + +Test precisions were evaluated on the exponential moving average $\bar { \theta }$ using $\alpha = 0 . 9 9 9 9$ . + +# A.3 INCEPTION, SECTION 4.2 + +For experiments comparing Async-Opt and Sync-Opt on the Inception model in Section 4.2, each worker is equipped with a k40 GPU. For $N + b = 5 3$ workers, 17 parameter servers were used; for $N + b = 1 0 6$ workers, we used 27 parameter servers; and 37 parameter servers were used for $N + b = 2 1 2$ . + +In the asynchronous training mode, gradient clipping is also needed for stabilization, which requires each worker to collect the gradient across all layers of the deep model, compute the global norm $| | G | |$ and then clip all gradient accordingly. However, synchronization turns out to be very stable so gradient clipping is no longer needed, which means that we can pipeline the update of parameters in different layers: the gradient of top layers’ parameters can be sent to parameter servers while concurrently computing gradients for the lower layers. + +The underlying optimizer is RMSProp with momentum, with decay of 0.9 and momentum of 0.9. Mini-batch size $B = 3 2$ was used. Initial learning rates $\gamma _ { 0 }$ for Async-Opt were set to 0.045; for Sync-Opt, we found as a rule-of-thumb that a learning rate of $0 . 0 4 5 N$ worked well for this model. Learning rates were then exponentially decayed with decay rate $\beta = 0 . 9 4$ as $\gamma _ { 0 } \beta ^ { t / ( 2 T ) }$ for AsyncOpt, where $T = | { \mathcal { X } } | / B$ is the number of mini-batches in the dataset. For Sync-Opt, we learning rates were also exponentially decreased at rate of $\gamma _ { 0 } \beta ^ { t N / ( 2 T ) }$ , so that the learning rates after computing the same number of datapoints are comparable for Async-Opt and Sync-Opt. + +Test precisions were evaluated on the exponential moving average $\bar { \theta }$ using $\alpha = 0 . 9 9 9 9$ + +# A.4 PIXELCNN, SECTION 4.3 + +The PixelCNN (Oord et al., 2016) model was trained on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset. Configurations of $N + b = 1 , 8 , 1 6$ workers each with a $\mathbf { k } 8 0$ GPU, and 10 parameter servers were used. For Sync-Opt, we always used $b = 1$ backup worker. The underlying optimizer is RMSProp with momentum, using decay of 0.95 and momentum of 0.9. Initial learning rates $\gamma _ { 0 }$ were set to $1 e - 4$ and slowly decreased to $3 e - 6$ after 200,000 iterations. Mini-batch size $B = 4$ was used. \ No newline at end of file diff --git a/parse/train/HyAddcLge/HyAddcLge_content_list.json b/parse/train/HyAddcLge/HyAddcLge_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..428adb4f4cdf156ff2291f0a5e3a7ab2385704ac --- /dev/null +++ b/parse/train/HyAddcLge/HyAddcLge_content_list.json @@ -0,0 +1,1357 @@ +[ + { + "type": "text", + "text": "REVISITING DISTRIBUTED SYNCHRONOUS SGD ", + "text_level": 1, + "bbox": [ + 173, + 99, + 754, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jianmin Chen∗, Xinghao Pan∗†, Rajat Monga, Samy Bengio \nGoogle Brain \nMountain View, CA, USA \n{jmchen,xinghao,rajatmonga,bengio}@google.com ", + "bbox": [ + 184, + 143, + 622, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Rafal Jozefowicz OpenAI San Francisco, CA, USA rafal@openai.com ", + "bbox": [ + 643, + 145, + 808, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 238, + 544, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Distributed training of deep learning models on large-scale training data is typically conducted with asynchronous stochastic optimization to maximize the rate of updates, at the cost of additional noise introduced from asynchrony. In contrast, the synchronous approach is often thought to be impractical due to idle time wasted on waiting for straggling workers. We revisit these conventional beliefs in this paper, and examine the weaknesses of both approaches. We demonstrate that a third approach, synchronous optimization with backup workers, can avoid asynchronous noise while mitigating for the worst stragglers. Our approach is empirically validated and shown to converge faster and to better test accuracies. ", + "bbox": [ + 233, + 270, + 764, + 395 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 422, + 336, + 438 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The recent success of deep learning approaches for domains like speech recognition (Hinton et al., 2012) and computer vision (Ioffe & Szegedy, 2015) stems from many algorithmic improvements but also from the fact that the size of available training data has grown significantly over the years, together with the computing power, in terms of both CPUs and GPUs. While a single GPU often provides algorithmic simplicity and speed up to a given scale of data and model, there exist an operating point where a distributed implementation of training algorithms for deep architectures becomes necessary. ", + "bbox": [ + 174, + 454, + 825, + 551 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Currently, popular distributed training algorithms include mini-batch versions of stochastic gradient descent (SGD) and other stochastic optimization algorithms such as AdaGrad (Duchi et al., 2011), RMSProp (Tieleman & Hinton, 2012), and ADAM (Kingma & Ba, 2014). Unfortunately, bulksynchronous implementations of stochastic optimization are often slow in practice due to the need to wait for the slowest machine in each synchronous batch. To circumvent this problem, practitioners have resorted to asynchronous approaches which emphasize speed by using potentially stale information for computation. While asynchronous training have proven to be faster than their synchronous counterparts, they often result in convergence to poorer results. ", + "bbox": [ + 174, + 559, + 825, + 670 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper1, we revisit synchronous learning, and propose a method for mitigating stragglers in synchronous stochastic optimization. Specifically, we synchronously compute a mini-batch gradient with only a subset of worker machines, thus alleviating the straggler effect while avoiding any staleness in our gradients. The primary contributions of our paper are: ", + "bbox": [ + 176, + 676, + 823, + 733 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• Illustration of how gradient staleness in asynchronous training negatively impacts test accuracy and is exacerbated by deep models. \nMeasurement of machine response times for synchronous stochastic optimization in a large deployment of 100 GPUs, showing how stragglers in the tail end affect convergence speed. Proposal of synchronous stochastic optimization with backup workers to mitigate straggler effects without gradient staleness. \n• Establishing the need to measure both speed of convergence and test accuracy of optimum for empirical validation. ", + "bbox": [ + 215, + 744, + 825, + 872 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• Empirical demonstration that our proposed synchronous training method outperforms asynchronous training by converging faster and to better test accuracies. ", + "bbox": [ + 209, + 103, + 821, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The remainder of this paper is organized as follows. We briefly present preliminaries and notation in Section 1.1. Section 2 describes asynchronous stochastic optimization and presents experimental evidence of gradient staleness in deep neural network models. We present our approach in Section 3, and exhibit straggler effects that motivate the approach. We then empirically evaluate our approach in Sections 4. Related work is discussed in Section 5, and we conclude in Section 6. ", + "bbox": [ + 174, + 142, + 825, + 213 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 PRELIMINARIES AND NOTATION ", + "text_level": 1, + "bbox": [ + 176, + 229, + 437, + 244 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given a dataset $\\mathcal { X } = \\{ x _ { i } : i = 1 , \\ldots , | \\mathcal { X } | \\}$ , our goal is to learn the parameters $\\theta$ of a model with respect to an empirical loss function $f$ , defined as $\\begin{array} { r } { f ( \\theta ) \\stackrel { \\Delta } { = } \\frac { 1 } { | \\mathcal { X } | } \\sum _ { i = 1 } ^ { | \\mathcal { X } | } F ( x _ { i } ; \\theta ) } \\end{array}$ , where $F ( x _ { i } ; \\theta )$ is the loss with respect to a datapoint $x _ { i }$ and the model $\\theta$ . ", + "bbox": [ + 174, + 255, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A first-order stochastic optimization algorithm achieves this by iteratively updating $\\theta$ using a stochastic gradient $G \\overset { \\Delta } { = } \\nabla F ( x _ { i } ; \\theta )$ computed at a randomly sampled $x _ { i }$ , producing a sequence of models $\\theta ^ { ( 0 ) } , \\theta ^ { ( 1 ) } , \\ldots$ . Stochastic optimization algorithms differ in their update equations. For example, the update of SGD is $\\bar { \\theta ^ { ( t + 1 ) } } \\bar { \\bf \\Phi } = \\theta ^ { ( t ) } - \\gamma _ { t } \\bar { G } ^ { ( t ) } = \\theta ^ { ( t ) } - \\gamma _ { t } \\nabla F ( x _ { i } ; \\bar { \\theta } ^ { ( t ) } )$ , where $\\gamma _ { t }$ is the learning rate or step size at iteration $t$ . A mini-batch version of the stochastic optimization algorithm computes the stochastic gradient over mini-batch of size $B$ instead of a single datapoint, i.e., $\\begin{array} { r } { G \\stackrel { \\Delta } { = } \\frac { 1 } { B } \\sum _ { i = 1 } ^ { B } \\nabla F ( \\widetilde { x } _ { i } ; \\theta ^ { ( t ) } ) } \\end{array}$ , where $\\widetilde { x } _ { i }$ ’s are randomly sampled from $\\mathcal { X }$ . We will often evaluate performance on an exponential moving average $\\bar { \\theta } ^ { ( t ) } = \\alpha \\bar { \\theta } ^ { ( t - 1 ) } + ( 1 - \\alpha ) \\theta ^ { ( t ) }$ with decay rate $\\alpha$ . ", + "bbox": [ + 173, + 311, + 825, + 440 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our interest is in distributed stochastic optimization using $N$ worker machines in charge of computing stochastic gradients that are sent to $M$ parameter servers. Each parameter server $j$ is responsible for storing a subset $\\theta [ j ]$ of the model, and performing updates on $\\theta [ j ]$ . In the synchronous setting, we will also introduce additional $b$ backup workers for straggler mitigation. ", + "bbox": [ + 174, + 445, + 825, + 502 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 ASYNCHRONOUS STOCHASTIC OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 174, + 521, + 591, + 537 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "An approach for a distributed stochastic gradient descent algorithm was presented in Dean et al. (2012), consisting of two main ingredients. First, the parameters of the model are distributed on multiple servers, depending on the architecture. This set of servers are called the parameter servers. Second, there can be multiple workers processing data in parallel and communicating with the parameter servers. Each worker processes a mini-batch of data independently of the others, as follows: ", + "bbox": [ + 176, + 553, + 825, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• The worker fetches from the parameter servers the most up-to-date parameters of the model needed to process the current mini-batch; \n• It then computes gradients of the loss with respect to these parameters; \n• Finally, these gradients are sent back to the parameter servers, which then updates the model accordingly. ", + "bbox": [ + 215, + 633, + 823, + 714 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Since each worker communicates with the parameter servers independently of the others, this is called Asynchronous Stochastic Gradient Descent (Async-SGD), or more generally, Asynchronous Stochastic Optimization (Async-Opt). A similar approach was later proposed by Chilimbi et al. (2014). Async-Opt is presented in Algorithms 1 and 2. ", + "bbox": [ + 174, + 724, + 825, + 781 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In practice, the updates of Async-Opt are different than those of serially running the stochastic optimization algorithm for two reasons. Firstly, the read operation (Algo 1 Line 2) on a worker may be interleaved with updates by other workers to different parameter servers, so the resultant $\\widehat { \\theta } _ { k }$ may not be consistent with any parameter incarnation $\\boldsymbol { \\theta } ^ { ( t ) }$ . Secondly, model updates may have occurred while a worker is computing its stochastic gradient; hence, the resultant gradients are typically computed with respect to outdated parameters. We refer to these as stale gradients, and its staleness as the number of updates that have occurred between its corresponding read and update operations. ", + "bbox": [ + 173, + 786, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Understanding the theoretical impact of staleness is difficult work and the topic of many recent papers, e.g. Recht et al. (2011); Duchi et al. (2013); Leblond et al. (2016); Reddi et al. (2015); ", + "bbox": [ + 176, + 895, + 821, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithm 1: Async-SGD worker $k$ ", + "text_level": 1, + "bbox": [ + 174, + 108, + 413, + 122 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Input: Dataset $\\mathcal { X }$ \nInput: $B$ mini-batch size \n1 while True do \n2 Read $\\widehat { \\theta } _ { k } = ( \\theta [ 0 ] , \\ldots , \\theta [ M ] )$ from PSs. \n3 $G _ { k } ^ { ( t ) } : = 0$ . \n4 for $i = 1 , \\ldots , B$ do \n5 Sample datapoint $\\widetilde { x } _ { i }$ from $\\mathcal { X }$ . \n6 $\\begin{array} { r } { G _ { k } ^ { ( t ) } G _ { k } ^ { ( t ) } + \\frac { 1 } { B } \\nabla F ( \\widetilde { x } _ { i } ; \\widehat { \\theta } _ { k } ) . } \\end{array}$ \n7 end \n8 Send $G _ { k } ^ { ( t ) }$ to parameter servers. \n9 end ", + "bbox": [ + 160, + 126, + 431, + 276 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 2: Async-SGD Parameter Server j ", + "text_level": 1, + "bbox": [ + 508, + 107, + 808, + 122 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Input: $\\gamma _ { 0 } , \\gamma _ { 1 } , \\ldots$ learning rates. Input: $\\alpha$ decay rate. Input: $\\theta ^ { ( 0 ) }$ model initialization. 1 for $t = 0 , 1 , \\ldots$ do 2 Wait for gradient $G$ from any worker. 3 $\\theta ^ { ( t + 1 ) } [ j ] \\theta ^ { ( t ) } [ j ] - \\gamma _ { t } G [ j ]$ . 4 $\\bar { \\theta } ^ { ( t ) } [ j ] = \\alpha \\bar { \\theta } ^ { ( t - 1 ) } [ j ] + ( 1 - \\alpha ) \\theta ^ { ( t ) } [ j ] ,$ . 5 end ", + "bbox": [ + 491, + 126, + 761, + 238 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 161, + 268, + 199, + 279 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/869a74e931a0a7ff1e7216146dd0c69e43230ff7820846197c9038f2307065dd.jpg", + "image_caption": [ + "Figure 1: Gradient staleness dependence on model layer. Gradients are computed in a bottom-up forward propagation step followed by a top-down back propagation step. Parameters are read from servers in the forward prop, but gradients are sent to servers during the back prop. Thus, gradients of lower layers are more stale than top layers. " + ], + "image_footnote": [], + "bbox": [ + 183, + 327, + 581, + 430 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/067a6882cd5a5473a6949076399e8dc649d5d54797b44d7ec5ef733add6828af.jpg", + "image_caption": [ + "Figure 2: Degradation of test classification error with increasing average gradient staleness in MNIST CNN model. " + ], + "image_footnote": [], + "bbox": [ + 602, + 321, + 820, + 446 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "De Sa et al. (2015); Mania et al. (2015), most of which focus on individual algorithms, under strong assumptions that may not hold up in practice. This is further complicated by deep models with multiple layers, since the times at which model parameters are read and which gradients are computed and sent are dependent on the depth of the layers (Figure 1). To better understand this dependence in real models, we collected staleness statistics on a Async-Opt run with 40 workers on a 18-layer Inception model (Szegedy et al., 2016) trained on the ImageNet Challenge dataset (Russakovsky et al., 2015), as shown in Table 1. ", + "bbox": [ + 173, + 546, + 825, + 645 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Despite the abovementioned problems, Async-Opt has been shown to be scale well up to a few dozens of workers for some models. However, at larger scales, increasing the number of machines (and thus staleness of gradients) can result in poorer trained models. ", + "bbox": [ + 173, + 652, + 415, + 763 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/02bec36451fb35c343bccde0a8bb365f5fa1c3f48434be8a9f82c3f421e4e645.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
LayerMinMeanMedianMaxStd DevCount
18414.5413.94293.8310908
12511.3511.3233.0944478
11819.819.59343.65187
02438.9738.43615.43178
", + "bbox": [ + 426, + 642, + 826, + 708 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Table 1: Staleness of gradients in a 18-layer Inception model. Gradients were collected in a run of asynchronous training using 40 machines. Staleness of a gradient is measured as the number of updates that have occurred between its corresponding read and update operations. The staleness of gradients increases from a mean of ${ \\sim } 1 4 . 5$ in the top layer (Layer 18) to ${ \\sim } 3 9 . 0 $ in the bottom layer (Layer 0). ", + "bbox": [ + 426, + 719, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 IMPACT OF STALENESS ON TEST ACCURACY ", + "text_level": 1, + "bbox": [ + 174, + 790, + 397, + 818 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We explore how increased staleness contributes to training of poorer models. In order to mimic the setting on a smaller scale, we trained a state-of-the-art MNIST CNN model but simulated staleness by using old gradients for the parameter updates. Details of the model and training are provided in Appendix A.1. ", + "bbox": [ + 174, + 832, + 825, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The best final classification error on a test set was $0 . 3 6 \\%$ , which increases to $0 . 4 7 \\%$ with average gradient staleness of 20 steps, and up to $0 . 7 9 \\%$ with 50 steps (see Figure 2). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Once the average simulated staleness was chosen to be more than 15 steps, the results started to significantly deteriorate and the training itself became much less stable. We had to employ following tricks to prevent the results from blowing up: ", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Slowly increase the staleness over the first 3 epochs of training. This mimics increasing the number of asynchronous workers and is also very important in practice for some of the models we experimented with (e.g. large word-level language models). The trick was not relevant with a simulated staleness less than 15 but became crucial for larger values. Use lower initial learning rates when staleness is at least 20, which reduces a frequency of explosions (train error goes to $90 \\%$ ). This observation is similar to what we found in other experiments - we were able to use much larger learning rates with synchronous training and the results were also more stable. \nEven with above tricks the divergence occurs occasionally and we found that restarting training from random weights can lead to more successful runs. The best results were then chosen based on validation set performance. ", + "bbox": [ + 215, + 160, + 825, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 REVISTING SYNCHRONOUS STOCHASTIC OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 358, + 676, + 376 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Both Dean et al. (2012) and Chilimbi et al. (2014) use versions of Async-SGD where the main potential problem is that each worker computes gradients over a potentially old version of the model. In order to remove this discrepancy, we propose here to reconsider a synchronous version of distributed stochastic gradient descent (Sync-SGD), or more generally, Synchronous Stochastic Optimization (Sync-Opt), where the parameter servers wait for all workers to send their gradients, aggregate them, and send the updated parameters to all workers afterward. This ensures that the actual algorithm is a true mini-batch stochastic gradient descent, with an effective batch size equal to the sum of all the mini-batch sizes of the workers. ", + "bbox": [ + 174, + 393, + 825, + 506 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While this approach solves the staleness problem, it also introduces the potential problem that the actual update time now depends on the slowest worker. Although workers have equivalent computation and network communication workload, slow stragglers may result from failing hardware, or contention on shared underlying hardware resources in data centers, or even due to preemption by other jobs. ", + "bbox": [ + 174, + 512, + 823, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To alleviate the straggler problem, we introduce backup workers (Dean & Barroso, 2013) as follows: instead of having only $N$ workers, we add $b$ extra workers, but as soon as the parameter servers receive gradients from any $N$ workers, they stop waiting and update their parameters using the $N$ gradients. The slowest $b$ workers’ gradients will be dropped when they arrive. Our method is presented in Algorithms 3, 4. ", + "bbox": [ + 173, + 589, + 825, + 659 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 3: Sync-SGD worker $k$ , where $k =$ $1 , \\ldots , N + b$ ", + "text_level": 1, + "bbox": [ + 173, + 684, + 490, + 712 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Input: Dataset $\\mathcal { X }$ Input: $B$ mini-batch size 1 for $t = 0 , 1 , \\ldots$ do 2 Wait to read ${ \\boldsymbol { \\theta } } ^ { ( t ) } = ( { \\boldsymbol { \\theta } } ^ { ( t ) } [ 0 ] , \\dots , { \\boldsymbol { \\theta } } ^ { ( t ) } [ M ] )$ from parameter servers. 3 $G _ { k } ^ { ( t ) } : = 0$ 4 for $i = 1 , \\ldots , B$ do 5 Sample datapoint $\\widetilde { x } _ { k , i }$ from $\\mathcal { X }$ . 6 $\\begin{array} { r } { G _ { k } ^ { ( t ) } G _ { k } ^ { ( t ) } + \\frac { 1 } { B } \\nabla F ( \\widetilde { x } _ { k , i } ; \\theta ^ { ( t ) } ) } \\end{array}$ . 7 end 8 Send $( G _ { k } ^ { ( t ) } , t )$ to parameter servers. 9 end ", + "bbox": [ + 160, + 714, + 485, + 886 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 4: Sync-SGD Parameter Server $j$ ", + "text_level": 1, + "bbox": [ + 508, + 684, + 797, + 698 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Input: $\\gamma _ { 0 } , \\gamma _ { 1 } , \\ldots$ learning rates. Input: $\\alpha$ decay rate. Input: $N$ number of mini-batches to aggregate. Input: $\\theta ^ { ( 0 ) }$ model initialization. for $t = 0 , 1 , \\ldots$ do 2 $\\mathcal { G } = \\{ \\}$ 3 while $| \\mathcal { G } | < N$ do 4 Wait for $( G , t ^ { \\prime } )$ from any worker. 5 if $t ^ { \\prime } = = t$ then ${ \\mathcal { G } } { \\mathcal { G } } \\cup \\{ G \\}$ . 6 else Drop gradient $G$ . 7 end 8 $\\begin{array} { r } { \\theta ^ { ( t + 1 ) } [ j ] \\theta ^ { ( t ) } [ j ] - \\frac { \\gamma _ { t } } { N } \\sum _ { G \\in \\mathcal { G } } G [ j ] . } \\\\ { \\bar { \\theta } ^ { ( t ) } [ j ] = \\alpha \\bar { \\theta } ^ { ( t - 1 ) } [ j ] + ( 1 - \\alpha ) \\theta ^ { ( t ) } [ j ] . } \\end{array}$ 9 10 end ", + "bbox": [ + 493, + 702, + 787, + 890 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 STRAGGLER EFFECTS", + "text_level": 1, + "bbox": [ + 176, + 103, + 362, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The use of backup workers is motivated by the need to mitigate slow stragglers while maximizing computation. We investigate the effect of stragglers on Sync-Opt model training here. ", + "bbox": [ + 171, + 130, + 823, + 159 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We ran Sync-Opt with $N = 1 0 0$ workers, $b = 0$ backups, and 19 parameter servers on the Inception model. Using one variable as a proxy, we collected for each iteration both the start time of the iteration and the time when the $k$ th gradient of that variable arrived at the parameter server. These times are presented in Figure 3 for $k = 1$ , 50, 90, 97, 98, 99, 100. Note that $80 \\%$ of the 98th gradient arrives in under 2s, whereas only $30 \\%$ of the final gradient do. Furthermore, the time to collect the final few gradients grows exponentially, resulting in wasted idle resources and time expended to wait for the slowest gradients. This exponential increase is also seen in Figure 4. ", + "bbox": [ + 173, + 165, + 826, + 263 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/f142cfb5431917c4256dc4d3a33158daba7d846ffc4487f811dfdda3fff7a939.jpg", + "image_caption": [ + "Figure 3: CDF of time taken to aggregate gradients from $N$ machines. For clarity, we only show times of $\\leq 6 \\mathrm { s }$ ; the maximum observed time is 310s. " + ], + "image_footnote": [], + "bbox": [ + 178, + 279, + 486, + 458 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/1faa87f637ff8939e5bbdb8a66acd6d420b49338648b740f8e467c1caef2ee75.jpg", + "image_caption": [ + "Figure 4: Mean and median times, across all iterations, to collect $k$ gradients on $N = 1 0 0$ workers and $b = 0$ backups. Most mean times fall between 1.4s and 1.8s, except of final few gradients. " + ], + "image_footnote": [], + "bbox": [ + 509, + 284, + 818, + 458 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Thus, one might choose to drop slow stragglers to decrease the iteration time. However, using fewer machines implies a smaller effective mini-batch size and thus greater gradient variance, which in turn could require more iterations for convergence. We examine this relationship by running Sync-Opt2 with $N = 5 0$ , 70, 80, 90, 100 and $b = 6$ , and note the number of iterations required for convergence in Figure 5. Additional details of this training are provided in Appendix A.2. As $N$ is doubled from 50 to 100, the number of iterations to converge nearly halves from $1 3 7 . 5 e 3$ to $7 6 . 2 e 3$ . ", + "bbox": [ + 173, + 540, + 825, + 625 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/808b4545e00ca62fea4a8ae6f8ace30f28b4dffb9d1dabc8f1fdf079679549b0.jpg", + "image_caption": [ + "Figure 5: Number of iterations to converge when aggregating gradient from $N$ machines. " + ], + "image_footnote": [], + "bbox": [ + 178, + 643, + 486, + 818 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/9b0b18171010653451f924f726f76139bfecb911a4efdc1c7e6e56ca49a25b7b.jpg", + "image_caption": [ + "Figure 6: Estimated time to converge when aggregating gradients from $N$ machines on a $N + b = \\bar { 1 } 0 0$ machine configuration. Convergence is fastest when choosing $N = 9 6$ , $b = 4$ . " + ], + "image_footnote": [], + "bbox": [ + 509, + 641, + 816, + 818 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Hence, there is a trade-off between dropping more stragglers to reduce iteration time, and waiting for more gradients to improve the gradient quality. Consider a hypothetical setting where we have $N + b = 1 0 0$ machines, and we wish to choose the best configuration of $N$ and $b$ to minimize running time to convergence3. For each configuration, we can estimate the iterations required from Figure 5 (linearly interpolating for values of $N$ for which we did not collect data). We can multiply this with the mean iteration times (Figure 4) to obtain the running time required to converge for each setting of $N$ and $b$ . These results are shown in Figure 6, indicating that $N = 9 6$ , $b = 4$ converges fastest. Therefore, this motivates our choice to use a few backup workers for mitigating stragglers. ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 236, + 326, + 251 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we present our empirical comparisons of synchronous and asynchronous distributed stochastic optimization algorithms as applied to models such as Inception and PixelCNN. All experiments in this paper are using the TensorFlow system (Abadi et al., 2015). ", + "bbox": [ + 174, + 267, + 825, + 309 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 METRICS OF COMPARISON: FASTER CONVERGENCE, BETTER OPTIMUM ", + "text_level": 1, + "bbox": [ + 174, + 325, + 709, + 340 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We are interested in two metrics of comparison for our empirical validation: (1) test error or accuracy, and (2) speed of convergence3. We point out that for non-convex deep learning models, it is possible to converge faster to a poorer local optimum. Here we show a simple example with Inception using different learning rates. ", + "bbox": [ + 173, + 352, + 825, + 409 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/a5721568a454d9eb4d6b58098695145714eed7c25b8fae79b2905aa7d541c982.jpg", + "table_caption": [ + "Table 2: Test accuracies at convergence and number of epochs to converge for different initial learning rates $\\gamma _ { 0 }$ . Low initial learning rates result in faster convergence to poorer local optimum. " + ], + "table_footnote": [], + "table_body": "
Initial rate 20Test precision at convergenceEpochs to converge
1.125 2.2577.29% 77.75%52628 65811
4.5
9.078.15%76209
78.17%77235
", + "bbox": [ + 173, + 446, + 401, + 551 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/cf7dd8a2918112ad1401df8582e7cb61ac45e2309e2c05a587c225cae4d880a5.jpg", + "image_caption": [ + "Figure 7: Convergence of Sync-Opt on Inception model using $N =$ 100 workers and $b = 6$ backups, with varying initial learning rates $\\gamma _ { 0 }$ . To reach a lower $\\epsilon$ test precision, small $\\gamma _ { 0 }$ ’s require fewer epochs than large $\\gamma _ { 0 }$ ’s. However, small $\\gamma _ { 0 }$ ’s either fail to attain high $\\epsilon$ precision, or take more epochs than higher $\\gamma _ { 0 }$ ’s. " + ], + "image_footnote": [], + "bbox": [ + 408, + 422, + 815, + 563 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We ran Sync-Opt on Inception with $N = 1 0 0$ and $b = 6$ , but varied the initial learning rate $\\gamma _ { 0 }$ between 1.125 and 9.0. (Learning rates are exponentially decreased with iterations.) Table 2 shows that smaller $\\gamma _ { 0 }$ converge faster, but to poorer test precisions. Focusing on speed on an early phase of training could lead to misleading conclusions if we fail to account for eventual convergence. For example, Figure 3b shows that using $\\gamma _ { 0 } = 1 . 1 2 5$ reaches $\\epsilon = 7 5 \\%$ precision $1 . 5 \\times$ faster than $\\gamma _ { 0 } = 4 . 5$ , but is slower for $\\epsilon = 7 7 . 7 5 \\%$ , and fails to reach higher precisions. ", + "bbox": [ + 174, + 654, + 825, + 738 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 INCEPTION ", + "text_level": 1, + "bbox": [ + 174, + 755, + 292, + 768 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We conducted experiments on the Inception model (Szegedy et al., 2016) trained on ImageNet Challenge dataset (Russakovsky et al., 2015), where the task is to classify images out of 1000 categories. We used several configurations, varying $N + b$ from 53 to 212 workers. Additional details of the training are provided in Appendix A.3. An epoch is a synchronous iteration for Sync-Opt, or a full pass of $N$ updates for Async-Opt, which represent similar amounts of computation. Results of this experiment are presented in Figure 8. ", + "bbox": [ + 174, + 780, + 825, + 864 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 8b shows that Sync-Opt outperforms Async-Opt in test precision: Sync-Opt attains ${ \\sim } 0 . 5 \\%$ better test precision than Async-Opt for comparable $N + b$ workers. Furthermore, Sync-Opt converges 6h and 18h faster than Async-Opt for 106 and 212 workers respectively, and is 3h slower when 53 workers are used, as seen in Figure 8d. This difference in speed is largely due to the fewer epochs (Figure 8c) needed by Sync-Opt, but comparable or better epoch time (Figure 8e). ", + "bbox": [ + 176, + 871, + 825, + 900 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/4a2795f1f53e652a98aa43d466d16efa8ffdf2c2db7dc3f06c9317fb0d07f46a.jpg", + "image_caption": [ + "Figure 8: Convergence of Sync-Opt and Async-Opt on Inception model using varying number of machines. Sync-Opt with backup workers converge faster, with fewer epochs, to higher test accuracies. " + ], + "image_footnote": [], + "bbox": [ + 179, + 101, + 816, + 435 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 507, + 825, + 550 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 PIXELCNN EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 574, + 398, + 588 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The second model we experimented on is PixelCNN (Oord et al., 2016), a conditional image generation deep neural network, which we train on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset. Configurations of $N + b = 1 , 8 , 1 6$ workers were used; for Sync-Opt, we always used $b = 1$ backup worker. Additional details are provided in Appendix A.4. ", + "bbox": [ + 174, + 603, + 825, + 660 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c0603f807e6115ff2ee2ef1f99ab3da81f0e47d7e4e735a38c20bae4c9c188e1.jpg", + "image_caption": [ + "Figure 9: Convergence of synchronous and asynchronous training on PixelCNN model. Sync-Opt achieves lower negative log likelihood in less time than Async-Opt. " + ], + "image_footnote": [], + "bbox": [ + 178, + 683, + 818, + 881 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Convergence of the test negative log likelihood (NLL) on PixelCNN is shown in Figure 9a, where lower is better. Observe that Sync-Opt obtains lower NLL than Async-Opt; in fact, Async-Opt is even outperformed by serial RMSProp with $N = 1$ worker, with degrading performance as $N$ increases from 8 to 16. Figure 9b further shows the time taken to reach $\\epsilon$ test NLL. Sync-Opt reduces the time to reach $\\epsilon = 2 . 1 4 5$ from 247h to $5 8 . 3 \\mathrm { h }$ ; this NLL is not even achieved by Async-Opt. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 195, + 344, + 210 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Multicore and distributed optimization algorithms have received much attention in recent years. Asynchronous algorithms include Recht et al. (2011); Duchi et al. (2013); Zhang et al. (2015a); Reddi et al. (2015); Leblond et al. (2016). Implementations of asynchronous optimization include Xing et al. (2015); Li et al. (2014); Chilimbi et al. (2014). Attempts have also been made in Zinkevich et al. (2010) and Zhang & Jordan (2015) to algorithmically improve synchronous SGD. ", + "bbox": [ + 174, + 227, + 825, + 297 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "An alternative solution, “softsync”, was presented in Zhang et al. (2015b), which proposed batching gradients from multiple machines before performing an asynchronous SGD update, thereby reducing the effective staleness of gradients. Similar to our proposal, softsync avoids stragglers by not forcing updates to wait for the slowest worker. However, softsync allows the use of stale gradients but we do not. The two solutions provide different explorations of the trade-off between high accuracy (by minimizing staleness) and fast throughput (by avoiding stragglers). ", + "bbox": [ + 174, + 304, + 823, + 387 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Watcharapichat et al. (2016) introduces a distributed deep learning system without parameter servers, by having workers interleave gradient computation and communication in a round-robin pattern. Like Async-Opt, this approach suffers from staleness. We also note that in principle, workers in Sync-Opt can double as parameter servers and execute the update operations and avoid the need to partition hardware resources between workers and servers. ", + "bbox": [ + 174, + 395, + 823, + 464 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Das et al. (2016) analyzes distributed stochastic optimization and optimizes the system by solving detailed system balance equations. We believe this approach is complimentary to our work, and could potentially be applied to guide the choice of systems configurations for Sync-Opt. ", + "bbox": [ + 174, + 472, + 825, + 513 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Keskar et al. (2016) suggests that large batch sizes for synchronous stochastic optimization leads to poorer generalization. Our effective batch size increases linearly with the number of workers $N$ . However, we did not observe this effect in our experiments; we believe we are not yet in the large batch size regime examined by Keskar et al. (2016). ", + "bbox": [ + 174, + 520, + 825, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 598, + 495, + 614 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Distributed training strategies for deep learning architectures will become ever more important as the size of datasets increases. In this work, we have shown how both synchronous and asynchronous distributed stochastic optimization suffer from their respective weaknesses of stragglers and staleness. This has motivated our development of synchronous stochastic optimization with backup workers, which we show to be a viable and scalable strategy. ", + "bbox": [ + 174, + 631, + 823, + 700 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We are currently experimenting with different kinds of datasets, including word-level language models where parts of the model (the embedding layers) are often very sparse, which involves very different communication constraints. We are also working on further improving the performance of synchronous training like combining gradients from multiple workers sharing the same machine before sending them to the parameter servers to reduce the communication overhead. An alternative of using time-outs instead of backup workers is also being explored. ", + "bbox": [ + 174, + 707, + 825, + 791 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 813, + 285, + 827 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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", + "bbox": [ + 169, + 102, + 826, + 404 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A DETAILS OF MODELS AND TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 102, + 521, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.1 MNIST CNN, SECTION 2.1 ", + "text_level": 1, + "bbox": [ + 176, + 132, + 411, + 148 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The model used in our experiments is a 4-layer CNN that have $3 { \\tt X } 3$ filters with max-pooling and weight normalization in every layer. We trained the model with SGD for 25 epochs and evaluated performance on the exponential moving average $\\bar { \\theta }$ using a decay rate of $\\alpha = 0 . 9 9 9 9$ . Initial learning rate was set to be 0.1 and linearly annealed to 0 in the last 10 epochs. We also used small image rotations and zooms as a data augmentation scheme. ", + "bbox": [ + 174, + 160, + 825, + 229 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.2 INCEPTION, SECTION 3.1 ", + "text_level": 1, + "bbox": [ + 176, + 246, + 392, + 261 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For our straggler experiments, we trained the Inception (Szegedy et al., 2016) model on the ImageNet Challenge dataset (Russakovsky et al., 2015). 10 parameter servers were used, and each worker was equipped with a k40 GPU. ", + "bbox": [ + 176, + 272, + 823, + 314 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The underlying optimizer was RMSProp with momentum, with decay of 0.9 and momentum of 0.9. Mini-batch size $B = 3 2$ was used. Initial learning rates $\\gamma _ { 0 }$ were set at $0 . 0 4 5 N$ , which we found to provide good test precisions for Inception. Learning rates were also exponentially decreased with decay rate $\\beta = 0 . 9 4$ as $\\gamma _ { 0 } \\beta ^ { t N / ( 2 T ) }$ , where $T = | { \\mathcal { X } } | / B$ is the number of mini-batches in the dataset. ", + "bbox": [ + 174, + 320, + 825, + 378 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Test precisions were evaluated on the exponential moving average $\\bar { \\theta }$ using $\\alpha = 0 . 9 9 9 9$ . ", + "bbox": [ + 173, + 385, + 743, + 400 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.3 INCEPTION, SECTION 4.2 ", + "text_level": 1, + "bbox": [ + 176, + 416, + 393, + 431 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For experiments comparing Async-Opt and Sync-Opt on the Inception model in Section 4.2, each worker is equipped with a k40 GPU. For $N + b = 5 3$ workers, 17 parameter servers were used; for $N + b = 1 0 6$ workers, we used 27 parameter servers; and 37 parameter servers were used for $N + b = 2 1 2$ . ", + "bbox": [ + 174, + 443, + 825, + 498 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the asynchronous training mode, gradient clipping is also needed for stabilization, which requires each worker to collect the gradient across all layers of the deep model, compute the global norm $| | G | |$ and then clip all gradient accordingly. However, synchronization turns out to be very stable so gradient clipping is no longer needed, which means that we can pipeline the update of parameters in different layers: the gradient of top layers’ parameters can be sent to parameter servers while concurrently computing gradients for the lower layers. ", + "bbox": [ + 174, + 506, + 825, + 589 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The underlying optimizer is RMSProp with momentum, with decay of 0.9 and momentum of 0.9. Mini-batch size $B = 3 2$ was used. Initial learning rates $\\gamma _ { 0 }$ for Async-Opt were set to 0.045; for Sync-Opt, we found as a rule-of-thumb that a learning rate of $0 . 0 4 5 N$ worked well for this model. Learning rates were then exponentially decayed with decay rate $\\beta = 0 . 9 4$ as $\\gamma _ { 0 } \\beta ^ { t / ( 2 T ) }$ for AsyncOpt, where $T = | { \\mathcal { X } } | / B$ is the number of mini-batches in the dataset. For Sync-Opt, we learning rates were also exponentially decreased at rate of $\\gamma _ { 0 } \\beta ^ { t N / ( 2 T ) }$ , so that the learning rates after computing the same number of datapoints are comparable for Async-Opt and Sync-Opt. ", + "bbox": [ + 174, + 595, + 825, + 698 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Test precisions were evaluated on the exponential moving average $\\bar { \\theta }$ using $\\alpha = 0 . 9 9 9 9$ ", + "bbox": [ + 176, + 704, + 741, + 718 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.4 PIXELCNN, SECTION 4.3 ", + "text_level": 1, + "bbox": [ + 176, + 734, + 397, + 750 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The PixelCNN (Oord et al., 2016) model was trained on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset. Configurations of $N + b = 1 , 8 , 1 6$ workers each with a $\\mathbf { k } 8 0$ GPU, and 10 parameter servers were used. For Sync-Opt, we always used $b = 1$ backup worker. The underlying optimizer is RMSProp with momentum, using decay of 0.95 and momentum of 0.9. Initial learning rates $\\gamma _ { 0 }$ were set to $1 e - 4$ and slowly decreased to $3 e - 6$ after 200,000 iterations. Mini-batch size $B = 4$ was used. 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Our approach is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 302, + 464, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 464, + 314 + ], + "score": 1.0, + "content": "empirically validated and shown to converge faster and to better test accuracies.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 213, + 470, + 314 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 335, + 206, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 208, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 208, + 351 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 374 + ], + "score": 1.0, + "content": "The recent success of deep learning approaches for domains like speech recognition (Hinton et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "2012) and computer vision (Ioffe & Szegedy, 2015) stems from many algorithmic improvements", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 395 + ], + "score": 1.0, + "content": "but also from the fact that the size of available training data has grown significantly over the years,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "together with the computing power, in terms of both CPUs and GPUs. While a single GPU often", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "provides algorithmic simplicity and speed up to a given scale of data and model, there exist an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "operating point where a distributed implementation of training algorithms for deep architectures", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 426, + 187, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 187, + 439 + ], + "score": 1.0, + "content": "becomes necessary.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 359, + 506, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "Currently, popular distributed training algorithms include mini-batch versions of stochastic gradient", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "descent (SGD) and other stochastic optimization algorithms such as AdaGrad (Duchi et al., 2011),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "RMSProp (Tieleman & Hinton, 2012), and ADAM (Kingma & Ba, 2014). Unfortunately, bulk-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "synchronous implementations of stochastic optimization are often slow in practice due to the need", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "to wait for the slowest machine in each synchronous batch. To circumvent this problem, practi-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "tioners have resorted to asynchronous approaches which emphasize speed by using potentially stale", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "information for computation. While asynchronous training have proven to be faster than their syn-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 520, + 399, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 399, + 532 + ], + "score": 1.0, + "content": "chronous counterparts, they often result in convergence to poorer results.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 442, + 506, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 536, + 504, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "In this paper1, we revisit synchronous learning, and propose a method for mitigating stragglers in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "synchronous stochastic optimization. Specifically, we synchronously compute a mini-batch gradient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "with only a subset of worker machines, thus alleviating the straggler effect while avoiding any", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 570, + 388, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 388, + 582 + ], + "score": 1.0, + "content": "staleness in our gradients. 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We will often evaluate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 334, + 495, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 299, + 351 + ], + "score": 1.0, + "content": "performance on an exponential moving average", + "type": "text" + }, + { + "bbox": [ + 299, + 335, + 417, + 349 + ], + "score": 0.92, + "content": "\\bar { \\theta } ^ { ( t ) } = \\alpha \\bar { \\theta } ^ { ( t - 1 ) } + ( 1 - \\alpha ) \\theta ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 334, + 482, + 351 + ], + "score": 1.0, + "content": "with decay rate", + "type": "text" + }, + { + "bbox": [ + 482, + 339, + 489, + 347 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 334, + 495, + 351 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 338, + 366 + ], + "score": 1.0, + "content": "Our interest is in distributed stochastic optimization using", + "type": "text" + }, + { + "bbox": [ + 339, + 354, + 349, + 363 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "worker machines in charge of comput-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 262, + 377 + ], + "score": 1.0, + "content": "ing stochastic gradients that are sent to", + "type": "text" + }, + { + "bbox": [ + 262, + 365, + 274, + 374 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 364, + 441, + 377 + ], + "score": 1.0, + "content": "parameter servers. Each parameter server", + "type": "text" + }, + { + "bbox": [ + 441, + 365, + 447, + 376 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "is responsible", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 186, + 389 + ], + "score": 1.0, + "content": "for storing a subset", + "type": "text" + }, + { + "bbox": [ + 187, + 375, + 203, + 387 + ], + "score": 0.9, + "content": "\\theta [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 374, + 372, + 389 + ], + "score": 1.0, + "content": "of the model, and performing updates on", + "type": "text" + }, + { + "bbox": [ + 372, + 375, + 388, + 387 + ], + "score": 0.91, + "content": "\\theta [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 374, + 505, + 389 + ], + "score": 1.0, + "content": ". In the synchronous setting,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 386, + 409, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 240, + 399 + ], + "score": 1.0, + "content": "we will also introduce additional", + "type": "text" + }, + { + "bbox": [ + 240, + 387, + 245, + 396 + ], + "score": 0.53, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 386, + 409, + 399 + ], + "score": 1.0, + "content": "backup workers for straggler mitigation.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 413, + 362, + 426 + ], + "lines": [ + { + "bbox": [ + 104, + 412, + 363, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 363, + 428 + ], + "score": 1.0, + "content": "2 ASYNCHRONOUS STOCHASTIC OPTIMIZATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 438, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "score": 1.0, + "content": "An approach for a distributed stochastic gradient descent algorithm was presented in Dean et al.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "(2012), consisting of two main ingredients. First, the parameters of the model are distributed on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "multiple servers, depending on the architecture. This set of servers are called the parameter servers.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 486 + ], + "score": 1.0, + "content": "Second, there can be multiple workers processing data in parallel and communicating with the pa-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "rameter servers. Each worker processes a mini-batch of data independently of the others, as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 132, + 502, + 504, + 566 + ], + "lines": [ + { + "bbox": [ + 132, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 132, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "• The worker fetches from the parameter servers the most up-to-date parameters of the model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 513, + 309, + 525 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 309, + 525 + ], + "score": 1.0, + "content": "needed to process the current mini-batch;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 132, + 527, + 427, + 541 + ], + "spans": [ + { + "bbox": [ + 132, + 527, + 427, + 541 + ], + "score": 1.0, + "content": "• It then computes gradients of the loss with respect to these parameters;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 131, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 131, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "• Finally, these gradients are sent back to the parameter servers, which then updates the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 553, + 221, + 567 + ], + "spans": [ + { + "bbox": [ + 141, + 553, + 221, + 567 + ], + "score": 1.0, + "content": "model accordingly.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "Since each worker communicates with the parameter servers independently of the others, this is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "score": 1.0, + "content": "called Asynchronous Stochastic Gradient Descent (Async-SGD), or more generally, Asynchronous", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "Stochastic Optimization (Async-Opt). A similar approach was later proposed by Chilimbi et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 327, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 327, + 620 + ], + "score": 1.0, + "content": "(2014). Async-Opt is presented in Algorithms 1 and 2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "In practice, the updates of Async-Opt are different than those of serially running the stochastic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "score": 1.0, + "content": "optimization algorithm for two reasons. Firstly, the read operation (Algo 1 Line 2) on a worker may", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 646, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 473, + 661 + ], + "score": 1.0, + "content": "be interleaved with updates by other workers to different parameter servers, so the resultant", + "type": "text" + }, + { + "bbox": [ + 473, + 646, + 484, + 659 + ], + "score": 0.9, + "content": "\\widehat { \\theta } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 646, + 505, + 661 + ], + "score": 1.0, + "content": "may", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 303, + 673 + ], + "score": 1.0, + "content": "not be consistent with any parameter incarnation", + "type": "text" + }, + { + "bbox": [ + 304, + 659, + 319, + 670 + ], + "score": 0.89, + "content": "\\boldsymbol { \\theta } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". Secondly, model updates may have occurred", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "while a worker is computing its stochastic gradient; hence, the resultant gradients are typically", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "computed with respect to outdated parameters. We refer to these as stale gradients, and its staleness", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 693, + 504, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 504, + 705 + ], + "score": 1.0, + "content": "as the number of updates that have occurred between its corresponding read and update operations.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Understanding the theoretical impact of staleness is difficult work and the topic of many recent", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "papers, e.g. Recht et al. (2011); Duchi et al. (2013); Leblond et al. (2016); Reddi et al. 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We briefly present preliminaries and notation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "score": 1.0, + "content": "in Section 1.1. Section 2 describes asynchronous stochastic optimization and presents experimental", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "evidence of gradient staleness in deep neural network models. We present our approach in Section 3,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "and exhibit straggler effects that motivate the approach. We then empirically evaluate our approach", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 443, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 443, + 170 + ], + "score": 1.0, + "content": "in Sections 4. Related work is discussed in Section 5, and we conclude in Section 6.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 113, + 505, + 170 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 182, + 268, + 194 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 269, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 269, + 195 + ], + "score": 1.0, + "content": "1.1 PRELIMINARIES AND NOTATION", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 202, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 171, + 216 + ], + "score": 1.0, + "content": "Given a dataset", + "type": "text" + }, + { + "bbox": [ + 172, + 203, + 285, + 215 + ], + "score": 0.92, + "content": "\\mathcal { X } = \\{ x _ { i } : i = 1 , \\ldots , | \\mathcal { X } | \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 201, + 430, + 216 + ], + "score": 1.0, + "content": ", our goal is to learn the parameters", + "type": "text" + }, + { + "bbox": [ + 430, + 203, + 437, + 213 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "of a model with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 101, + 210, + 509, + 237 + ], + "spans": [ + { + "bbox": [ + 101, + 210, + 253, + 237 + ], + "score": 1.0, + "content": "respect to an empirical loss function", + "type": "text" + }, + { + "bbox": [ + 253, + 218, + 260, + 230 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 210, + 306, + 237 + ], + "score": 1.0, + "content": ", defined as", + "type": "text" + }, + { + "bbox": [ + 307, + 214, + 415, + 232 + ], + "score": 0.94, + "content": "\\begin{array} { r } { f ( \\theta ) \\stackrel { \\Delta } { = } \\frac { 1 } { | \\mathcal { X } | } \\sum _ { i = 1 } ^ { | \\mathcal { X } | } F ( x _ { i } ; \\theta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 210, + 445, + 237 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 445, + 217, + 480, + 230 + ], + "score": 0.93, + "content": "F ( x _ { i } ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 210, + 509, + 237 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 230, + 313, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 232, + 244 + ], + "score": 1.0, + "content": "loss with respect to a datapoint", + "type": "text" + }, + { + "bbox": [ + 232, + 232, + 243, + 242 + ], + "score": 0.85, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 230, + 302, + 244 + ], + "score": 1.0, + "content": "and the model", + "type": "text" + }, + { + "bbox": [ + 303, + 231, + 309, + 241 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 230, + 313, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 101, + 201, + 509, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 247, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 462, + 261 + ], + "score": 1.0, + "content": "A first-order stochastic optimization algorithm achieves this by iteratively updating", + "type": "text" + }, + { + "bbox": [ + 462, + 248, + 469, + 258 + ], + "score": 0.76, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "using a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 259, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 186, + 277 + ], + "score": 1.0, + "content": "stochastic gradient", + "type": "text" + }, + { + "bbox": [ + 186, + 259, + 255, + 275 + ], + "score": 0.93, + "content": "G \\overset { \\Delta } { = } \\nabla F ( x _ { i } ; \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 259, + 398, + 277 + ], + "score": 1.0, + "content": "computed at a randomly sampled", + "type": "text" + }, + { + "bbox": [ + 398, + 264, + 408, + 273 + ], + "score": 0.82, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 259, + 506, + 277 + ], + "score": 1.0, + "content": ", producing a sequence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 272, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 272, + 150, + 288 + ], + "score": 1.0, + "content": "of models", + "type": "text" + }, + { + "bbox": [ + 150, + 273, + 205, + 286 + ], + "score": 0.9, + "content": "\\theta ^ { ( 0 ) } , \\theta ^ { ( 1 ) } , \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 272, + 506, + 288 + ], + "score": 1.0, + "content": ". Stochastic optimization algorithms differ in their update equations. For", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 284, + 507, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 235, + 300 + ], + "score": 1.0, + "content": "example, the update of SGD is", + "type": "text" + }, + { + "bbox": [ + 235, + 285, + 436, + 298 + ], + "score": 0.89, + "content": "\\bar { \\theta ^ { ( t + 1 ) } } \\bar { \\bf \\Phi } = \\theta ^ { ( t ) } - \\gamma _ { t } \\bar { G } ^ { ( t ) } = \\theta ^ { ( t ) } - \\gamma _ { t } \\nabla F ( x _ { i } ; \\bar { \\theta } ^ { ( t ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 284, + 468, + 300 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 468, + 288, + 479, + 298 + ], + "score": 0.84, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 284, + 507, + 300 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 504, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 258, + 310 + ], + "score": 1.0, + "content": "learning rate or step size at iteration", + "type": "text" + }, + { + "bbox": [ + 258, + 299, + 263, + 308 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 297, + 504, + 310 + ], + "score": 1.0, + "content": ". A mini-batch version of the stochastic optimization algo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 359, + 322 + ], + "score": 1.0, + "content": "rithm computes the stochastic gradient over mini-batch of size", + "type": "text" + }, + { + "bbox": [ + 359, + 309, + 369, + 318 + ], + "score": 0.82, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "instead of a single datapoint, i.e.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 507, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 221, + 336 + ], + "score": 0.92, + "content": "\\begin{array} { r } { G \\stackrel { \\Delta } { = } \\frac { 1 } { B } \\sum _ { i = 1 } ^ { B } \\nabla F ( \\widetilde { x } _ { i } ; \\theta ^ { ( t ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 320, + 254, + 337 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 254, + 323, + 265, + 334 + ], + "score": 0.86, + "content": "\\widetilde { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 320, + 392, + 337 + ], + "score": 1.0, + "content": "’s are randomly sampled from", + "type": "text" + }, + { + "bbox": [ + 392, + 324, + 402, + 333 + ], + "score": 0.78, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 320, + 507, + 337 + ], + "score": 1.0, + "content": ". We will often evaluate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 334, + 495, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 334, + 299, + 351 + ], + "score": 1.0, + "content": "performance on an exponential moving average", + "type": "text" + }, + { + "bbox": [ + 299, + 335, + 417, + 349 + ], + "score": 0.92, + "content": "\\bar { \\theta } ^ { ( t ) } = \\alpha \\bar { \\theta } ^ { ( t - 1 ) } + ( 1 - \\alpha ) \\theta ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 334, + 482, + 351 + ], + "score": 1.0, + "content": "with decay rate", + "type": "text" + }, + { + "bbox": [ + 482, + 339, + 489, + 347 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 334, + 495, + 351 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 246, + 507, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 338, + 366 + ], + "score": 1.0, + "content": "Our interest is in distributed stochastic optimization using", + "type": "text" + }, + { + "bbox": [ + 339, + 354, + 349, + 363 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "worker machines in charge of comput-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 262, + 377 + ], + "score": 1.0, + "content": "ing stochastic gradients that are sent to", + "type": "text" + }, + { + "bbox": [ + 262, + 365, + 274, + 374 + ], + "score": 0.81, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 364, + 441, + 377 + ], + "score": 1.0, + "content": "parameter servers. Each parameter server", + "type": "text" + }, + { + "bbox": [ + 441, + 365, + 447, + 376 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "is responsible", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 186, + 389 + ], + "score": 1.0, + "content": "for storing a subset", + "type": "text" + }, + { + "bbox": [ + 187, + 375, + 203, + 387 + ], + "score": 0.9, + "content": "\\theta [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 374, + 372, + 389 + ], + "score": 1.0, + "content": "of the model, and performing updates on", + "type": "text" + }, + { + "bbox": [ + 372, + 375, + 388, + 387 + ], + "score": 0.91, + "content": "\\theta [ j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 374, + 505, + 389 + ], + "score": 1.0, + "content": ". In the synchronous setting,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 386, + 409, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 240, + 399 + ], + "score": 1.0, + "content": "we will also introduce additional", + "type": "text" + }, + { + "bbox": [ + 240, + 387, + 245, + 396 + ], + "score": 0.53, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 386, + 409, + 399 + ], + "score": 1.0, + "content": "backup workers for straggler mitigation.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 353, + 505, + 399 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 413, + 362, + 426 + ], + "lines": [ + { + "bbox": [ + 104, + 412, + 363, + 428 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 363, + 428 + ], + "score": 1.0, + "content": "2 ASYNCHRONOUS STOCHASTIC OPTIMIZATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 438, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 450 + ], + "score": 1.0, + "content": "An approach for a distributed stochastic gradient descent algorithm was presented in Dean et al.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "(2012), consisting of two main ingredients. First, the parameters of the model are distributed on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "multiple servers, depending on the architecture. This set of servers are called the parameter servers.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 486 + ], + "score": 1.0, + "content": "Second, there can be multiple workers processing data in parallel and communicating with the pa-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "rameter servers. Each worker processes a mini-batch of data independently of the others, as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 438, + 506, + 494 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 502, + 504, + 566 + ], + "lines": [ + { + "bbox": [ + 132, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 132, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "• The worker fetches from the parameter servers the most up-to-date parameters of the model", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 513, + 309, + 525 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 309, + 525 + ], + "score": 1.0, + "content": "needed to process the current mini-batch;", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 527, + 427, + 541 + ], + "spans": [ + { + "bbox": [ + 132, + 527, + 427, + 541 + ], + "score": 1.0, + "content": "• It then computes gradients of the loss with respect to these parameters;", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 131, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "• Finally, these gradients are sent back to the parameter servers, which then updates the", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 553, + 221, + 567 + ], + "spans": [ + { + "bbox": [ + 141, + 553, + 221, + 567 + ], + "score": 1.0, + "content": "model accordingly.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + } + ], + "index": 31, + "bbox_fs": [ + 131, + 502, + 505, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "Since each worker communicates with the parameter servers independently of the others, this is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "score": 1.0, + "content": "called Asynchronous Stochastic Gradient Descent (Async-SGD), or more generally, Asynchronous", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "Stochastic Optimization (Async-Opt). A similar approach was later proposed by Chilimbi et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 607, + 327, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 327, + 620 + ], + "score": 1.0, + "content": "(2014). Async-Opt is presented in Algorithms 1 and 2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 573, + 506, + 620 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "In practice, the updates of Async-Opt are different than those of serially running the stochastic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 649 + ], + "score": 1.0, + "content": "optimization algorithm for two reasons. Firstly, the read operation (Algo 1 Line 2) on a worker may", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 646, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 473, + 661 + ], + "score": 1.0, + "content": "be interleaved with updates by other workers to different parameter servers, so the resultant", + "type": "text" + }, + { + "bbox": [ + 473, + 646, + 484, + 659 + ], + "score": 0.9, + "content": "\\widehat { \\theta } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 646, + 505, + 661 + ], + "score": 1.0, + "content": "may", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 303, + 673 + ], + "score": 1.0, + "content": "not be consistent with any parameter incarnation", + "type": "text" + }, + { + "bbox": [ + 304, + 659, + 319, + 670 + ], + "score": 0.89, + "content": "\\boldsymbol { \\theta } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". Secondly, model updates may have occurred", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "while a worker is computing its stochastic gradient; hence, the resultant gradients are typically", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "computed with respect to outdated parameters. 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(2015); Mania et al. (2015), most of which focus on individual algorithms, under strong", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "assumptions that may not hold up in practice. This is further complicated by deep models with mul-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 468 + ], + "score": 1.0, + "content": "tiple layers, since the times at which model parameters are read and which gradients are computed", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "and sent are dependent on the depth of the layers (Figure 1). 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The trick was not", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 160, + 479, + 173 + ], + "spans": [ + { + "bbox": [ + 141, + 160, + 479, + 173 + ], + "score": 1.0, + "content": "relevant with a simulated staleness less than 15 but became crucial for larger values.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 141, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "Use lower initial learning rates when staleness is at least 20, which reduces a frequency of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 141, + 190, + 264, + 202 + ], + "score": 1.0, + "content": "explosions (train error goes to", + "type": "text" + }, + { + "bbox": [ + 264, + 190, + 284, + 201 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "). This observation is similar to what we found in other", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 201, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 141, + 201, + 505, + 215 + ], + "score": 1.0, + "content": "experiments - we were able to use much larger learning rates with synchronous training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 213, + 294, + 223 + ], + "spans": [ + { + "bbox": [ + 142, + 213, + 294, + 223 + ], + "score": 1.0, + "content": "and the results were also more stable.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 139, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 139, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "Even with above tricks the divergence occurs occasionally and we found that restarting", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 243, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 142, + 243, + 505, + 254 + ], + "score": 1.0, + "content": "training from random weights can lead to more successful runs. 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Most mean times fall between 1.4s", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 308, + 405, + 449, + 416 + ], + "spans": [ + { + "bbox": [ + 308, + 405, + 449, + 416 + ], + "score": 1.0, + "content": "and 1.8s, except of final few gradients.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "index": 31.25 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "Thus, one might choose to drop slow stragglers to decrease the iteration time. 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Additional details of this training are provided in Appendix A.2. 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We investigate the effect of stragglers on Sync-Opt model training here.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 101, + 505, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 506, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 197, + 144 + ], + "score": 1.0, + "content": "We ran Sync-Opt with", + "type": "text" + }, + { + "bbox": [ + 198, + 132, + 236, + 142 + ], + "score": 0.89, + "content": "N = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 131, + 273, + 144 + ], + "score": 1.0, + "content": "workers,", + "type": "text" + }, + { + "bbox": [ + 274, + 132, + 298, + 142 + ], + "score": 0.9, + "content": "b = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "backups, and 19 parameter servers on the Inception", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "model. Using one variable as a proxy, we collected for each iteration both the start time of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 235, + 166 + ], + "score": 1.0, + "content": "iteration and the time when the", + "type": "text" + }, + { + "bbox": [ + 235, + 154, + 241, + 163 + ], + "score": 0.65, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 153, + 506, + 166 + ], + "score": 1.0, + "content": "th gradient of that variable arrived at the parameter server. These", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 245, + 177 + ], + "score": 1.0, + "content": "times are presented in Figure 3 for", + "type": "text" + }, + { + "bbox": [ + 245, + 164, + 270, + 174 + ], + "score": 0.6, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 164, + 404, + 177 + ], + "score": 1.0, + "content": ", 50, 90, 97, 98, 99, 100. Note that", + "type": "text" + }, + { + "bbox": [ + 405, + 164, + 424, + 175 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "of the 98th gradient", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 175, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 241, + 187 + ], + "score": 1.0, + "content": "arrives in under 2s, whereas only", + "type": "text" + }, + { + "bbox": [ + 241, + 175, + 261, + 186 + ], + "score": 0.84, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 176, + 505, + 187 + ], + "score": 1.0, + "content": "of the final gradient do. Furthermore, the time to collect the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "final few gradients grows exponentially, resulting in wasted idle resources and time expended to wait", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 197, + 411, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 411, + 209 + ], + "score": 1.0, + "content": "for the slowest gradients. This exponential increase is also seen in Figure 4.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 131, + 506, + 209 + ] + }, + { + "type": "image", + "bbox": [ + 109, + 221, + 298, + 363 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 221, + 298, + 363 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 221, + 298, + 363 + ], + "spans": [ + { + "bbox": [ + 109, + 221, + 298, + 363 + ], + "score": 0.967, + "type": "image", + "image_path": "f142cfb5431917c4256dc4d3a33158daba7d846ffc4487f811dfdda3fff7a939.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 109, + 221, + 298, + 233.9090909090909 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 109, + 233.9090909090909, + 298, + 246.8181818181818 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 109, + 246.8181818181818, + 298, + 259.72727272727275 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 109, + 259.72727272727275, + 298, + 272.6363636363637 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 109, + 272.6363636363637, + 298, + 285.5454545454546 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 109, + 285.5454545454546, + 298, + 298.45454545454555 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 109, + 298.45454545454555, + 298, + 311.3636363636365 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 109, + 311.3636363636365, + 298, + 324.2727272727274 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 109, + 324.2727272727274, + 298, + 337.18181818181836 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 109, + 337.18181818181836, + 298, + 350.0909090909093 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 109, + 350.0909090909093, + 298, + 363.0000000000002 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 375, + 302, + 406 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 374, + 303, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 303, + 387 + ], + "score": 1.0, + "content": "Figure 3: CDF of time taken to aggregate gradients", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 384, + 303, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 126, + 396 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 127, + 385, + 136, + 394 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 384, + 303, + 396 + ], + "score": 1.0, + "content": "machines. For clarity, we only show times of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 395, + 263, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 126, + 406 + ], + "score": 0.72, + "content": "\\leq 6 \\mathrm { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 395, + 263, + 406 + ], + "score": 1.0, + "content": "; the maximum observed time is 310s.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "index": 24.0 + }, + { + "type": "image", + "bbox": [ + 312, + 225, + 501, + 363 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 312, + 225, + 501, + 363 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 312, + 225, + 501, + 363 + ], + "spans": [ + { + "bbox": [ + 312, + 225, + 501, + 363 + ], + "score": 0.966, + "type": "image", + "image_path": "1faa87f637ff8939e5bbdb8a66acd6d420b49338648b740f8e467c1caef2ee75.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 312, + 225, + 501, + 237.54545454545453 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 312, + 237.54545454545453, + 501, + 250.09090909090907 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 312, + 250.09090909090907, + 501, + 262.6363636363636 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 312, + 262.6363636363636, + 501, + 275.1818181818182 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 312, + 275.1818181818182, + 501, + 287.72727272727275 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 312, + 287.72727272727275, + 501, + 300.2727272727273 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 312, + 300.2727272727273, + 501, + 312.81818181818187 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 312, + 312.81818181818187, + 501, + 325.36363636363643 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 312, + 325.36363636363643, + 501, + 337.909090909091 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 312, + 337.909090909091, + 501, + 350.45454545454555 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 312, + 350.45454545454555, + 501, + 363.0000000000001 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 308, + 375, + 505, + 416 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 308, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 308, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "Figure 4: Mean and median times, across all itera-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 308, + 385, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 308, + 385, + 366, + 395 + ], + "score": 1.0, + "content": "tions, to collect", + "type": "text" + }, + { + "bbox": [ + 366, + 385, + 373, + 394 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 385, + 420, + 395 + ], + "score": 1.0, + "content": "gradients on", + "type": "text" + }, + { + "bbox": [ + 421, + 385, + 457, + 394 + ], + "score": 0.9, + "content": "N = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 385, + 505, + 395 + ], + "score": 1.0, + "content": "workers and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 308, + 395, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 308, + 395, + 334, + 405 + ], + "score": 0.89, + "content": "b = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 395, + 505, + 406 + ], + "score": 1.0, + "content": "backups. Most mean times fall between 1.4s", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 308, + 405, + 449, + 416 + ], + "spans": [ + { + "bbox": [ + 308, + 405, + 449, + 416 + ], + "score": 1.0, + "content": "and 1.8s, except of final few gradients.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "index": 31.25 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "Thus, one might choose to drop slow stragglers to decrease the iteration time. However, using fewer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "machines implies a smaller effective mini-batch size and thus greater gradient variance, which in turn", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "could require more iterations for convergence. We examine this relationship by running Sync-Opt2", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 126, + 474 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 462, + 160, + 472 + ], + "score": 0.83, + "content": "N = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 461, + 241, + 474 + ], + "score": 1.0, + "content": ", 70, 80, 90, 100 and", + "type": "text" + }, + { + "bbox": [ + 241, + 462, + 265, + 472 + ], + "score": 0.9, + "content": "b = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 461, + 505, + 474 + ], + "score": 1.0, + "content": ", and note the number of iterations required for convergence", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 428, + 486 + ], + "score": 1.0, + "content": "in Figure 5. Additional details of this training are provided in Appendix A.2. As", + "type": "text" + }, + { + "bbox": [ + 428, + 473, + 438, + 483 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "is doubled from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 483, + 450, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 372, + 496 + ], + "score": 1.0, + "content": "50 to 100, the number of iterations to converge nearly halves from", + "type": "text" + }, + { + "bbox": [ + 373, + 484, + 406, + 494 + ], + "score": 0.49, + "content": "1 3 7 . 5 e 3", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 483, + 418, + 496 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 418, + 484, + 446, + 494 + ], + "score": 0.7, + "content": "7 6 . 2 e 3", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 483, + 450, + 496 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + 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Consider a hypothetical setting where we have", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 167, + 115 + ], + "score": 0.91, + "content": "N + b = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 104, + 415, + 117 + ], + "score": 1.0, + "content": "machines, and we wish to choose the best configuration of", + "type": "text" + }, + { + "bbox": [ + 415, + 105, + 426, + 114 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 104, + 445, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 105, + 452, + 114 + ], + "score": 0.69, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "to minimize", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "running time to convergence3. For each configuration, we can estimate the iterations required from", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 285, + 140 + ], + "score": 1.0, + "content": "Figure 5 (linearly interpolating for values of", + "type": "text" + }, + { + "bbox": [ + 285, + 127, + 295, + 136 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "for which we did not collect data). We can multiply", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "this with the mean iteration times (Figure 4) to obtain the running time required to converge for each", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 147, + 162 + ], + "score": 1.0, + "content": "setting of", + "type": "text" + }, + { + "bbox": [ + 147, + 149, + 158, + 158 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 147, + 176, + 162 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 176, + 149, + 182, + 158 + ], + "score": 0.65, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 147, + 397, + 162 + ], + "score": 1.0, + "content": ". These results are shown in Figure 6, indicating that", + "type": "text" + }, + { + "bbox": [ + 397, + 149, + 432, + 159 + ], + "score": 0.85, + "content": "N = 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 147, + 436, + 162 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 436, + 149, + 462, + 159 + ], + "score": 0.82, + "content": "b = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "converges", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 501, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 501, + 172 + ], + "score": 1.0, + "content": "fastest. Therefore, this motivates our choice to use a few backup workers for mitigating stragglers.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 187, + 200, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 201, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 201, + 201 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 212, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "In this section, we present our empirical comparisons of synchronous and asynchronous distributed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "stochastic optimization algorithms as applied to models such as Inception and PixelCNN. All exper-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 403, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 403, + 246 + ], + "score": 1.0, + "content": "iments in this paper are using the TensorFlow system (Abadi et al., 2015).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 258, + 434, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 436, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 436, + 271 + ], + "score": 1.0, + "content": "4.1 METRICS OF COMPARISON: FASTER CONVERGENCE, BETTER OPTIMUM", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "We are interested in two metrics of comparison for our empirical validation: (1) test error or ac-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "curacy, and (2) speed of convergence3. We point out that for non-convex deep learning models,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 300, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 315 + ], + "score": 1.0, + "content": "it is possible to converge faster to a poorer local optimum. Here we show a simple example with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 312, + 266, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 266, + 325 + ], + "score": 1.0, + "content": "Inception using different learning rates.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "table", + "bbox": [ + 106, + 354, + 246, + 437 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 354, + 246, + 437 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 354, + 246, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 246, + 437 + ], + "score": 0.955, + "html": "
Initial rate 20Test precision at convergenceEpochs to converge
1.125 2.2577.29% 77.75%52628 65811
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Focusing on speed on an early phase", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "of training could lead to misleading conclusions if we fail to account for eventual convergence.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 273, + 575 + ], + "score": 1.0, + "content": "For example, Figure 3b shows that using", + "type": "text" + }, + { + "bbox": [ + 274, + 563, + 323, + 574 + ], + "score": 0.9, + "content": "\\gamma _ { 0 } = 1 . 1 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 562, + 357, + 575 + ], + "score": 1.0, + "content": "reaches", + "type": "text" + }, + { + "bbox": [ + 357, + 562, + 396, + 573 + ], + "score": 0.91, + "content": "\\epsilon = 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 562, + 437, + 575 + ], + "score": 1.0, + "content": "precision", + "type": "text" + }, + { + "bbox": [ + 437, + 563, + 459, + 573 + ], + "score": 0.86, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "faster than", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 573, + 412, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 143, + 585 + ], + "score": 0.91, + "content": "\\gamma _ { 0 } = 4 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 573, + 214, + 586 + ], + "score": 1.0, + "content": ", but is slower for", + "type": "text" + }, + { + "bbox": [ + 214, + 573, + 264, + 585 + ], + "score": 0.91, + "content": "\\epsilon = 7 7 . 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 573, + 412, + 586 + ], + "score": 1.0, + "content": ", and fails to reach higher precisions.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 107, + 598, + 179, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 180, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 180, + 611 + ], + "score": 1.0, + "content": "4.2 INCEPTION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 618, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "We conducted experiments on the Inception model (Szegedy et al., 2016) trained on ImageNet Chal-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 629, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 643 + ], + "score": 1.0, + "content": "lenge dataset (Russakovsky et al., 2015), where the task is to classify images out of 1000 categories.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 271, + 654 + ], + "score": 1.0, + "content": "We used several configurations, varying", + "type": "text" + }, + { + "bbox": [ + 271, + 641, + 299, + 651 + ], + "score": 0.9, + "content": "N + b", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "from 53 to 212 workers. Additional details of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "training are provided in Appendix A.3. An epoch is a synchronous iteration for Sync-Opt, or a full", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 662, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 137, + 676 + ], + "score": 1.0, + "content": "pass of", + "type": "text" + }, + { + "bbox": [ + 137, + 663, + 148, + 673 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 662, + 506, + 676 + ], + "score": 1.0, + "content": "updates for Async-Opt, which represent similar amounts of computation. 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Furthermore, Sync-Opt con-", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 721, + 474, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 474, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 474, + 734 + ], + "score": 1.0, + "content": "3Convergence is defined as the point where maximum test accuracy or lowest test error is reached.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "Hence, there is a trade-off between dropping more stragglers to reduce iteration time, and waiting", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "for more gradients to improve the gradient quality. Consider a hypothetical setting where we have", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 167, + 115 + ], + "score": 0.91, + "content": "N + b = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 104, + 415, + 117 + ], + "score": 1.0, + "content": "machines, and we wish to choose the best configuration of", + "type": "text" + }, + { + "bbox": [ + 415, + 105, + 426, + 114 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 104, + 445, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 105, + 452, + 114 + ], + "score": 0.69, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "to minimize", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "running time to convergence3. 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Therefore, this motivates our choice to use a few backup workers for mitigating stragglers.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 172 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 187, + 200, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 201, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 201, + 201 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 212, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "In this section, we present our empirical comparisons of synchronous and asynchronous distributed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "stochastic optimization algorithms as applied to models such as Inception and PixelCNN. All exper-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 234, + 403, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 403, + 246 + ], + "score": 1.0, + "content": "iments in this paper are using the TensorFlow system (Abadi et al., 2015).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 211, + 506, + 246 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 258, + 434, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 436, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 436, + 271 + ], + "score": 1.0, + "content": "4.1 METRICS OF COMPARISON: FASTER CONVERGENCE, BETTER OPTIMUM", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "We are interested in two metrics of comparison for our empirical validation: (1) test error or ac-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "curacy, and (2) speed of convergence3. We point out that for non-convex deep learning models,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 300, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 315 + ], + "score": 1.0, + "content": "it is possible to converge faster to a poorer local optimum. Here we show a simple example with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 312, + 266, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 266, + 325 + ], + "score": 1.0, + "content": "Inception using different learning rates.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 279, + 506, + 325 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 354, + 246, + 437 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 354, + 246, + 437 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 354, + 246, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 246, + 437 + ], + "score": 0.955, + "html": "
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Observe that Sync-Opt obtains lower NLL than Async-Opt; in fact, Async-Opt", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 297, + 117 + ], + "score": 1.0, + "content": "is even outperformed by serial RMSProp with", + "type": "text" + }, + { + "bbox": [ + 298, + 105, + 329, + 114 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 104, + 493, + 117 + ], + "score": 1.0, + "content": "worker, with degrading performance as", + "type": "text" + }, + { + "bbox": [ + 493, + 105, + 504, + 114 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 385, + 128 + ], + "score": 1.0, + "content": "increases from 8 to 16. Figure 9b further shows the time taken to reach", + "type": "text" + }, + { + "bbox": [ + 386, + 117, + 392, + 125 + ], + "score": 0.33, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "test NLL. 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(2011); Duchi et al. (2013); Zhang et al. (2015a);", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "Reddi et al. (2015); Leblond et al. (2016). Implementations of asynchronous optimization include", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "Xing et al. (2015); Li et al. (2014); Chilimbi et al. (2014). Attempts have also been made in Zinke-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 477, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 477, + 237 + ], + "score": 1.0, + "content": "vich et al. (2010) and Zhang & Jordan (2015) to algorithmically improve synchronous SGD.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 504, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 255 + ], + "score": 1.0, + "content": "An alternative solution, “softsync”, was presented in Zhang et al. (2015b), which proposed batching", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "gradients from multiple machines before performing an asynchronous SGD update, thereby reducing", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "score": 1.0, + "content": "the effective staleness of gradients. Similar to our proposal, softsync avoids stragglers by not forcing", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "updates to wait for the slowest worker. However, softsync allows the use of stale gradients but we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "score": 1.0, + "content": "do not. The two solutions provide different explorations of the trade-off between high accuracy (by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 296, + 376, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 376, + 309 + ], + "score": 1.0, + "content": "minimizing staleness) and fast throughput (by avoiding stragglers).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 504, + 368 + ], + "lines": [ + { + "bbox": [ + 107, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 107, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "Watcharapichat et al. (2016) introduces a distributed deep learning system without parameter", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "servers, by having workers interleave gradient computation and communication in a round-robin", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "pattern. Like Async-Opt, this approach suffers from staleness. We also note that in principle, work-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "ers in Sync-Opt can double as parameter servers and execute the update operations and avoid the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 374, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 374, + 369 + ], + "score": 1.0, + "content": "need to partition hardware resources between workers and servers.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "Das et al. (2016) analyzes distributed stochastic optimization and optimizes the system by solving", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "detailed system balance equations. We believe this approach is complimentary to our work, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 459, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 459, + 408 + ], + "score": 1.0, + "content": "could potentially be applied to guide the choice of systems configurations for Sync-Opt.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "Keskar et al. (2016) suggests that large batch sizes for synchronous stochastic optimization leads to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 491, + 436 + ], + "score": 1.0, + "content": "poorer generalization. Our effective batch size increases linearly with the number of workers", + "type": "text" + }, + { + "bbox": [ + 491, + 424, + 501, + 434 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 424, + 505, + 436 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "However, we did not observe this effect in our experiments; we believe we are not yet in the large", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 315, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 315, + 457 + ], + "score": 1.0, + "content": "batch size regime examined by Keskar et al. 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In this work, we have shown how both synchronous and asynchronous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "distributed stochastic optimization suffer from their respective weaknesses of stragglers and stal-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 547 + ], + "score": 1.0, + "content": "eness. This has motivated our development of synchronous stochastic optimization with backup", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 351, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 351, + 558 + ], + "score": 1.0, + "content": "workers, which we show to be a viable and scalable strategy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "We are currently experimenting with different kinds of datasets, including word-level language mod-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "els where parts of the model (the embedding layers) are often very sparse, which involves very", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "different communication constraints. We are also working on further improving the performance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "of synchronous training like combining gradients from multiple workers sharing the same machine", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "before sending them to the parameter servers to reduce the communication overhead. 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TensorFlow:", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 115, + 721, + 489, + 733 + ], + "spans": [ + { + "bbox": [ + 115, + 721, + 489, + 733 + ], + "score": 1.0, + "content": "Large-scale machine learning on heterogeneous systems, 2015. URL http://tensorflow.org/.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Convergence of the test negative log likelihood (NLL) on PixelCNN is shown in Figure 9a, where", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "lower is better. Observe that Sync-Opt obtains lower NLL than Async-Opt; in fact, Async-Opt", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 297, + 117 + ], + "score": 1.0, + "content": "is even outperformed by serial RMSProp with", + "type": "text" + }, + { + "bbox": [ + 298, + 105, + 329, + 114 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 104, + 493, + 117 + ], + "score": 1.0, + "content": "worker, with degrading performance as", + "type": "text" + }, + { + "bbox": [ + 493, + 105, + 504, + 114 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 385, + 128 + ], + "score": 1.0, + "content": "increases from 8 to 16. Figure 9b further shows the time taken to reach", + "type": "text" + }, + { + "bbox": [ + 386, + 117, + 392, + 125 + ], + "score": 0.33, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "test NLL. Sync-Opt reduces", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 482, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 175, + 140 + ], + "score": 1.0, + "content": "the time to reach", + "type": "text" + }, + { + "bbox": [ + 176, + 127, + 217, + 137 + ], + "score": 0.84, + "content": "\\epsilon = 2 . 1 4 5", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 125, + 272, + 140 + ], + "score": 1.0, + "content": "from 247h to", + "type": "text" + }, + { + "bbox": [ + 273, + 127, + 297, + 137 + ], + "score": 0.43, + "content": "5 8 . 3 \\mathrm { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 125, + 482, + 140 + ], + "score": 1.0, + "content": "; this NLL is not even achieved by Async-Opt.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 140 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 155, + 211, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 213, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 213, + 169 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "Multicore and distributed optimization algorithms have received much attention in recent years.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "Asynchronous algorithms include Recht et al. (2011); Duchi et al. (2013); Zhang et al. (2015a);", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "Reddi et al. (2015); Leblond et al. (2016). Implementations of asynchronous optimization include", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "Xing et al. (2015); Li et al. (2014); Chilimbi et al. (2014). Attempts have also been made in Zinke-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 477, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 477, + 237 + ], + "score": 1.0, + "content": "vich et al. (2010) and Zhang & Jordan (2015) to algorithmically improve synchronous SGD.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 180, + 505, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 504, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 255 + ], + "score": 1.0, + "content": "An alternative solution, “softsync”, was presented in Zhang et al. (2015b), which proposed batching", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "gradients from multiple machines before performing an asynchronous SGD update, thereby reducing", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "score": 1.0, + "content": "the effective staleness of gradients. Similar to our proposal, softsync avoids stragglers by not forcing", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "updates to wait for the slowest worker. However, softsync allows the use of stale gradients but we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "score": 1.0, + "content": "do not. The two solutions provide different explorations of the trade-off between high accuracy (by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 296, + 376, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 376, + 309 + ], + "score": 1.0, + "content": "minimizing staleness) and fast throughput (by avoiding stragglers).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 239, + 506, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 504, + 368 + ], + "lines": [ + { + "bbox": [ + 107, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 107, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "Watcharapichat et al. (2016) introduces a distributed deep learning system without parameter", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "servers, by having workers interleave gradient computation and communication in a round-robin", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "pattern. Like Async-Opt, this approach suffers from staleness. We also note that in principle, work-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "ers in Sync-Opt can double as parameter servers and execute the update operations and avoid the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 374, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 374, + 369 + ], + "score": 1.0, + "content": "need to partition hardware resources between workers and servers.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 313, + 506, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "Das et al. (2016) analyzes distributed stochastic optimization and optimizes the system by solving", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "detailed system balance equations. We believe this approach is complimentary to our work, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 459, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 459, + 408 + ], + "score": 1.0, + "content": "could potentially be applied to guide the choice of systems configurations for Sync-Opt.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 372, + 505, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "Keskar et al. (2016) suggests that large batch sizes for synchronous stochastic optimization leads to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 491, + 436 + ], + "score": 1.0, + "content": "poorer generalization. Our effective batch size increases linearly with the number of workers", + "type": "text" + }, + { + "bbox": [ + 491, + 424, + 501, + 434 + ], + "score": 0.76, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 424, + 505, + 436 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "However, we did not observe this effect in our experiments; we believe we are not yet in the large", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 315, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 315, + 457 + ], + "score": 1.0, + "content": "batch size regime examined by Keskar et al. 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In this work, we have shown how both synchronous and asynchronous", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "distributed stochastic optimization suffer from their respective weaknesses of stragglers and stal-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 531, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 547 + ], + "score": 1.0, + "content": "eness. This has motivated our development of synchronous stochastic optimization with backup", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 351, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 351, + 558 + ], + "score": 1.0, + "content": "workers, which we show to be a viable and scalable strategy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 500, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "We are currently experimenting with different kinds of datasets, including word-level language mod-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "els where parts of the model (the embedding layers) are often very sparse, which involves very", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "different communication constraints. 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training costs remain daunting. Conventional posttraining pruning often incurs higher training budgets. In contrast, this paper aims to trim down both the training memory overhead and the inference complexity, without sacrificing the achievable accuracy. We carry out the first-of-its-kind comprehensive exploration, on taking a unified approach of integrating sparsity in ViTs “from end to end”. Specifically, instead of training full ViTs, we dynamically extract and train sparse subnetworks, while sticking to a fixed small parameter budget. Our approach jointly optimizes model parameters and explores connectivity throughout training, ending up with one sparse network as the final output. The approach is seamlessly extended from unstructured to structured sparsity, the latter by considering to guide the prune-and-grow of self-attention heads inside ViTs. We further co-explore data and architecture sparsity for additional efficiency gains by plugging in a novel learnable token selector to adaptively determine the currently most vital patches. Extensive results on ImageNet with diverse ViT backbones validate the effectiveness of our proposals which obtain significantly reduced computational cost and almost unimpaired generalization. Perhaps most surprisingly, we find that the proposed sparse (co-)training can sometimes improve the ViT accuracy rather than compromising it, making sparsity a tantalizing “free lunch”. For example, our sparsified DeiT-Small at $( 5 \%$ , $5 0 \%$ ) sparsity for (data, architecture), improves $\mathbf { 0 . 2 8 \% }$ top-1 accuracy, and meanwhile enjoys ${ \bf 4 9 . 3 2 \% }$ FLOPs and ${ \bf 4 . 4 \bar { 0 } \% }$ running time savings. Our codes are available at https: //github.com/VITA-Group/SViTE. + +# 1 Introduction + +Recent years have seen substantial efforts devoted to scaling deep networks to enormous sizes. Parameter counts are frequently measured in billions rather than millions, with the time and financial outlay necessary to train these models growing in concert. The trend undoubtedly continues with the recent forefront of transformers [1–3] for computer vision tasks. By leveraging self-attention, reducing weight sharing such as convolutions, and feeding massive training data, vision transformers have established many new state-of-the-art (SOTA) records in image classification [1, 2], object detection [4–7], image enhancement [8, 9], and image generation [10–12]. Existing vision transformers and variants, despite the impressive empirical performance, have in general suffered from gigantic parameter-counts, heavy run-time memory usages, and tedious training. That naturally calls for the next step research of slimming their inference and training, without compromising the performance. + +Model compression and efficient learning are no strangers to deep learning researchers, although their exploration in the emerging vision transformer field remains scarce [13]. Among the large variety of compression means [14], sparsity has been one of the central themes since the beginning [15]. + +Conventional approaches first train dense networks, and then prune a large portion of parameters in the trained networks to zero. Those methods significantly reduce the inference complexity. However, the price is to cost even more significant computational resources and memory footprints at training, since they commonly require (multiple rounds of) re-training to restore the accuracy loss [15–17]. That price becomes particularly prohibitive for vision transformers, whose vanilla one-pass training is already much more tedious, slow, and unstable compared to training standard convolutional networks. + +An emerging subfield has explored the prospect of directly training smaller, sparse subnetworks in place of the full networks without sacrificing performance. The key idea is to reuse the sparsity pattern found through pruning and train a sparse network from scratch. The seminal work of lottery ticket hypothesis (LTH) [18] demonstrated that standard dense networks contain sparse matching subnetworks (sometimes called “winning tickets”) capable of training in isolation to full accuracy. In other words, we could have trained smaller networks from the start if only we had known which subnetworks to choose. Unfortunately, LTH requires to empirically find these intriguing subnetworks by an iterative pruning procedure [18–27] , which still cannot get rid of the expensiveness of posttraining pruning. In view of that, follow-up works reveal that sparsity patterns might emerge at the initialization [28, 29], the early stage of training [30, 31], or in dynamic forms throughout training [32–34] by updating model parameters and architecture typologies simultaneously. These efforts shed light on the appealing prospect of “end to end” efficiency from training to inference, by involving sparsity throughout the full learning lifecycle. + +This paper presents the first-of-its-kind comprehensive exploration of integrating sparsity in vision transformers (ViTs) “from end to end”. With (dynamic) sparsity as the unified tool, we can improve the inference efficiency from both model and data perspectives, while also saving training memory costs. Our innovative efforts are unfolded along with the following three thrusts: + +• From Dense to (Dynamic) Sparse: Our primary quest is to find sparse ViTs without sacrificing the achievable accuracy, and meanwhile trimming down the training memory overhead. To meet this challenging demand, we draw inspirations from the latest sparse training works [34, 35] that dynamically extract and train sparse subnetworks instead of training the full models. Sticking to a fixed small parameter budget, our technique jointly optimizes model parameters and explores connectivity throughout the entire training process. We term our first basic approach as Sparse Vision Transformer Exploration (SViTE). + +• From Unstructured to Structured: Most sparse training works [32, 33, 36–39, 38, 34, 40, 41, 35] restricted discussion to unstructured sparsity. To attain structured sparsity which is more hardware-friendly, unlike classical channel pruning available for convolutional networks, we customize a first-order importance approximation [16, 42] to guide the pruneand-grow of self-attention heads inside ViTs. This seamlessly extends SViTE to its second variant of Structured Sparse Vision Transformer Exploration $\mathbf { \left( S ^ { 2 } V i T E \right) }$ . + +• From Model to Data: We further conduct a unified co-exploration towards joint data and architecture sparsity. That is by plugging in a novel learnable token selector to determine the most vital patch embeddings in the current input sample. The resultant framework of Sparse Vision Transformer Co-Exploration $\mathbf { \eta } ( \mathbf { S } \mathbf { V i T E } +$ ) remains to be end-to-end trainable and can gain additional efficiency. + +Extensive experiments are conducted on ImageNet with DeiT-Tiny/Small/Base. Results of substantial computation savings and nearly undamaged accuracies consistently endorse our proposals’ effectiveness. Perhaps most impressively, we find that the sparse (co-)training can even improve the ViT accuracy rather than compromising it, making sparsity a tantalizing “free lunch”. For example, applying $\mathrm { S V i T E { + } }$ on DeiT-Small produces superior compressed ViTs at $5 0 \%$ model sparsity plus $5 \%$ data sparsity, saving $4 9 . 3 2 \%$ FLOPs and $4 . 4 \mathrm { { \bar { 0 } } \% }$ running time, while attaining a surprising improvement of $0 . 2 8 \%$ accuracy; even when the data sparsity increases to $1 0 \%$ (the model sparsity unchanged), there is still no accuracy degradation, meanwhile saving $5 2 . 3 8 \%$ FLOPs and $7 . { \bar { 6 3 \% } }$ running time. + +# 2 Related Work + +Vision Transformer. Transformer [43] stems from natural language processing (NLP) applications. The Vision Transformer (ViT) [1] pioneered to leverage a pure transformer, to encode an image by splitting it into a sequence of patches, projecting them into token embeddings, and feeding them to transformer encoders. With sufficient training data, ViT is able to outperform convolution neural networks on various image classification benchmarks [1, 44]. Many ViT variants have been proposed since then. For example, DeiT [2] and T2T-ViT [45] are proposed to enhance ViT’s training data efficiency, by leveraging teacher-student and better crafted architectures respectively. In addition to image classification, ViT has attracted wide attention in diverse computer vision tasks, including object detection [4–7], segmentation [46, 47], enhancement [8, 9], image generation [10–12], video understanding [48, 49], vision-language [50–57] and 3D point cloud [58]. + +Despite the impressive empirical performance, ViTs are generally heavy to train, and the trained models remain massive. That naturally motivates the study to reduce ViT inference and training costs, by considering model compression means. Model compression has been well studied in both computer vision and NLP applications [59–61, 42, 62, 21]. Two concurrent works [13, 63] made initial attempts towards ViT post-training compression by pruning the intermediate features and tokens respectively, but did not jointly consider weight pruning nor efficient training. Another loosely related field is the study of efficient attention mechanisms [64, 10, 52, 65–75]. They mainly reduce the calculation complexity for self-attention modules via various approximations such as low-rank decomposition. Our proposed techniques represent an orthogonal direction and can be potentially combined with them, which we leave as future work. Another latest concurrent work [76] introduced an interpretable module to dynamically and gracefully drop the redundant patches, gaining not only inference efficiency but also interpretability. Being a unique and orthogonal effort from ours, their method did not consider the training efficiency yet. + +Pruning and Sparse Training. Pruning is well-known to effectively reduce deep network inference costs [77, 15]. It can be roughly categorized into two groups: $( i )$ unstructured pruning by removing insignificant weight elements per certain criterion, such as weight magnitude [78, 15], gradient [16] and hessian [79]; $( i i )$ structured pruning [80–82] by remove model sub-structures, e.g., channels [80, 81] and attention heads [42], which are often more aligned with hardware efficiency. All above require training the full dense model first, usually for several train-prune-retrain rounds. + +The recent surge of sparse training seeks to adaptively identify high-quality sparse subnetworks and train only them. Starting from scratch, those methods learn to optimize the model weights together with sparse connectivity simultaneously. [32, 33] first introduced the Sparse Evolutionary Training (SET) technique [32], reaching superior performance compared to training with fixed sparse connectivity [83, 36]. [37–39] leverages “weight reallocation" to improve performance of obtained sparse subnetworks. Furthermore, gradient information from the backward pass is utilized to guide the update of the dynamic sparse connectivity [38, 34], which produces substantial performance gains. The latest investigations [40, 41, 35] demonstrate that more exhaustive exploration in the connectivity space plays a crucial role in the quality of found sparse subnetworks. Current sparse training methods mostly focus on convolutional networks. Most of them discuss unstructured sparsity, except a handful [84, 30] considering training convolutional networks with structured sparsity. + +# 3 Methodology + +Our SViTE method (and its variants $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ and $\mathrm { S V i T E { + } }$ ) is inspired from state-of-the-art sparse training approaches [34, 35] in CNNs. This section presents the sparse exploration of ViT architectures, then shows the detailed procedure of input token selection for extra efficiency gains. + +# 3.1 Sparse ViT Exploration + +Revisiting sparse training. Sparse training starts from a randomly sparsified model; after optimizing several iterations, it shrinks a portion of parameters based on pre-defined pruning criterion, and activates new connections w.r.t. grow indicators. After upgrading the sparse topology, it trains the new subnetwork until the next update of the connectivity. An illustration of the overall procedure is shown in Figure 1. The key factors of sparse training are $\bullet$ sparsity distribution, $\otimes$ update schedule, $\otimes$ pruning and $\bullet$ grow criterion. + +Notations. For a consistent description, we follow the standard notations in [34, 35]. Let $\mathcal { D }$ be the training dataset. $b _ { t } \sim \mathcal { D }$ is a randomly sampled data batch for iteration $t$ . $f _ { W } ( \cdot )$ represents the model with parameters $W = ( W ^ { ( 1 ) } , \cdots , W ^ { ( L ) } )$ , where $W ^ { ( l ) } \in \mathbb { R } ^ { N _ { l } } , 1 \le l \le L , N _ { l }$ is the number of prunable parameters in the $l _ { \mathrm { t h } }$ layer, and $L$ denotes the number of transformer layers. Note that the first linear projection layer and the classifier of ViT [1, 2] are not sparsified in our framework. As illust W (l)Q rated in Figure 1(bottom-left), are the weights of the self-atte $W _ { Q } ^ { ( l ) } = \{ W _ { Q } ^ { ( l , h ) } \} _ { h = 1 } ^ { H }$ , W (l)K $W _ { K } ^ { ( l ) } = \{ W _ { K } ^ { ( l , h ) } \} _ { h = 1 } ^ { H }$ , $W _ { V } ^ { ( l ) } = \{ W _ { V } ^ { ( l , h ) } \} _ { h = 1 } ^ { H }$ $l _ { \mathrm { t h } }$ $W ^ { ( l , 1 ) }$ $W ^ { ( l , 2 ) }$ $W ^ { ( l , 3 ) }$ perceptron (MLP) module in the collectively represent all the para $l _ { \mathrm { t h } }$ layer, andters in the $W ^ { ( l ) } =$ $( W _ { Q } ^ { ( l ) } , W _ { K } ^ { ( l ) } , W _ { V } ^ { ( l ) } , \bar { W } ^ { ( l , 1 ) } , W ^ { ( l , 2 ) } , W ^ { ( \bar { l } , 3 ) } )$ $l _ { \mathrm { t h } }$ where $H$ denotes the number of attention heads, and $Q ^ { ( l ) }$ , , and $V ^ { ( l ) }$ are the corresponding input and intermediate features, respectively. Each sparse layer only maintains a fraction $s _ { l } \in ( 0 , 1 )$ of its connections, and the overall sparsity of a sparse subnetwork is calculated as the ratio of pruned elements to the total parameter counts, i.e., $\frac { \sum _ { l } s _ { l } \times N _ { l } } { \sum _ { l } N _ { l } }$ + +![](images/03c337fe2dadbbacbee28ecd5811945c8d6e72c346d559e7ef61f5a292efe60e.jpg) +Figure 1: The overall procedure of our proposed sparse ViT exploration framework. Upper Figure: first training ViT for $\Delta \mathrm { T }$ iterations, then performing prune-and-grow strategies to explore critical sparse connectivities, repreating until convergence. Bottom Left Figure: enforcing either structured or unstructured sparsity to transformer layers in ViT. Bottom Right Figure: first scoring each input embedding and applying the learnable top- $k$ selection to identify the most informative tokens. + +Sparse Vision Transformer Exploration (SViTE). SViTE explores the unstructured sparse topology in vision transformers. To be specific, we adopt Erdo¨s-Re´nyi [32] as our $\bullet$ sparsity distribution. The number of parameters in the sparse layer is scaled by $\begin{array} { r } { 1 - \frac { n _ { l - 1 } + n _ { l } } { n _ { l - 1 } \times n _ { l } } } \end{array}$ , where $n _ { l }$ is the number of neurons at layer l. This distribution allocates higher sparsities to the layers with more parameters by scaling the portion of remaining weights with the sum of the number of output and input neurons/channels. For the $\otimes$ update schedule, it contains: (i) the update interval $\Delta \mathrm { T }$ , which is the number of training iterations between two sparse topology updates; $( i i )$ the end iteration $\mathrm { T _ { e n d } }$ , indicating when to stop updating the sparsity connectivity, and we set $\mathrm { T _ { e n d } }$ to $8 0 \%$ of total training iterations in our experiments; $( i i i )$ the initial fraction $\alpha$ of connections that can be pruned or $5 0 \%$ $( i v )$ a decay schedule of the fraction of changeable connections, where a cosine annealing is used, following [34, 35]. During $\begin{array} { r } { \dot { f } _ { \mathrm { d e c a y } } ( t , \alpha , \mathrm { T } _ { \mathrm { e n d } } ) = \frac { \alpha } { 2 } ( 1 + \cos ( \frac { t \times \pi } { \mathrm { T } _ { \mathrm { e n d } } } ) ) } \end{array}$ +each connectivity update, we choose the weight magnitude as the pruning indicator, and gradient magnitude as $\bullet$ the grow indicator. Specifically, we eliminate the parameters with the layer-wise smallest weight values by applying a binary mask $m _ { \mathrm { p r u n e } }$ , then grow new connections with the highest magnitude gradients by generating a new binary mask $m _ { \mathrm { g r o w } }$ . Both masks are employed to $W ^ { ( l ) }$ via the element-wise dot product, and note that the number of non-zero elements in $m _ { \mathrm { p r u n e } }$ and $m _ { \mathrm { g r o w } }$ are equal and fixed across the overall procedure. Newly added connections are not activated in the last sparse topology, and are initialized to zero since it produces better performance as demonstrated in [34, 35]. + +Infrequent gradient calculation [34] is adopted in our case, which computes the gradients in an online manner and only stores the top gradient values. As illustrated in [34], such fashion amortizes the extra effort of gradient calculation, and makes it still proportional to $1 - s$ as long as $\Delta \mathrm { T } \geq { \frac { 1 } { 1 - s } }$ where $s$ is the overall sparsity. + +Structured Sparse Vision Transformer Exploration $\mathbf { ( S ^ { 2 } V i T E ) }$ . Although models with unstructured sparsity achieve superior performance, structured sparsity [80–82] is much more hardware friendly and brings practical efficiency on realistic platforms, which motivates us to propose Structured Sparse ViT Exploration $( \mathrm { S ^ { 2 } V i T E } )$ . We inherit the design of $\bullet$ sparsity distribution and $\otimes$ update schedule from the unstructured SViTE, and a round-up function is used to eliminate decimals in the parameter counting. The key differences lie in the new $\otimes$ pruning and $\bullet$ grow strategies. + +Pruning criterion: Let $A _ { ( l , h ) }$ denote features computed from the self-attention head {W (l,h)Q , $\{ W _ { Q } ^ { ( \mathit { l } , h ) } , W _ { K } ^ { ( \mathit { l } , h ) } , W _ { V } ^ { ( \mathit { l } , h ) } \}$ and input embeddings $X ^ { ( l ) }$ , as shown in Figure 1. We perform the Taylor expansion to the loss function [16, 42], and derive a proxy score for head importance blow: + +# Algorithm 1 Sparse ViT Co-Exploration $\mathrm { ( S V i T E + ) }$ . + +$$ +\mathcal { T } _ { p } ^ { ( l , h ) } = \bigg | A _ { ( l , h ) } ^ { \mathrm { T } } \cdot \frac { \partial \mathcal { L } ( X ^ { ( l ) } ) } { \partial A _ { ( l , h ) } } \bigg | , +$$ + +where $\mathcal { L } ( \cdot )$ is the cross-entropy loss as used in ViT. During each topology update, we remove attention heads with the smallest $\mathcal { T } _ { p } ^ { ( l , h ) }$ . For MLPs, we score neurons with $\ell _ { 1 }$ -norm of their associated weight vectors [85], and drop insignificant neurons. For example, the $j _ { \mathrm { t h } }$ neuron of $W ^ { ( l , 1 ) }$ in Figure 1 has an importance score kW (l,1)j,· k $W _ { j , \cdot } ^ { ( l , 1 ) }$ is the $j _ { \mathrm { t h } }$ row $W ^ { ( l , 1 ) }$ , where + +Initialize: ViT model $f _ { W }$ , Dataset $\mathcal { D }$ , Sparsity distribution $\begin{array} { r c l } { \mathbb { S } } & { = } & { \{ s _ { 1 } , \dots , s _ { L } \} } \end{array}$ , Update schedule $\{ \Delta \mathrm { T } , \mathrm { T _ { e n d } } , \alpha , f _ { \mathrm { d e c a y } } \}$ , Learning rate $\eta$ +1: Initialize $f _ { W }$ with random sparsity $\mathbb { S }$ . Highly reduced parameter count. +2: for each training iteration $t$ do +3: Sampling a batch $b _ { t } \sim \mathcal { D }$ +4: Scoring the input token embeddings and selecting the top- $k$ informative tokens . Token selection +5: if $\mathbf { \chi } _ { t }$ mod $\Delta \mathrm { T } = = 0 \ \mathrm { \Omega }$ ) and $t < \mathrm { T _ { e n d } }$ then +6: for each layer $l$ do +7: $\rho = f _ { \mathrm { d e c a y } } ( t , \alpha , \mathrm { T } _ { \mathrm { e n d } } ) \cdot ( 1 - s _ { l } ) \cdot N _ { l }$ +8: Performing prune-and-grow with portion $\rho$ w.r.t. certain criterion, generating masks $m _ { \mathrm { p r u n e } }$ and $m _ { \mathrm { g r o w } }$ to update $f _ { W }$ ’s sparsity patterns $\triangleright$ Connectivity exploration +9: end for +10: else +11: $\begin{array} { r l r l } { W = W - \eta \cdot \nabla _ { W } \mathcal { L } _ { t } } & { { } } & { \triangleright U p d a t i n g \ W e i g h t s } \end{array}$ +12: end if +13: end for +14: return a sparse ViT with a trained token selector + +Grow criterion: Similar to [34, 35], we active the new units with the highest + +magnitude gradients, such as $\| \frac { \partial \bar { \mathcal { L } } ( X ^ { ( l ) } ) } { \partial A _ { ( l , h ) } } \| _ { \ell _ { 1 } }$ k\`1 and $\| \frac { \partial \mathcal { L } ( X ^ { ( l ) } ) } { \partial W _ { j , \cdot } ^ { ( l , 1 ) } } \| _ { \ell _ { 1 } }$ for the $h _ { \mathrm { t h } }$ attention head and the $j _ { \mathrm { t h } }$ neuron of the MLP $( W ^ { ( l , 1 ) } )$ , respectively. The gradients are calculated in the same manner as the one in unstructured SViTE, and newly added units are also initialized to zero. + +# 3.2 Data and Architecture Sparsity Co-Exploration for Higher Efficiency + +Besides exploring sparse transformer architectures, we further slim the dimension of input token embeddings for extra efficiency bonus by leveraging a learnable token selector, as presented in Figure 1. Meanwhile, the introduced data sparsity also serves as an implicit regularization for ViT training, which potentially leads to improved generalization ability, as evidenced in Table 6. Note that, due to skip connections, the number of input tokens actually determines + +Algorithm 2 The top- $k$ selector in a PyTorch-like style. + +def topk_selector(logits, k, tau, ${ \dot { \mathsf { d i m } } } = - 1 \cdot$ ): + +# Maintain tokens with the top-\$k\$ highest scores gumbels $=$ -torch.empty_like(logits).exponential_().log() gumbels $=$ (logits $^ +$ gumbels) / tau # tau is the temperature y_soft $=$ gumbels.softmax(dim) # Straight through index $=$ y_soft.topk(k, dim $\mathbf { \Psi } _ { 1 } =$ dim)[1] y_hard $=$ scatter(logits, index, k) ret $=$ y_hard - y_soft.detach() $^ +$ y_soft return ret + +the dimension of intermediate features, which substantially contributes to the overall computation cost. In other words, the slimmed input token embeddings directly result in compressed intermediate features, and bring substantial efficiency gains. + +Table 1: Details of training configurations in our experiments, mainly following the settings in [2]. + +
BackboneUpdate Schedule{△T,Tend,α,fdecay}Batch SizeEpochsInherited Settings from DeiT[2]
DeiT-Tiny{20000,1200000,0.5,cosine}512600AdamW, 0.0005 × batchsize,cosine decay
DeiT-Small{15000,1200000,0.5,cosine}512600warmup 5 epochs,0.05 weight decay
DeiT-Base{7000,600000,0.5,cosine}10246000.1 label smoothing,augmentations, etc.
+ +For the input tokens $X ^ { ( 1 ) } \in \mathbb { R } ^ { n \times d }$ , where $n$ denotes the number of tokens to be shrunk, and $d$ is the dimension of each token embedding that keeps unchanged. As shown in Figure 1, all token embeddings are passed through a learnable scorer function which is parameterized by an MLP in our experiments. Then, a selection of the top- $k$ importance scores $( 1 \leq k \leq d )$ is applied on top of it, aiming to preserve the significant tokens and remove the useless ones. To optimize parameters of the scorer function, we introduce the popular Gumbel-Softmax [86, 87] and straight-through tricks [88] to enable gradient back-propagation through the top- $k$ selection, which provides an efficient solution to draw samples from a discrete probability distribution. A detailed implementation is in Algorithm 2. + +The full pipeline of data and architecture co-exploration is summarized in Algorithm 1. We term this approach $\mathrm { S V i T E { + } }$ . We first feed the randomly sampled data batch to the token selector and pick the top- $k$ informative token embeddings. Then, we alternatively train the sparse ViT for $\Delta \mathrm { T }$ iterations and perform prune-and-grow to explore the sparse connectivity in ViTs dynamically. In the end, a sparse ViT model with a trained token selector is returned and ready for evaluation. + +# 4 Experiments + +Baseline pruning methods. We extend several effective pruning methods from CNN compression as our strong baselines. Unstructured pruning: $( i )$ One-shot weight Magnitude Pruning (OMP) [15], which removes insignificant parameters with the globally smallest weight values; (ii) Gradually Magnitude Pruning (GMP) [17], which seamlessly incorporates gradual pruning techniques within the training process by eliminating a few small magnitude weights per iteration; and (iii) Taylor Pruning (TP) [16], which utilizes the first-order approximation of the training loss to estimate units’ importance for model sparsification. Structured pruning: Salience-based Structured Pruning (SSP). We draw inspiration from [42, 85], and remove sub-modules in ViT (e.g., self-attention heads) by leveraging their weight, activation, and gradient information. Moreover, due to the repetitive architecture of ViT, we can easily reduce the number of transformer layers to create a smaller dense ViT (Small-Dense) baseline that has similar parameter counts to the pruned ViT model. + +Implementation details. Our experiments are conducted on ImageNet with DeiTTiny/Small/Base backbones. The detailed training configurations are listed in Table 1, which mainly follows the default setups in [2]. All involved customized hyperparameters are tuned via grid search (later shown in Figure 3). For a better exploration of sparsity connectivities, we increase training epochs to 600 for all experiments. GMP [17] has an additional hyperparameter, i.e., the pruning schedule, which starts from $\frac { 1 } { 6 }$ and ends at ${ \frac { \overline { { \frac { 1 } { 2 } } } } { 2 } }$ of the training epochs with 20 times pruning in total. More details are referred to Appendix A1. + +Training time measuring protocol. We strictly measure the running time saving of + +![](images/a5ad90174ff1547fe598c114e7c4b77c1f965a37ed3695b0c6fcde1a09ee9c95.jpg) +The Overall Performance of SViTE, S 2ViTE, and SViTE+ +Figure 2: Top-1 accuracy $( \% )$ over FLOPs $( \times 1 0 ^ { 1 0 } )$ on ImageNet of our methods, i.e., SViTE, $\mathrm { S ^ { 2 } V i T E }$ , and ${ \mathrm { S V i T E } } +$ compared to DeiT baselines, trained on Imagenet-1K only. + +(sparse) vision transformers on the ImageNet-1K task using CUDA benchmark mode. To be specific, we separately calculate the time elapsed during each iteration, to eliminate the impact of the hardware environment as much as possible. Note that the time for the data I/O is excluded. + +Highlight of our findings. The overall performance of SViTE, $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ , and $\mathrm { S V i T E { + } }$ on DeiT backbones are summarized in Figure 2. We highlight some takeaways below. + +Takeaways: ❶ SViTE produces sparse DeiTs with enhanced generalization and substantial reduced FLOPs, compared to its dense counterpart $( { \star } )$ . ${ \mathrm { S V i T E } } +$ further improves the performance of SViTE by selecting the most vital patches. ❷ $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ achieves matched accuracy on DeiT-Small, and significantly enhances performance on DeiT-Base. Meanwhile, its structural sparsity brings considerable running time savings. $\otimes$ Appropriate data and architecture sparsities can effectively regularize ViT training, leading to a new SOTA win-win between ViT accuracy and efficiency. + +# 4.1 SViTE with Unstructured Sparsity + +We perform SViTE to mine vital unstructured sparsity in DeiTs [2]. Solid lines in Figure 2 record the top-1 test-set accuracy over FLOPs on ImageNet-1K of SViTE-Small and SViTE-Base with a range of sparsity from $3 0 \%$ to $7 0 \%$ . In general, we observe that SViTE generates superior sparse ViTs with both accuracy and efficiency gains. Table 2, 3, and 5 present the comparison between SViTE and various pruning baselines. From these extensive results, we draw several consistent observations. First, compared to the dense baselines, SViTE-Tiny, -Small, and -Base obtain $2 5 . 5 6 \% \sim 3 4 . 1 6 \%$ , $4 6 . 2 6 \% \sim 5 5 . 4 4 \%$ , and $4 7 . 9 5 \% \sim 5 7 . 5 0 \%$ FLOPs reduction, respectively, at $3 0 \% \sim 6 0 \%$ sparsity levels with only a negligible accuracy drop within $0 . 5 \%$ . It verifies the effectiveness of our proposal, and indicates severe parameter redundancy in ViT. Second, our SViTE models from dynamic explorations consistently surpass other competitive baseline methods, including OMP, GMP, TP, and Small-Dense by a substantial performance margin. Among all the baseline approaches, GMP that advocates a gradual pruning schedule achieves the best accuracy with all three DeiT backbones. Third, in Figure 2, both SViTE-Small (blue solid line) and SViTE-Base (green solid line) show an improved trade-off between accuracy and efficiency, compared to their dense DeiT counterparts. Interestingly, we also observe that with similar parameter counts, a large sparse ViT consistently outperforms the corresponding smaller dense ViT. A possible explanation is those appropriate sparse typologies regularize network training and lead to enhanced generalization, which coincides with recent findings of critical subnetworks (i.e., winning tickets) in dense CNNs [89, 22] and NLP transformer [21, 90] models. + +Table 2: Results of SViTE-Tiny on ImageNet-1K. Accuracies $( \% )$ within/out of parenthesis are the reproduced/reported [2] performance. + +
ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Tiny0% (5.72M)0%72.20 (71.80)
SViTE-Tiny30% (4.02M)25.56%71.78
OMP30% (4.02M)25.56%68.35
GMP30% (4.02M)25.56%69.56
TP30% (4.02M)25.56%68.38
SViTE-Tiny40% (3.46M)34.16%71.75
OMP40% (3.46M)34.16%66.52
GMP40% (3.46M)34.15%68.36
TP40% (3.46M)34.17%65.45
Small-Dense0% (3.94M)32.54%67.33
+ +Table 3: Results of SViTE-Small on ImageNet-1K. Accuracies $( \% )$ within/out of parenthesis are the reproduced/reported [2] performance. + +
ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Small0% (22.1M)0%79.90 (79.78)
SViTE-Small50% (11.1M)46.26%79.72
OMP50% (11.1M)46.25%76.32
GMP50% (11.1M)46.26%76.88
TP50% (11.1M)46.26%76.30
SViTE-Small60% (8.9M)55.44%79.41
OMP60% (8.9M)55.44%75.32
GMP60% (8.9M)55.44%76.79
TP60% (8.9M)55.44%74.50
Small-Dense0% (11.4M)49.32%73.93
+ +Table 4: Results of $\mathrm { S ^ { 2 } V i T E }$ with structured sparsity on ImageNet-1K with DeiT-Tiny/Small/Base. Accuracies $( \% )$ within/out of parenthesis are the reproduced/reported [2] performance. + +
ModelsSparsity (%)ParametersFLOPs SavingRunning Time Reduced|Top-1 Accuracy (%)
DeiT-Tiny (Dense)0%5.72M0%0%72.20 (71.80)
SViTE-Tiny (Unstructured)30%4.02M25.56%0%71.78
SSP-Tiny (Structured)30%4.21M23.69%10.57%68.59
S2ViTE-Tiny (Structured)30%4.21M23.69%10.57%70.12
DeiT-Small (Dense)0%22.1M0%0%79.90 (79.78)
SViTE-Small (Unstructured)40%13.3M36.73%0%80.26
SSP-Small (Structured)40%14.6M31.63%22.65%77.74
S²ViTE-Small (Structured)40%14.6M31.63%22.65%79.22
DeiT-Base (Dense)0%86.6M0%0%81.80 (80.98)
SViTE-Base (Unstructured)40%52.0M38.30%0%81.56
SSP-Base (Structured)40%56.8M33.13%24.70%80.08
S2ViTE-Base (Structured)40%56.8M33.13%24.70%82.22
+ +# 4.2 $\mathbf { S } ^ { 2 }$ ViTE with Structured Sparsity + +For more practical benefits, we investigate sparse DeiTs with structured sparsity. Results are summarized in Table 4. Besides the obtained $2 3 . { \bar { 7 } } 9 \% \sim 3 3 . 6 3 \%$ FLOPs savings, ${ \mathsf { S } } ^ { \tilde { 2 } }$ ViTE-Tiny, $S ^ { 2 }$ ViTE + +Small, and $S ^ { 2 }$ ViTE-Base enjoy an extra $1 0 . 5 7 \%$ , $2 2 . 6 5 \%$ , and $2 4 . 7 0 \%$ running time reduction, respectively, from $3 0 \% \sim \hat { 4 } 0 \%$ structured sparsity with competitive top-1 accuracies. Furthermore, $\mathrm { S ^ { 2 } V i T E }$ consistently outperforms the baseline structured pruning method (SSP), which again demonstrates the superior sparse connectivity learned from dynamic sparse training. + +The most impressive results come from $S ^ { 2 }$ ViTE-Base at $4 0 \%$ structured sparsity. It even surpasses the dense DeiT base model by $0 . 4 2 \% \sim 1 . 2 4 \%$ accuracy with $3 4 . 4 1 \%$ parameter counts, $3 3 . 1 3 \%$ FLOPs, and $2 4 . 7 0 \%$ running time reductions. We conclude that $( i )$ an adequate sparsity from $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ boosts ViT’s generalization ability, which can be regarded as an implicit regularization; $( i i )$ larger ViTs (e.g., DeiT-Base) tend to have more superfluous self-attention heads, and are more amenable to structural sparsification from $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ , based on Figure 2 where dash lines denote the overall performance of $\hat { \mathbf { S } ^ { 2 } }$ ViTE-Small and $S ^ { 2 }$ ViTE-Base with a range of sparsity from $3 0 \%$ to $7 0 \%$ . + +Table 5: Results of SViTE-Base on ImageNet-1K. Accuracies $( \% )$ within/out of parenthesis are the reproduced/reported [2] performance. + +
ModelsSparsity (#Para.)FLOPs Saving|Accuracy (%)
DeiT-Base0% (86.6M)0%81.80 (80.98)
SViTE-Base50% (43.4M)47.95%81.51
OMP50% (43.4M)47.94%80.26
GMP50% (43.4M)47.95%80.79
TP50% (43.4M)47.94%80.55
SViTE-Base60% (34.8M)57.50%81.28
OMP60% (34.8M)57.50%80.25
GMP60% (34.8M)57.50%80.44
TP60% (34.8M)57.49%80.37
Small-Dense0% (44.0M)49.46%78.59
+ +Table 6: Results of SViTE $^ +$ -Small on ImageNet-1K. Accuracies $( \% )$ within/out of parenthesis are the reproduced/reported [2] performance. + +
#Tokens (%)|Time ReducedFLOPs Saving|Accuracy (%)
SViTE+-Small 50% Unstructured Sparsity
100%0%46.26%79.72
95%4.40%49.32%80.18
90%7.63%52.38%79.91
70%19.77%63.95%77.90
S²ViTE+-Small 40% Structured Sparsity
100%22.65%31.63%79.22
95%27.17%37.76%78.44
90%29.21%41.50%78.16
70%39.10%54.96%74.77
+ +# 4.3 $\mathbf { S V i T E { + } }$ with Data and Architecture Sparsity Co-Exploration + +In this section, we study data and architecture sparsity co-exploration for ViTs, i.e., $\mathrm { S V i T E { + } }$ . Blessed by the reduced input token embeddings, even ViTs with unstructured sparsity can have running time savings. The benefits are mainly from the shrunk input and intermediate feature dimensions. Without loss of generality, we consider SViTE $+ \cdot$ -Small with $5 0 \%$ unstructured sparsity and ${ \mathrm { S ^ { 2 } V i T E } } +$ -Small with $4 0 \%$ structured sparsity as examples. As shown in Table 6 and Figure 2, SViTE+-Small at $5 0 \%$ unstructured sparsity is capable of abandoning $5 \% \sim 1 0 \%$ tokens while achieving $4 . 4 0 \% \sim 7 . 6 3 \%$ running time and $4 9 . 3 2 \% \sim 5 2 . 3 8 \%$ FLOPs savings, with even improved top-1 testing accuracy. It again demonstrates that data sparsity as an implicit regularizer plays a beneficial role in ViT training. However, slimming input and intermediate embedding is less effective when incorporated with $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ , suggesting that aggressively removing structural sub-modules hurts ViT’s generalization. + +# 4.4 Ablation and Generalization Study of SViTEs + +Update interval in SViTE. The length of the update interval $\Delta \mathrm { T }$ controls one of the essential tradeoffs in our proposed dynamic sparse exploration, since $\Delta \mathrm { T }$ multiplying the number of updates is the pre-defined $\mathrm { T _ { e n d } }$ . On the one hand, a larger updated interval (i.e., smaller update frequency) produces a more well-trained model for improved estimation of units’ importance. On the other hand, a larger update frequency (i.e., smaller $\Delta \mathrm { T }$ ) allows more sufficient exploration of sparse connectivities, which potentially generates higher-quality sparse subnetworks, as demonstrated in [35]. We evaluate this factor in our SViTE context, and collect the results in Figure 3 (Left). We observe that $\Delta \mathrm { T } = 2 0 0 0 0$ works the best for SViTE-Tiny, and both larger and smaller $\Delta \mathrm { T }$ degrade the performance. + +![](images/a4655f90e93b2c9a04681fc021da3282c731aba6f4cba193b459acd0b25f0153.jpg) +Figure 3: Accuracy of SViTE-Tiny with $5 0 \%$ unstructured sparsity. Left: ablation studies of the update interval $( \bar { \Delta \mathrm { T } } )$ ; Right: ablations studies of the adopted batch size $( b )$ . + +![](images/20204951e060810529cbad9a007da6259808df546d2c9563a4a457ffbdf0a012.jpg) +Dense DeiT-Base + +![](images/33e5e064c5dba74ca99e27db03615dbf08bda7f2b81b68715098af3c25c6d603.jpg) +-Base with $4 0 \%$ Structured Sparsity + +![](images/fcf8ae89303d80f050af5c5ed88ce6e7bc4eee6a2db8431edf4a7768d4fc9049.jpg) +SViTE-Base with $4 0 \%$ Unstructured Sparsity + +![](images/a0e0619f2f06fe582065a11779901de10112cfc6477c8e3ec5b96052b7f80f29.jpg) +Figure 4: Attention probabilities for DeiT-Base, $S ^ { 2 }$ ViTE-Base, and SViTE-Base models with 12 layers (rows) and 12 heads (columns) using visualization tools provided in [94]. Attention maps are averaged over 100 test samples from ImageNet-1K to present head behavior and remove the dependence on the input content. The black square is the query pixel. indicates pruned attention heads. Zoom-in for better visibility. +Figure 5: Learned patch selection patterns of SViTE $+$ -Small at $1 0 \%$ data and $5 0 \%$ architecture sparsity levels. $\widehat { \mathbb { I } }$ indicates removed inessential patches. + +Batch size in SViTE. Besides the update interval $\Delta \mathrm { T }$ , batch size (b) also affects the aforementioned trade-off, especially for the data-hungry ViT training. We investigate different batch sizes in Figure 3 (Right), and find that $b = 5 1 2$ outperforms other common options for SViTE-Tiny. + +Generalization study of SViTE and its variants. It is worth mentioning that our proposed frameworks (SViTE, $\mathrm { S ^ { 2 } V i T E }$ , $\mathrm { S V i T E { + } }$ ) are independent of the backbone architectures, and can be easily plugged in other vision transformer models [91, 45, 92, 93]. We implemented both SViTE and ${ \bar { \mathbf { S } } } ^ { 2 }$ ViTE on TNT-S [91]. SViTE-TNT-S gains 0.13 accuracy improvements (Ours: 81.63 v.s. TNT-S: 81.50) and $3 7 . 5 4 \%$ FLOPs savings at $4 0 \%$ unstructured sparsity; $S ^ { 2 }$ ViTE-TNT-S obtains $3 2 . 9 6 \%$ FLOPs and $2 3 . 7 1 \%$ running time reductions at $4 0 \%$ structured sparsity with almost unimpaired accuracy (Ours: 81.34 v.s. TNT-S:81.50). + +# 4.5 Visualization + +Sparse connectivity patterns. We provide unit-wise and element-wise heatmap visualizations for SViTE-Base with $4 0 \%$ structured sparsity in Figure A7 (in Appendix). Similarly, element-wise heatmap visualizations of SViTE-Base with $5 0 \%$ unstructured sparsity are displayed in Figure A6. We find that even unstructured sparsity exploration can develop obvious structural patterns (i.e., “vertical lines” in mask heatmaps), which implies a stronger potential for hardware speedup [95]. + +Self-attention heatmaps. As shown in Figure 4, we utilize tools in [94] to visualize attention maps of (sparse) ViTs. Multiple attention heads show similar behaviors, which implies the structural redundancy. Fortunately, $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ eliminates unnecessary heads to some extent. With regard to SViTE-Base’s visual results, it seems to activate fewer attention heads for predictions (darker colors mean larger values), compared to the ones of dense DeiT-Base. We also observe that in the bottom layers, the attention probabilities are more centered at several heads; while in the top layers, the attention probabilities are more uniformly distributed. This kind of tendency is well preserved by our sparse ViT (SViTE) from Dense ViTs. + +Learned patch selection patterns. Figure 5 presents the learned behaviors of our token selector in SViTE+. We observe that the useless removed patches are typically distributed around the main object or in the background. Meanwhile, the patches within the objects of interest are largely persevered, which evidences the effectiveness of our learned patch token selector. + +# 5 Conclusion and Discussion of Broader Impact + +In this work, we introduce sparse ViT exploration algorithms, SViTE, and its variants $\mathbf { S } ^ { \mathrm { 2 } } \mathbf { V i T E }$ and ${ \mathrm { S V i T E } } +$ , to explore high-quality sparse patterns in both ViT’s architecture and input token embeddings, alleviating training memory bottleneck and pursuing inference ultra-efficiency (e.g., running time and FLOPs). Comprehensive experiments on ImageNet validate the effectiveness of our proposal. Our informative visualizations further demonstrate that $\mathrm { S V i T E { + } }$ is capable of mining crucial connections and input tokens by eliminating redundant units and dropping useless token embeddings. Future work includes examining the performance of our sparse ViTs on incoming hardware accelerators [96–100], which will provide better supports for sparsity. + +This work is scientific in nature, and we do not believe it has immediate negative societal impacts. Our findings of sparse vision transformers are highly likely to reduce both memory and energy costs substantially, leading to economic deployment in real-world applications (e.g., on smartphones). + +# Acknowledgment + +Z.W. is in part supported by an NSF RTML project (#2053279). + +# References + +[1] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. +[2] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. 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IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 9(2):292–308, 2019. \ No newline at end of file diff --git a/parse/train/LKoMTwTuQnC/LKoMTwTuQnC_content_list.json b/parse/train/LKoMTwTuQnC/LKoMTwTuQnC_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..685b0e7b69882f67db76ec7ebd484365b6ecb6a5 --- /dev/null +++ b/parse/train/LKoMTwTuQnC/LKoMTwTuQnC_content_list.json @@ -0,0 +1,1115 @@ +[ + { + "type": "text", + "text": "Chasing Sparsity in Vision Transformers: An End-to-End Exploration ", + "text_level": 1, + "bbox": [ + 251, + 122, + 753, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tianlong Chen1, Yu Cheng2, Zhe $\\mathbf { G a n } ^ { 2 }$ , Lu Yuan2, Lei Zhang3, Zhangyang Wang1 1University of Texas at Austin, 2Microsoft Corporation, 3International Digital Economy Academy {tianlong.chen,atlaswang}@utexas.edu,{yu.cheng,zhe.gan,luyuan}@microsoft.com leizhangcn@ieee.org ", + "bbox": [ + 186, + 223, + 825, + 282 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 335 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Vision transformers (ViTs) have recently received explosive popularity, but their enormous model sizes and training costs remain daunting. Conventional posttraining pruning often incurs higher training budgets. In contrast, this paper aims to trim down both the training memory overhead and the inference complexity, without sacrificing the achievable accuracy. We carry out the first-of-its-kind comprehensive exploration, on taking a unified approach of integrating sparsity in ViTs “from end to end”. Specifically, instead of training full ViTs, we dynamically extract and train sparse subnetworks, while sticking to a fixed small parameter budget. Our approach jointly optimizes model parameters and explores connectivity throughout training, ending up with one sparse network as the final output. The approach is seamlessly extended from unstructured to structured sparsity, the latter by considering to guide the prune-and-grow of self-attention heads inside ViTs. We further co-explore data and architecture sparsity for additional efficiency gains by plugging in a novel learnable token selector to adaptively determine the currently most vital patches. Extensive results on ImageNet with diverse ViT backbones validate the effectiveness of our proposals which obtain significantly reduced computational cost and almost unimpaired generalization. Perhaps most surprisingly, we find that the proposed sparse (co-)training can sometimes improve the ViT accuracy rather than compromising it, making sparsity a tantalizing “free lunch”. For example, our sparsified DeiT-Small at $( 5 \\%$ , $5 0 \\%$ ) sparsity for (data, architecture), improves $\\mathbf { 0 . 2 8 \\% }$ top-1 accuracy, and meanwhile enjoys ${ \\bf 4 9 . 3 2 \\% }$ FLOPs and ${ \\bf 4 . 4 \\bar { 0 } \\% }$ running time savings. Our codes are available at https: //github.com/VITA-Group/SViTE. ", + "bbox": [ + 232, + 349, + 764, + 667 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 688, + 312, + 705 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent years have seen substantial efforts devoted to scaling deep networks to enormous sizes. Parameter counts are frequently measured in billions rather than millions, with the time and financial outlay necessary to train these models growing in concert. The trend undoubtedly continues with the recent forefront of transformers [1–3] for computer vision tasks. By leveraging self-attention, reducing weight sharing such as convolutions, and feeding massive training data, vision transformers have established many new state-of-the-art (SOTA) records in image classification [1, 2], object detection [4–7], image enhancement [8, 9], and image generation [10–12]. Existing vision transformers and variants, despite the impressive empirical performance, have in general suffered from gigantic parameter-counts, heavy run-time memory usages, and tedious training. That naturally calls for the next step research of slimming their inference and training, without compromising the performance. ", + "bbox": [ + 174, + 713, + 826, + 852 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Model compression and efficient learning are no strangers to deep learning researchers, although their exploration in the emerging vision transformer field remains scarce [13]. Among the large variety of compression means [14], sparsity has been one of the central themes since the beginning [15]. ", + "bbox": [ + 176, + 858, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Conventional approaches first train dense networks, and then prune a large portion of parameters in the trained networks to zero. Those methods significantly reduce the inference complexity. However, the price is to cost even more significant computational resources and memory footprints at training, since they commonly require (multiple rounds of) re-training to restore the accuracy loss [15–17]. That price becomes particularly prohibitive for vision transformers, whose vanilla one-pass training is already much more tedious, slow, and unstable compared to training standard convolutional networks. ", + "bbox": [ + 174, + 92, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "An emerging subfield has explored the prospect of directly training smaller, sparse subnetworks in place of the full networks without sacrificing performance. The key idea is to reuse the sparsity pattern found through pruning and train a sparse network from scratch. The seminal work of lottery ticket hypothesis (LTH) [18] demonstrated that standard dense networks contain sparse matching subnetworks (sometimes called “winning tickets”) capable of training in isolation to full accuracy. In other words, we could have trained smaller networks from the start if only we had known which subnetworks to choose. Unfortunately, LTH requires to empirically find these intriguing subnetworks by an iterative pruning procedure [18–27] , which still cannot get rid of the expensiveness of posttraining pruning. In view of that, follow-up works reveal that sparsity patterns might emerge at the initialization [28, 29], the early stage of training [30, 31], or in dynamic forms throughout training [32–34] by updating model parameters and architecture typologies simultaneously. These efforts shed light on the appealing prospect of “end to end” efficiency from training to inference, by involving sparsity throughout the full learning lifecycle. ", + "bbox": [ + 174, + 181, + 825, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This paper presents the first-of-its-kind comprehensive exploration of integrating sparsity in vision transformers (ViTs) “from end to end”. With (dynamic) sparsity as the unified tool, we can improve the inference efficiency from both model and data perspectives, while also saving training memory costs. Our innovative efforts are unfolded along with the following three thrusts: ", + "bbox": [ + 176, + 367, + 825, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• From Dense to (Dynamic) Sparse: Our primary quest is to find sparse ViTs without sacrificing the achievable accuracy, and meanwhile trimming down the training memory overhead. To meet this challenging demand, we draw inspirations from the latest sparse training works [34, 35] that dynamically extract and train sparse subnetworks instead of training the full models. Sticking to a fixed small parameter budget, our technique jointly optimizes model parameters and explores connectivity throughout the entire training process. We term our first basic approach as Sparse Vision Transformer Exploration (SViTE). ", + "bbox": [ + 217, + 434, + 825, + 531 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• From Unstructured to Structured: Most sparse training works [32, 33, 36–39, 38, 34, 40, 41, 35] restricted discussion to unstructured sparsity. To attain structured sparsity which is more hardware-friendly, unlike classical channel pruning available for convolutional networks, we customize a first-order importance approximation [16, 42] to guide the pruneand-grow of self-attention heads inside ViTs. This seamlessly extends SViTE to its second variant of Structured Sparse Vision Transformer Exploration $\\mathbf { \\left( S ^ { 2 } V i T E \\right) }$ . ", + "bbox": [ + 217, + 536, + 826, + 619 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• From Model to Data: We further conduct a unified co-exploration towards joint data and architecture sparsity. That is by plugging in a novel learnable token selector to determine the most vital patch embeddings in the current input sample. The resultant framework of Sparse Vision Transformer Co-Exploration $\\mathbf { \\eta } ( \\mathbf { S } \\mathbf { V i T E } +$ ) remains to be end-to-end trainable and can gain additional efficiency. ", + "bbox": [ + 218, + 625, + 825, + 694 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Extensive experiments are conducted on ImageNet with DeiT-Tiny/Small/Base. Results of substantial computation savings and nearly undamaged accuracies consistently endorse our proposals’ effectiveness. Perhaps most impressively, we find that the sparse (co-)training can even improve the ViT accuracy rather than compromising it, making sparsity a tantalizing “free lunch”. For example, applying $\\mathrm { S V i T E { + } }$ on DeiT-Small produces superior compressed ViTs at $5 0 \\%$ model sparsity plus $5 \\%$ data sparsity, saving $4 9 . 3 2 \\%$ FLOPs and $4 . 4 \\mathrm { { \\bar { 0 } } \\% }$ running time, while attaining a surprising improvement of $0 . 2 8 \\%$ accuracy; even when the data sparsity increases to $1 0 \\%$ (the model sparsity unchanged), there is still no accuracy degradation, meanwhile saving $5 2 . 3 8 \\%$ FLOPs and $7 . { \\bar { 6 3 \\% } }$ running time. ", + "bbox": [ + 174, + 707, + 826, + 818 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 837, + 321, + 854 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Vision Transformer. Transformer [43] stems from natural language processing (NLP) applications. The Vision Transformer (ViT) [1] pioneered to leverage a pure transformer, to encode an image by splitting it into a sequence of patches, projecting them into token embeddings, and feeding them to transformer encoders. With sufficient training data, ViT is able to outperform convolution neural networks on various image classification benchmarks [1, 44]. Many ViT variants have been proposed since then. For example, DeiT [2] and T2T-ViT [45] are proposed to enhance ViT’s training data efficiency, by leveraging teacher-student and better crafted architectures respectively. In addition to image classification, ViT has attracted wide attention in diverse computer vision tasks, including object detection [4–7], segmentation [46, 47], enhancement [8, 9], image generation [10–12], video understanding [48, 49], vision-language [50–57] and 3D point cloud [58]. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 189 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Despite the impressive empirical performance, ViTs are generally heavy to train, and the trained models remain massive. That naturally motivates the study to reduce ViT inference and training costs, by considering model compression means. Model compression has been well studied in both computer vision and NLP applications [59–61, 42, 62, 21]. Two concurrent works [13, 63] made initial attempts towards ViT post-training compression by pruning the intermediate features and tokens respectively, but did not jointly consider weight pruning nor efficient training. Another loosely related field is the study of efficient attention mechanisms [64, 10, 52, 65–75]. They mainly reduce the calculation complexity for self-attention modules via various approximations such as low-rank decomposition. Our proposed techniques represent an orthogonal direction and can be potentially combined with them, which we leave as future work. Another latest concurrent work [76] introduced an interpretable module to dynamically and gracefully drop the redundant patches, gaining not only inference efficiency but also interpretability. Being a unique and orthogonal effort from ours, their method did not consider the training efficiency yet. ", + "bbox": [ + 174, + 194, + 825, + 375 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Pruning and Sparse Training. Pruning is well-known to effectively reduce deep network inference costs [77, 15]. It can be roughly categorized into two groups: $( i )$ unstructured pruning by removing insignificant weight elements per certain criterion, such as weight magnitude [78, 15], gradient [16] and hessian [79]; $( i i )$ structured pruning [80–82] by remove model sub-structures, e.g., channels [80, 81] and attention heads [42], which are often more aligned with hardware efficiency. All above require training the full dense model first, usually for several train-prune-retrain rounds. ", + "bbox": [ + 174, + 390, + 825, + 473 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The recent surge of sparse training seeks to adaptively identify high-quality sparse subnetworks and train only them. Starting from scratch, those methods learn to optimize the model weights together with sparse connectivity simultaneously. [32, 33] first introduced the Sparse Evolutionary Training (SET) technique [32], reaching superior performance compared to training with fixed sparse connectivity [83, 36]. [37–39] leverages “weight reallocation\" to improve performance of obtained sparse subnetworks. Furthermore, gradient information from the backward pass is utilized to guide the update of the dynamic sparse connectivity [38, 34], which produces substantial performance gains. The latest investigations [40, 41, 35] demonstrate that more exhaustive exploration in the connectivity space plays a crucial role in the quality of found sparse subnetworks. Current sparse training methods mostly focus on convolutional networks. Most of them discuss unstructured sparsity, except a handful [84, 30] considering training convolutional networks with structured sparsity. ", + "bbox": [ + 174, + 479, + 825, + 632 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Methodology ", + "text_level": 1, + "bbox": [ + 174, + 651, + 313, + 670 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our SViTE method (and its variants $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ and $\\mathrm { S V i T E { + } }$ ) is inspired from state-of-the-art sparse training approaches [34, 35] in CNNs. This section presents the sparse exploration of ViT architectures, then shows the detailed procedure of input token selection for extra efficiency gains. ", + "bbox": [ + 176, + 683, + 825, + 726 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Sparse ViT Exploration ", + "text_level": 1, + "bbox": [ + 174, + 742, + 379, + 757 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Revisiting sparse training. Sparse training starts from a randomly sparsified model; after optimizing several iterations, it shrinks a portion of parameters based on pre-defined pruning criterion, and activates new connections w.r.t. grow indicators. After upgrading the sparse topology, it trains the new subnetwork until the next update of the connectivity. An illustration of the overall procedure is shown in Figure 1. The key factors of sparse training are $\\bullet$ sparsity distribution, $\\otimes$ update schedule, $\\otimes$ pruning and $\\bullet$ grow criterion. ", + "bbox": [ + 174, + 768, + 825, + 852 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Notations. For a consistent description, we follow the standard notations in [34, 35]. Let $\\mathcal { D }$ be the training dataset. $b _ { t } \\sim \\mathcal { D }$ is a randomly sampled data batch for iteration $t$ . $f _ { W } ( \\cdot )$ represents the model with parameters $W = ( W ^ { ( 1 ) } , \\cdots , W ^ { ( L ) } )$ , where $W ^ { ( l ) } \\in \\mathbb { R } ^ { N _ { l } } , 1 \\le l \\le L , N _ { l }$ is the number of prunable parameters in the $l _ { \\mathrm { t h } }$ layer, and $L$ denotes the number of transformer layers. Note that the first linear projection layer and the classifier of ViT [1, 2] are not sparsified in our framework. As illust W (l)Q rated in Figure 1(bottom-left), are the weights of the self-atte $W _ { Q } ^ { ( l ) } = \\{ W _ { Q } ^ { ( l , h ) } \\} _ { h = 1 } ^ { H }$ , W (l)K $W _ { K } ^ { ( l ) } = \\{ W _ { K } ^ { ( l , h ) } \\} _ { h = 1 } ^ { H }$ , $W _ { V } ^ { ( l ) } = \\{ W _ { V } ^ { ( l , h ) } \\} _ { h = 1 } ^ { H }$ $l _ { \\mathrm { t h } }$ $W ^ { ( l , 1 ) }$ $W ^ { ( l , 2 ) }$ $W ^ { ( l , 3 ) }$ perceptron (MLP) module in the collectively represent all the para $l _ { \\mathrm { t h } }$ layer, andters in the $W ^ { ( l ) } =$ $( W _ { Q } ^ { ( l ) } , W _ { K } ^ { ( l ) } , W _ { V } ^ { ( l ) } , \\bar { W } ^ { ( l , 1 ) } , W ^ { ( l , 2 ) } , W ^ { ( \\bar { l } , 3 ) } )$ $l _ { \\mathrm { t h } }$ where $H$ denotes the number of attention heads, and $Q ^ { ( l ) }$ , , and $V ^ { ( l ) }$ are the corresponding input and intermediate features, respectively. Each sparse layer only maintains a fraction $s _ { l } \\in ( 0 , 1 )$ of its connections, and the overall sparsity of a sparse subnetwork is calculated as the ratio of pruned elements to the total parameter counts, i.e., $\\frac { \\sum _ { l } s _ { l } \\times N _ { l } } { \\sum _ { l } N _ { l } }$ ", + "bbox": [ + 176, + 867, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/03c337fe2dadbbacbee28ecd5811945c8d6e72c346d559e7ef61f5a292efe60e.jpg", + "image_caption": [ + "Figure 1: The overall procedure of our proposed sparse ViT exploration framework. Upper Figure: first training ViT for $\\Delta \\mathrm { T }$ iterations, then performing prune-and-grow strategies to explore critical sparse connectivities, repreating until convergence. Bottom Left Figure: enforcing either structured or unstructured sparsity to transformer layers in ViT. Bottom Right Figure: first scoring each input embedding and applying the learnable top- $k$ selection to identify the most informative tokens. " + ], + "image_footnote": [], + "bbox": [ + 174, + 90, + 821, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 453, + 826, + 618 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Sparse Vision Transformer Exploration (SViTE). SViTE explores the unstructured sparse topology in vision transformers. To be specific, we adopt Erdo¨s-Re´nyi [32] as our $\\bullet$ sparsity distribution. The number of parameters in the sparse layer is scaled by $\\begin{array} { r } { 1 - \\frac { n _ { l - 1 } + n _ { l } } { n _ { l - 1 } \\times n _ { l } } } \\end{array}$ , where $n _ { l }$ is the number of neurons at layer l. This distribution allocates higher sparsities to the layers with more parameters by scaling the portion of remaining weights with the sum of the number of output and input neurons/channels. For the $\\otimes$ update schedule, it contains: (i) the update interval $\\Delta \\mathrm { T }$ , which is the number of training iterations between two sparse topology updates; $( i i )$ the end iteration $\\mathrm { T _ { e n d } }$ , indicating when to stop updating the sparsity connectivity, and we set $\\mathrm { T _ { e n d } }$ to $8 0 \\%$ of total training iterations in our experiments; $( i i i )$ the initial fraction $\\alpha$ of connections that can be pruned or $5 0 \\%$ $( i v )$ a decay schedule of the fraction of changeable connections, where a cosine annealing is used, following [34, 35]. During $\\begin{array} { r } { \\dot { f } _ { \\mathrm { d e c a y } } ( t , \\alpha , \\mathrm { T } _ { \\mathrm { e n d } } ) = \\frac { \\alpha } { 2 } ( 1 + \\cos ( \\frac { t \\times \\pi } { \\mathrm { T } _ { \\mathrm { e n d } } } ) ) } \\end{array}$ \neach connectivity update, we choose the weight magnitude as the pruning indicator, and gradient magnitude as $\\bullet$ the grow indicator. Specifically, we eliminate the parameters with the layer-wise smallest weight values by applying a binary mask $m _ { \\mathrm { p r u n e } }$ , then grow new connections with the highest magnitude gradients by generating a new binary mask $m _ { \\mathrm { g r o w } }$ . Both masks are employed to $W ^ { ( l ) }$ via the element-wise dot product, and note that the number of non-zero elements in $m _ { \\mathrm { p r u n e } }$ and $m _ { \\mathrm { g r o w } }$ are equal and fixed across the overall procedure. Newly added connections are not activated in the last sparse topology, and are initialized to zero since it produces better performance as demonstrated in [34, 35]. ", + "bbox": [ + 173, + 637, + 826, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Infrequent gradient calculation [34] is adopted in our case, which computes the gradients in an online manner and only stores the top gradient values. As illustrated in [34], such fashion amortizes the extra effort of gradient calculation, and makes it still proportional to $1 - s$ as long as $\\Delta \\mathrm { T } \\geq { \\frac { 1 } { 1 - s } }$ where $s$ is the overall sparsity. ", + "bbox": [ + 174, + 90, + 825, + 148 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Structured Sparse Vision Transformer Exploration $\\mathbf { ( S ^ { 2 } V i T E ) }$ . Although models with unstructured sparsity achieve superior performance, structured sparsity [80–82] is much more hardware friendly and brings practical efficiency on realistic platforms, which motivates us to propose Structured Sparse ViT Exploration $( \\mathrm { S ^ { 2 } V i T E } )$ . We inherit the design of $\\bullet$ sparsity distribution and $\\otimes$ update schedule from the unstructured SViTE, and a round-up function is used to eliminate decimals in the parameter counting. The key differences lie in the new $\\otimes$ pruning and $\\bullet$ grow strategies. ", + "bbox": [ + 173, + 162, + 826, + 247 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Pruning criterion: Let $A _ { ( l , h ) }$ denote features computed from the self-attention head {W (l,h)Q , $\\{ W _ { Q } ^ { ( \\mathit { l } , h ) } , W _ { K } ^ { ( \\mathit { l } , h ) } , W _ { V } ^ { ( \\mathit { l } , h ) } \\}$ and input embeddings $X ^ { ( l ) }$ , as shown in Figure 1. We perform the Taylor expansion to the loss function [16, 42], and derive a proxy score for head importance blow: ", + "bbox": [ + 173, + 253, + 433, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 Sparse ViT Co-Exploration $\\mathrm { ( S V i T E + ) }$ . ", + "text_level": 1, + "bbox": [ + 446, + 258, + 787, + 273 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/58ba181dad254fa7766be0d4985c1d4c51271ec9a0e6d2401c857e9e157799ed.jpg", + "text": "$$\n\\mathcal { T } _ { p } ^ { ( l , h ) } = \\bigg | A _ { ( l , h ) } ^ { \\mathrm { T } } \\cdot \\frac { \\partial \\mathcal { L } ( X ^ { ( l ) } ) } { \\partial A _ { ( l , h ) } } \\bigg | ,\n$$", + "text_format": "latex", + "bbox": [ + 194, + 376, + 390, + 412 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathcal { L } ( \\cdot )$ is the cross-entropy loss as used in ViT. During each topology update, we remove attention heads with the smallest $\\mathcal { T } _ { p } ^ { ( l , h ) }$ . For MLPs, we score neurons with $\\ell _ { 1 }$ -norm of their associated weight vectors [85], and drop insignificant neurons. For example, the $j _ { \\mathrm { t h } }$ neuron of $W ^ { ( l , 1 ) }$ in Figure 1 has an importance score kW (l,1)j,· k $W _ { j , \\cdot } ^ { ( l , 1 ) }$ is the $j _ { \\mathrm { t h } }$ row $W ^ { ( l , 1 ) }$ , where ", + "bbox": [ + 173, + 420, + 433, + 574 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Initialize: ViT model $f _ { W }$ , Dataset $\\mathcal { D }$ , Sparsity distribution $\\begin{array} { r c l } { \\mathbb { S } } & { = } & { \\{ s _ { 1 } , \\dots , s _ { L } \\} } \\end{array}$ , Update schedule $\\{ \\Delta \\mathrm { T } , \\mathrm { T _ { e n d } } , \\alpha , f _ { \\mathrm { d e c a y } } \\}$ , Learning rate $\\eta$ \n1: Initialize $f _ { W }$ with random sparsity $\\mathbb { S }$ . Highly reduced parameter count. \n2: for each training iteration $t$ do \n3: Sampling a batch $b _ { t } \\sim \\mathcal { D }$ \n4: Scoring the input token embeddings and selecting the top- $k$ informative tokens . Token selection \n5: if $\\mathbf { \\chi } _ { t }$ mod $\\Delta \\mathrm { T } = = 0 \\ \\mathrm { \\Omega }$ ) and $t < \\mathrm { T _ { e n d } }$ then \n6: for each layer $l$ do \n7: $\\rho = f _ { \\mathrm { d e c a y } } ( t , \\alpha , \\mathrm { T } _ { \\mathrm { e n d } } ) \\cdot ( 1 - s _ { l } ) \\cdot N _ { l }$ \n8: Performing prune-and-grow with portion $\\rho$ w.r.t. certain criterion, generating masks $m _ { \\mathrm { p r u n e } }$ and $m _ { \\mathrm { g r o w } }$ to update $f _ { W }$ ’s sparsity patterns $\\triangleright$ Connectivity exploration \n9: end for \n10: else \n11: $\\begin{array} { r l r l } { W = W - \\eta \\cdot \\nabla _ { W } \\mathcal { L } _ { t } } & { { } } & { \\triangleright U p d a t i n g \\ W e i g h t s } \\end{array}$ \n12: end if \n13: end for \n14: return a sparse ViT with a trained token selector ", + "bbox": [ + 449, + 279, + 830, + 582 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Grow criterion: Similar to [34, 35], we active the new units with the highest ", + "bbox": [ + 174, + 578, + 431, + 606 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "magnitude gradients, such as $\\| \\frac { \\partial \\bar { \\mathcal { L } } ( X ^ { ( l ) } ) } { \\partial A _ { ( l , h ) } } \\| _ { \\ell _ { 1 } }$ k\\`1 and $\\| \\frac { \\partial \\mathcal { L } ( X ^ { ( l ) } ) } { \\partial W _ { j , \\cdot } ^ { ( l , 1 ) } } \\| _ { \\ell _ { 1 } }$ for the $h _ { \\mathrm { t h } }$ attention head and the $j _ { \\mathrm { t h } }$ neuron of the MLP $( W ^ { ( l , 1 ) } )$ , respectively. The gradients are calculated in the same manner as the one in unstructured SViTE, and newly added units are also initialized to zero. ", + "bbox": [ + 174, + 606, + 825, + 662 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 Data and Architecture Sparsity Co-Exploration for Higher Efficiency ", + "text_level": 1, + "bbox": [ + 173, + 678, + 694, + 694 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Besides exploring sparse transformer architectures, we further slim the dimension of input token embeddings for extra efficiency bonus by leveraging a learnable token selector, as presented in Figure 1. Meanwhile, the introduced data sparsity also serves as an implicit regularization for ViT training, which potentially leads to improved generalization ability, as evidenced in Table 6. Note that, due to skip connections, the number of input tokens actually determines ", + "bbox": [ + 173, + 705, + 400, + 898 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 2 The top- $k$ selector in a PyTorch-like style. ", + "bbox": [ + 415, + 710, + 741, + 724 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "def topk_selector(logits, k, tau, ${ \\dot { \\mathsf { d i m } } } = - 1 \\cdot$ ): ", + "bbox": [ + 415, + 729, + 735, + 742 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "# Maintain tokens with the top-\\$k\\$ highest scores gumbels $=$ -torch.empty_like(logits).exponential_().log() gumbels $=$ (logits $^ +$ gumbels) / tau # tau is the temperature y_soft $=$ gumbels.softmax(dim) # Straight through index $=$ y_soft.topk(k, dim $\\mathbf { \\Psi } _ { 1 } =$ dim)[1] y_hard $=$ scatter(logits, index, k) ret $=$ y_hard - y_soft.detach() $^ +$ y_soft return ret ", + "bbox": [ + 415, + 743, + 828, + 883 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "the dimension of intermediate features, which substantially contributes to the overall computation cost. In other words, the slimmed input token embeddings directly result in compressed intermediate features, and bring substantial efficiency gains. ", + "bbox": [ + 181, + 897, + 821, + 911 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/0f89d381009b3bfa2398de990ca8bf70baef49b3898f0730f7187c6217692cb4.jpg", + "table_caption": [ + "Table 1: Details of training configurations in our experiments, mainly following the settings in [2]. " + ], + "table_footnote": [], + "table_body": "
BackboneUpdate Schedule{△T,Tend,α,fdecay}Batch SizeEpochsInherited Settings from DeiT[2]
DeiT-Tiny{20000,1200000,0.5,cosine}512600AdamW, 0.0005 × batchsize,cosine decay
DeiT-Small{15000,1200000,0.5,cosine}512600warmup 5 epochs,0.05 weight decay
DeiT-Base{7000,600000,0.5,cosine}10246000.1 label smoothing,augmentations, etc.
", + "bbox": [ + 176, + 113, + 823, + 167 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 178, + 823, + 207 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For the input tokens $X ^ { ( 1 ) } \\in \\mathbb { R } ^ { n \\times d }$ , where $n$ denotes the number of tokens to be shrunk, and $d$ is the dimension of each token embedding that keeps unchanged. As shown in Figure 1, all token embeddings are passed through a learnable scorer function which is parameterized by an MLP in our experiments. Then, a selection of the top- $k$ importance scores $( 1 \\leq k \\leq d )$ is applied on top of it, aiming to preserve the significant tokens and remove the useless ones. To optimize parameters of the scorer function, we introduce the popular Gumbel-Softmax [86, 87] and straight-through tricks [88] to enable gradient back-propagation through the top- $k$ selection, which provides an efficient solution to draw samples from a discrete probability distribution. A detailed implementation is in Algorithm 2. ", + "bbox": [ + 173, + 212, + 825, + 325 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The full pipeline of data and architecture co-exploration is summarized in Algorithm 1. We term this approach $\\mathrm { S V i T E { + } }$ . We first feed the randomly sampled data batch to the token selector and pick the top- $k$ informative token embeddings. Then, we alternatively train the sparse ViT for $\\Delta \\mathrm { T }$ iterations and perform prune-and-grow to explore the sparse connectivity in ViTs dynamically. In the end, a sparse ViT model with a trained token selector is returned and ready for evaluation. ", + "bbox": [ + 174, + 330, + 825, + 400 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 414, + 312, + 431 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baseline pruning methods. We extend several effective pruning methods from CNN compression as our strong baselines. Unstructured pruning: $( i )$ One-shot weight Magnitude Pruning (OMP) [15], which removes insignificant parameters with the globally smallest weight values; (ii) Gradually Magnitude Pruning (GMP) [17], which seamlessly incorporates gradual pruning techniques within the training process by eliminating a few small magnitude weights per iteration; and (iii) Taylor Pruning (TP) [16], which utilizes the first-order approximation of the training loss to estimate units’ importance for model sparsification. Structured pruning: Salience-based Structured Pruning (SSP). We draw inspiration from [42, 85], and remove sub-modules in ViT (e.g., self-attention heads) by leveraging their weight, activation, and gradient information. Moreover, due to the repetitive architecture of ViT, we can easily reduce the number of transformer layers to create a smaller dense ViT (Small-Dense) baseline that has similar parameter counts to the pruned ViT model. ", + "bbox": [ + 173, + 439, + 826, + 592 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Implementation details. Our experiments are conducted on ImageNet with DeiTTiny/Small/Base backbones. The detailed training configurations are listed in Table 1, which mainly follows the default setups in [2]. All involved customized hyperparameters are tuned via grid search (later shown in Figure 3). For a better exploration of sparsity connectivities, we increase training epochs to 600 for all experiments. GMP [17] has an additional hyperparameter, i.e., the pruning schedule, which starts from $\\frac { 1 } { 6 }$ and ends at ${ \\frac { \\overline { { \\frac { 1 } { 2 } } } } { 2 } }$ of the training epochs with 20 times pruning in total. More details are referred to Appendix A1. ", + "bbox": [ + 174, + 602, + 472, + 795 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training time measuring protocol. We strictly measure the running time saving of ", + "bbox": [ + 173, + 805, + 472, + 832 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/a5ad90174ff1547fe598c114e7c4b77c1f965a37ed3695b0c6fcde1a09ee9c95.jpg", + "image_caption": [ + "The Overall Performance of SViTE, S 2ViTE, and SViTE+ ", + "Figure 2: Top-1 accuracy $( \\% )$ over FLOPs $( \\times 1 0 ^ { 1 0 } )$ on ImageNet of our methods, i.e., SViTE, $\\mathrm { S ^ { 2 } V i T E }$ , and ${ \\mathrm { S V i T E } } +$ compared to DeiT baselines, trained on Imagenet-1K only. " + ], + "image_footnote": [], + "bbox": [ + 488, + 607, + 818, + 789 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(sparse) vision transformers on the ImageNet-1K task using CUDA benchmark mode. To be specific, we separately calculate the time elapsed during each iteration, to eliminate the impact of the hardware environment as much as possible. Note that the time for the data I/O is excluded. ", + "bbox": [ + 174, + 833, + 826, + 873 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Highlight of our findings. The overall performance of SViTE, $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ , and $\\mathrm { S V i T E { + } }$ on DeiT backbones are summarized in Figure 2. We highlight some takeaways below. ", + "bbox": [ + 174, + 882, + 821, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Takeaways: ❶ SViTE produces sparse DeiTs with enhanced generalization and substantial reduced FLOPs, compared to its dense counterpart $( { \\star } )$ . ${ \\mathrm { S V i T E } } +$ further improves the performance of SViTE by selecting the most vital patches. ❷ $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ achieves matched accuracy on DeiT-Small, and significantly enhances performance on DeiT-Base. Meanwhile, its structural sparsity brings considerable running time savings. $\\otimes$ Appropriate data and architecture sparsities can effectively regularize ViT training, leading to a new SOTA win-win between ViT accuracy and efficiency. ", + "bbox": [ + 176, + 90, + 828, + 178 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 SViTE with Unstructured Sparsity ", + "text_level": 1, + "bbox": [ + 174, + 199, + 452, + 214 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We perform SViTE to mine vital unstructured sparsity in DeiTs [2]. Solid lines in Figure 2 record the top-1 test-set accuracy over FLOPs on ImageNet-1K of SViTE-Small and SViTE-Base with a range of sparsity from $3 0 \\%$ to $7 0 \\%$ . In general, we observe that SViTE generates superior sparse ViTs with both accuracy and efficiency gains. Table 2, 3, and 5 present the comparison between SViTE and various pruning baselines. From these extensive results, we draw several consistent observations. First, compared to the dense baselines, SViTE-Tiny, -Small, and -Base obtain $2 5 . 5 6 \\% \\sim 3 4 . 1 6 \\%$ , $4 6 . 2 6 \\% \\sim 5 5 . 4 4 \\%$ , and $4 7 . 9 5 \\% \\sim 5 7 . 5 0 \\%$ FLOPs reduction, respectively, at $3 0 \\% \\sim 6 0 \\%$ sparsity levels with only a negligible accuracy drop within $0 . 5 \\%$ . It verifies the effectiveness of our proposal, and indicates severe parameter redundancy in ViT. Second, our SViTE models from dynamic explorations consistently surpass other competitive baseline methods, including OMP, GMP, TP, and Small-Dense by a substantial performance margin. Among all the baseline approaches, GMP that advocates a gradual pruning schedule achieves the best accuracy with all three DeiT backbones. Third, in Figure 2, both SViTE-Small (blue solid line) and SViTE-Base (green solid line) show an improved trade-off between accuracy and efficiency, compared to their dense DeiT counterparts. Interestingly, we also observe that with similar parameter counts, a large sparse ViT consistently outperforms the corresponding smaller dense ViT. A possible explanation is those appropriate sparse typologies regularize network training and lead to enhanced generalization, which coincides with recent findings of critical subnetworks (i.e., winning tickets) in dense CNNs [89, 22] and NLP transformer [21, 90] models. ", + "bbox": [ + 173, + 224, + 825, + 486 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/08efc9ac71a6d77f0025d863cb9df9efc84a9ec33de6ce08abe47b7a987eef3b.jpg", + "table_caption": [ + "Table 2: Results of SViTE-Tiny on ImageNet-1K. Accuracies $( \\% )$ within/out of parenthesis are the reproduced/reported [2] performance. " + ], + "table_footnote": [], + "table_body": "
ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Tiny0% (5.72M)0%72.20 (71.80)
SViTE-Tiny30% (4.02M)25.56%71.78
OMP30% (4.02M)25.56%68.35
GMP30% (4.02M)25.56%69.56
TP30% (4.02M)25.56%68.38
SViTE-Tiny40% (3.46M)34.16%71.75
OMP40% (3.46M)34.16%66.52
GMP40% (3.46M)34.15%68.36
TP40% (3.46M)34.17%65.45
Small-Dense0% (3.94M)32.54%67.33
", + "bbox": [ + 176, + 529, + 493, + 657 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/816fcda684feef4194c0cc774f5f2ee9b3971d35729300758bcabab40dd55fe8.jpg", + "table_caption": [ + "Table 3: Results of SViTE-Small on ImageNet-1K. Accuracies $( \\% )$ within/out of parenthesis are the reproduced/reported [2] performance. " + ], + "table_footnote": [], + "table_body": "
ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Small0% (22.1M)0%79.90 (79.78)
SViTE-Small50% (11.1M)46.26%79.72
OMP50% (11.1M)46.25%76.32
GMP50% (11.1M)46.26%76.88
TP50% (11.1M)46.26%76.30
SViTE-Small60% (8.9M)55.44%79.41
OMP60% (8.9M)55.44%75.32
GMP60% (8.9M)55.44%76.79
TP60% (8.9M)55.44%74.50
Small-Dense0% (11.4M)49.32%73.93
", + "bbox": [ + 508, + 529, + 825, + 656 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/5d5ed755d309b5b29b419455ac8775093c6ae72f8244f04c130e06cbeeff4bea.jpg", + "table_caption": [ + "Table 4: Results of $\\mathrm { S ^ { 2 } V i T E }$ with structured sparsity on ImageNet-1K with DeiT-Tiny/Small/Base. Accuracies $( \\% )$ within/out of parenthesis are the reproduced/reported [2] performance. " + ], + "table_footnote": [], + "table_body": "
ModelsSparsity (%)ParametersFLOPs SavingRunning Time Reduced|Top-1 Accuracy (%)
DeiT-Tiny (Dense)0%5.72M0%0%72.20 (71.80)
SViTE-Tiny (Unstructured)30%4.02M25.56%0%71.78
SSP-Tiny (Structured)30%4.21M23.69%10.57%68.59
S2ViTE-Tiny (Structured)30%4.21M23.69%10.57%70.12
DeiT-Small (Dense)0%22.1M0%0%79.90 (79.78)
SViTE-Small (Unstructured)40%13.3M36.73%0%80.26
SSP-Small (Structured)40%14.6M31.63%22.65%77.74
S²ViTE-Small (Structured)40%14.6M31.63%22.65%79.22
DeiT-Base (Dense)0%86.6M0%0%81.80 (80.98)
SViTE-Base (Unstructured)40%52.0M38.30%0%81.56
SSP-Base (Structured)40%56.8M33.13%24.70%80.08
S2ViTE-Base (Structured)40%56.8M33.13%24.70%82.22
", + "bbox": [ + 176, + 689, + 825, + 849 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 $\\mathbf { S } ^ { 2 }$ ViTE with Structured Sparsity ", + "text_level": 1, + "bbox": [ + 174, + 857, + 442, + 872 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For more practical benefits, we investigate sparse DeiTs with structured sparsity. Results are summarized in Table 4. Besides the obtained $2 3 . { \\bar { 7 } } 9 \\% \\sim 3 3 . 6 3 \\%$ FLOPs savings, ${ \\mathsf { S } } ^ { \\tilde { 2 } }$ ViTE-Tiny, $S ^ { 2 }$ ViTE", + "bbox": [ + 174, + 883, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Small, and $S ^ { 2 }$ ViTE-Base enjoy an extra $1 0 . 5 7 \\%$ , $2 2 . 6 5 \\%$ , and $2 4 . 7 0 \\%$ running time reduction, respectively, from $3 0 \\% \\sim \\hat { 4 } 0 \\%$ structured sparsity with competitive top-1 accuracies. Furthermore, $\\mathrm { S ^ { 2 } V i T E }$ consistently outperforms the baseline structured pruning method (SSP), which again demonstrates the superior sparse connectivity learned from dynamic sparse training. ", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The most impressive results come from $S ^ { 2 }$ ViTE-Base at $4 0 \\%$ structured sparsity. It even surpasses the dense DeiT base model by $0 . 4 2 \\% \\sim 1 . 2 4 \\%$ accuracy with $3 4 . 4 1 \\%$ parameter counts, $3 3 . 1 3 \\%$ FLOPs, and $2 4 . 7 0 \\%$ running time reductions. We conclude that $( i )$ an adequate sparsity from $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ boosts ViT’s generalization ability, which can be regarded as an implicit regularization; $( i i )$ larger ViTs (e.g., DeiT-Base) tend to have more superfluous self-attention heads, and are more amenable to structural sparsification from $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ , based on Figure 2 where dash lines denote the overall performance of $\\hat { \\mathbf { S } ^ { 2 } }$ ViTE-Small and $S ^ { 2 }$ ViTE-Base with a range of sparsity from $3 0 \\%$ to $7 0 \\%$ . ", + "bbox": [ + 173, + 152, + 825, + 251 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/facf8afddb44ba3a8d72c8cb6b4b90dee343464081b964fc00623ae2fe190bc3.jpg", + "table_caption": [ + "Table 5: Results of SViTE-Base on ImageNet-1K. Accuracies $( \\% )$ within/out of parenthesis are the reproduced/reported [2] performance. " + ], + "table_footnote": [], + "table_body": "
ModelsSparsity (#Para.)FLOPs Saving|Accuracy (%)
DeiT-Base0% (86.6M)0%81.80 (80.98)
SViTE-Base50% (43.4M)47.95%81.51
OMP50% (43.4M)47.94%80.26
GMP50% (43.4M)47.95%80.79
TP50% (43.4M)47.94%80.55
SViTE-Base60% (34.8M)57.50%81.28
OMP60% (34.8M)57.50%80.25
GMP60% (34.8M)57.50%80.44
TP60% (34.8M)57.49%80.37
Small-Dense0% (44.0M)49.46%78.59
", + "bbox": [ + 176, + 296, + 500, + 428 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2882166a73083b210a64df1d75c08fbb72b1f4c341db6ad565dde75f4e195c5c.jpg", + "table_caption": [ + "Table 6: Results of SViTE $^ +$ -Small on ImageNet-1K. Accuracies $( \\% )$ within/out of parenthesis are the reproduced/reported [2] performance. " + ], + "table_footnote": [], + "table_body": "
#Tokens (%)|Time ReducedFLOPs Saving|Accuracy (%)
SViTE+-Small 50% Unstructured Sparsity
100%0%46.26%79.72
95%4.40%49.32%80.18
90%7.63%52.38%79.91
70%19.77%63.95%77.90
S²ViTE+-Small 40% Structured Sparsity
100%22.65%31.63%79.22
95%27.17%37.76%78.44
90%29.21%41.50%78.16
70%39.10%54.96%74.77
", + "bbox": [ + 508, + 294, + 825, + 428 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 $\\mathbf { S V i T E { + } }$ with Data and Architecture Sparsity Co-Exploration ", + "text_level": 1, + "bbox": [ + 174, + 436, + 637, + 452 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we study data and architecture sparsity co-exploration for ViTs, i.e., $\\mathrm { S V i T E { + } }$ . Blessed by the reduced input token embeddings, even ViTs with unstructured sparsity can have running time savings. The benefits are mainly from the shrunk input and intermediate feature dimensions. Without loss of generality, we consider SViTE $+ \\cdot$ -Small with $5 0 \\%$ unstructured sparsity and ${ \\mathrm { S ^ { 2 } V i T E } } +$ -Small with $4 0 \\%$ structured sparsity as examples. As shown in Table 6 and Figure 2, SViTE+-Small at $5 0 \\%$ unstructured sparsity is capable of abandoning $5 \\% \\sim 1 0 \\%$ tokens while achieving $4 . 4 0 \\% \\sim 7 . 6 3 \\%$ running time and $4 9 . 3 2 \\% \\sim 5 2 . 3 8 \\%$ FLOPs savings, with even improved top-1 testing accuracy. It again demonstrates that data sparsity as an implicit regularizer plays a beneficial role in ViT training. However, slimming input and intermediate embedding is less effective when incorporated with $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ , suggesting that aggressively removing structural sub-modules hurts ViT’s generalization. ", + "bbox": [ + 173, + 462, + 826, + 601 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Ablation and Generalization Study of SViTEs ", + "text_level": 1, + "bbox": [ + 176, + 616, + 529, + 631 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Update interval in SViTE. The length of the update interval $\\Delta \\mathrm { T }$ controls one of the essential tradeoffs in our proposed dynamic sparse exploration, since $\\Delta \\mathrm { T }$ multiplying the number of updates is the pre-defined $\\mathrm { T _ { e n d } }$ . On the one hand, a larger updated interval (i.e., smaller update frequency) produces a more well-trained model for improved estimation of units’ importance. On the other hand, a larger update frequency (i.e., smaller $\\Delta \\mathrm { T }$ ) allows more sufficient exploration of sparse connectivities, which potentially generates higher-quality sparse subnetworks, as demonstrated in [35]. We evaluate this factor in our SViTE context, and collect the results in Figure 3 (Left). We observe that $\\Delta \\mathrm { T } = 2 0 0 0 0$ works the best for SViTE-Tiny, and both larger and smaller $\\Delta \\mathrm { T }$ degrade the performance. ", + "bbox": [ + 173, + 641, + 826, + 753 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/a4655f90e93b2c9a04681fc021da3282c731aba6f4cba193b459acd0b25f0153.jpg", + "image_caption": [ + "Figure 3: Accuracy of SViTE-Tiny with $5 0 \\%$ unstructured sparsity. Left: ablation studies of the update interval $( \\bar { \\Delta \\mathrm { T } } )$ ; Right: ablations studies of the adopted batch size $( b )$ . " + ], + "image_footnote": [], + "bbox": [ + 181, + 758, + 813, + 886 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/20204951e060810529cbad9a007da6259808df546d2c9563a4a457ffbdf0a012.jpg", + "image_caption": [ + "Dense DeiT-Base " + ], + "image_footnote": [], + "bbox": [ + 178, + 90, + 388, + 250 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/33e5e064c5dba74ca99e27db03615dbf08bda7f2b81b68715098af3c25c6d603.jpg", + "image_caption": [ + "-Base with $4 0 \\%$ Structured Sparsity " + ], + "image_footnote": [], + "bbox": [ + 395, + 92, + 604, + 251 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/fcf8ae89303d80f050af5c5ed88ce6e7bc4eee6a2db8431edf4a7768d4fc9049.jpg", + "image_caption": [ + "SViTE-Base with $4 0 \\%$ Unstructured Sparsity " + ], + "image_footnote": [], + "bbox": [ + 609, + 90, + 820, + 251 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/a0e0619f2f06fe582065a11779901de10112cfc6477c8e3ec5b96052b7f80f29.jpg", + "image_caption": [ + "Figure 4: Attention probabilities for DeiT-Base, $S ^ { 2 }$ ViTE-Base, and SViTE-Base models with 12 layers (rows) and 12 heads (columns) using visualization tools provided in [94]. Attention maps are averaged over 100 test samples from ImageNet-1K to present head behavior and remove the dependence on the input content. The black square is the query pixel. indicates pruned attention heads. Zoom-in for better visibility. ", + "Figure 5: Learned patch selection patterns of SViTE $+$ -Small at $1 0 \\%$ data and $5 0 \\%$ architecture sparsity levels. $\\widehat { \\mathbb { I } }$ indicates removed inessential patches. " + ], + "image_footnote": [], + "bbox": [ + 179, + 318, + 820, + 439 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Batch size in SViTE. Besides the update interval $\\Delta \\mathrm { T }$ , batch size (b) also affects the aforementioned trade-off, especially for the data-hungry ViT training. We investigate different batch sizes in Figure 3 (Right), and find that $b = 5 1 2$ outperforms other common options for SViTE-Tiny. ", + "bbox": [ + 176, + 479, + 823, + 522 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Generalization study of SViTE and its variants. It is worth mentioning that our proposed frameworks (SViTE, $\\mathrm { S ^ { 2 } V i T E }$ , $\\mathrm { S V i T E { + } }$ ) are independent of the backbone architectures, and can be easily plugged in other vision transformer models [91, 45, 92, 93]. We implemented both SViTE and ${ \\bar { \\mathbf { S } } } ^ { 2 }$ ViTE on TNT-S [91]. SViTE-TNT-S gains 0.13 accuracy improvements (Ours: 81.63 v.s. TNT-S: 81.50) and $3 7 . 5 4 \\%$ FLOPs savings at $4 0 \\%$ unstructured sparsity; $S ^ { 2 }$ ViTE-TNT-S obtains $3 2 . 9 6 \\%$ FLOPs and $2 3 . 7 1 \\%$ running time reductions at $4 0 \\%$ structured sparsity with almost unimpaired accuracy (Ours: 81.34 v.s. TNT-S:81.50). ", + "bbox": [ + 174, + 537, + 825, + 635 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 Visualization ", + "text_level": 1, + "bbox": [ + 174, + 651, + 303, + 666 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sparse connectivity patterns. We provide unit-wise and element-wise heatmap visualizations for SViTE-Base with $4 0 \\%$ structured sparsity in Figure A7 (in Appendix). Similarly, element-wise heatmap visualizations of SViTE-Base with $5 0 \\%$ unstructured sparsity are displayed in Figure A6. We find that even unstructured sparsity exploration can develop obvious structural patterns (i.e., “vertical lines” in mask heatmaps), which implies a stronger potential for hardware speedup [95]. ", + "bbox": [ + 173, + 678, + 825, + 747 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Self-attention heatmaps. As shown in Figure 4, we utilize tools in [94] to visualize attention maps of (sparse) ViTs. Multiple attention heads show similar behaviors, which implies the structural redundancy. Fortunately, $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ eliminates unnecessary heads to some extent. With regard to SViTE-Base’s visual results, it seems to activate fewer attention heads for predictions (darker colors mean larger values), compared to the ones of dense DeiT-Base. We also observe that in the bottom layers, the attention probabilities are more centered at several heads; while in the top layers, the attention probabilities are more uniformly distributed. This kind of tendency is well preserved by our sparse ViT (SViTE) from Dense ViTs. ", + "bbox": [ + 174, + 760, + 825, + 871 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learned patch selection patterns. Figure 5 presents the learned behaviors of our token selector in SViTE+. We observe that the useless removed patches are typically distributed around the main object or in the background. Meanwhile, the patches within the objects of interest are largely persevered, which evidences the effectiveness of our learned patch token selector. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 92, + 825, + 119 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 Conclusion and Discussion of Broader Impact ", + "text_level": 1, + "bbox": [ + 174, + 135, + 588, + 151 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this work, we introduce sparse ViT exploration algorithms, SViTE, and its variants $\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }$ and ${ \\mathrm { S V i T E } } +$ , to explore high-quality sparse patterns in both ViT’s architecture and input token embeddings, alleviating training memory bottleneck and pursuing inference ultra-efficiency (e.g., running time and FLOPs). Comprehensive experiments on ImageNet validate the effectiveness of our proposal. Our informative visualizations further demonstrate that $\\mathrm { S V i T E { + } }$ is capable of mining crucial connections and input tokens by eliminating redundant units and dropping useless token embeddings. Future work includes examining the performance of our sparse ViTs on incoming hardware accelerators [96–100], which will provide better supports for sparsity. ", + "bbox": [ + 174, + 162, + 825, + 273 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work is scientific in nature, and we do not believe it has immediate negative societal impacts. Our findings of sparse vision transformers are highly likely to reduce both memory and energy costs substantially, leading to economic deployment in real-world applications (e.g., on smartphones). ", + "bbox": [ + 176, + 280, + 825, + 321 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgment ", + "text_level": 1, + "bbox": [ + 174, + 335, + 321, + 352 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Z.W. is in part supported by an NSF RTML project (#2053279). ", + "bbox": [ + 174, + 359, + 593, + 376 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 395, + 266, + 410 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. \n[2] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. 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We further co-explore data and architecture sparsity for additional efficiency", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 470, + 433 + ], + "score": 1.0, + "content": "gains by plugging in a novel learnable token selector to adaptively determine the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 431, + 469, + 442 + ], + "spans": [ + { + "bbox": [ + 141, + 431, + 469, + 442 + ], + "score": 1.0, + "content": "currently most vital patches. Extensive results on ImageNet with diverse ViT", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 442, + 469, + 454 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 469, + 454 + ], + "score": 1.0, + "content": "backbones validate the effectiveness of our proposals which obtain significantly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 452, + 470, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 470, + 465 + ], + "score": 1.0, + "content": "reduced computational cost and almost unimpaired generalization. Perhaps most", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 464, + 469, + 476 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 469, + 476 + ], + "score": 1.0, + "content": "surprisingly, we find that the proposed sparse (co-)training can sometimes improve", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 474, + 470, + 487 + ], + "spans": [ + { + "bbox": [ + 141, + 474, + 470, + 487 + ], + "score": 1.0, + "content": "the ViT accuracy rather than compromising it, making sparsity a tantalizing “free", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 485, + 470, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 352, + 498 + ], + "score": 1.0, + "content": "lunch”. For example, our sparsified DeiT-Small at", + "type": "text" + }, + { + "bbox": [ + 352, + 485, + 367, + 496 + ], + "score": 0.64, + "content": "( 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 486, + 371, + 498 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 371, + 485, + 392, + 496 + ], + "score": 0.66, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 486, + 470, + 498 + ], + "score": 1.0, + "content": ") sparsity for (data,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 496, + 469, + 508 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 239, + 508 + ], + "score": 1.0, + "content": "architecture), improves", + "type": "text" + }, + { + "bbox": [ + 239, + 496, + 269, + 507 + ], + "score": 0.89, + "content": "\\mathbf { 0 . 2 8 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 496, + 433, + 508 + ], + "score": 1.0, + "content": "top-1 accuracy, and meanwhile enjoys", + "type": "text" + }, + { + "bbox": [ + 433, + 496, + 469, + 507 + ], + "score": 0.89, + "content": "{ \\bf 4 9 . 3 2 \\% }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 506, + 471, + 520 + ], + "spans": [ + { + "bbox": [ + 141, + 506, + 194, + 520 + ], + "score": 1.0, + "content": "FLOPs and", + "type": "text" + }, + { + "bbox": [ + 194, + 507, + 225, + 518 + ], + "score": 0.88, + "content": "{ \\bf 4 . 4 \\bar { 0 } \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 506, + 471, + 520 + ], + "score": 1.0, + "content": "running time savings. Our codes are available at https:", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 518, + 299, + 530 + ], + "spans": [ + { + "bbox": [ + 142, + 518, + 299, + 530 + ], + "score": 1.0, + "content": "//github.com/VITA-Group/SViTE.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 18, + "bbox_fs": [ + 141, + 278, + 471, + 530 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 545, + 191, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 192, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 192, + 561 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 506, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 507, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 507, + 578 + ], + "score": 1.0, + "content": "Recent years have seen substantial efforts devoted to scaling deep networks to enormous sizes.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "Parameter counts are frequently measured in billions rather than millions, with the time and financial", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "outlay necessary to train these models growing in concert. The trend undoubtedly continues with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "the recent forefront of transformers [1–3] for computer vision tasks. By leveraging self-attention,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 610, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 621 + ], + "score": 1.0, + "content": "reducing weight sharing such as convolutions, and feeding massive training data, vision transformers", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "have established many new state-of-the-art (SOTA) records in image classification [1, 2], object de-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "tection [4–7], image enhancement [8, 9], and image generation [10–12]. Existing vision transformers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "and variants, despite the impressive empirical performance, have in general suffered from gigantic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "parameter-counts, heavy run-time memory usages, and tedious training. That naturally calls for the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "next step research of slimming their inference and training, without compromising the performance.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 565, + 507, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 680, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "Model compression and efficient learning are no strangers to deep learning researchers, although their", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "exploration in the emerging vision transformer field remains scarce [13]. Among the large variety", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 701, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 506, + 715 + ], + "score": 1.0, + "content": "of compression means [14], sparsity has been one of the central themes since the beginning [15].", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 679, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Conventional approaches first train dense networks, and then prune a large portion of parameters in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 97 + ], + "score": 1.0, + "content": "the trained networks to zero. Those methods significantly reduce the inference complexity. However,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 109 + ], + "score": 1.0, + "content": "the price is to cost even more significant computational resources and memory footprints at training,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "since they commonly require (multiple rounds of) re-training to restore the accuracy loss [15–17].", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "That price becomes particularly prohibitive for vision transformers, whose vanilla one-pass training is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "score": 1.0, + "content": "already much more tedious, slow, and unstable compared to training standard convolutional networks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "An emerging subfield has explored the prospect of directly training smaller, sparse subnetworks in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "place of the full networks without sacrificing performance. The key idea is to reuse the sparsity", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 164, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 180 + ], + "score": 1.0, + "content": "pattern found through pruning and train a sparse network from scratch. The seminal work of lottery", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "ticket hypothesis (LTH) [18] demonstrated that standard dense networks contain sparse matching", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "score": 1.0, + "content": "subnetworks (sometimes called “winning tickets”) capable of training in isolation to full accuracy.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "In other words, we could have trained smaller networks from the start if only we had known which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "subnetworks to choose. Unfortunately, LTH requires to empirically find these intriguing subnetworks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "by an iterative pruning procedure [18–27] , which still cannot get rid of the expensiveness of post-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "training pruning. In view of that, follow-up works reveal that sparsity patterns might emerge at the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 239, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 256 + ], + "score": 1.0, + "content": "initialization [28, 29], the early stage of training [30, 31], or in dynamic forms throughout training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 251, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 266 + ], + "score": 1.0, + "content": "[32–34] by updating model parameters and architecture typologies simultaneously. These efforts shed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "light on the appealing prospect of “end to end” efficiency from training to inference, by involving", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 275, + 291, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 291, + 287 + ], + "score": 1.0, + "content": "sparsity throughout the full learning lifecycle.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 108, + 291, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "This paper presents the first-of-its-kind comprehensive exploration of integrating sparsity in vision", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "transformers (ViTs) “from end to end”. With (dynamic) sparsity as the unified tool, we can improve", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "the inference efficiency from both model and data perspectives, while also saving training memory", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 429, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 429, + 336 + ], + "score": 1.0, + "content": "costs. Our innovative efforts are unfolded along with the following three thrusts:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 133, + 344, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 132, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 132, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "• From Dense to (Dynamic) Sparse: Our primary quest is to find sparse ViTs without", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "sacrificing the achievable accuracy, and meanwhile trimming down the training memory", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "overhead. To meet this challenging demand, we draw inspirations from the latest sparse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "training works [34, 35] that dynamically extract and train sparse subnetworks instead of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "training the full models. Sticking to a fixed small parameter budget, our technique jointly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 399, + 507, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 507, + 412 + ], + "score": 1.0, + "content": "optimizes model parameters and explores connectivity throughout the entire training process.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 409, + 483, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 483, + 423 + ], + "score": 1.0, + "content": "We term our first basic approach as Sparse Vision Transformer Exploration (SViTE).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 133, + 425, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 132, + 424, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 132, + 424, + 507, + 438 + ], + "score": 1.0, + "content": "• From Unstructured to Structured: Most sparse training works [32, 33, 36–39, 38, 34, 40,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 140, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 140, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "41, 35] restricted discussion to unstructured sparsity. To attain structured sparsity which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "is more hardware-friendly, unlike classical channel pruning available for convolutional", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 458, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 507, + 470 + ], + "score": 1.0, + "content": "networks, we customize a first-order importance approximation [16, 42] to guide the prune-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "and-grow of self-attention heads inside ViTs. This seamlessly extends SViTE to its second", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 479, + 430, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 387, + 492 + ], + "score": 1.0, + "content": "variant of Structured Sparse Vision Transformer Exploration", + "type": "text" + }, + { + "bbox": [ + 387, + 479, + 425, + 491 + ], + "score": 0.6, + "content": "\\mathbf { \\left( S ^ { 2 } V i T E \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 479, + 430, + 492 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 134, + 495, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 133, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 133, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "• From Model to Data: We further conduct a unified co-exploration towards joint data and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "architecture sparsity. That is by plugging in a novel learnable token selector to determine", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 141, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "the most vital patch embeddings in the current input sample. The resultant framework of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 321, + 540 + ], + "score": 1.0, + "content": "Sparse Vision Transformer Co-Exploration", + "type": "text" + }, + { + "bbox": [ + 321, + 529, + 359, + 539 + ], + "score": 0.76, + "content": "\\mathbf { \\eta } ( \\mathbf { S } \\mathbf { V i T E } +", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 528, + 505, + 540 + ], + "score": 1.0, + "content": ") remains to be end-to-end trainable", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 538, + 281, + 552 + ], + "spans": [ + { + "bbox": [ + 141, + 538, + 281, + 552 + ], + "score": 1.0, + "content": "and can gain additional efficiency.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 571 + ], + "score": 1.0, + "content": "Extensive experiments are conducted on ImageNet with DeiT-Tiny/Small/Base. Results of substantial", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 571, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 582 + ], + "score": 1.0, + "content": "computation savings and nearly undamaged accuracies consistently endorse our proposals’ effec-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "tiveness. Perhaps most impressively, we find that the sparse (co-)training can even improve the ViT", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "accuracy rather than compromising it, making sparsity a tantalizing “free lunch”. 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Transformer [43] stems from natural language processing (NLP) applications.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "The Vision Transformer (ViT) [1] pioneered to leverage a pure transformer, to encode an image by", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "splitting it into a sequence of patches, projecting them into token embeddings, and feeding them to", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Conventional approaches first train dense networks, and then prune a large portion of parameters in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 97 + ], + "score": 1.0, + "content": "the trained networks to zero. Those methods significantly reduce the inference complexity. However,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 507, + 109 + ], + "score": 1.0, + "content": "the price is to cost even more significant computational resources and memory footprints at training,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "since they commonly require (multiple rounds of) re-training to restore the accuracy loss [15–17].", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "That price becomes particularly prohibitive for vision transformers, whose vanilla one-pass training is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 139 + ], + "score": 1.0, + "content": "already much more tedious, slow, and unstable compared to training standard convolutional networks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 72, + 507, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "An emerging subfield has explored the prospect of directly training smaller, sparse subnetworks in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "place of the full networks without sacrificing performance. The key idea is to reuse the sparsity", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 164, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 180 + ], + "score": 1.0, + "content": "pattern found through pruning and train a sparse network from scratch. The seminal work of lottery", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "ticket hypothesis (LTH) [18] demonstrated that standard dense networks contain sparse matching", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "score": 1.0, + "content": "subnetworks (sometimes called “winning tickets”) capable of training in isolation to full accuracy.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "In other words, we could have trained smaller networks from the start if only we had known which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "subnetworks to choose. Unfortunately, LTH requires to empirically find these intriguing subnetworks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "by an iterative pruning procedure [18–27] , which still cannot get rid of the expensiveness of post-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "training pruning. In view of that, follow-up works reveal that sparsity patterns might emerge at the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 239, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 256 + ], + "score": 1.0, + "content": "initialization [28, 29], the early stage of training [30, 31], or in dynamic forms throughout training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 251, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 266 + ], + "score": 1.0, + "content": "[32–34] by updating model parameters and architecture typologies simultaneously. These efforts shed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "light on the appealing prospect of “end to end” efficiency from training to inference, by involving", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 275, + 291, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 291, + 287 + ], + "score": 1.0, + "content": "sparsity throughout the full learning lifecycle.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 144, + 507, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 291, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "This paper presents the first-of-its-kind comprehensive exploration of integrating sparsity in vision", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "transformers (ViTs) “from end to end”. With (dynamic) sparsity as the unified tool, we can improve", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "the inference efficiency from both model and data perspectives, while also saving training memory", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 429, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 429, + 336 + ], + "score": 1.0, + "content": "costs. Our innovative efforts are unfolded along with the following three thrusts:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 290, + 506, + 336 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 344, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 132, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 132, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "• From Dense to (Dynamic) Sparse: Our primary quest is to find sparse ViTs without", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "sacrificing the achievable accuracy, and meanwhile trimming down the training memory", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "overhead. To meet this challenging demand, we draw inspirations from the latest sparse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "training works [34, 35] that dynamically extract and train sparse subnetworks instead of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "training the full models. Sticking to a fixed small parameter budget, our technique jointly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 399, + 507, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 507, + 412 + ], + "score": 1.0, + "content": "optimizes model parameters and explores connectivity throughout the entire training process.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 409, + 483, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 483, + 423 + ], + "score": 1.0, + "content": "We term our first basic approach as Sparse Vision Transformer Exploration (SViTE).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 132, + 343, + 507, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 425, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 132, + 424, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 132, + 424, + 507, + 438 + ], + "score": 1.0, + "content": "• From Unstructured to Structured: Most sparse training works [32, 33, 36–39, 38, 34, 40,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 140, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 140, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "41, 35] restricted discussion to unstructured sparsity. To attain structured sparsity which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "is more hardware-friendly, unlike classical channel pruning available for convolutional", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 458, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 507, + 470 + ], + "score": 1.0, + "content": "networks, we customize a first-order importance approximation [16, 42] to guide the prune-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "and-grow of self-attention heads inside ViTs. This seamlessly extends SViTE to its second", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 479, + 430, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 387, + 492 + ], + "score": 1.0, + "content": "variant of Structured Sparse Vision Transformer Exploration", + "type": "text" + }, + { + "bbox": [ + 387, + 479, + 425, + 491 + ], + "score": 0.6, + "content": "\\mathbf { \\left( S ^ { 2 } V i T E \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 479, + 430, + 492 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 132, + 424, + 507, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 495, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 133, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 133, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "• From Model to Data: We further conduct a unified co-exploration towards joint data and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "architecture sparsity. That is by plugging in a novel learnable token selector to determine", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 141, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "the most vital patch embeddings in the current input sample. The resultant framework of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 321, + 540 + ], + "score": 1.0, + "content": "Sparse Vision Transformer Co-Exploration", + "type": "text" + }, + { + "bbox": [ + 321, + 529, + 359, + 539 + ], + "score": 0.76, + "content": "\\mathbf { \\eta } ( \\mathbf { S } \\mathbf { V i T E } +", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 528, + 505, + 540 + ], + "score": 1.0, + "content": ") remains to be end-to-end trainable", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 141, + 538, + 281, + 552 + ], + "spans": [ + { + "bbox": [ + 141, + 538, + 281, + 552 + ], + "score": 1.0, + "content": "and can gain additional efficiency.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 133, + 495, + 506, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 571 + ], + "score": 1.0, + "content": "Extensive experiments are conducted on ImageNet with DeiT-Tiny/Small/Base. Results of substantial", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 571, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 582 + ], + "score": 1.0, + "content": "computation savings and nearly undamaged accuracies consistently endorse our proposals’ effec-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 582, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 506, + 594 + ], + "score": 1.0, + "content": "tiveness. Perhaps most impressively, we find that the sparse (co-)training can even improve the ViT", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "accuracy rather than compromising it, making sparsity a tantalizing “free lunch”. For example, apply-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 121, + 616 + ], + "score": 1.0, + "content": "ing", + "type": "text" + }, + { + "bbox": [ + 122, + 604, + 156, + 614 + ], + "score": 0.68, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 603, + 370, + 616 + ], + "score": 1.0, + "content": "on DeiT-Small produces superior compressed ViTs at", + "type": "text" + }, + { + "bbox": [ + 370, + 603, + 390, + 614 + ], + "score": 0.89, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 603, + 470, + 616 + ], + "score": 1.0, + "content": "model sparsity plus", + "type": "text" + }, + { + "bbox": [ + 470, + 603, + 485, + 614 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 603, + 506, + 616 + ], + "score": 1.0, + "content": "data", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 613, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 170, + 628 + ], + "score": 1.0, + "content": "sparsity, saving", + "type": "text" + }, + { + "bbox": [ + 171, + 614, + 203, + 625 + ], + "score": 0.88, + "content": "4 9 . 3 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 613, + 252, + 628 + ], + "score": 1.0, + "content": "FLOPs and", + "type": "text" + }, + { + "bbox": [ + 252, + 614, + 280, + 625 + ], + "score": 0.9, + "content": "4 . 4 \\mathrm { { \\bar { 0 } } \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 613, + 506, + 628 + ], + "score": 1.0, + "content": "running time, while attaining a surprising improvement", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 117, + 639 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 625, + 145, + 636 + ], + "score": 0.9, + "content": "0 . 2 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 625, + 352, + 639 + ], + "score": 1.0, + "content": "accuracy; even when the data sparsity increases to", + "type": "text" + }, + { + "bbox": [ + 353, + 625, + 372, + 636 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "(the model sparsity unchanged),", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 635, + 498, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 331, + 649 + ], + "score": 1.0, + "content": "there is still no accuracy degradation, meanwhile saving", + "type": "text" + }, + { + "bbox": [ + 332, + 636, + 365, + 647 + ], + "score": 0.86, + "content": "5 2 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 635, + 413, + 649 + ], + "score": 1.0, + "content": "FLOPs and", + "type": "text" + }, + { + "bbox": [ + 413, + 636, + 441, + 647 + ], + "score": 0.88, + "content": "7 . { \\bar { 6 3 \\% } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 635, + 498, + 649 + ], + "score": 1.0, + "content": "running time.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 560, + 506, + 649 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 663, + 197, + 677 + ], + "lines": [ + { + "bbox": [ + 104, + 662, + 198, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 198, + 678 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 507, + 702 + ], + "score": 1.0, + "content": "Vision Transformer. Transformer [43] stems from natural language processing (NLP) applications.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "The Vision Transformer (ViT) [1] pioneered to leverage a pure transformer, to encode an image by", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "splitting it into a sequence of patches, projecting them into token embeddings, and feeding them to", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "transformer encoders. With sufficient training data, ViT is able to outperform convolution neural", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "networks on various image classification benchmarks [1, 44]. Many ViT variants have been proposed", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "since then. For example, DeiT [2] and T2T-ViT [45] are proposed to enhance ViT’s training data", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "efficiency, by leveraging teacher-student and better crafted architectures respectively. In addition", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "to image classification, ViT has attracted wide attention in diverse computer vision tasks, including", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "object detection [4–7], segmentation [46, 47], enhancement [8, 9], image generation [10–12], video", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 403, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 403, + 150 + ], + "score": 1.0, + "content": "understanding [48, 49], vision-language [50–57] and 3D point cloud [58].", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 687, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "transformer encoders. With sufficient training data, ViT is able to outperform convolution neural", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "networks on various image classification benchmarks [1, 44]. Many ViT variants have been proposed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "since then. For example, DeiT [2] and T2T-ViT [45] are proposed to enhance ViT’s training data", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "efficiency, by leveraging teacher-student and better crafted architectures respectively. In addition", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "to image classification, ViT has attracted wide attention in diverse computer vision tasks, including", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "object detection [4–7], segmentation [46, 47], enhancement [8, 9], image generation [10–12], video", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 403, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 403, + 150 + ], + "score": 1.0, + "content": "understanding [48, 49], vision-language [50–57] and 3D point cloud [58].", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Despite the impressive empirical performance, ViTs are generally heavy to train, and the trained", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "models remain massive. That naturally motivates the study to reduce ViT inference and training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "costs, by considering model compression means. Model compression has been well studied in both", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "computer vision and NLP applications [59–61, 42, 62, 21]. Two concurrent works [13, 63] made", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "initial attempts towards ViT post-training compression by pruning the intermediate features and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "tokens respectively, but did not jointly consider weight pruning nor efficient training. Another loosely", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "related field is the study of efficient attention mechanisms [64, 10, 52, 65–75]. They mainly reduce", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "the calculation complexity for self-attention modules via various approximations such as low-rank", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "decomposition. Our proposed techniques represent an orthogonal direction and can be potentially", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "combined with them, which we leave as future work. Another latest concurrent work [76] introduced", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "an interpretable module to dynamically and gracefully drop the redundant patches, gaining not only", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "inference efficiency but also interpretability. Being a unique and orthogonal effort from ours, their", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 312, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 312, + 299 + ], + "score": 1.0, + "content": "method did not consider the training efficiency yet.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "Pruning and Sparse Training. Pruning is well-known to effectively reduce deep network inference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 353, + 333 + ], + "score": 1.0, + "content": "costs [77, 15]. It can be roughly categorized into two groups:", + "type": "text" + }, + { + "bbox": [ + 353, + 320, + 365, + 332 + ], + "score": 0.48, + "content": "( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "unstructured pruning by removing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "insignificant weight elements per certain criterion, such as weight magnitude [78, 15], gradient [16]", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 177, + 354 + ], + "score": 1.0, + "content": "and hessian [79];", + "type": "text" + }, + { + "bbox": [ + 177, + 342, + 192, + 354 + ], + "score": 0.49, + "content": "( i i )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "structured pruning [80–82] by remove model sub-structures, e.g., channels [80,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "81] and attention heads [42], which are often more aligned with hardware efficiency. All above", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 364, + 459, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 459, + 376 + ], + "score": 1.0, + "content": "require training the full dense model first, usually for several train-prune-retrain rounds.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "The recent surge of sparse training seeks to adaptively identify high-quality sparse subnetworks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "and train only them. Starting from scratch, those methods learn to optimize the model weights", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "together with sparse connectivity simultaneously. [32, 33] first introduced the Sparse Evolutionary", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "Training (SET) technique [32], reaching superior performance compared to training with fixed sparse", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "connectivity [83, 36]. [37–39] leverages “weight reallocation\" to improve performance of obtained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "sparse subnetworks. Furthermore, gradient information from the backward pass is utilized to guide", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "the update of the dynamic sparse connectivity [38, 34], which produces substantial performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "gains. The latest investigations [40, 41, 35] demonstrate that more exhaustive exploration in the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "connectivity space plays a crucial role in the quality of found sparse subnetworks. Current sparse", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "training methods mostly focus on convolutional networks. Most of them discuss unstructured sparsity,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 490, + 484, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 484, + 502 + ], + "score": 1.0, + "content": "except a handful [84, 30] considering training convolutional networks with structured sparsity.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 516, + 192, + 531 + ], + "lines": [ + { + "bbox": [ + 104, + 514, + 194, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 194, + 534 + ], + "score": 1.0, + "content": "3 Methodology", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 541, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 247, + 554 + ], + "score": 1.0, + "content": "Our SViTE method (and its variants", + "type": "text" + }, + { + "bbox": [ + 247, + 541, + 280, + 552 + ], + "score": 0.6, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 541, + 296, + 554 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 542, + 330, + 553 + ], + "score": 0.57, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 541, + 506, + 554 + ], + "score": 1.0, + "content": ") is inspired from state-of-the-art sparse train-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 507, + 565 + ], + "score": 1.0, + "content": "ing approaches [34, 35] in CNNs. This section presents the sparse exploration of ViT architectures,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 563, + 444, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 444, + 577 + ], + "score": 1.0, + "content": "then shows the detailed procedure of input token selection for extra efficiency gains.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 588, + 232, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 232, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 232, + 604 + ], + "score": 1.0, + "content": "3.1 Sparse ViT Exploration", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "Revisiting sparse training. Sparse training starts from a randomly sparsified model; after optimiz-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 632 + ], + "score": 1.0, + "content": "ing several iterations, it shrinks a portion of parameters based on pre-defined pruning criterion, and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "activates new connections w.r.t. grow indicators. After upgrading the sparse topology, it trains the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "new subnetwork until the next update of the connectivity. An illustration of the overall procedure is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 652, + 507, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 333, + 666 + ], + "score": 1.0, + "content": "shown in Figure 1. The key factors of sparse training are", + "type": "text" + }, + { + "bbox": [ + 333, + 653, + 343, + 663 + ], + "score": 0.79, + "content": "\\bullet", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 652, + 427, + 666 + ], + "score": 1.0, + "content": "sparsity distribution,", + "type": "text" + }, + { + "bbox": [ + 427, + 653, + 437, + 663 + ], + "score": 0.77, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 652, + 507, + 666 + ], + "score": 1.0, + "content": "update schedule,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 663, + 239, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 116, + 673 + ], + "score": 0.74, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 663, + 167, + 676 + ], + "score": 1.0, + "content": "pruning and", + "type": "text" + }, + { + "bbox": [ + 167, + 664, + 177, + 674 + ], + "score": 0.75, + "content": "\\bullet", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 663, + 239, + 676 + ], + "score": 1.0, + "content": "grow criterion.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 482, + 700 + ], + "score": 1.0, + "content": "Notations. For a consistent description, we follow the standard notations in [34, 35]. Let", + "type": "text" + }, + { + "bbox": [ + 482, + 688, + 491, + 697 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 192, + 711 + ], + "score": 1.0, + "content": "the training dataset.", + "type": "text" + }, + { + "bbox": [ + 193, + 699, + 226, + 709 + ], + "score": 0.91, + "content": "b _ { t } \\sim \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 699, + 422, + 711 + ], + "score": 1.0, + "content": "is a randomly sampled data batch for iteration", + "type": "text" + }, + { + "bbox": [ + 422, + 699, + 427, + 708 + ], + "score": 0.56, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 699, + 433, + 711 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 434, + 699, + 459, + 710 + ], + "score": 0.86, + "content": "f _ { W } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "represents", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 220, + 723 + ], + "score": 1.0, + "content": "the model with parameters", + "type": "text" + }, + { + "bbox": [ + 220, + 709, + 324, + 723 + ], + "score": 0.92, + "content": "W = ( W ^ { ( 1 ) } , \\cdots , W ^ { ( L ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 708, + 357, + 723 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 357, + 709, + 478, + 722 + ], + "score": 0.89, + "content": "W ^ { ( l ) } \\in \\mathbb { R } ^ { N _ { l } } , 1 \\le l \\le L , N _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 150 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 72, + 505, + 150 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Despite the impressive empirical performance, ViTs are generally heavy to train, and the trained", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "models remain massive. That naturally motivates the study to reduce ViT inference and training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "costs, by considering model compression means. Model compression has been well studied in both", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "computer vision and NLP applications [59–61, 42, 62, 21]. Two concurrent works [13, 63] made", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "initial attempts towards ViT post-training compression by pruning the intermediate features and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "tokens respectively, but did not jointly consider weight pruning nor efficient training. Another loosely", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "related field is the study of efficient attention mechanisms [64, 10, 52, 65–75]. They mainly reduce", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "the calculation complexity for self-attention modules via various approximations such as low-rank", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "decomposition. Our proposed techniques represent an orthogonal direction and can be potentially", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "combined with them, which we leave as future work. Another latest concurrent work [76] introduced", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "an interpretable module to dynamically and gracefully drop the redundant patches, gaining not only", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "inference efficiency but also interpretability. Being a unique and orthogonal effort from ours, their", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 285, + 312, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 312, + 299 + ], + "score": 1.0, + "content": "method did not consider the training efficiency yet.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 154, + 506, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "Pruning and Sparse Training. Pruning is well-known to effectively reduce deep network inference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 353, + 333 + ], + "score": 1.0, + "content": "costs [77, 15]. It can be roughly categorized into two groups:", + "type": "text" + }, + { + "bbox": [ + 353, + 320, + 365, + 332 + ], + "score": 0.48, + "content": "( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "unstructured pruning by removing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "insignificant weight elements per certain criterion, such as weight magnitude [78, 15], gradient [16]", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 177, + 354 + ], + "score": 1.0, + "content": "and hessian [79];", + "type": "text" + }, + { + "bbox": [ + 177, + 342, + 192, + 354 + ], + "score": 0.49, + "content": "( i i )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "structured pruning [80–82] by remove model sub-structures, e.g., channels [80,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "81] and attention heads [42], which are often more aligned with hardware efficiency. All above", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 364, + 459, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 459, + 376 + ], + "score": 1.0, + "content": "require training the full dense model first, usually for several train-prune-retrain rounds.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 308, + 506, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "The recent surge of sparse training seeks to adaptively identify high-quality sparse subnetworks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "and train only them. Starting from scratch, those methods learn to optimize the model weights", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "together with sparse connectivity simultaneously. [32, 33] first introduced the Sparse Evolutionary", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "Training (SET) technique [32], reaching superior performance compared to training with fixed sparse", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "connectivity [83, 36]. [37–39] leverages “weight reallocation\" to improve performance of obtained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "sparse subnetworks. Furthermore, gradient information from the backward pass is utilized to guide", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "the update of the dynamic sparse connectivity [38, 34], which produces substantial performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "gains. The latest investigations [40, 41, 35] demonstrate that more exhaustive exploration in the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 480 + ], + "score": 1.0, + "content": "connectivity space plays a crucial role in the quality of found sparse subnetworks. Current sparse", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "training methods mostly focus on convolutional networks. Most of them discuss unstructured sparsity,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 490, + 484, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 484, + 502 + ], + "score": 1.0, + "content": "except a handful [84, 30] considering training convolutional networks with structured sparsity.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 380, + 506, + 502 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 516, + 192, + 531 + ], + "lines": [ + { + "bbox": [ + 104, + 514, + 194, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 194, + 534 + ], + "score": 1.0, + "content": "3 Methodology", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 108, + 541, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 247, + 554 + ], + "score": 1.0, + "content": "Our SViTE method (and its variants", + "type": "text" + }, + { + "bbox": [ + 247, + 541, + 280, + 552 + ], + "score": 0.6, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 541, + 296, + 554 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 542, + 330, + 553 + ], + "score": 0.57, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 541, + 506, + 554 + ], + "score": 1.0, + "content": ") is inspired from state-of-the-art sparse train-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 507, + 565 + ], + "score": 1.0, + "content": "ing approaches [34, 35] in CNNs. This section presents the sparse exploration of ViT architectures,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 563, + 444, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 444, + 577 + ], + "score": 1.0, + "content": "then shows the detailed procedure of input token selection for extra efficiency gains.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 541, + 507, + 577 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 588, + 232, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 232, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 232, + 604 + ], + "score": 1.0, + "content": "3.1 Sparse ViT Exploration", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "Revisiting sparse training. Sparse training starts from a randomly sparsified model; after optimiz-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 620, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 632 + ], + "score": 1.0, + "content": "ing several iterations, it shrinks a portion of parameters based on pre-defined pruning criterion, and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "activates new connections w.r.t. grow indicators. After upgrading the sparse topology, it trains the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "new subnetwork until the next update of the connectivity. An illustration of the overall procedure is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 652, + 507, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 333, + 666 + ], + "score": 1.0, + "content": "shown in Figure 1. The key factors of sparse training are", + "type": "text" + }, + { + "bbox": [ + 333, + 653, + 343, + 663 + ], + "score": 0.79, + "content": "\\bullet", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 652, + 427, + 666 + ], + "score": 1.0, + "content": "sparsity distribution,", + "type": "text" + }, + { + "bbox": [ + 427, + 653, + 437, + 663 + ], + "score": 0.77, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 652, + 507, + 666 + ], + "score": 1.0, + "content": "update schedule,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 663, + 239, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 116, + 673 + ], + "score": 0.74, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 663, + 167, + 676 + ], + "score": 1.0, + "content": "pruning and", + "type": "text" + }, + { + "bbox": [ + 167, + 664, + 177, + 674 + ], + "score": 0.75, + "content": "\\bullet", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 663, + 239, + 676 + ], + "score": 1.0, + "content": "grow criterion.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 609, + 507, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 482, + 700 + ], + "score": 1.0, + "content": "Notations. For a consistent description, we follow the standard notations in [34, 35]. 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Upper Figure:", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 192, + 312 + ], + "score": 1.0, + "content": "first training ViT for", + "type": "text" + }, + { + "bbox": [ + 192, + 300, + 209, + 310 + ], + "score": 0.74, + "content": "\\Delta \\mathrm { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "iterations, then performing prune-and-grow strategies to explore critical", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "sparse connectivities, repreating until convergence. 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BackboneUpdate Schedule{△T,Tend,α,fdecay}Batch SizeEpochsInherited Settings from DeiT[2]
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To optimize parameters of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "scorer function, we introduce the popular Gumbel-Softmax [86, 87] and straight-through tricks [88]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 314, + 248 + ], + "score": 1.0, + "content": "to enable gradient back-propagation through the top-", + "type": "text" + }, + { + "bbox": [ + 315, + 236, + 321, + 245 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "selection, which provides an efficient solution", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "to draw samples from a discrete probability distribution. A detailed implementation is in Algorithm 2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "The full pipeline of data and architecture co-exploration is summarized in Algorithm 1. We term this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 145, + 286 + ], + "score": 1.0, + "content": "approach", + "type": "text" + }, + { + "bbox": [ + 146, + 273, + 179, + 284 + ], + "score": 0.59, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 272, + 506, + 286 + ], + "score": 1.0, + "content": ". We first feed the randomly sampled data batch to the token selector and pick the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 123, + 297 + ], + "score": 1.0, + "content": "top-", + "type": "text" + }, + { + "bbox": [ + 123, + 285, + 129, + 294 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 284, + 447, + 297 + ], + "score": 1.0, + "content": "informative token embeddings. Then, we alternatively train the sparse ViT for", + "type": "text" + }, + { + "bbox": [ + 447, + 285, + 464, + 295 + ], + "score": 0.71, + "content": "\\Delta \\mathrm { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "iterations", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "and perform prune-and-grow to explore the sparse connectivity in ViTs dynamically. In the end, a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 306, + 441, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 441, + 319 + ], + "score": 1.0, + "content": "sparse ViT model with a trained token selector is returned and ready for evaluation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 328, + 191, + 342 + ], + "lines": [ + { + "bbox": [ + 104, + 326, + 193, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 193, + 345 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 506, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "Baseline pruning methods. We extend several effective pruning methods from CNN compression", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 294, + 373 + ], + "score": 1.0, + "content": "as our strong baselines. Unstructured pruning:", + "type": "text" + }, + { + "bbox": [ + 294, + 360, + 306, + 371 + ], + "score": 0.38, + "content": "( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "One-shot weight Magnitude Pruning (OMP) [15],", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "which removes insignificant parameters with the globally smallest weight values; (ii) Gradually", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "Magnitude Pruning (GMP) [17], which seamlessly incorporates gradual pruning techniques within", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "the training process by eliminating a few small magnitude weights per iteration; and (iii) Taylor", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "Pruning (TP) [16], which utilizes the first-order approximation of the training loss to estimate units’", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "importance for model sparsification. Structured pruning: Salience-based Structured Pruning (SSP).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "We draw inspiration from [42, 85], and remove sub-modules in ViT (e.g., self-attention heads)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "by leveraging their weight, activation, and gradient information. 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The overall performance of SViTE,", + "type": "text" + }, + { + "bbox": [ + 376, + 699, + 410, + 711 + ], + "score": 0.64, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 699, + 432, + 712 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 432, + 699, + 467, + 711 + ], + "score": 0.74, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 699, + 504, + 712 + ], + "score": 1.0, + "content": "on DeiT", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 711, + 416, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 416, + 723 + ], + "score": 1.0, + "content": "backbones are summarized in Figure 2. We highlight some takeaways below.", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 65.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 90, + 504, + 133 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 78, + 502, + 88 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 75, + 502, + 91 + ], + "spans": [ + { + "bbox": [ + 106, + 75, + 502, + 91 + ], + "score": 1.0, + "content": "Table 1: Details of training configurations in our experiments, mainly following the settings in [2].", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 90, + 504, + 133 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 90, + 504, + 133 + ], + "spans": [ + { + "bbox": [ + 108, + 90, + 504, + 133 + ], + "score": 0.975, + "html": "
BackboneUpdate Schedule{△T,Tend,α,fdecay}Batch SizeEpochsInherited Settings from DeiT[2]
DeiT-Tiny{20000,1200000,0.5,cosine}512600AdamW, 0.0005 × batchsize,cosine decay
DeiT-Small{15000,1200000,0.5,cosine}512600warmup 5 epochs,0.05 weight decay
DeiT-Base{7000,600000,0.5,cosine}10246000.1 label smoothing,augmentations, etc.
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As shown in Figure 1, all token", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "embeddings are passed through a learnable scorer function which is parameterized by an MLP in our", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 276, + 215 + ], + "score": 1.0, + "content": "experiments. 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To optimize parameters of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "scorer function, we introduce the popular Gumbel-Softmax [86, 87] and straight-through tricks [88]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 314, + 248 + ], + "score": 1.0, + "content": "to enable gradient back-propagation through the top-", + "type": "text" + }, + { + "bbox": [ + 315, + 236, + 321, + 245 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "selection, which provides an efficient solution", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "to draw samples from a discrete probability distribution. A detailed implementation is in Algorithm 2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 167, + 506, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "The full pipeline of data and architecture co-exploration is summarized in Algorithm 1. We term this", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 145, + 286 + ], + "score": 1.0, + "content": "approach", + "type": "text" + }, + { + "bbox": [ + 146, + 273, + 179, + 284 + ], + "score": 0.59, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 272, + 506, + 286 + ], + "score": 1.0, + "content": ". We first feed the randomly sampled data batch to the token selector and pick the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 123, + 297 + ], + "score": 1.0, + "content": "top-", + "type": "text" + }, + { + "bbox": [ + 123, + 285, + 129, + 294 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 284, + 447, + 297 + ], + "score": 1.0, + "content": "informative token embeddings. Then, we alternatively train the sparse ViT for", + "type": "text" + }, + { + "bbox": [ + 447, + 285, + 464, + 295 + ], + "score": 0.71, + "content": "\\Delta \\mathrm { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "iterations", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "and perform prune-and-grow to explore the sparse connectivity in ViTs dynamically. In the end, a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 306, + 441, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 441, + 319 + ], + "score": 1.0, + "content": "sparse ViT model with a trained token selector is returned and ready for evaluation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 262, + 506, + 319 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 328, + 191, + 342 + ], + "lines": [ + { + "bbox": [ + 104, + 326, + 193, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 193, + 345 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 506, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "Baseline pruning methods. We extend several effective pruning methods from CNN compression", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 294, + 373 + ], + "score": 1.0, + "content": "as our strong baselines. Unstructured pruning:", + "type": "text" + }, + { + "bbox": [ + 294, + 360, + 306, + 371 + ], + "score": 0.38, + "content": "( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "One-shot weight Magnitude Pruning (OMP) [15],", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "which removes insignificant parameters with the globally smallest weight values; (ii) Gradually", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "Magnitude Pruning (GMP) [17], which seamlessly incorporates gradual pruning techniques within", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "the training process by eliminating a few small magnitude weights per iteration; and (iii) Taylor", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "Pruning (TP) [16], which utilizes the first-order approximation of the training loss to estimate units’", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "importance for model sparsification. Structured pruning: Salience-based Structured Pruning (SSP).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "We draw inspiration from [42, 85], and remove sub-modules in ViT (e.g., self-attention heads)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "by leveraging their weight, activation, and gradient information. 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To be specific,", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "we separately calculate the time elapsed during each iteration, to eliminate the impact of the hardware", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 681, + 431, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 431, + 693 + ], + "score": 1.0, + "content": "environment as much as possible. Note that the time for the data I/O is excluded.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 63, + "bbox_fs": [ + 105, + 659, + 506, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 376, + 712 + ], + "score": 1.0, + "content": "Highlight of our findings. The overall performance of SViTE,", + "type": "text" + }, + { + "bbox": [ + 376, + 699, + 410, + 711 + ], + "score": 0.64, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 699, + 432, + 712 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 432, + 699, + 467, + 711 + ], + "score": 0.74, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 699, + 504, + 712 + ], + "score": 1.0, + "content": "on DeiT", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 711, + 416, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 416, + 723 + ], + "score": 1.0, + "content": "backbones are summarized in Figure 2. We highlight some takeaways below.", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 65.5, + "bbox_fs": [ + 105, + 699, + 504, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 72, + 507, + 141 + ], + "lines": [ + { + "bbox": [ + 109, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 109, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "Takeaways: ❶ SViTE produces sparse DeiTs with enhanced generalization and substantial reduced", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 109, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 109, + 83, + 291, + 97 + ], + "score": 1.0, + "content": "FLOPs, compared to its dense counterpart", + "type": "text" + }, + { + "bbox": [ + 292, + 85, + 304, + 96 + ], + "score": 0.78, + "content": "( { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 83, + 312, + 97 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 313, + 85, + 348, + 95 + ], + "score": 0.72, + "content": "{ \\mathrm { S V i T E } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "further improves the performance of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 110, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 110, + 95, + 295, + 108 + ], + "score": 1.0, + "content": "SViTE by selecting the most vital patches. ❷", + "type": "text" + }, + { + "bbox": [ + 295, + 95, + 329, + 106 + ], + "score": 0.58, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 95, + 506, + 108 + ], + "score": 1.0, + "content": "achieves matched accuracy on DeiT-Small,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 109, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 109, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "and significantly enhances performance on DeiT-Base. Meanwhile, its structural sparsity brings", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 110, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 110, + 117, + 254, + 130 + ], + "score": 1.0, + "content": "considerable running time savings.", + "type": "text" + }, + { + "bbox": [ + 254, + 118, + 264, + 128 + ], + "score": 0.59, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "Appropriate data and architecture sparsities can effectively", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 109, + 127, + 488, + 141 + ], + "spans": [ + { + "bbox": [ + 109, + 127, + 488, + 141 + ], + "score": 1.0, + "content": "regularize ViT training, leading to a new SOTA win-win between ViT accuracy and efficiency.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 158, + 277, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 279, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 279, + 172 + ], + "score": 1.0, + "content": "4.1 SViTE with Unstructured Sparsity", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "We perform SViTE to mine vital unstructured sparsity in DeiTs [2]. Solid lines in Figure 2 record", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "the top-1 test-set accuracy over FLOPs on ImageNet-1K of SViTE-Small and SViTE-Base with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 213, + 213 + ], + "score": 1.0, + "content": "a range of sparsity from", + "type": "text" + }, + { + "bbox": [ + 213, + 200, + 233, + 211 + ], + "score": 0.87, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 199, + 247, + 213 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 247, + 200, + 267, + 210 + ], + "score": 0.89, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 199, + 506, + 213 + ], + "score": 1.0, + "content": ". In general, we observe that SViTE generates superior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "score": 1.0, + "content": "sparse ViTs with both accuracy and efficiency gains. Table 2, 3, and 5 present the comparison", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "between SViTE and various pruning baselines. From these extensive results, we draw several", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 244 + ], + "score": 1.0, + "content": "consistent observations. 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It verifies the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 278 + ], + "score": 1.0, + "content": "effectiveness of our proposal, and indicates severe parameter redundancy in ViT. Second, our SViTE", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "models from dynamic explorations consistently surpass other competitive baseline methods, including", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "OMP, GMP, TP, and Small-Dense by a substantial performance margin. Among all the baseline", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "approaches, GMP that advocates a gradual pruning schedule achieves the best accuracy with all three", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "score": 1.0, + "content": "DeiT backbones. Third, in Figure 2, both SViTE-Small (blue solid line) and SViTE-Base (green", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "solid line) show an improved trade-off between accuracy and efficiency, compared to their dense", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "DeiT counterparts. Interestingly, we also observe that with similar parameter counts, a large sparse", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "ViT consistently outperforms the corresponding smaller dense ViT. A possible explanation is those", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "appropriate sparse typologies regularize network training and lead to enhanced generalization, which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "coincides with recent findings of critical subnetworks (i.e., winning tickets) in dense CNNs [89, 22]", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 262, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 262, + 386 + ], + "score": 1.0, + "content": "and NLP transformer [21, 90] models.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 16 + }, + { + "type": "table", + "bbox": [ + 108, + 419, + 302, + 521 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 389, + 302, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 388, + 303, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 303, + 400 + ], + "score": 1.0, + "content": "Table 2: Results of SViTE-Tiny on ImageNet-1K.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 398, + 303, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 148, + 410 + ], + "score": 1.0, + "content": "Accuracies", + "type": "text" + }, + { + "bbox": [ + 148, + 399, + 162, + 408 + ], + "score": 0.67, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 398, + 303, + 410 + ], + "score": 1.0, + "content": "within/out of parenthesis are the repro-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 408, + 225, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 225, + 420 + ], + "score": 1.0, + "content": "duced/reported [2] performance.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 108, + 419, + 302, + 521 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 419, + 302, + 521 + ], + "spans": [ + { + "bbox": [ + 108, + 419, + 302, + 521 + ], + "score": 0.975, + "html": "
ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Tiny0% (5.72M)0%72.20 (71.80)
SViTE-Tiny30% (4.02M)25.56%71.78
OMP30% (4.02M)25.56%68.35
GMP30% (4.02M)25.56%69.56
TP30% (4.02M)25.56%68.38
SViTE-Tiny40% (3.46M)34.16%71.75
OMP40% (3.46M)34.16%66.52
GMP40% (3.46M)34.15%68.36
TP40% (3.46M)34.17%65.45
Small-Dense0% (3.94M)32.54%67.33
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ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Small0% (22.1M)0%79.90 (79.78)
SViTE-Small50% (11.1M)46.26%79.72
OMP50% (11.1M)46.25%76.32
GMP50% (11.1M)46.26%76.88
TP50% (11.1M)46.26%76.30
SViTE-Small60% (8.9M)55.44%79.41
OMP60% (8.9M)55.44%75.32
GMP60% (8.9M)55.44%76.79
TP60% (8.9M)55.44%74.50
Small-Dense0% (11.4M)49.32%73.93
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ModelsSparsity (%)ParametersFLOPs SavingRunning Time Reduced|Top-1 Accuracy (%)
DeiT-Tiny (Dense)0%5.72M0%0%72.20 (71.80)
SViTE-Tiny (Unstructured)30%4.02M25.56%0%71.78
SSP-Tiny (Structured)30%4.21M23.69%10.57%68.59
S2ViTE-Tiny (Structured)30%4.21M23.69%10.57%70.12
DeiT-Small (Dense)0%22.1M0%0%79.90 (79.78)
SViTE-Small (Unstructured)40%13.3M36.73%0%80.26
SSP-Small (Structured)40%14.6M31.63%22.65%77.74
S²ViTE-Small (Structured)40%14.6M31.63%22.65%79.22
DeiT-Base (Dense)0%86.6M0%0%81.80 (80.98)
SViTE-Base (Unstructured)40%52.0M38.30%0%81.56
SSP-Base (Structured)40%56.8M33.13%24.70%80.08
S2ViTE-Base (Structured)40%56.8M33.13%24.70%82.22
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Results are summa-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 264, + 723 + ], + "score": 1.0, + "content": "rized in Table 4. Besides the obtained", + "type": "text" + }, + { + "bbox": [ + 264, + 711, + 342, + 722 + ], + "score": 0.91, + "content": "2 3 . { \\bar { 7 } } 9 \\% \\sim 3 3 . 6 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 709, + 410, + 723 + ], + "score": 1.0, + "content": "FLOPs savings,", + "type": "text" + }, + { + "bbox": [ + 410, + 710, + 421, + 721 + ], + "score": 0.62, + "content": "{ \\mathsf { S } } ^ { \\tilde { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 709, + 468, + 723 + ], + "score": 1.0, + "content": "ViTE-Tiny,", + "type": "text" + }, + { + "bbox": [ + 469, + 710, + 480, + 721 + ], + "score": 0.6, + "content": "S ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "ViTE-", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 72, + 507, + 141 + ], + "lines": [ + { + "bbox": [ + 109, + 73, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 109, + 73, + 506, + 86 + ], + "score": 1.0, + "content": "Takeaways: ❶ SViTE produces sparse DeiTs with enhanced generalization and substantial reduced", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 109, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 109, + 83, + 291, + 97 + ], + "score": 1.0, + "content": "FLOPs, compared to its dense counterpart", + "type": "text" + }, + { + "bbox": [ + 292, + 85, + 304, + 96 + ], + "score": 0.78, + "content": "( { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 83, + 312, + 97 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 313, + 85, + 348, + 95 + ], + "score": 0.72, + "content": "{ \\mathrm { S V i T E } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "further improves the performance of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 110, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 110, + 95, + 295, + 108 + ], + "score": 1.0, + "content": "SViTE by selecting the most vital patches. ❷", + "type": "text" + }, + { + "bbox": [ + 295, + 95, + 329, + 106 + ], + "score": 0.58, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 95, + 506, + 108 + ], + "score": 1.0, + "content": "achieves matched accuracy on DeiT-Small,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 109, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 109, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "and significantly enhances performance on DeiT-Base. Meanwhile, its structural sparsity brings", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 110, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 110, + 117, + 254, + 130 + ], + "score": 1.0, + "content": "considerable running time savings.", + "type": "text" + }, + { + "bbox": [ + 254, + 118, + 264, + 128 + ], + "score": 0.59, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "Appropriate data and architecture sparsities can effectively", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 109, + 127, + 488, + 141 + ], + "spans": [ + { + "bbox": [ + 109, + 127, + 488, + 141 + ], + "score": 1.0, + "content": "regularize ViT training, leading to a new SOTA win-win between ViT accuracy and efficiency.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 109, + 73, + 506, + 141 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 158, + 277, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 279, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 279, + 172 + ], + "score": 1.0, + "content": "4.1 SViTE with Unstructured Sparsity", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "We perform SViTE to mine vital unstructured sparsity in DeiTs [2]. Solid lines in Figure 2 record", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "the top-1 test-set accuracy over FLOPs on ImageNet-1K of SViTE-Small and SViTE-Base with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 213, + 213 + ], + "score": 1.0, + "content": "a range of sparsity from", + "type": "text" + }, + { + "bbox": [ + 213, + 200, + 233, + 211 + ], + "score": 0.87, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 199, + 247, + 213 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 247, + 200, + 267, + 210 + ], + "score": 0.89, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 199, + 506, + 213 + ], + "score": 1.0, + "content": ". In general, we observe that SViTE generates superior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "score": 1.0, + "content": "sparse ViTs with both accuracy and efficiency gains. Table 2, 3, and 5 present the comparison", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "between SViTE and various pruning baselines. From these extensive results, we draw several", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 244 + ], + "score": 1.0, + "content": "consistent observations. First, compared to the dense baselines, SViTE-Tiny, -Small, and -Base obtain", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 241, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 187, + 254 + ], + "score": 0.87, + "content": "2 5 . 5 6 \\% \\sim 3 4 . 1 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 241, + 191, + 257 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 191, + 243, + 272, + 254 + ], + "score": 0.9, + "content": "4 6 . 2 6 \\% \\sim 5 5 . 4 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 241, + 294, + 257 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 294, + 243, + 375, + 254 + ], + "score": 0.91, + "content": "4 7 . 9 5 \\% \\sim 5 7 . 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 241, + 506, + 257 + ], + "score": 1.0, + "content": "FLOPs reduction, respectively,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 117, + 267 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 254, + 171, + 265 + ], + "score": 0.9, + "content": "3 0 \\% \\sim 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 254, + 418, + 267 + ], + "score": 1.0, + "content": "sparsity levels with only a negligible accuracy drop within", + "type": "text" + }, + { + "bbox": [ + 419, + 254, + 441, + 265 + ], + "score": 0.87, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 254, + 505, + 267 + ], + "score": 1.0, + "content": ". It verifies the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 278 + ], + "score": 1.0, + "content": "effectiveness of our proposal, and indicates severe parameter redundancy in ViT. Second, our SViTE", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "models from dynamic explorations consistently surpass other competitive baseline methods, including", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "OMP, GMP, TP, and Small-Dense by a substantial performance margin. Among all the baseline", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "approaches, GMP that advocates a gradual pruning schedule achieves the best accuracy with all three", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "score": 1.0, + "content": "DeiT backbones. Third, in Figure 2, both SViTE-Small (blue solid line) and SViTE-Base (green", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "solid line) show an improved trade-off between accuracy and efficiency, compared to their dense", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "DeiT counterparts. Interestingly, we also observe that with similar parameter counts, a large sparse", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "ViT consistently outperforms the corresponding smaller dense ViT. A possible explanation is those", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "appropriate sparse typologies regularize network training and lead to enhanced generalization, which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "coincides with recent findings of critical subnetworks (i.e., winning tickets) in dense CNNs [89, 22]", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 262, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 262, + 386 + ], + "score": 1.0, + "content": "and NLP transformer [21, 90] models.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 178, + 506, + 386 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 419, + 302, + 521 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 389, + 302, + 419 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 388, + 303, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 303, + 400 + ], + "score": 1.0, + "content": "Table 2: Results of SViTE-Tiny on ImageNet-1K.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 398, + 303, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 148, + 410 + ], + "score": 1.0, + "content": "Accuracies", + "type": "text" + }, + { + "bbox": [ + 148, + 399, + 162, + 408 + ], + "score": 0.67, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 398, + 303, + 410 + ], + "score": 1.0, + "content": "within/out of parenthesis are the repro-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 408, + 225, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 225, + 420 + ], + "score": 1.0, + "content": "duced/reported [2] performance.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "table_body", + "bbox": [ + 108, + 419, + 302, + 521 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 419, + 302, + 521 + ], + "spans": [ + { + "bbox": [ + 108, + 419, + 302, + 521 + ], + "score": 0.975, + "html": "
ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Tiny0% (5.72M)0%72.20 (71.80)
SViTE-Tiny30% (4.02M)25.56%71.78
OMP30% (4.02M)25.56%68.35
GMP30% (4.02M)25.56%69.56
TP30% (4.02M)25.56%68.38
SViTE-Tiny40% (3.46M)34.16%71.75
OMP40% (3.46M)34.16%66.52
GMP40% (3.46M)34.15%68.36
TP40% (3.46M)34.17%65.45
Small-Dense0% (3.94M)32.54%67.33
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ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Small0% (22.1M)0%79.90 (79.78)
SViTE-Small50% (11.1M)46.26%79.72
OMP50% (11.1M)46.25%76.32
GMP50% (11.1M)46.26%76.88
TP50% (11.1M)46.26%76.30
SViTE-Small60% (8.9M)55.44%79.41
OMP60% (8.9M)55.44%75.32
GMP60% (8.9M)55.44%76.79
TP60% (8.9M)55.44%74.50
Small-Dense0% (11.4M)49.32%73.93
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ModelsSparsity (%)ParametersFLOPs SavingRunning Time Reduced|Top-1 Accuracy (%)
DeiT-Tiny (Dense)0%5.72M0%0%72.20 (71.80)
SViTE-Tiny (Unstructured)30%4.02M25.56%0%71.78
SSP-Tiny (Structured)30%4.21M23.69%10.57%68.59
S2ViTE-Tiny (Structured)30%4.21M23.69%10.57%70.12
DeiT-Small (Dense)0%22.1M0%0%79.90 (79.78)
SViTE-Small (Unstructured)40%13.3M36.73%0%80.26
SSP-Small (Structured)40%14.6M31.63%22.65%77.74
S²ViTE-Small (Structured)40%14.6M31.63%22.65%79.22
DeiT-Base (Dense)0%86.6M0%0%81.80 (80.98)
SViTE-Base (Unstructured)40%52.0M38.30%0%81.56
SSP-Base (Structured)40%56.8M33.13%24.70%80.08
S2ViTE-Base (Structured)40%56.8M33.13%24.70%82.22
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ModelsSparsity (#Para.)FLOPs Saving|Accuracy (%)
DeiT-Base0% (86.6M)0%81.80 (80.98)
SViTE-Base50% (43.4M)47.95%81.51
OMP50% (43.4M)47.94%80.26
GMP50% (43.4M)47.95%80.79
TP50% (43.4M)47.94%80.55
SViTE-Base60% (34.8M)57.50%81.28
OMP60% (34.8M)57.50%80.25
GMP60% (34.8M)57.50%80.44
TP60% (34.8M)57.49%80.37
Small-Dense0% (44.0M)49.46%78.59
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#Tokens (%)|Time ReducedFLOPs Saving|Accuracy (%)
SViTE+-Small 50% Unstructured Sparsity
100%0%46.26%79.72
95%4.40%49.32%80.18
90%7.63%52.38%79.91
70%19.77%63.95%77.90
S²ViTE+-Small 40% Structured Sparsity
100%22.65%31.63%79.22
95%27.17%37.76%78.44
90%29.21%41.50%78.16
70%39.10%54.96%74.77
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ModelsSparsity (#Para.)FLOPs Saving|Accuracy (%)
DeiT-Base0% (86.6M)0%81.80 (80.98)
SViTE-Base50% (43.4M)47.95%81.51
OMP50% (43.4M)47.94%80.26
GMP50% (43.4M)47.95%80.79
TP50% (43.4M)47.94%80.55
SViTE-Base60% (34.8M)57.50%81.28
OMP60% (34.8M)57.50%80.25
GMP60% (34.8M)57.50%80.44
TP60% (34.8M)57.49%80.37
Small-Dense0% (44.0M)49.46%78.59
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#Tokens (%)|Time ReducedFLOPs Saving|Accuracy (%)
SViTE+-Small 50% Unstructured Sparsity
100%0%46.26%79.72
95%4.40%49.32%80.18
90%7.63%52.38%79.91
70%19.77%63.95%77.90
S²ViTE+-Small 40% Structured Sparsity
100%22.65%31.63%79.22
95%27.17%37.76%78.44
90%29.21%41.50%78.16
70%39.10%54.96%74.77
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We provide unit-wise and element-wise heatmap visualizations for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 182, + 561 + ], + "score": 1.0, + "content": "SViTE-Base with", + "type": "text" + }, + { + "bbox": [ + 182, + 549, + 202, + 559 + ], + "score": 0.88, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "structured sparsity in Figure A7 (in Appendix). 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As shown in Figure 4, we utilize tools in [94] to visualize attention maps", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "of (sparse) ViTs. Multiple attention heads show similar behaviors, which implies the structural", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 213, + 636 + ], + "score": 1.0, + "content": "redundancy. Fortunately,", + "type": "text" + }, + { + "bbox": [ + 213, + 623, + 247, + 635 + ], + "score": 0.53, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "eliminates unnecessary heads to some extent. 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It is worth mentioning that our proposed frame-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 168, + 450 + ], + "score": 1.0, + "content": "works (SViTE,", + "type": "text" + }, + { + "bbox": [ + 169, + 437, + 202, + 448 + ], + "score": 0.42, + "content": "\\mathrm { S ^ { 2 } V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 436, + 205, + 450 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 438, + 240, + 448 + ], + "score": 0.28, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 436, + 505, + 450 + ], + "score": 1.0, + "content": ") are independent of the backbone architectures, and can be easily", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "plugged in other vision transformer models [91, 45, 92, 93]. 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TNT-S:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 153, + 483 + ], + "score": 1.0, + "content": "81.50) and", + "type": "text" + }, + { + "bbox": [ + 153, + 470, + 185, + 480 + ], + "score": 0.87, + "content": "3 7 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 469, + 261, + 483 + ], + "score": 1.0, + "content": "FLOPs savings at", + "type": "text" + }, + { + "bbox": [ + 261, + 470, + 281, + 481 + ], + "score": 0.88, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 469, + 373, + 483 + ], + "score": 1.0, + "content": "unstructured sparsity;", + "type": "text" + }, + { + "bbox": [ + 374, + 469, + 385, + 481 + ], + "score": 0.61, + "content": "S ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 469, + 471, + 483 + ], + "score": 1.0, + "content": "ViTE-TNT-S obtains", + "type": "text" + }, + { + "bbox": [ + 472, + 470, + 505, + 481 + ], + "score": 0.87, + "content": "3 2 . 9 6 \\%", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 156, + 494 + ], + "score": 1.0, + "content": "FLOPs and", + "type": "text" + }, + { + "bbox": [ + 157, + 481, + 189, + 492 + ], + "score": 0.89, + "content": "2 3 . 7 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 480, + 303, + 494 + ], + "score": 1.0, + "content": "running time reductions at", + "type": "text" + }, + { + "bbox": [ + 304, + 481, + 324, + 492 + ], + "score": 0.88, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "structured sparsity with almost unimpaired", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 492, + 275, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 275, + 504 + ], + "score": 1.0, + "content": "accuracy (Ours: 81.34 v.s. TNT-S:81.50).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 425, + 507, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 516, + 186, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 187, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 187, + 530 + ], + "score": 1.0, + "content": "4.5 Visualization", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Sparse connectivity patterns. We provide unit-wise and element-wise heatmap visualizations for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 182, + 561 + ], + "score": 1.0, + "content": "SViTE-Base with", + "type": "text" + }, + { + "bbox": [ + 182, + 549, + 202, + 559 + ], + "score": 0.88, + "content": "4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "structured sparsity in Figure A7 (in Appendix). Similarly, element-wise", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 286, + 572 + ], + "score": 1.0, + "content": "heatmap visualizations of SViTE-Base with", + "type": "text" + }, + { + "bbox": [ + 286, + 559, + 306, + 570 + ], + "score": 0.89, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "unstructured sparsity are displayed in Figure A6.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 569, + 507, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 507, + 584 + ], + "score": 1.0, + "content": "We find that even unstructured sparsity exploration can develop obvious structural patterns (i.e.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 580, + 496, + 594 + ], + "spans": [ + { + "bbox": [ + 104, + 580, + 496, + 594 + ], + "score": 1.0, + "content": "“vertical lines” in mask heatmaps), which implies a stronger potential for hardware speedup [95].", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 537, + 507, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "Self-attention heatmaps. As shown in Figure 4, we utilize tools in [94] to visualize attention maps", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "of (sparse) ViTs. Multiple attention heads show similar behaviors, which implies the structural", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 213, + 636 + ], + "score": 1.0, + "content": "redundancy. Fortunately,", + "type": "text" + }, + { + "bbox": [ + 213, + 623, + 247, + 635 + ], + "score": 0.53, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "eliminates unnecessary heads to some extent. With regard to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "SViTE-Base’s visual results, it seems to activate fewer attention heads for predictions (darker colors", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "mean larger values), compared to the ones of dense DeiT-Base. We also observe that in the bottom", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "layers, the attention probabilities are more centered at several heads; while in the top layers, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "attention probabilities are more uniformly distributed. This kind of tendency is well preserved by our", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 679, + 261, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 261, + 690 + ], + "score": 1.0, + "content": "sparse ViT (SViTE) from Dense ViTs.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 601, + 506, + 690 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Learned patch selection patterns. Figure 5 presents the learned behaviors of our token selector in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "SViTE+. We observe that the useless removed patches are typically distributed around the main object", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "or in the background. Meanwhile, the patches within the objects of interest are largely persevered,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 386, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 386, + 96 + ], + "score": 1.0, + "content": "which evidences the effectiveness of our learned patch token selector.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 73, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "or in the background. Meanwhile, the patches within the objects of interest are largely persevered,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 386, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 386, + 96 + ], + "score": 1.0, + "content": "which evidences the effectiveness of our learned patch token selector.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 107, + 107, + 360, + 120 + ], + "lines": [ + { + "bbox": [ + 104, + 106, + 361, + 123 + ], + "spans": [ + { + "bbox": [ + 104, + 106, + 361, + 123 + ], + "score": 1.0, + "content": "5 Conclusion and Discussion of Broader Impact", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 504, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 470, + 142 + ], + "score": 1.0, + "content": "In this work, we introduce sparse ViT exploration algorithms, SViTE, and its variants", + "type": "text" + }, + { + "bbox": [ + 470, + 129, + 504, + 140 + ], + "score": 0.71, + "content": "\\mathbf { S } ^ { \\mathrm { 2 } } \\mathbf { V i T E }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 124, + 153 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 141, + 159, + 151 + ], + "score": 0.76, + "content": "{ \\mathrm { S V i T E } } +", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 140, + 505, + 153 + ], + "score": 1.0, + "content": ", to explore high-quality sparse patterns in both ViT’s architecture and input token", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 507, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 507, + 165 + ], + "score": 1.0, + "content": "embeddings, alleviating training memory bottleneck and pursuing inference ultra-efficiency (e.g.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 174 + ], + "score": 1.0, + "content": "running time and FLOPs). Comprehensive experiments on ImageNet validate the effectiveness of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 385, + 187 + ], + "score": 1.0, + "content": "our proposal. Our informative visualizations further demonstrate that", + "type": "text" + }, + { + "bbox": [ + 386, + 173, + 420, + 184 + ], + "score": 0.57, + "content": "\\mathrm { S V i T E { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 172, + 506, + 187 + ], + "score": 1.0, + "content": "is capable of mining", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "score": 1.0, + "content": "crucial connections and input tokens by eliminating redundant units and dropping useless token", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 209 + ], + "score": 1.0, + "content": "embeddings. Future work includes examining the performance of our sparse ViTs on incoming", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 425, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 425, + 219 + ], + "score": 1.0, + "content": "hardware accelerators [96–100], which will provide better supports for sparsity.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 108, + 222, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "This work is scientific in nature, and we do not believe it has immediate negative societal impacts.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "Our findings of sparse vision transformers are highly likely to reduce both memory and energy costs", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 492, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 492, + 257 + ], + "score": 1.0, + "content": "substantially, leading to economic deployment in real-world applications (e.g., on smartphones).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 266, + 197, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 199, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 199, + 282 + ], + "score": 1.0, + "content": "Acknowledgment", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 363, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 363, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 363, + 299 + ], + "score": 1.0, + "content": "Z.W. is in part supported by an NSF RTML project (#2053279).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 313, + 163, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 165, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 165, + 327 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 111, + 330, + 506, + 724 + ], + "lines": [ + { + "bbox": [ + 116, + 331, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 116, + 331, + 506, + 345 + ], + "score": 1.0, + "content": "[1] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 132, + 342, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 132, + 342, + 506, + 357 + ], + "score": 1.0, + "content": "Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 132, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 132, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "et al. 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BackboneUpdate Schedule{△T,Tend,α,fdecay}Batch SizeEpochsInherited Settings from DeiT[2]
DeiT-Tiny{20000,1200000,0.5,cosine}512600AdamW, 0.0005 × batchsize,cosine decay
DeiT-Small{15000,1200000,0.5,cosine}512600warmup 5 epochs,0.05 weight decay
DeiT-Base{7000,600000,0.5,cosine}10246000.1 label smoothing,augmentations, etc.
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ModelsSparsity (%)ParametersFLOPs SavingRunning Time Reduced|Top-1 Accuracy (%)
DeiT-Tiny (Dense)0%5.72M0%0%72.20 (71.80)
SViTE-Tiny (Unstructured)30%4.02M25.56%0%71.78
SSP-Tiny (Structured)30%4.21M23.69%10.57%68.59
S2ViTE-Tiny (Structured)30%4.21M23.69%10.57%70.12
DeiT-Small (Dense)0%22.1M0%0%79.90 (79.78)
SViTE-Small (Unstructured)40%13.3M36.73%0%80.26
SSP-Small (Structured)40%14.6M31.63%22.65%77.74
S²ViTE-Small (Structured)40%14.6M31.63%22.65%79.22
DeiT-Base (Dense)0%86.6M0%0%81.80 (80.98)
SViTE-Base (Unstructured)40%52.0M38.30%0%81.56
SSP-Base (Structured)40%56.8M33.13%24.70%80.08
S2ViTE-Base (Structured)40%56.8M33.13%24.70%82.22
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ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Tiny0% (5.72M)0%72.20 (71.80)
SViTE-Tiny30% (4.02M)25.56%71.78
OMP30% (4.02M)25.56%68.35
GMP30% (4.02M)25.56%69.56
TP30% (4.02M)25.56%68.38
SViTE-Tiny40% (3.46M)34.16%71.75
OMP40% (3.46M)34.16%66.52
GMP40% (3.46M)34.15%68.36
TP40% (3.46M)34.17%65.45
Small-Dense0% (3.94M)32.54%67.33
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ModelsSparsity (#Para.)FLOPs SavingAccuracy (%)
DeiT-Small0% (22.1M)0%79.90 (79.78)
SViTE-Small50% (11.1M)46.26%79.72
OMP50% (11.1M)46.25%76.32
GMP50% (11.1M)46.26%76.88
TP50% (11.1M)46.26%76.30
SViTE-Small60% (8.9M)55.44%79.41
OMP60% (8.9M)55.44%75.32
GMP60% (8.9M)55.44%76.79
TP60% (8.9M)55.44%74.50
Small-Dense0% (11.4M)49.32%73.93
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ModelsSparsity (#Para.)FLOPs Saving|Accuracy (%)
DeiT-Base0% (86.6M)0%81.80 (80.98)
SViTE-Base50% (43.4M)47.95%81.51
OMP50% (43.4M)47.94%80.26
GMP50% (43.4M)47.95%80.79
TP50% (43.4M)47.94%80.55
SViTE-Base60% (34.8M)57.50%81.28
OMP60% (34.8M)57.50%80.25
GMP60% (34.8M)57.50%80.44
TP60% (34.8M)57.49%80.37
Small-Dense0% (44.0M)49.46%78.59
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#Tokens (%)|Time ReducedFLOPs Saving|Accuracy (%)
SViTE+-Small 50% Unstructured Sparsity
100%0%46.26%79.72
95%4.40%49.32%80.18
90%7.63%52.38%79.91
70%19.77%63.95%77.90
S²ViTE+-Small 40% Structured Sparsity
100%22.65%31.63%79.22
95%27.17%37.76%78.44
90%29.21%41.50%78.16
70%39.10%54.96%74.77
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newline at end of file diff --git a/parse/train/rkeMHjR9Ym/rkeMHjR9Ym.md b/parse/train/rkeMHjR9Ym/rkeMHjR9Ym.md new file mode 100644 index 0000000000000000000000000000000000000000..69a375d7ae985b3125b0b400f5a2bf8bfac1a07e --- /dev/null +++ b/parse/train/rkeMHjR9Ym/rkeMHjR9Ym.md @@ -0,0 +1,1036 @@ +# STOCHASTIC GRADIENT DESCENT LEARNS STATE EQUATIONS WITH NONLINEAR ACTIVATIONS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +We study discrete time dynamical systems governed by the state equation $h _ { t + 1 } =$ $\phi ( A h _ { t } + B u _ { t } )$ . Here $A , B$ are weight matrices, $\phi$ is an activation function, and $\mathbf { \Delta } \mathbf { u } _ { t }$ is the input data. This relation is the backbone of recurrent neural networks (e.g. LSTMs) which have broad applications in sequential learning tasks. We utilize stochastic gradient descent to learn the weight matrices from a finite input/state trajectory $\mathbf { \bar { \{ u } } _ { t } , h _ { t } \} _ { t = 0 } ^ { N }$ . We prove that SGD estimate linearly converges to the ground truth weights while using near-optimal sample size. Our results apply to increasing activations whose derivatives are bounded away from zero. The analysis is based on i) a novel SGD convergence result with nonlinear activations and ii) careful statistical characterization of the state vector. Numerical experiments verify the fast convergence of SGD on ReLU and leaky ReLU in consistence with our theory. + +# 1 INTRODUCTION + +A wide range of problems involve sequential data with a natural temporal ordering. Examples include natural language processing, time series prediction, system identification, and control design, among others. State-of-the-art algorithms for sequential problems often stem from dynamical systems theory and are tailored to learn from temporally dependent data. Linear models and algorithms; such as Kalman filter, PID controller, and linear dynamical systems, have a long history and are utilized in control theory since 1960’s with great success (Brown et al. (1992); Ho & Kalman (1966); Åström & Hägglund (1995)). More recently, nonlinear models such as recurrent neural networks (RNN) found applications in complex tasks such as machine translation and speech recognition (Bahdanau et al. (2014); Graves et al. (2013); Hochreiter & Schmidhuber (1997)). Unlike feedforward neural networks, RNNs are dynamical systems that use their internal state to process inputs. The goal of this work is to shed light on the inner workings of RNNs from a theoretical point of view. In particular, we focus on the RNN state equation which is characterized by a nonlinear activation function $\phi$ , state weight matrix $\pmb { A }$ , and input weight matrix $\textbf { { B } }$ as follows + +$$ +h _ { t + 1 } = \phi ( A h _ { t } + B u _ { t } ) , +$$ + +Here $h _ { t }$ is the state vector and $\mathbf { \pmb { u } } _ { t }$ is the input data at timestamp $t$ . This equation is the source of dynamic behavior of RNNs and distinguishes RNN from feedforward networks. The weight matrices $\pmb { A }$ and $\textbf { { B } }$ govern the dynamics of the state equation and are inferred from data. We will explore the statistical and computational efficiency of stochastic gradient descent (SGD) for learning these weight matrices. + +Contributions: Suppose we are given a finite trajectory of input/state pairs $( { \mathbf { } } u _ { t } , h _ { t } ) _ { t = 0 } ^ { N }$ generated from the state equation (1.1). We consider a least-squares regression obtained from $N$ equations; with inputs $( { \mathbf { } } u _ { t } , \mathbf { \dot { \boldsymbol { h } } } _ { t } ) _ { t = 1 } ^ { N }$ and outputs $( h _ { t + 1 } ) _ { t = 1 } ^ { N }$ . For a class of activation functions including leaky ReLU and for stable systems1, we show that SGD linearly converges to the ground truth weight matrices while requiring near-optimal trajectory length $N$ . In particular, the required sample size is $\mathcal { O } ( n + p )$ where $n$ and $p$ are the dimensions of the state and input vectors respectively. The results are extended to unstable systems when the samples are collected from multiple independent RNN trajectories rather than a single trajectory. Our theory applies to increasing activation functions whose derivatives are bounded away from zero, which includes leaky ReLU, and Gaussian input data. Numerical experiments on ReLU and leaky ReLU corroborate our theory and demonstrate that SGD converges faster as the activation slope increases. To obtain our results, we i) characterize the statistical properties of the state vector (e.g. well-conditioned covariance) and ii) derive a novel SGD convergence result with nonlinear activations; which may be of independent interest. As a whole, this paper provides a step towards foundational understanding of RNN training via SGD. + +# 1.1 RELATED WORK + +Our work is related to the recent optimization and statistics literature on linear dynamical systems (LDS) and neural networks. + +Linear dynamical systems: The state-equation (1.1) reduces to a LDS when $\phi$ is the linear activation $( \phi ( x ) = { \dot { x } } )$ ). Identifying the weight matrices is a core problem in linear system identification and is related to the optimal control problem (e.g. linear quadratic regulator) with unknown system dynamics. While these problems are studied since 1950’s (Ljung (1998; 1987); Åström & Eykhoff (1971)), our work is closer to the recent literature that provides data dependent bounds and characterize the non-asymptotic learning performance. Recht and coauthors have a series of papers exploring optimal control problem (Simchowitz et al. (2018); Tu et al. (2018; 2017)). In particular, Hardt et al. (2016) shows gradient descent learns single-input-single-output (SISO) LDS with polynomial guarantees. Oymak & Ozay (2018) and Faradonbeh et al. (2018) provide sample complexity bounds for learning LDS. Sanandaji et al. (2011b;a); Pereira et al. (2010) study the identification of sparse systems. + +Neural networks: There is a growing literature on the theoretical aspects of deep learning and provable algorithms for training neural networks. Most of the existing results are concerned with feedforward networks. Ge et al. (2017); Li & Yuan (2017); Mei et al. (2018b); Soltanolkotabi (2017); Janzamin et al. (2015); Zhong et al. (2017b) consider learning fully-connected shallow networks with gradient descent. Mei et al. (2018a); Soltanolkotabi et al. (2017); Foster et al. (2018) analyze empirical landscape of related nonlinear learning problems. Brutzkus & Globerson (2017); Zhong et al. (2017a); Du et al. (2017); Goel et al. (2018) address convolutional neural networks; which is an efficient weight-sharing architecture. Brutzkus et al. (2017); Wang et al. (2018) studies over-parameterized networks when data is linearly separable. Janzamin et al. (2015); Oymak & Soltanolkotabi (2018) utilize tensor decomposition techniques for learning feedforward neural nets. For recurrent networks, Sedghi & Anandkumar (2016) proposed tensor algorithms with polynomial guarantees and Khrulkov et al. (2017) studied their expressive power. More recently, Miller & Hardt (2018) showed that stable RNNs can be approximated by feed-forward networks. + +# 2 PROBLEM SETUP + +We first introduce the notation. $\| \cdot \|$ returns the spectral norm of a matrix and $s _ { \mathrm { m i n } } ( \cdot )$ returns the minimum singular value. The activation $\phi : \mathbb { R } \mathbb { R }$ applies entry-wise if its input is a vector. Throughout, $\phi$ is assumed to be a 1-Lipschitz function. With proper scaling of its parameters, the system (1.1) with a Lipschitz activation can be transformed into a system with 1-Lipschitz activation. The functions $\Sigma [ \cdot ]$ and var $[ \cdot ]$ return the covariance of a random vector and variance of a random variable respectively. ${ { I } _ { n } }$ is the identity matrix of size $n \times n$ . Normal distribution with mean $\pmb { \mu }$ and covariance $\pmb { \Sigma }$ is denoted by $\mathcal { N } ( \boldsymbol { \mu } , \boldsymbol { \Sigma } )$ . Throughout, $c , C , c _ { 0 } , c _ { 1 } , \ldots$ denote positive absolute constants. + +Setup: We consider the dynamical system parametrized by an activation function $\phi ( \cdot )$ and weight matrices $\pmb { A } \in \mathbb { R } ^ { n \times n } , \pmb { B } \in \mathbf { \bar { \mathbb { R } } } ^ { n \times p }$ as described in (1.1). Here, $\boldsymbol { h } _ { t }$ is the $n$ dimensional state-vector and $\mathbf { \pmb { u } } _ { t }$ is the $p$ dimensional input to the system at time $t$ . As mentioned previously, (1.1) corresponds to the state equation of a recurrent neural network. For most RNNs of interest, the state $\boldsymbol { h } _ { t }$ is hidden and we only get to interact with $\boldsymbol { h } _ { t }$ via an additional output equation. For Elman networks Elman (1990), this equation is characterized by some output activation $\phi _ { y }$ and output weights ${ C , D }$ as follows + +$$ +\mathbf { \nabla } y _ { t } = \phi _ { y } ( C h _ { t } + D \mathbf { u } _ { t } ) . +$$ + +In this work, our attention is restricted to the state equation (1.1); which corresponds to setting $\pmb { y } _ { t } = \pmb { h } _ { t + 1 }$ in the output equation. To analyze (1.1) in a non-asymptotic data-dependent setup, we assume a finite input/state trajectory of $\{ u _ { t } , h _ { t } \} _ { t = 0 } ^ { N }$ generated by some ground truth weight matrices + +# Algorithm 1 Learning state equations with nonlinear activations + +1: Inputs: $( \mathbf { \boldsymbol { y } } _ { t } , h _ { t } , \mathbf { \boldsymbol { u } } _ { t } ) _ { t = 1 } ^ { N }$ sampled from a trajectory. Scaling $\mu$ , learning rate $\eta$ . Initialization +$A _ { 0 } , B _ { 0 }$ . +2: Outputs: Estimates ${ \hat { A } } , { \hat { B } }$ of the weight matrices $A , B$ . +3: $\mathbf { \pmb { x } } _ { t } \bar { } [ \mu \pmb { h } _ { t } ^ { T } \mathbf { \pmb { u } } _ { t } ^ { T } ] ^ { T }$ for $1 \leq t \leq N$ . +4: $\Theta _ { 0 } [ \mu ^ { - 1 } A _ { 0 } \ : B _ { 0 } ]$ +5: for $\tau$ from 1 to END do +6: Pick $\gamma _ { \tau }$ from $\{ 1 , 2 , \ldots , N \}$ uniformly at random. +7: $\Theta _ { \tau } \gets \Theta _ { \tau - 1 } - \eta \nabla \mathcal { L } _ { \gamma _ { \tau } } ( \Theta _ { \tau - 1 } )$ +8: end for +9: return $[ \hat { A } \hat { B } ] \Theta _ { \mathrm { E N D } } [ \begin{array} { c c } { \mu I _ { n } } & { 0 } \\ { 0 } & { I _ { p } } \end{array} ] .$ + +$( A , B )$ . Our goal is learning the unknown weights $\pmb { A }$ and $\textbf { { B } }$ in a data and computationally efficient way. In essence, we will show that, if the trajectory length satisfies $N \gtrsim n + p$ , SGD can quickly and provably accomplish this goal using a constant step size. + +Appoach: Our approach is described in Algorithm 1. It takes two hyperparameters; the scaling factor $\mu$ and learning rate $\eta$ . Using the RNN trajectory, we construct $N$ triples of the form $\{ u _ { t } , h _ { t } , h _ { t + 1 } \} _ { t = 1 } ^ { N }$ We formulate a regression problem by defining the output vector ${ \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \mathbf { } } _ { \mathbf { } } \mathbf { \Xi } _ { \mathbf { } } \mathbf { \Lambda } _ { \mathbf { } } \mathbf { \Lambda } _ { \mathbf { } } \textbf { } _ { \mathbf { } } \textbf { } \textbf { } _ { \mathrm { } }$ , input vector $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , and the target parameter $C$ as follows + +$$ +\begin{array} { r } { y _ { t } = h _ { t + 1 } \quad , \quad x _ { t } = \left[ \begin{array} { l } { \mu h _ { t } } \\ { u _ { t } } \end{array} \right] \in \mathbb { R } ^ { n + p } \quad , \quad C = [ \mu ^ { - 1 } A B ] \in \mathbb { R } ^ { n \times ( n + p ) } . } \end{array} +$$ + +With this reparameterization, we find the input/output identity ${ \pmb y } _ { t } = \phi ( { \pmb C } { \pmb x } _ { t } )$ . We will consider the least-squares regression given by + +$$ +\mathcal { L } ( \boldsymbol { \Theta } ) = \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \mathcal { L } _ { t } ( \boldsymbol { \Theta } ) \quad \mathrm { w h e r e } \quad \mathcal { L } _ { t } ( \boldsymbol { \Theta } ) = \frac { 1 } { 2 } \| y _ { t } - \phi ( \boldsymbol { \Theta } x _ { t } ) \| _ { \ell _ { 2 } } ^ { 2 } . +$$ + +For learning the ground truth parameter $C$ , we utilize SGD on the loss function (2.3) with a constant learning rate $\eta$ . Starting from an initial point $\Theta _ { 0 }$ , after END SGD iterations, Algrorithm 1 returns an estimate $\hat { C } = \Theta _ { \mathrm { E N D } }$ . Estimates of $\pmb { A }$ and $\textbf { { B } }$ are decoded from the left and right submatrices of $\hat { C }$ respectively. + +# 3 MAIN RESULTS + +# 3.1 PRELIMINARIES + +The analysis of the state equation naturally depends on the choice of the activation function; which is the source of nonlinearity. We first define a class of Lipschitz and increasing activation functions. + +Definition 3.1 ( $\beta$ -increasing activation). Given $1 \geq \beta \geq 0$ , the activation function $\phi$ satisfies $\phi ( 0 ) = 0$ and $1 \geq \phi ^ { \prime } ( x ) \geq \beta$ for all $x \in \mathbb { R }$ . + +Our results will apply to strictly increasing activations where $\phi$ is $\beta$ -increasing for some $\beta > 0$ . Observe that, this excludes ReLU activation which has zero derivative for negative values. However, it includes Leaky ReLU which is a generalization of ReLU. Parameterized by $1 \ge \beta \ge 0$ , Leaky ReLU is a $\beta$ -increasing function given by + +$$ +{ \mathrm { L R e L U } } ( x ) = \operatorname* { m a x } ( \beta x , x ) . +$$ + +In general, given an increasing and 1-Lipschitz activation $\phi$ , a $\beta$ -increasing function $\phi _ { \beta }$ can be obtained by blending $\phi$ with the linear activation, i.e. $\phi _ { \beta } ( x ) = ( 1 - \beta ) \phi ( x ) + \beta x$ . + +A critical property that enables SGD is that the state-vector covariance $\pmb { \Sigma } [ h _ { t } ]$ is well-conditioned under proper assumptions. The lemma below provides upper and lower bounds on this covariance matrix in terms of problem variables. + +Lemma 3.2 (State vector covariance). Consider the state equation (1.1) where $h _ { 0 } = 0$ and ${ \mathbf { } } u _ { t } \stackrel { i . i . d . } { \sim }$ $\mathcal { N } ( 0 , I _ { p } )$ . Define the upper bound term $B _ { t }$ as + +$$ +B _ { t } = \| B \| \sqrt { \frac { 1 - \| A \| ^ { 2 t } } { 1 - \| A \| ^ { 2 } } } . +$$ + +• Suppose $\phi$ is 1-Lipschitz and $\phi ( 0 ) = 0$ . Then, for all $t \ge 0 , \Sigma [ h _ { t } ] \preceq B _ { t } ^ { 2 } I _ { n }$ . + +• Suppose $\phi$ is a $\beta$ -increasing function and $p \geq n$ . Then, $\Sigma [ h _ { t } ] \succeq \beta ^ { 2 } s _ { \mathrm { m i n } } ( B ) ^ { 2 } I _ { n }$ + +As a natural extension from linear dynamical systems, we will say the system is stable if $\| A \| < 1$ and unstable otherwise. For activations we consider, stability implies that if the input is set to 0, state vector $\boldsymbol { h } _ { t }$ will exponentially converge to 0 i.e. the system forgets the past states quickly. This is also the reason $( B _ { t } ) _ { t \geq 0 }$ sequence converges for stable systems and diverges otherwise. The condition number of the covariance will play a critical role in our analysis. Using Lemma 3.2, this number can be upper bounded by $\rho$ defined as + +$$ +\rho = \left( \frac { B _ { \infty } } { \beta s _ { \operatorname* { m i n } } ( B ) } \right) ^ { 2 } = \left( \frac { \| B \| } { s _ { \operatorname* { m i n } } ( B ) } \right) ^ { 2 } \frac { 1 } { \beta ^ { 2 } ( 1 - \| A \| ^ { 2 } ) } . +$$ + +Observe that, the condition number of $\textbf { { B } }$ appears inside the $\rho$ term. + +# 3.2 LEARNING FROM SINGLE TRAJECTORY + +Our main result applies to stable systems $( \left. A \right. < 1 )$ and provides a non-asymptotic convergence guarantee for SGD in terms of the upper bound on the state vector covariance. This result characterizes the sample complexity and the rate of convergence of SGD; and also provides insights into the role of activation function and the spectral norm of $\pmb { A }$ . + +Theorem 3.3 (Main result). Let $\{ u _ { t } , h _ { t + 1 } \} _ { t = 1 } ^ { N }$ be a finite trajectory generated from the state equation (1.1). Suppose $\| A \| < 1$ , $\phi$ is $\beta$ -increasing, $h _ { 0 } = 0$ , $p \geq n$ , and ${ \mathbf { \boldsymbol { u } } _ { t } } \overset { i . i . d . } { \sim } \mathcal { N } ( 0 , \pmb { I } _ { p } )$ . Let $\rho$ be same as (3.3) and $c , C , c _ { 0 }$ be properly chosen absolute constants. Pick the trajectory length $N$ to satisfy + +$$ +N \geq C L \rho ^ { 2 } ( n + p ) , +$$ + +where loss fu $\begin{array} { r } { L = 1 - \frac { \log ( c n \rho ) } { \log \| A \| } } \end{array}$ ) g µ = 1/B∞, learning rate η = c0 β2ρn(n+p) . Pick scalinh probability , and consider the, starting from an $1 - 4 N \exp ( - 1 0 0 n ) - 8 L \exp ( - \mathcal { O } ( \frac { N } { L \rho ^ { 2 } } ) )$ initial point $\Theta _ { 0 }$ , for all $\tau \geq 0$ , the SGD iterations described in Algorithm $^ { l }$ satisfies + +$$ +\mathbb { E } [ \| \Theta _ { \tau } - C \| _ { F } ^ { 2 } ] \le ( 1 - c _ { 0 } \frac { \beta ^ { 4 } } { 2 \rho ^ { 2 } n ( n + p ) } ) ^ { \tau } \| \Theta _ { 0 } - C \| _ { F } ^ { 2 } . +$$ + +Here the expectation is over the randomness of the SGD updates. + +Sample complexity: Theorem 3.3 essentially requires $N \gtrsim ( n + p ) / \beta ^ { 4 }$ samples for learning. This can be seen by unpacking (3.3) and ignoring the logarithmic $L$ term and the condition number of $\textbf { { B } }$ . Observe that $\bar { \mathcal { O } } ( n + \bar { p } )$ growth achieves near-optimal sample size for our problem. Each state equation (1.1) consists of $n$ sub-equations (one for each entry of $h _ { t + 1 }$ ). We collect $N$ state equations to obtain a system of $N n$ equations. On the other hand, the total number of unknown parameters in $\pmb { A }$ and $\textbf { { B } }$ are $n ( n + p )$ . This implies Theorem 3.3 is applicable as soon as the problem is mildly overdetermined i.e. $N n \gtrsim n ( n + p )$ . + +Computational complexity: Theorem 3.3 requires $\begin{array} { r } { \mathcal { O } ( n ( n + p ) \log \frac { 1 } { \varepsilon } ) } \end{array}$ iterations to reach $\varepsilon$ - neighborhood of the ground truth. Our analysis reveals that, this rate can be accelerated if the state vector is zero-mean. This happens for odd activation functions satisfying $\phi ( - x ) = - \phi ( x )$ (e.g. linear activation). The result below is a corollary and requires $\times n$ less iterations. + +Theorem 3.4 (Faster learning for odd activations). Consider the same setup provided in Theorem 3.3. Additionally, assume that φ is an odd function. Pick scaling µ = 1/B∞, learning rate η = c0 β2ρ(n+p) , and consider the loss function (2.3). With probability $\begin{array} { r } { 1 - 4 N \exp ( - 1 0 0 n ) - 8 L \exp ( - \mathcal { O } ( \frac { N } { L \rho ^ { 2 } } ) ) } \end{array}$ , starting from an initial point $\Theta _ { 0 }$ , for all $\tau \geq 0$ , the SGD iterations described in Algorithm $^ { l }$ satisfies + +$$ +\mathbb { E } [ \| \Theta _ { \tau } - C \| _ { F } ^ { 2 } ] \le ( 1 - c _ { 0 } \frac { \beta ^ { 4 } } { 2 \rho ^ { 2 } ( n + p ) } ) ^ { \tau } \| \Theta _ { 0 } - C \| _ { F } ^ { 2 } , +$$ + +where the expectation is over the randomness of the SGD updates. + +Another aspect of the convergence rate is the dependence on $\beta$ . In terms of $\beta$ , the SGD error (3.4) decays as $( \bar { 1 } - \mathcal { O } ( \beta ^ { 8 } ) ) ^ { \tau }$ . While it is not clear how optimal is the exponent 8, numerical experiments in Section 6 demonstrate that larger $\beta$ indeed results in drastically faster convergence. + +# 4 MAIN IDEAS AND PROOF STRATEGY + +We first outline our high-level proof strategy for Theorem 3.3; which brings together ideas from statistics and optimization. + +1. We first show that input data is well-behaved by proving that state-vector $h _ { t }$ has a wellconditioned covariance as discussed in Lemma 3.2 and shown in Appendix B. The key idea is if $\phi$ is $\beta$ -increasing, then the random input data $\mathbf { \Delta } \mathbf { u } _ { t }$ provides sufficient excitation for the output state $\boldsymbol { h } _ { t + 1 }$ . +2. Even if individual samples are well-behaved, analyzing (2.3) is still challenging due to temporal dependencies between the samples. These dependencies prevent us from directly using statistical learning results that typically assume i.i.d. samples. We show that the dependency between samples at time $t$ and $t + T$ decay exponentially fast in separation $T$ (for stable systems). This is outlined in Appendix C. +3. This observation allows us to obtain nearly independent data by subsampling the original trajectory to get $( h _ { i T } , \pmb { u } _ { i T } ) _ { i \geq 0 }$ . Thanks to exponential decay, a logarithmically small $T$ can be chosen to generate large subtrajectories of size $N / T$ . Appendix D uses additional perturbation arguments to establish the well-behavedness of the overall data matrix. +4. To conclude, we obtain a deterministic result which establishes fast convergence result for $\beta$ -increasing activations and well-behaved dataset. This is provided in Theorem 4.1 and proved in Appendix A. + +The first three steps are related to the statistical nature of the problem which can be decoupled from the last step. Specifically, the last step derives a deterministic result that establishes the linear convergence of SGD for $\beta$ -increasing functions. For linear convergence proofs, a typical strategy is showing the strong convexity of the loss function i.e. showing that, for some $\alpha > 0$ and all points $\mathbf { \nabla } _ { v } , \mathbf { \nabla } _ { u }$ , the gradient satisfies + +$$ +\begin{array} { r } { \langle \nabla \mathcal { L } ( \pmb { v } ) - \nabla \mathcal { L } ( \pmb { u } ) , \pmb { v } - \pmb { u } \rangle \geq \alpha \| \pmb { v } - \pmb { u } \| _ { \ell _ { 2 } } ^ { 2 } . } \end{array} +$$ + +The core idea of our convergence result is that the strong convexity parameter of the loss function with $\beta$ -increasing activations can be connected to the loss function with linear activations. In particular, recalling (2.3), set $\pmb { y } _ { t } ^ { \mathrm { l i n } } = \pmb { C } \pmb { x } _ { t }$ and define the linear loss to be + +$$ +\mathcal { L } ^ { \mathrm { l i n } } ( \Theta ) = \frac { 1 } { 2 N } \sum _ { i = 1 } ^ { N } \Vert \pmb { y } _ { t } ^ { \mathrm { l i n } } - \Theta \pmb { x } _ { t } \Vert _ { \ell _ { 2 } } ^ { 2 } . +$$ + +Denoting the strong convexity parameter of the original loss by $\alpha _ { \phi }$ and that of linear loss by $\alpha _ { \mathrm { l i n } }$ , we argue that $\alpha _ { \phi } \geq \beta ^ { 2 } \alpha _ { \mathrm { l i n } }$ ; which allows us to establish a convergence result as soon as $\alpha _ { \mathrm { l i n } }$ is strictly positive. Next result is our SGD convergence theorem which follows from this discussion. + +Theorem 4.1 (Deterministic convergence). Suppose a data set $\{ \pmb { x } _ { i } , \pmb { y } _ { i } \} _ { i = 1 } ^ { N }$ is given; where output $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \psi _ { i }$ is related to input $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ via $\pmb { y } _ { i } = \phi ( \langle \pmb { x } _ { i } , \pmb { \theta } \rangle )$ for some $\pmb { \theta } \in \mathbb { R } ^ { n }$ . Suppose $\beta > 0$ and $\phi$ is a $\beta$ -increasing. Le $\ : \gamma _ { + } \geq \gamma _ { - } > 0 \ :$ be scalars. Assume that input samples satisfy the bounds + +$$ +\gamma _ { + } I _ { n } \succeq \frac { 1 } { N } \sum _ { i = 1 } ^ { N } x _ { i } x _ { i } ^ { T } \succeq \gamma _ { - } I _ { n } \quad , \quad \| x _ { i } \| _ { \ell _ { 2 } } ^ { 2 } \leq B f o r a l l i . +$$ + +Let $\{ r _ { \tau } \} _ { \tau = 0 } ^ { \infty }$ be a sequence of i.i.d. integers uniformly distributed between 1 to $N$ . Then, starting from an arbitrary point $\pmb { \theta } _ { 0 }$ , setting learning rate $\begin{array} { r } { \eta = \frac { \beta ^ { 2 } \gamma _ { - } } { \gamma _ { + } B } } \end{array}$ β2γ−γ+B , for all τ ≥ 0, the SGD iterations for quadratic loss + +$$ +\pmb { \theta } _ { \tau + 1 } = \pmb { \theta } _ { \tau } - \eta ( \phi ( \mathbf { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } ) - \pmb { y } _ { r _ { \tau } } ) \phi ^ { \prime } ( \mathbf { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } ) \pmb { x } _ { r _ { \tau } } , +$$ + +satisfies the error bound + +$$ +\mathbb { E } [ \| \pmb { \theta } _ { \tau } - \pmb { \theta } \| _ { \ell _ { 2 } } ^ { 2 } ] \le \| \pmb { \theta } _ { 0 } - \pmb { \theta } \| _ { \ell _ { 2 } } ^ { 2 } \big ( 1 - \frac { \beta ^ { 4 } \gamma _ { - } ^ { 2 } } { \gamma _ { + } B } \big ) ^ { \tau } , +$$ + +where the expectation is over the random selection of the SGD iterations $\{ r _ { \tau } \} _ { \tau = 0 } ^ { \infty }$ + +This theorem provides a clean convergence rate for SGD for $\beta$ -increasing activations and naturally generalizes standard results on linear regression which corresponds to $\beta = 1$ . We remark that related results appear in the literature on generalized linear models. Kakade et al. (2011); Foster et al. (2018); Mei et al. (2018a) provide learning theoretic loss/gradient/hessian convergence results for isotonic regression, robust regression, and $\beta$ -increasing activations. Goel et al. (2018) establishes a similar result for leaky ReLU activations under the assumption of symmetric input distribution and infinitely many samples (i.e. in population limit). Compared to these, we establish a deterministic linear convergence guarantee for SGD that works whenever the data matrix is full rank. We believe extensions to proximal gradient methods might be beneficial for high-dimensional nonlinear problems (e.g. sparse/low-rank approximation, manifold constraints Cai et al. (2010); Beck & Teboulle (2009); Oymak et al. (2018); Agarwal et al. (2010); Pereira et al. (2010)) and is left as a future work. + +To derive our main results in Section 3, we need to address the first three steps outlined earlier and determine the conditions under which Theorem 4.1 is applicable to the data obtained from RNN state equation with high probability. Below we provide desirable characteristics of the state vector; which enables our statistical results. + +Assumption 1 (Well-behaved state vector). Let $L > 1$ be an integer. There exists positive scalars√ $\gamma _ { + } , \gamma _ { - } , \theta$ and an absolute constant $C > 0$ such that $\theta \leq 3 \sqrt { n }$ and the following holds + +• Lower bound: $\Sigma [ h _ { L - 1 } ] \succeq \gamma _ { - } I _ { n }$ , • Upper bound: for all $t ,$ , the state vector satisfies + +$$ +\Sigma [ h _ { t } ] \preceq \gamma _ { + } I _ { n } \quad , \quad \| h _ { t } - \mathbb { E } [ h _ { t } ] \| _ { \psi _ { 2 } } \leq C \sqrt { \gamma _ { + } } \quad a n d \quad \| \mathbb { E } [ h _ { t } ] \| _ { \ell _ { 2 } } \leq \theta \sqrt { \gamma _ { + } } . +$$ + +Here $\lVert \cdot \rVert _ { \psi _ { 2 } }$ returns the subgaussian norm of a vector (see Def. 5.22 of Vershynin (2010)). + +Assumption 1 ensures that covariance is well-conditioned, state vector is well-concentrated, and it has a reasonably small expectation. Our next theorem establishes statistical guarantees for learning the RNN state equation based on this assumption. + +Theorem 4.2 (General result). Let $\{ u _ { t } , h _ { t + 1 } \} _ { t = 1 } ^ { N }$ be a length $N$ trajectory of the state equation (1.1). Suppose $\| A \| < 1$ , $\phi$ is $\beta$ -increasing, $h _ { 0 } = 0$ , and ${ \mathbf { \boldsymbol { u } } _ { t } } \overset { i . i . d . } { \sim } \mathcal { N } ( 0 , \pmb { I } _ { p } )$ . Given scalars $\gamma _ { + } \geq \gamma _ { - } > 0$ , set the condition number as $\rho = \gamma _ { + } / \gamma _ { - }$ . For absolute constants $C , c , c _ { 0 } > 0$ , choose trajectory length $N$ to satisfy + +$$ +N \geq C L \rho ^ { 2 } ( n + p ) w h e r e L = \lceil 1 - \frac { \log \left( c n \rho \right) } { \log \| A \| } \rceil . +$$ + +Suppose Assumption $^ { l }$ holds with $L , \gamma _ { + } , \gamma _ { - } , \theta$ . Pick scaling to be $\mu = 1 / \sqrt { \gamma _ { + } }$ and learning rate to be η = c0 ρ(θ+√2)2(n+p) . β2 With probability $1 - 4 N \exp ( - 1 0 0 n ) - 8 L \exp ( - \mathcal { O } ( \frac { N } { L \rho ^ { 2 } } ) )$ , starting from $\Theta _ { 0 }$ , for all $\tau \geq 0$ , the SGD iterations on loss (2.3) as described in Algorithm $^ { l }$ satisfies + +$$ +\mathbb { E } [ \| \Theta _ { \tau } - C \| _ { F } ^ { 2 } ] \le ( 1 - c _ { 0 } \frac { \beta ^ { 4 } } { 2 \rho ^ { 2 } ( \theta + \sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \tau } \| \Theta _ { 0 } - C \| _ { F } ^ { 2 } , +$$ + +where the expectation is over the randomness of SGD updates. + +The advantage of this theorem is that, it isolates the optimization problem from the statistical properties of state vector. If one can prove tighter bounds on achievable $( \gamma _ { + } , \gamma _ { - } , \theta )$ , it will immediately imply improved performance for SGD. In particular, Theorems 3.3 and 3.4 are simple corollaries of Theorem 4.2 with proper choices. + +• Theorem 3.3 follows by setting $\gamma _ { + } = B _ { \infty } ^ { 2 }$ , $\gamma _ { - } = \beta ^ { 2 } s _ { \mathrm { m i n } } ( B ) ^ { 2 }$ , and $\theta = { \sqrt { n } }$ . +• Theorem 3.4 follows by setting $\gamma _ { + } = B _ { \infty } ^ { 2 }$ , $\gamma _ { - } = \beta ^ { 2 } s _ { \mathrm { m i n } } ( B ) ^ { 2 }$ , and $\theta = 0$ . + +# 5 LEARNING UNSTABLE SYSTEMS + +So far, we considered learning from a single RNN trajectory for stable systems $( \left. A \right. < 1 )$ . For such systems, as the time goes on, the impact of the earlier states disappear. In our analysis, this allows us to split a single trajectory into multiple nearly-independent trajectories. This approach will not work for unstable systems $\mathbf { A }$ is arbitrary) where the impact of older states may be amplified over time. To address this, we consider a model where the data is sampled from multiple independent trajectories. + +Suppose $N$ independent trajectories of the state-equation (1.1) are available. Pick some integer $T _ { 0 } \geq 1$ . Denoting the ith trajectory by the triple $( { h } _ { t + 1 } ^ { ( i ) } , { h } _ { t } ^ { ( i ) } , { u } _ { t } ^ { ( i ) } ) _ { t \geq 0 }$ , we collect a single sample from each trajectory at time $T _ { 0 }$ r he triple $( \boldsymbol { h } _ { T _ { 0 } + 1 } ^ { ( i ) } , \boldsymbol { h } _ { T _ { 0 } } ^ { ( i ) } , \boldsymbol { u } _ { T _ { 0 } } ^ { ( i ) } )$ To utilize the existing $1 \leq i \leq N$ + +$$ +( { \bf y } _ { i } , { \bf h } _ { i } , { \bf u } _ { i } ) = ( { \bf h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \bf h } _ { T _ { 0 } } ^ { ( i ) } , { \bf u } _ { T _ { 0 } } ^ { ( i ) } ) . +$$ + +With this setup, we can again use the SGD Algorithm 1 to learn the weights $\pmb { A }$ and $\textbf { { B } }$ . The crucial difference compared to Section 3 is that, the samples $( { \bf y } _ { i } , h _ { i } , { \bf u } _ { i } ) _ { i = 1 } ^ { N }$ are now independent of each other; hence, the analysis is simplified. As previously, having an upper bound on the condition number of the state-vector covariance is critical. This upper bound can be shown to be $\rho$ defined as + +$$ +\rho = \left\{ \begin{array} { l l } { \bar { \rho } } & { \mathrm { i f } \ n > 1 } \\ { \bar { \rho } \frac { \ 1 - \beta ^ { 2 } \vert A \vert ^ { 2 } } { 1 - ( \beta \vert A \vert ) ^ { 2 T _ { 0 } } } } & { \mathrm { i f } \ n = 1 } \end{array} \right. \quad \mathrm { w h e r e } \quad \bar { \rho } = \frac { B _ { T _ { 0 } } ^ { 2 } } { \beta ^ { 2 } s _ { \mathrm { m i n } } ( B ) ^ { 2 } } . +$$ + +The mod $\bar { \rho }$ term is similar to the earlier decation is indeed necessary since 3); ho when nvolves . On the $B _ { T _ { 0 } }$ rather than er hand, not $B _ { \infty }$ . t, $B _ { \infty } = \infty$ $\| A \| > 1$ $B _ { T _ { 0 } } ^ { 2 }$ grows proportional to $\| A \| ^ { 2 T _ { 0 } }$ ; which results in exponentially bad condition number in $T _ { 0 }$ . Our $\rho$ definition remedies this issue for single-output systems; where $n = 1$ and $\pmb { A }$ is a scalar. In particular, when $\beta = 1$ (e.g. $\phi$ is linear) $\rho$ becomes equal to the correct value $1 ^ { 2 }$ . The next theorem provides our result on unstable systems in terms of this condition number and other model parameters. + +Theorem 5.1 (Unstable systems). Suppose we are given $N$ independent trajectories $( h _ { t } ^ { ( i ) } , u _ { t } ^ { ( i ) } ) _ { t \geq 0 }$ for $1 \leq i \leq N$ . Each trajectory is sampled at time $T _ { 0 }$ to obtain $N$ samples $( { \bf y } _ { i } , h _ { i } , { \bf u } _ { i } ) _ { i = 1 } ^ { N }$ where the ith sample is given by (5.1). Suppose the sample size satisfies + +$$ +N \geq C \rho ^ { 2 } ( n + p ) +$$ + +where $\rho$ is given by (5.2). Assume the initial states are 0, $\phi$ is $\beta$ -increasing, $p \geq n ,$ , and ${ \pmb u } _ { t } \stackrel { i . i . d . } { \sim }$ $\mathcal { N } ( 0 , { I } _ { p } )$ . Set scaling $\mu = 1 / \sqrt { B _ { T _ { 0 } } }$ , learning rate $\begin{array} { r } { \eta = c _ { 0 } \frac { \beta ^ { 2 } } { \rho n ( n + p ) } } \end{array}$ , and run SGD over the equations described in (2.2) and (2.3). Starting from $\Theta _ { 0 }$ , with probability $1 - 2 N \exp ( - 1 0 0 ( n + p ) ) -$ $4 \exp ( - \mathcal { O } ( \frac { N } { \rho ^ { 2 } } ) )$ , all SGD iterations satisfy + +$$ +\mathbb { E } [ \| \Theta _ { \tau } - C \| _ { F } ^ { 2 } ] \le ( 1 - c _ { 0 } \frac { \beta ^ { 4 } } { 2 \rho ^ { 2 } n ( n + p ) } ) ^ { \tau } \| \Theta _ { 0 } - C \| _ { F } ^ { 2 } , +$$ + +where the expectation is over the randomness of the SGD updates. + +# 6 NUMERICAL EXPERIMENTS + +We conducted experiments on ReLU and Leaky ReLU activations. Let us first describe the experimental setup. We pick the state dimension $n = 5 0$ and the input dimension $p = 1 0 0$ . We choose the ground truth matrix $\pmb { A }$ to be a scaled random unitary matrix; which ensures that all singular values of $\pmb { A }$ are equal. $\textbf { { B } }$ is generated with i.i.d. $\mathcal { N } ( 0 , 1 )$ entries. Instead of using the theoretical scaling choice, we determine the scaling $\mu$ from empirical covariance matrices outlined in Algorithm 2. Similar to our proof strategy, this algorithm equalizes the spectral norms of the input and state covariances to speed up convergence. We also empirically determined the learning rate and used $\eta = 1 / 1 0 0$ in all experiments. + +# Algorithm 2 Empirical hyperparameter selection. + +1: Inputs: $( h _ { t } , u _ { t } ) _ { t = 1 } ^ { N }$ sampled from a trajectory. +2: Outputs: Scaling 3: Form the empirica $\mu$ .covariance matrix from +$\Sigma _ { h }$ $\{ h _ { t } \} _ { t = 1 } ^ { N }$ +4: Form the empirical covariance matrix $\Sigma _ { u }$ $\{ u _ { t } \} _ { t = 1 } ^ { N }$ . +5: return $\sqrt { \| \Sigma _ { u } \| / \| \Sigma _ { h } \| }$ . + +![](images/0bf1b15a51fed848e839755197683e302f9a9ddcb50642893bcbe2352a4d0bde.jpg) +Figure 1: SGD convergence behavior for Leaky ReLUs with varying minimum slope $\beta$ . Figures a) and b) repeat the same experiments. The difference is the spectral norm of the ground truth state matrix $\pmb { A }$ . + +Evaluation: We consider two performance measures in the experiments. Let $\hat { C }$ be an estimate of the ground truth parameter $\boldsymbol { C } \dot { = } [ \mu ^ { - 1 } \boldsymbol { A } \boldsymbol { B } ]$ . The first measure is the normalized error defined as $\| \hat { C } - C \| _ { F } ^ { 2 } / \| C \| _ { F } ^ { 2 }$ . The second measure is the normalized loss defined as + +$$ +\frac { \sum _ { i = 1 } ^ { N } | | \pmb { y } _ { t } - \phi ( \hat { C } \pmb { x } _ { t } ) | | _ { \ell _ { 2 } } ^ { 2 } } { \sum _ { i = 1 } ^ { N } | | \pmb { y } _ { t } | | _ { \ell _ { 2 } } ^ { 2 } } . +$$ + +In all experiments, we run Algorithm 1 for 50000 SGD iterations and plot these measures as a function of $\tau$ ; by using the estimate available at the end of the $\tau$ th SGD iteration for $0 \leq \tau \leq 5 0 0 0 0$ . Each curve is obtained by averaging the outcomes of 20 independent realizations.Our first experiments use $N = 5 0 0$ ; which is mildly larger than the total dimension $n + p = 1 5 0$ . In Figure 1, we plot the Leaky ReLU errors with varying slopes as described in (3.1). Here $\beta = 0$ corresponds to ReLU and $\beta = 1$ is the linear model. In consistence with our theory, SGD achieves linear convergence and as $\beta$ increases, the rate of convergence drastically improves3. The improvement is more visible for less stable systems driven by $\pmb { A }$ with a larger spectral norm. In particular, while ReLU converges for small $\| A \|$ , SGD gets stuck before reaching the ground truth when $\| A \| = 0 . 8$ . + +To understand, how well SGD fits the training data, in Figure 2a, we plotted the normalized loss for ReLU activation. For more unstable system $\left. \left. A \right. \right. = \bar { 0 } . 9 )$ , training loss stagnates in a similar fashion to the parameter error. We also verified that the norm of the overall gradient $\lVert \nabla \mathcal { L } ( \Theta _ { \tau } ) \rVert _ { F }$ continues to decay (where $\Theta _ { \tau }$ is the $\tau$ th SGD iterate); which implies that SGD converges before reaching a global minima. As $\pmb { A }$ becomes more stable, rate of convergence improves and linear rate is visible. Finally, to better understand the population landscape of the quadratic loss with ReLU activations, Figure 2b repeats the same ReLU experiments while increasing the sample size five times to $N = 2 5 0 0$ . For this more overdetermined problem, SGD converges even for $\mathbf { | } \mathbf { \boldsymbol { A } } | \mathbf { | } = 0 . 9$ ; indicating that + +• population landscape of loss with ReLU activation is well-behaved, • however ReLU problem requires more data compared to the Leaky ReLU for finding global minima. + +Overall, as predicted by our theory, experiments verify that SGD indeed quickly finds the optimal weight matrices of the state equation (1.1) and as the activation slope $\beta$ increases, the convergence rate improves. + +![](images/49e542a82fdc668eda12f703902eb01c8a18f213a3ab21f6cde72a6a3d9f2842.jpg) +Figure 2: SGD convergence behavior for ReLU with varying spectral norm of the state matrix $\pmb { A }$ . Figures a) and b) repeats the same experiments. The difference is that a) uses $N = 5 0 0$ trajectory length whereas b) uses $N = 2 5 0 0$ (i.e. $\times 5$ more data). Shaded regions highlight the one standard deviation around the mean. + +# 7 CONCLUSIONS + +This work showed that SGD can learn the nonlinear dynamical system (1.1); which is characterized by weight matrices and an activation function. This problem is of interest for recurrent neural networks as well as nonlinear system identification. We showed that efficient learning is possible with optimal sample complexity and good computational performance. Our results apply to strictly increasing activations such as Leaky ReLU. We empirically showed that Leaky ReLU converges faster than ReLU and requires less samples; in consistence with our theory. We list a few unanswered problems that would provide further insights into recurrent neural networks. + +• Covariance of the state-vector: Our results depend on the covariance of the state-vector and requires it to be positive definite. One might be able to improve the current bounds on the condition number and relax the assumptions on the activation function. Deriving similar performance bounds for ReLU is particularly interesting. + +• Hidden state: For RNNs, the state vector is hidden and is observed through an additional equation (2.1); which further complicates the optimization landscape. Even for linear dynamical systems, learning the $( A , B , C , D { \bar { ) } }$ system ((1.1), (2.1)) is a non-trivial task Ho & Kalman (1966); Hardt et al. (2016). What can be said when we add the nonlinear activations? + +• Classification task: In this work, we used normally distributed input and least-squares regression for our theoretical guarantees. 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Learning non-overlapping convolutional neural networks with multiple kernels. arXiv preprint arXiv:1711.03440, 2017a. + +Kai Zhong, Zhao Song, Prateek Jain, Peter L Bartlett, and Inderjit S Dhillon. Recovery guarantees for onehidden-layer neural networks. arXiv preprint arXiv:1706.03175, 2017b. + +# A DETERMINISTIC CONVERGENCE RESULT FOR SGD + +Proof of Theorem 4.1. Given two distinct scalars $a , b$ ; define $\begin{array} { r } { \phi ^ { \prime } ( a , b ) = \frac { \phi ( a ) - \phi ( b ) } { a - b } } \end{array}$ . $\phi ^ { \prime } ( a , b ) \ge \beta$ since $\phi$ is $\beta$ -increasing. Define $\pmb { w } _ { \tau }$ to be the residual ${ \pmb w } _ { \tau } = { \pmb \theta } _ { \tau } - { \pmb \theta }$ . Observing + +$$ +\phi ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } ) - \pmb { y } _ { r _ { \tau } } = \phi ^ { \prime } ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } , \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } ) \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { w } _ { \tau } , +$$ + +the SGD recursion obeys + +$$ +\begin{array} { r l } & { \| \pmb { w } _ { \tau + 1 } \| _ { \ell _ { 2 } } ^ { 2 } = \| \pmb { w } _ { \tau } - \eta \big ( \phi ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } ) - \pmb { y } _ { r _ { \tau } } \big ) \phi ^ { \prime } \big ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } \big ) \pmb { x } _ { r _ { \tau } } \| _ { \ell _ { 2 } } ^ { 2 } . } \\ & { \qquad = \| \pmb { w } _ { \tau } - \eta \pmb { x } _ { r _ { \tau } } \phi ^ { \prime } \big ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } \big ) \phi ^ { \prime } \big ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } , \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } \big ) \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } } \\ & { \qquad = \| \big ( \pmb { I } - \eta \pmb { G } _ { r _ { \tau } } \big ) \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } } \end{array} +$$ + +where $G _ { r _ { \tau } } = \pmb { x } _ { r _ { \tau } } \phi ^ { \prime } ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } ) \phi ^ { \prime } ( \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } _ { \tau } , \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { \theta } ) \pmb { x } _ { r _ { \tau } } ^ { T }$ . Since $\phi$ is 1-Lipschitz and $\beta$ -increasing, $\mathbf { { G } } _ { r _ { \tau } }$ is a positivesemidefinite matrix satisfying + +$$ +\begin{array} { r l } & { \pmb { x } _ { r _ { \tau } } \pmb { x } _ { r _ { \tau } } ^ { T } \succeq \pmb { G } _ { r _ { \tau } } \succeq \beta ^ { 2 } \pmb { x } _ { r _ { \tau } } \pmb { x } _ { r _ { \tau } } ^ { T } , } \\ & { \pmb { G } _ { r _ { \tau } } ^ { T } \pmb { G } _ { r _ { \tau } } \preceq \pmb { x } _ { r _ { \tau } } \pmb { x } _ { r _ { \tau } } ^ { T } \pmb { x } _ { r _ { \tau } } \pmb { x } _ { r _ { \tau } } ^ { T } \preceq B \pmb { x } _ { r _ { \tau } } \pmb { x } _ { r _ { \tau } } ^ { T } . } \end{array} +$$ + +Consequently, we find the following bounds in expectation + +$$ +\begin{array} { r l } & { \gamma _ { + } I _ { n } \succeq \mathbb { E } [ G _ { r _ { \tau } } ] \succeq \beta ^ { 2 } \gamma _ { - } I _ { n } , } \\ & { \mathbb { E } [ G _ { r _ { \tau } } ^ { T } G _ { r _ { \tau } } ] \preceq B \gamma _ { + } I _ { n } . } \end{array} +$$ + +Observe that (A.1) essentially lower bounds the strong convexity parameter of the problem with $\beta ^ { 2 } \gamma _ { - }$ ; which is the strong convexity of the identical problem with the linear activation (i.e. $\beta = 1$ ). However, we only consider strong convexity around the ground truth parameter $\pmb \theta$ i.e. we restricted our attention to $( \pmb \theta , \pmb \theta _ { \tau } )$ pairs. With this, ${ \pmb w } _ { \tau + 1 }$ can be controlled as, + +$$ +\begin{array} { r l } & { \mathbb { E } [ \| \pmb { w } _ { \tau + 1 } \| _ { \ell _ { 2 } } ^ { 2 } ] = \mathbb { E } [ \| ( \pmb { I } - \eta \pmb { G } _ { r _ { \tau } } ) \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } ] } \\ & { \quad \quad \quad = \| \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } - 2 \eta \mathbb { E } [ \pmb { w } _ { \tau } ^ { T } \pmb { G } _ { r _ { \tau } } \pmb { w } _ { \tau } ] + \eta ^ { 2 } \mathbb { E } [ \pmb { w } _ { \tau } ^ { T } \pmb { G } _ { r _ { \tau } } ^ { T } \pmb { G } _ { r _ { \tau } } \pmb { w } _ { \tau } ] } \\ & { \quad \quad \quad \le \| \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } ( 1 - 2 \eta \beta ^ { 2 } \gamma _ { - } + \eta ^ { 2 } B \gamma _ { + } ) . } \end{array} +$$ + +Setting η = $\begin{array} { r } { \eta = \frac { \beta ^ { 2 } \gamma _ { - } } { \gamma _ { + } B } } \end{array}$ , we find the advertised bound + +$$ +\mathbb { E } [ \| \pmb { w } _ { \tau + 1 } \| _ { \ell _ { 2 } } ^ { 2 } ] \le \mathbb { E } [ \| \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } ] ( 1 - \frac { \beta ^ { 4 } \gamma _ { - } ^ { 2 } } { \gamma _ { + } B } ) . +$$ + +Applying induction over the iterations $\tau$ , we find the advertised bound (4.2) + +$$ +\mathbb { E } [ \| \pmb { w } _ { \tau } \| _ { \ell _ { 2 } } ^ { 2 } ] \le \| \pmb { w } _ { 0 } \| _ { \ell _ { 2 } } ^ { 2 } ( 1 - \frac { \beta ^ { 4 } \gamma _ { - } ^ { 2 } } { \gamma _ { + } B } ) ^ { \tau } . +$$ + +Lemma A.1 (Merging $L$ splits). Assume matrices √ $\pmb { X } ^ { ( i ) } \in \mathbb { R } ^ { N _ { i } \times q }$ are given for $1 \leq i \leq L$ . Suppose for all $1 \leq i \leq L$ , rows of $\pmb { X } ^ { ( i ) }$ has $\ell _ { 2 }$ norm at most $\sqrt { B }$ and each $\pmb { X } ^ { ( i ) }$ satisfies + +$$ +\gamma _ { + } I _ { n } \succeq \frac { { X ^ { ( i ) } } ^ { T } X ^ { ( i ) } } { N _ { i } } \succeq \gamma _ { - } I _ { n } . +$$ + +Set $\begin{array} { r } { N = \sum _ { i = 1 } ^ { L } N _ { i } } \end{array}$ and form the concatenated matrix $\begin{array} { r } { \pmb { X } = \left[ \pmb { X } ^ { ( 1 ) } \right] . } \\ { \quad } \\ { \quad } \\ { \pmb { X } ^ { ( L ) } } \end{array}$ Denote ith row of $\pmb { X }$ by $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ . Then, for + +each i, $\| \pmb { x } _ { i } \| _ { \ell _ { 2 } } ^ { 2 } \leq B$ and + +$$ +\gamma _ { + } I _ { n } \varrho \overset { \mathbf { \substack { \textstyle X } } } { = } \frac { \mathbf { { X } } ^ { T } \mathbf { { X } } } { N } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } x _ { i } \mathbf { { x } } _ { i } ^ { T } \succeq \gamma _ { - } I _ { n } . +$$ + +Proof. The bound on the rows $\| \pmb { x } _ { i } \| _ { \ell _ { 2 } }$ directly follows by assumption. For the remaining result, first observe that $\begin{array} { r } { { \pmb X } ^ { T } { \pmb X } = \sum _ { i = 1 } ^ { L } { \pmb X } ^ { ( i ) ^ { T } } { \pmb X } ^ { ( i ) } } \end{array}$ . Next, we have + +$$ +N \gamma _ { + } I _ { n } = \sum _ { i = 1 } ^ { L } N _ { i } \gamma _ { + } I _ { n } \succeq \sum _ { i = 1 } ^ { L } { X ^ { ( i ) } } ^ { T } X ^ { ( i ) } \succeq \sum _ { i = 1 } ^ { L } N _ { i } \gamma _ { - } I _ { n } = N \gamma _ { - } I _ { n } . +$$ + +Combining these two yields the desired upper/lower bounds on $X ^ { T } X / N$ . + +# B PROPERTIES OF THE NONLINEAR STATE EQUATIONS + +This section characterizes the properties of the state vector $\pmb { h } _ { t }$ when input sequence is normally distributed. These bounds will be crucial for obtaining upper and lower bounds for the singular values of the data matrix $\pmb { X } = [ \pmb { x } _ { 1 } ~ \ldots ~ \pmb { x } _ { N } ] ^ { T }$ described in (2.2). For probabilistic arguments, we will use the properties of subgaussian random variables. Orlicz norm provides a general framework that subsumes subgaussianity. + +Definition B.1 (Orlicz norms). For a scalar random variable Orlicz-a norm is defined as + +$$ +\| X \| _ { \psi _ { a } } = \operatorname* { s u p } _ { k \geq 1 } k ^ { - 1 / a } ( \mathbb { E } [ | X | ^ { k } ] ) ^ { 1 / k } +$$ + +Orlicz-a norm of a vector $\pmb { x } \in \mathbb { R } ^ { p }$ is defined as $\begin{array} { r } { \| \pmb { x } \| _ { \psi _ { a } } = \operatorname* { s u p } _ { \pmb { v } \in B ^ { p } } \| \pmb { v } ^ { T } \pmb { x } \| _ { \psi _ { a } } } \end{array}$ where $B ^ { p }$ is the unit $\ell _ { 2 }$ ball. The subexponential norm is the Orlicz-1 norm $\lVert \cdot \rVert _ { \psi _ { 1 } }$ and the subgaussian norm is the Orlicz-2 norm $\lVert \cdot \rVert _ { \psi _ { 2 } }$ . + +Lemma B.2 (Lipschitz properties of the state vector). Consider the state equation (1.1). Suppose activation $\phi$ is 1-Lipschitz. Observe that $\boldsymbol { h } _ { t + 1 }$ is a deterministic function of the input sequence $\{ { \pmb u } _ { \tau } \} _ { \tau = 0 } ^ { t }$ . Fixing all vectors $\{ { \pmb u } _ { i } \} _ { i \neq \tau }$ (i.e. all except ${ \bf { u } } _ { \tau }$ ), $\boldsymbol { h } _ { t + 1 }$ is $\| A \| ^ { t - \tau } \| B \|$ Lipschitz function of ${ \pmb u } _ { \tau }$ for $0 \leq \tau \leq t$ . + +Proof. Fixing $\{ { \pmb u } _ { i } \} _ { i \neq \tau }$ , denote $\boldsymbol { h } _ { t + 1 }$ as a function of ${ \pmb u } _ { \tau }$ by $\pmb { h } _ { t + 1 } ( \pmb { u } _ { \tau } )$ . Given a pair of vectors ${ \bf { \boldsymbol { u } } } _ { \tau } , { \bf { \boldsymbol { u } } } _ { \tau } ^ { \prime }$ using 1-Lipschitzness of $\phi$ , for any $t > \tau$ , we have + +$$ +\begin{array} { r l } & { \| h _ { t + 1 } ( \boldsymbol { u } _ { \tau } ) - h _ { t + 1 } ( \boldsymbol { u } _ { \tau } ^ { \prime } ) \| _ { \ell _ { 2 } } \leq \| \phi ( A h _ { t } ( \boldsymbol { u } _ { \tau } ) + B \boldsymbol { u } _ { t } ) - \phi ( A h _ { t } ( \boldsymbol { u } _ { \tau } ^ { \prime } ) + B \boldsymbol { u } _ { t } ) \| _ { \ell _ { 2 } } } \\ & { \qquad \leq \| A ( h _ { t } ( \boldsymbol { u } _ { \tau } ) - h _ { t } ( \boldsymbol { u } _ { \tau } ^ { \prime } ) ) \| _ { \ell _ { 2 } } } \\ & { \qquad \leq \| A \| \| h _ { t } ( \boldsymbol { u } _ { \tau } ) - h _ { t } ( \boldsymbol { u } _ { \tau } ^ { \prime } ) \| _ { \ell _ { 2 } } . } \end{array} +$$ + +Proceeding with this recursion until $t = \tau$ , we find + +$$ +\begin{array} { r l } & { \| h _ { t + 1 } ( \boldsymbol { u } _ { \tau } ) - h _ { t + 1 } ( \boldsymbol { u } _ { \tau } ^ { \prime } ) \| _ { \ell _ { 2 } } \leq \| \boldsymbol { A } \| ^ { t - \tau } \| h _ { \tau + 1 } ( \boldsymbol { u } _ { \tau } ) - h _ { \tau + 1 } ( \boldsymbol { u } _ { \tau } ^ { \prime } ) \| _ { \ell _ { 2 } } } \\ & { \qquad \leq \| \boldsymbol { A } \| ^ { t - \tau } \| \phi ( \boldsymbol { A } h _ { \tau } + \boldsymbol { B } \boldsymbol { u } _ { \tau } ) - \phi ( \boldsymbol { A } h _ { \tau } + \boldsymbol { B } \boldsymbol { u } _ { \tau } ^ { \prime } ) \| _ { \ell _ { 2 } } } \\ & { \qquad \leq \| \boldsymbol { A } \| ^ { t - \tau } \| \boldsymbol { B } \| \| \boldsymbol { u } _ { \tau } - \boldsymbol { u } _ { \tau } ^ { \prime } \| _ { \ell _ { 2 } } . } \end{array} +$$ + +This bound implies $\pmb { h } _ { t + 1 } ( \pmb { u } _ { \tau } )$ is $\| A \| ^ { t - \tau } \| B \|$ Lipschitz function of ${ \pmb u } _ { \tau }$ + +Lemma B.3 (Upper bound). Consider the state equation governed by equation (1.1). Suppose ${ \pmb u } _ { t } \overset { i . i . d . } { \sim } \mathcal { N } ( 0 , { \pmb I } _ { p } )$ , $\phi$ is 1-Lipschitz, $\bar { \phi } ( 0 ) = 0$ and $h _ { 0 } = 0$ . Recall the definition (3.2) of $B _ { t }$ . We have the following properties + +• $\pmb { h } _ { t }$ is a $B _ { t }$ -Lipschitz function of the vector $\begin{array} { r } { \pmb q _ { t } = [ \pmb { u } _ { 0 } ^ { T } \ . . . \ \pmb { u } _ { t - 1 } ^ { T } ] ^ { T } \in \mathbb { R } ^ { t p } . } \end{array}$ . + +• There exists an absolute constant $c > 0$ such that $\| h _ { t } - \mathbb { E } [ h _ { t } ] \| _ { \psi _ { 2 } } \leq c B _ { t }$ and $\Sigma [ h _ { t } ] \preceq B _ { t } ^ { 2 } { \cal I } _ { n }$ . + +• $\pmb { h } _ { t }$ satisfies + +$$ +\mathbb { E } [ \| \pmb { h } _ { t } \| _ { \ell _ { 2 } } ^ { 2 } ] \le t r ( \pmb { B } \pmb { B } ^ { T } ) \frac { 1 - \| \pmb { A } \| ^ { 2 t } } { 1 - \| \pmb { A } \| ^ { 2 } } \le \operatorname* { m i n } \{ n , p \} B _ { t } ^ { 2 } . +$$ + +Also, there exists an absolute constant √ $c > 0$ such that for any $m \geq n$ , with probability $1 -$ $2 \exp ( - 1 0 0 m )$ , $\| h _ { t } \| _ { \ell _ { 2 } } \leq c \sqrt { m } B _ { t }$ . + +Proof. i) Bounding Lipschitz constant: Observe that $\pmb { h } _ { t }$ is a deterministic function of $\pmb q _ { t }$ i.e. $h _ { t } = f ( q _ { t } )$ for some function $f$ . To bound Lipschitz constant of $f$ , for all (deterministic) vector pairs $\pmb q _ { t }$ and $\hat { \pmb q } _ { t }$ , we find a scalar $L _ { f }$ satisfying, + +$$ +\| f ( \pmb q _ { t } ) - f ( \hat { \pmb q } _ { t } ) \| _ { \ell _ { 2 } } \leq L _ { f } \| \pmb q _ { t } - \hat { \pmb q } _ { t } \| _ { \ell _ { 2 } } . +$$ + +Define the vectors, $\{ a _ { i } \} _ { i = 0 } ^ { t }$ , as follows + +$$ +\mathbf { \delta } \mathbf { \delta } \mathbf { \tilde { a } } _ { i } = [ \hat { \mathbf { u } } _ { 0 } ^ { T } \dots \hat { \mathbf { u } } _ { i - 1 } ^ { T } \mathbf { \delta } \mathbf { u } _ { i } ^ { T } \dots \mathbf { \delta } \mathbf { u } _ { t - 1 } ^ { T } ] ^ { T } . +$$ + +Observing that ${ \pmb a } _ { 0 } = { \pmb q } _ { t } , { \pmb a } _ { t } = \hat { { \pmb q } } _ { t }$ , we write the telescopic sum, + +$$ +\lVert f ( \pmb { q } _ { t } ) - f ( \pmb { \hat { q } } _ { t } ) \rVert _ { \ell _ { 2 } } \leq \sum _ { i = 0 } ^ { t - 1 } \lVert f ( \pmb { a } _ { i + 1 } ) - f ( \pmb { a } _ { i } ) \rVert _ { \ell _ { 2 } } . +$$ + +Focusing on the individual terms $f ( \pmb { a } _ { i + 1 } ) - f ( \pmb { a } _ { i } )$ , observe that the only difference is the ${ \mathbf { } } { \mathbf { } } u _ { i } , \hat { { \mathbf { } } } { \mathbf { } } \bar { { \mathbf { } } } u _ { i }$ terms. Viewing $\pmb { h } _ { t }$ as a function of ${ \bf { u } } _ { i }$ and applying Lemma B.2, + +$$ +\| f ( \pmb { a } _ { i + 1 } ) - f ( \pmb { a } _ { i } ) \| _ { \ell _ { 2 } } \leq \| \pmb { A } \| ^ { t - 1 - i } \| \pmb { B } \| \| \pmb { u } _ { i } - \hat { \pmb { u } } _ { i } \| _ { \ell _ { 2 } } . +$$ + +To bound the sum, we apply the Cauchy-Schwarz inequality; which yields + +$$ +\begin{array} { r l } { | f ( q _ { t } ) - f ( \hat { q } _ { t } ) | \le \displaystyle \sum _ { i = 0 } ^ { t - 1 } \| A \| ^ { t - 1 - i } \| B \| \| u _ { i } - \hat { u } _ { t } \| _ { \ell _ { 2 } } } & { } \\ { \quad \quad \le ( \displaystyle \sum _ { i = 0 } ^ { t - 1 } \| A \| ^ { 2 ( t - 1 - i ) } \| B \| ^ { 2 } ) ^ { 1 / 2 } \displaystyle \underbrace { ( \displaystyle \sum _ { i = 0 } ^ { t - 1 } \| u _ { i } - \hat { u } _ { i } \| _ { \ell _ { 2 } } ^ { 2 } ) ^ { 1 / 2 } } _ { \| q _ { t } - \hat { q } _ { t } \| _ { \ell _ { 2 } } } } \\ { \quad \quad \le \sqrt { \displaystyle \frac { \| B \| ^ { 2 } ( 1 - \| A \| ^ { 2 / t } ) } { 1 - \| A \| ^ { 2 } } } \| q _ { t } - \hat { q } _ { t } \| _ { \ell _ { 2 } } } & { } \\ { \quad \quad = B _ { t } \| q _ { t } - \hat { q } _ { t } \| _ { \ell _ { 2 } } . } \end{array} +$$ + +The final line achieves the inequality (B.1) with $L _ { f } = B _ { t }$ hence $\pmb { h } _ { t }$ is $B _ { t }$ Lipschitz function of $\mathbf { \nabla } _ { \mathbf { \eta } } \mathbf { q } _ { t }$ + +ii) Bounding subgaussian norm: When ${ \pmb u } _ { t } \overset { \mathrm { i . i . d . } } { \sim } \mathcal { N } ( 0 , { \pmb I } _ { p } )$ , the vector $\pmb q _ { t }$ is distributed as $\mathcal { N } ( 0 , { I } _ { t p } )$ . Since $\pmb { h } _ { t }$ a $B _ { t }$ Lipschitz function of $\pmb q _ { t }$ , for any fixed unit length vector $_ { v }$ , $\alpha _ { \pmb { v } } : = \pmb { v } ^ { T } h _ { t } = \pmb { v } ^ { T } f ( \pmb { q } _ { t } )$ is still $B _ { t }$ -Lipschitz function of $\pmb q _ { t }$ . Hence, using Gaussian concentration of Lipschitz functions, $\alpha _ { v }$ satisfies + +$$ +\mathbb { P } ( | \alpha _ { v } - \mathbb { E } [ \alpha _ { v } ] | \geq t ) \leq 2 \exp ( - \frac { t ^ { 2 } } { 2 B _ { t } ^ { 2 } } ) . +$$ + +This implies that for any $\pmb { v }$ , $\alpha _ { v } \mathrm { ~ - ~ } \mathbb { E } [ \alpha _ { v } ]$ is $\mathcal { O } ( B _ { t } )$ subgaussian Vershynin (2010). This is true for all unit $_ { v }$ , hence using Definition B.1, the vector $\pmb { h } _ { t }$ satisfies $\| h _ { t } - \mathbb { E } [ h _ { t } ] \| _ { \psi _ { 2 } } \leq \mathcal { O } ( B _ { t } )$ as well. Secondly, $B _ { t }$ -Lipschitz function of a Gaussian vector obeys the variance inequality v $\mathbf { a r } [ \alpha _ { v } ] \leq B _ { t } ^ { 2 }$ (page 49 of Ledoux (2001)), which implies the covariance bound + +$$ +\Sigma [ h _ { t } ] \preceq B _ { t } ^ { 2 } I _ { n } . +$$ + +iii) Bounding $\ell _ { 2 }$ -norm: To obtain this result, we first bound $\mathbb { E } [ \| h _ { t } \| _ { \ell _ { 2 } } ^ { 2 } ]$ . Since $\phi$ is 1-Lipschitz and $\phi ( 0 ) = 0$ we have the deterministic relation + +$$ +\lVert h _ { t + 1 } \rVert _ { \ell _ { 2 } } \leq \lVert A h _ { t } + B u _ { t } \rVert _ { \ell _ { 2 } } . +$$ + +Taking squares of both sides, expanding the right hand side, and using the independence of $\mathbf { } h _ { t } , \mathbf { } u _ { t }$ and the covariance information of $\mathbf { \Delta } \mathbf { u } _ { t }$ , we obtain + +$$ +\begin{array} { r l } & { \mathbb { E } [ \| h _ { t + 1 } \| _ { \ell _ { 2 } } ^ { 2 } ] \leq \mathbb { E } [ \| A h _ { t } + B u _ { t } \| _ { \ell _ { 2 } } ^ { 2 } ] = \mathbb { E } [ \| A h _ { t } \| _ { \ell _ { 2 } } ^ { 2 } ] + \mathbb { E } [ \| B u _ { t } \| _ { \ell _ { 2 } } ^ { 2 } ] } \\ & { \qquad \leq \| A \| ^ { 2 } \mathbb { E } [ \| h _ { t } \| _ { \ell _ { 2 } } ^ { 2 } ] + \mathrm { t r } ( B B ^ { T } ) . } \end{array} +$$ + +Now that the recursion is established, expanding $\pmb { h } _ { t }$ on the right hand side until $h _ { 0 } = 0$ , we obtain + +$$ +\mathbb { E } [ \| \pmb { h } _ { t + 1 } \| _ { \ell _ { 2 } } ^ { 2 } ] \le \sum _ { i = 0 } ^ { t } \| \pmb { A } \| ^ { 2 i } \mathrm { t r } ( \pmb { B } \pmb { B } ^ { T } ) \le \mathrm { t r } ( \pmb { B } \pmb { B } ^ { T } ) \frac { 1 - \| \pmb { A } \| ^ { 2 ( t + 1 ) } } { 1 - \| \pmb { A } \| ^ { 2 } } . +$$ + +Now using the fact that $\mathrm { t r } ( B B ^ { T } ) \leq \mathrm { r a n k } ( B ) \| B \| ^ { 2 } \leq \mathrm { m i n } \{ n , p \} \| B \| ^ { 2 }$ , we find + +$$ +\begin{array} { r } { \mathbb { E } [ \| \boldsymbol { h } _ { t + 1 } \| _ { \ell _ { 2 } } ] ^ { 2 } \le \mathbb { E } [ \| \boldsymbol { h } _ { t + 1 } \| _ { \ell _ { 2 } } ^ { 2 } ] \le \operatorname* { m i n } \{ n , p \} B _ { t + 1 } ^ { 2 } . } \end{array} +$$ + +Finally, using the fact that $\pmb { h } _ { t }$ is $B _ { t }$ -Lipschitz function and utilizing Gaussian concentration of $\mathbf { \boldsymbol { q } } _ { t } \sim \mathcal { N } ( 0 , I _ { t p } )$ we find + +$$ +\mathbb { P } ( \| h _ { t + 1 } \| _ { \ell _ { 2 } } - \mathbb { E } [ \| h _ { t + 1 } \| _ { \ell _ { 2 } } ] \ge t ) \le \exp ( - \frac { t ^ { 2 } } { 2 B _ { t } ^ { 2 } } ) . +$$ + +Setting $t = ( c - 1 ) \sqrt { m } B _ { t }$ for sufficiently large $c > 0$ , we find $\mathbb { P } ( \| h _ { t } \| _ { \ell _ { 2 } } \geq \sqrt { n } B _ { t } + ( c - 1 ) \sqrt { m } B _ { t } ) \leq$ $\exp ( - 1 0 0 m )$ . + +Lemma B.4 (Odd activations). Suppose $\phi$ is strictly increasing and obeys $\phi ( x ) = - \phi ( - x )$ for all $_ x$ and $h _ { 0 } = 0$ . Consider the state equation (1.1) driven ${ \pmb u } _ { t } \overset { i . i . d . } { \sim } \mathcal { N } ( 0 , { \pmb I } _ { p } )$ . We have that $\mathbb { E } [ h _ { t } ] = 0$ . + +Proof. We will inductively show that $\{ h _ { t } \} _ { t \ge 0 }$ has a symmetric distribution around 0. Suppose the vector $\pmb { h } _ { t }$ satisfies this assumption. Let $S \subset \mathbb { R } ^ { n }$ be a set. We will argue that $\mathbb { P } ( h _ { t + 1 } \subset S ) = \mathbb { P } ( h _ { t + 1 } \subset - S )$ . Since $\phi$ is strictly increasing, it is bijective on vectors, and we can define the unique inverse set $S ^ { \prime } = \phi ^ { - 1 } ( S )$ . Also since $\phi$ is odd, $\phi ( - S ^ { \prime } ) = - S$ . Since $\mathbf { } h _ { t } , \mathbf { } u _ { t }$ are independent and symmetric, we reach the desired conclusion as follows + +$$ +\begin{array} { r l } & { \mathbb { P } ( h _ { t + 1 } \subset S ) = \mathbb { P } ( A h _ { t } + B u _ { t } \subset S ^ { \prime } ) = \mathbb { P } ( A ( - h _ { t } ) + B ( - u _ { t } ) \subset S ^ { \prime } ) } \\ & { \qquad = \mathbb { P } ( A h _ { t } + B u _ { t } \subset - S ^ { \prime } ) = \mathbb { P } ( \phi ( A h _ { t } + B u _ { t } ) \subset \phi ( - S ^ { \prime } ) ) = \mathbb { P } ( h _ { t + 1 } \subset - S ) . } \end{array} +$$ + +Theorem B.5 (State-vector lower bound). Consider the nonlinear state equation (1.1) with $\{ { \pmb u } _ { t } \} _ { t \ge 0 }$ i.i.d. ∼ $\mathcal { N } ( 0 , { \pmb I } _ { p } )$ . Suppose $\phi$ is a $\beta$ -increasing function for some constant $\beta > 0$ . For any $t \geq 1$ , the state vector obeys + +$$ +\begin{array} { r } { \Sigma [ h _ { t } ] \succeq \beta ^ { 2 } s _ { \mathrm { m i n } } ( B B ^ { T } ) I _ { n } . } \end{array} +$$ + +Proof. The proof is an application of Lemma B.7. The main idea is to write $\pmb { h } _ { t }$ as sum of two independent vectors, one of which has independent entries. Consider a multivariate Gaussian vector $\mathbf { \sigma } _ { \mathbf { \mathcal { g } } } \sim \mathcal { N } ( \bar { 0 , } \Sigma )$ . $\pmb { g }$ is statistically identical to ${ \pmb g } _ { 1 } + { \pmb g } _ { 2 }$ where ${ \pmb g } _ { 1 } \sim \mathcal { N } ( 0 , s _ { \mathrm { m i n } } ( { \pmb \Sigma } ) { \pmb I } _ { d } )$ and $g _ { 2 } \sim \mathcal { N } ( 0 , \pmb { \Sigma } - s _ { \mathrm { m i n } } ( \pmb { \Sigma } ) \pmb { I } _ { d } )$ are independent multivariate Gaussians. + +Since $B { \boldsymbol { u } } _ { t } \sim \mathcal { N } ( 0 , B B ^ { T } )$ , setting $\pmb { \Sigma } = \pmb { B } \pmb { B } ^ { T }$ and $s _ { \mathrm { m i n } } = s _ { \mathrm { m i n } } ( \Sigma )$ , we have that $B u _ { t } \sim g _ { 1 } + g _ { 2 }$ where ${ \bf { \mathit { g } } } _ { 1 } , { \bf { \mathit { g } } } _ { 2 }$ are independent and ${ \pmb g } _ { 1 } \sim \mathcal { N } ( 0 , s _ { \mathrm { m i n } } { \pmb I } _ { n } )$ and $\pmb { g } _ { 2 } \sim \mathcal { N } ( 0 , \pmb { \Sigma } - s _ { \mathrm { m i n } } \pmb { I } _ { n } )$ . Consequently, we may write + +$$ +B u _ { t } + A h _ { t } \sim g _ { 1 } + g _ { 2 } + A h _ { t } . +$$ + +For lower bound, the crucial component will be the $\pmb { g } _ { 1 }$ term; which has i.i.d. entries. Applying Lemma B.7 by setting ${ \pmb x } = { \pmb g } _ { 1 }$ and $\pmb { y } = \pmb { g } _ { 2 } + \pmb { A } \hat { h } _ { t }$ , and using the fact that $h _ { t } , g _ { 1 } , g _ { 2 }$ are all independent of each other, we find the advertised bound, for all $t \geq 0$ , via + +$$ +\Sigma [ h _ { t + 1 } ] = \Sigma [ \phi ( g _ { 1 } + g _ { 2 } + A h _ { t } ) ] \succeq \beta ^ { 2 } s _ { \mathrm { m i n } } I _ { n } . +$$ + +The next theorem applies to multiple-input-single-output (MISO) systems where $\pmb { A }$ is a scalar and $_ B$ is a row vector. The goal is refining the lower bound of Theorem B.5. + +Theorem B.6 (MISO lower bound). Consider the setup of Theorem B.5 with single output i.e. $n = 1$ . For any $t \geq 1$ , the state vector obeys + +$$ +\mathbf { v a r } [ \pmb { h } _ { t } ] \geq \beta ^ { 2 } \| \pmb { B } \| _ { \ell _ { 2 } } ^ { 2 } \frac { 1 - ( \beta | \pmb { A } | ) ^ { 2 t } } { 1 - \beta ^ { 2 } | \pmb { A } | ^ { 2 } } . +$$ + +Proof. For any random variable $X$ , applying Lemma B.7, we have va ${ \mathfrak { r } } [ \phi ( X ) ] \geq \beta ^ { 2 } \mathbf { v a r } [ X ]$ . Recursively, this yields + +$$ +\mathbf { v a r } [ h _ { t + 1 } ] = \mathbf { v a r } [ \phi ( A h _ { t } + B u _ { t } ) ] \geq \beta ^ { 2 } \mathbf { v a r } [ A h _ { t } + B u _ { t } ] = \beta ^ { 2 } ( | A | ^ { 2 } \mathbf { v a r } [ h _ { t } ] + \| B \| _ { \ell _ { 2 } } ^ { 2 } ) . +$$ + +Expanding these inequalities till $\scriptstyle h _ { 0 }$ , we obtain the desired bound + +$$ +\mathbf { v a r } [ \pmb { h } _ { t } ] \geq \sum _ { i = 1 } ^ { t } ( \beta ^ { i } | \pmb { A } | ^ { i - 1 } \| \pmb { B } \| _ { \ell _ { 2 } } ) ^ { 2 } . +$$ + +Lemma B.7 (Vector lower bound). Suppose $\phi$ is a $\beta$ -increasing function. Let ${ \pmb x } = [ { \pmb x } _ { 1 } ~ . ~ . ~ { \pmb x } _ { n } ] ^ { T }$ be a vector with i.i.d. entries distributed as $x _ { i } \sim X$ . Let $\textbf { { y } }$ be a random vector independent of $_ { \textbf { \em x } }$ . Then, + +$$ +\Sigma [ \phi ( { \pmb x } + { \pmb y } ) ] \succeq \beta ^ { 2 } { \mathbf { v a r } } [ X ] { \cal I } _ { n } . +$$ + +Proof. We first apply law of total covariance (e.g. Lemma B.8) to simplify the problem using the following lower bound based on the independence of $_ { \textbf { \em x } }$ and $\pmb { y }$ , + +$$ +\begin{array} { r } { \Sigma [ \phi ( { \pmb x } + { \pmb y } ) ] \succeq \mathbb { E } _ { { \pmb y } } [ \Sigma [ \phi ( { \pmb x } + { \pmb y } ) | { \pmb y } ] ] } \\ { = \mathbb { E } _ { { \pmb y } } [ \Sigma _ { \pmb x } [ \phi ( { \pmb x } + { \pmb y } ) ] ] . } \end{array} +$$ + +Now, focusing on the covariance $\Sigma _ { x } [ \phi ( { \pmb x } + { \pmb y } ) ]$ , fixing a realization of $\textbf { { y } }$ , and using the fact that $_ { \textbf { \em x } }$ has i.i.d. entries; $\phi ( { \pmb x } + { \pmb y } )$ has independent entries as $\phi$ applies entry-wise. This implies that $\Sigma _ { \pmb { x } } [ \phi ( \pmb { x } + \pmb { y } ) ]$ is a diagonal matrix. Consequently, its lowest eigenvalue is the minimum variance over all entries, + +$$ +\pmb { \Sigma } _ { \pmb { x } } [ \phi ( \pmb { x } + \pmb { y } ) ] \succeq \operatorname* { m i n } _ { 1 \leq i \leq n } \mathbf { v a r } [ \phi ( \pmb { x } _ { i } + \pmb { y } _ { i } ) ] \pmb { I } _ { n } . +$$ + +Fortunately, Lemma B.9 provides the lower bound $\mathbf { v a r } [ \phi ( \pmb { x } _ { i } + \pmb { y } _ { i } ) ] \geq \beta ^ { 2 } \mathbf { v a r } [ X ]$ . Since this lower bound holds for any fixed realization of $\textbf { { y } }$ , it still holds after taking expectation over $\textbf { { y } }$ ; which concludes the proof. □ + +The next two lemmas are helper results for Lemma B.7 and are provided for the sake of completeness. + +Lemma B.8 (Law of total covariance). Let $\mathbf { \nabla } _ { \mathbf { x } , \mathbf { y } }$ be two random vectors and assume $\textbf { { y } }$ has finite covariance. Then + +$$ +\pmb { \Sigma } [ \pmb { y } ] = \mathbb { E } [ \pmb { \Sigma } [ \pmb { y } \mid \pmb { x } ] ] + \pmb { \Sigma } [ \mathbb { E } [ \pmb { y } \mid \pmb { x } ] ] . +$$ + +Proof. First, write $\Sigma [ { \pmb y } ] = \mathbb { E } [ { \pmb y } { \pmb y } ^ { T } ] - \mathbb { E } [ { \pmb y } ] \mathbb { E } [ { \pmb y } ^ { T } ]$ . Then, applying the law of total expectation to each term, + +$$ +\pmb { \Sigma } [ \pmb { y } ] = \mathbb { E } [ \mathbb { E } [ \pmb { y } \pmb { y } ^ { T } \mid \pmb { x } ] ] - \mathbb { E } [ \mathbb { E } [ \pmb { y } \mid \pmb { x } ] ] \mathbb { E } [ \mathbb { E } [ \pmb { y } ^ { T } \mid \pmb { x } ] ] +$$ + +Next, we can write the conditional expectation as $\mathbb { E } [ \mathbb { E } [ \pmb { y } \pmb { y } ^ { T } \mid \pmb { x } ] ] = \mathbb { E } [ \pmb { \Sigma } [ \pmb { y } \mid \pmb { x } ] ] + \mathbb { E } [ \mathbb { E } [ \pmb { y } \mid \pmb { x } ] \mathbb { E } [ \pmb { y } \mid \pmb { x } ] ] ^ { T } .$ . To conclude, we obtain the covariance of $\mathbb { E } [ { \pmb y } \mid { \pmb x } ]$ via the difference, + +$$ +\begin{array} { r } { \mathbb { E } [ \mathbb { E } [ \pmb { y } \mid \pmb { x } ] \mathbb { E } [ \pmb { y } \mid \pmb { x } ] ] ^ { T } - \mathbb { E } [ \mathbb { E } [ \pmb { y } \mid \pmb { x } ] ] \mathbb { E } [ \mathbb { E } [ \pmb { y } ^ { T } \mid \pmb { x } ] ] = \pmb { \Sigma } [ \mathbb { E } [ \pmb { y } \mid \pmb { x } ] ] , } \end{array} +$$ + +which yields the desired bound. + +Lemma B.9 (Scalar lower bound). Suppose $\phi$ is a $\beta$ -increasing function with $\beta > 0$ as defined in Definition 3.1. Given a random variable $X$ and a scalar $_ y$ , we have + +$$ +\mathbf { v a r } [ { \phi } ( X + y ) ] \geq { \beta } ^ { 2 } \mathbf { v a r } [ X ] . +$$ + +Proof. Since $\phi$ is $\beta$ -increasing, it is invertible and $\phi ^ { - 1 }$ is strictly increasing. Additionally, $\phi ^ { - 1 }$ is $1 / \beta$ Lipschitz since, + +Using this observation and the fact that $\mathbb { E } [ X ]$ minimizes $\mathbb { E } ( X - \alpha ) ^ { 2 }$ over $\alpha$ , $\mathbf { v a r } [ \phi ( X + y ) ]$ can be lower bounded as follows + +$$ +\begin{array} { r l } & { \mathbf { v a r } [ \phi ( X + y ) ] = \mathbb { E } ( \phi ( X + y ) - \mathbb { E } [ \phi ( X + y ) ] ) ^ { 2 } } \\ & { \qquad \geq \beta ^ { 2 } \mathbb { E } ( ( X + y ) - \phi ^ { - 1 } ( \mathbb { E } [ \phi ( X + y ) ] ) ) ^ { 2 } } \\ & { \qquad \geq \beta ^ { 2 } \mathbb { E } ( X + y - \mathbb { E } [ X + y ] ) ^ { 2 } } \\ & { \qquad = \beta ^ { 2 } \mathbb { E } ( X - \mathbb { E } X ) ^ { 2 } = \beta ^ { 2 } \mathbf { v a r } [ X ] . } \end{array} +$$ + +Note that, the final line is the desired conclusion. + +# C TRUNCATING STABLE SYSTEMS + +One of the challenges in analyzing dynamical systems is the fact that samples from the same trajectory have temporal dependence. This section shows that, for stable systems, the impact of the past states decay exponentially fast and the system can be approximated by using the recent inputs only. We first define the truncation of the state vector. + +Definition C.1 (Truncated state vector). Suppose $\phi ( 0 ) = 0$ , initial condition $h _ { 0 } = 0$ , and consider the state equation (1.1). Given a timestamp $t$ , $L$ -truncation of the state vector $\pmb { h } _ { t }$ is denoted by $\bar { h } _ { t , L }$ and is equal to $\mathbf { \nabla } _ { \mathbf { \eta } } \mathbf { q } _ { t }$ where + +$$ +\pmb { q } \tau + 1 = \phi ( \pmb { A } \pmb { q } _ { \tau } + \pmb { B } \pmb { u } _ { \tau } ^ { \prime } ) \quad , \quad q _ { 0 } = 0 +$$ + +is the state vector generated by the inputs ${ \pmb u } _ { \tau } ^ { \prime }$ satisfying + +$$ +{ \pmb u } _ { \tau } ^ { \prime } = \left\{ \begin{array} { l l } { 0 i f \tau < t - L } \\ { { \pmb u } _ { \tau } e l s e } \end{array} \right. . +$$ + +In words, $L$ truncated state vector $\bar { h } _ { t , L }$ is obtained by unrolling $\mathbf { \delta } _ { h _ { t } }$ until time $t - L$ and setting the contribution of the state vector $\boldsymbol { h } _ { t - L }$ to 0. This way, $\bar { h } _ { t , L }$ depends only on the variables $\{ u _ { \tau } \} _ { \tau = t - L } ^ { t - 1 }$ . + +The following lemma states that impact of truncation can be made fairly small for stable systems $( \left. A \right. < 1 )$ . Lemma C.2 (Truncation impact – deterministic). Consider the state vector $\pmb { h } _ { t }$ and its $L$ -truncation $\bar { h } _ { t , L }$ from Definition C.1. Suppose $\phi$ is 1-Lipschitz. We have that + +$$ +\| h _ { t } - \bar { h } _ { t , L } \| _ { \ell _ { 2 } } \leq \left\{ \begin{array} { l l } { 0 i f t \leq L } \\ { \| A \| ^ { L } \| h _ { t - L } \| _ { \ell _ { 2 } } e l s e } \end{array} \right. . +$$ + +Proof. When $t \leq L$ , Definition C.1 implies ${ \pmb u } _ { \tau } ^ { \prime } = { \pmb u } _ { \tau }$ hence ${ \pmb h } _ { t } = { \pmb q } _ { t } = \bar { { \pmb h } } _ { t , L }$ . When $t > L$ , we again use Definition C.1 and recall that ${ \pmb u } _ { \tau } ^ { \prime } = 0$ until time $\tau = t - L - 1$ . For all $t - L < \tau \leq t$ , using 1-Lipschitzness of $\phi$ , we have that + +$$ +\begin{array} { r l } & { \| h _ { \tau } - { \pmb q } _ { \tau } \| _ { \ell _ { 2 } } = \| \phi ( { \pmb A } h _ { \tau - 1 } + { \pmb B } { \pmb u } _ { \tau - 1 } ) - \phi ( { \pmb A } { \pmb q } _ { \tau - 1 } + { \pmb B } { \pmb u } _ { \tau - 1 } ) \| _ { \ell _ { 2 } } } \\ & { \qquad \le \| ( { \pmb A } h _ { \tau - 1 } + { \pmb B } { \pmb u } _ { \tau - 1 } ) - ( { \pmb A } { \pmb q } _ { \tau - 1 } + { \pmb B } { \pmb u } _ { \tau - 1 } ) \| _ { \ell _ { 2 } } } \\ & { \qquad \le \| { \pmb A } ( h _ { \tau - 1 } - { \pmb q } _ { \tau - 1 } ) \| _ { \ell _ { 2 } } \le \| { \pmb A } \| \| h _ { \tau - 1 } - { \pmb q } _ { \tau - 1 } \| _ { \ell _ { 2 } } . } \end{array} +$$ + +Applying this recursion between $t - L < \tau \leq t$ and using the fact that $\pmb q _ { t - L } = 0$ implies the advertised result + +$$ +\begin{array} { r l } & { \| \pmb { h } _ { t } - \pmb { q } _ { t } \| _ { \ell _ { 2 } } \leq \| \pmb { A } \| ^ { L } \| \pmb { h } _ { t - L } - \pmb { q } _ { t - L } \| _ { \ell _ { 2 } } } \\ & { \qquad \leq \| \pmb { A } \| ^ { L } \| \pmb { h } _ { t - L } \| _ { \ell _ { 2 } } . } \end{array} +$$ + +We will now argue that, for stable systems, a single trajectory can be split into multiple nearly independent trajectories. First, we describe how the sub-trajectories are constructed. + +Definition C.3 (Sub-trajectory). Let sampling rate $L \geq 1$ and offset $1 \le \bar { \tau } \le L$ be two integers. Let $\bar { N } = \bar { N } _ { \bar { \tau } }$ be the largest integer obeying $( { \bar { N } } - 1 ) { \bar { L } } + { \bar { \tau } } \leq N$ . We sample the trajectory $\{ h _ { t } , \boldsymbol { u } _ { t } \} _ { t = 0 } ^ { N }$ at the points $\bar { \tau } , \bar { \tau } + L , \dots , \bar { \tau } + ( \bar { N } - 1 ) L + \bar { \tau }$ and define the τ¯th sub-trajectory as + +$$ +( { \pmb h } ^ { ( i ) } , { \pmb u } ^ { ( i ) } ) : = ( { \pmb h } ^ { ( i , \bar { \tau } ) } , { \pmb u } ^ { ( i , \bar { \tau } ) } ) = ( { \pmb h } _ { ( i - 1 ) L + \bar { \tau } } , { \pmb u } _ { ( i - 1 ) L + \bar { \tau } } ) . +$$ + +Definition C.4 (Truncated sub-trajectory). Consider the state equation (1.1) and recall Definition C.1. Given offset $\bar { \tau }$ and sampling rate $L$ , for $1 \leq i \leq \bar { N }$ , the ith truncated sub-trajectory states are $\{ \bar { h } ^ { ( i ) } \} _ { i = 1 } ^ { \bar { N } }$ where the ith state is defined as + +$$ +\bar { \pmb { h } } ^ { ( i ) } = \bar { \pmb { h } } _ { L ( i - 1 ) + \bar { \tau } , L - 1 } . +$$ + +The truncated samples are independent of each other as shown in the next lemma. + +Lemma C.5. Consider the truncated states of Definition C.4. If (1.1) is generated by independent vectors {ut}t≥0, for any offset τ¯ and sampling rate L, the vectors {h¯ (i)}N¯i=1 $\{ \bar { \pmb { h } } ^ { ( i ) } \} _ { i = 1 } ^ { \bar { N } } , \{ \pmb { u } ^ { ( i ) } \} _ { i = 1 } ^ { \bar { N } }$ are all independent of each other. + +Proof. By construction $\bar { \pmb { h } } ^ { ( i ) }$ only depends on the vectors $\left\{ \pmb { u } _ { \tau } \right\} _ { \tau = L ( i - 2 ) + \bar { \tau } + 1 } ^ { L ( i - 1 ) + \bar { \tau } - 1 }$ . Note that the dependence ranges $[ L ( i - 2 ) + \bar { \tau } + 1 , L ( i - 1 ) + \bar { \tau } - 1 ]$ are disjoint intervals for different $_ { i }$ ’s; hence $( \bar { \pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \bar { N } }$ are independent of each other. To show the independence of $\mathbf { \pmb { u } } ^ { ( i ) }$ and $\bar { \pmb { h } } ^ { ( i ) }$ ; observe that inputs $\pmb { u } ^ { ( i ) } = \pmb { u } _ { L ( i - 1 ) + \hat { \tau } }$ have timestamp $\bar { \tau }$ modulo $L$ ; which is not covered by the dependence range of $( \bar { \pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \bar { N } }$ . □ + +If the input is randomly generated, Lemma C.2 can be combined with a probabilistic bound on $\pmb { h } _ { t }$ , to show that truncated states $\bar { \pmb { h } } ^ { ( i ) }$ are fairly close to the actual states $\mathbf { \delta } _ { h } ( i )$ . + +Lemma C.6 (Truncation impact – random). Given offset $\bar { \tau }$ and sampling rate $L$ , consider the state vectors of the sub-trajectory $\{ h ^ { ( i ) } \} _ { i = 1 } ^ { \bar { N } }$ and $L - 1$ -truncations $( \bar { \pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \bar { N } }$ . Suppose $\{ \pmb { u } _ { t } \} _ { t \geq 0 } \stackrel { i . i . d . } { \sim } \mathcal { N } ( 0 , \pmb { I } _ { p } ) , \lVert \pmb { A } \rVert < 1$ $h _ { 0 } = 0$ , $\phi$ is 1-Lipschitz, and $\phi ( 0 ) = 0$ . Also suppose upper bound (4.3) of Assumption $^ { l }$ holds for some $\theta \leq$ ${ \sqrt { n } } , \gamma _ { + } > 0$ . There exists an absolute constant $c > 0$ such that with probability at least $1 - 2 \bar { N } \mathrm { \bar { e x p } } ( - 1 0 0 n )$ , for all $1 \leq i \leq \bar { N }$ , the following bound holds + +$$ +\| h ^ { ( i ) } - \bar { h } ^ { ( i ) } \| _ { \ell _ { 2 } } \leq c \sqrt { n } \| A \| ^ { L - 1 } \sqrt { \gamma _ { + } } . +$$ + +In particular, we can always pick $\gamma _ { + } = B _ { \infty } ^ { 2 }$ (via Lemma $B . 3$ ). + +Proof. Using Assumption 1, we can apply Lemma F.3 on vectors $\{ h _ { ( i - 2 ) L + \bar { \tau } + 1 } \} _ { i = 1 } ^ { \bar { N } }$ . Using a union bound, with desired probability, all vectors obey + +$$ +\begin{array} { r } { \| h _ { ( i - 2 ) L + \bar { \tau } + 1 } - \mathbb { E } \big [ h _ { ( i - 2 ) L + \bar { \tau } + 1 } \big ] \| _ { \ell _ { 2 } } \leq ( c - 1 ) \sqrt { n \gamma _ { + } } , } \end{array} +$$ + +for sufficiently large $c$ . Since $\theta \leq \sqrt { n }$ , triangle inequality implies $\| h _ { ( i - 2 ) L + \bar { \tau } + 1 } \| _ { \ell _ { 2 } } \leq c \sqrt { n \gamma _ { + } }$ . Now, applying Lemma C.2, for all $1 \leq i \leq \bar { N }$ , we find + +$$ +\begin{array} { r l } & { \| \pmb { h } ^ { ( i ) } - \bar { \pmb { h } } ^ { ( i ) } \| _ { \ell _ { 2 } } = \| \pmb { h } _ { ( i - 1 ) L + \bar { \tau } } - \bar { \pmb { h } } _ { ( i - 1 ) L + \bar { \tau } , L - 1 } \| _ { \ell _ { 2 } } } \\ & { \qquad \leq \| \pmb { A } \| ^ { L - 1 } \| \pmb { h } _ { ( i - 2 ) L + \bar { \tau } + 1 } \| _ { \ell _ { 2 } } } \\ & { \qquad \leq c \| \pmb { A } \| ^ { L - 1 } \sqrt { n \gamma + 1 } . } \end{array} +$$ + +# D PROPERTIES OF THE DATA MATRIX + +This section utilizes the probabilistic estimates from Section B to provide bounds on the condition number of data matrices obtained from the RNN trajectory (1.1). Following (2.2), these matrices $_ { H , U }$ and $\boldsymbol { x }$ are defined as + +$$ +\mathbf { { \cal H } } = [ h _ { 1 } \ \dots \ h _ { N } ] ^ { T } \quad , \quad \mathbf { { \cal U } } = \mathbf { { \cal H } } = [ u _ { 1 } \ \dots \ u _ { N } ] ^ { T } \quad , \quad \mathbf { { \cal X } } = [ x _ { 1 } \ \dots \ x _ { N } ] ^ { T } . +$$ + +The challenge is that, the state matrix $_ H$ has dependent rows; which will be addressed by carefully splitting the trajectory $\{ u _ { t } , h _ { t } \} _ { t = 0 } ^ { N }$ into multiple sub-trajectories which are internally weakly dependent as discussed in Section C. We first define the matrices obtained from these sub-trajectories. + +Definition D.1. Given sampling rate $L$ and offset $\bar { \tau }$ , consider the $L$ -subsampled trajectory $\{ \boldsymbol { h } ^ { ( i ) } , \boldsymbol { u } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ i) }Ni=1 as described in Definitions C.3 and C.4. Define the matrices $\bar { \pmb { H } } = \bar { \pmb { H } } ^ { ( \bar { \tau } ) } \in \mathbb { R } ^ { \bar { \pmb { N } } \times { n } }$ , $\tilde { \pmb { H } } = \tilde { \pmb { H } } ^ { ( \bar { \tau } ) } \in \mathbb { R } ^ { \bar { \cal N } \times n }$ , $\tilde { \pmb { U } } =$ $\tilde { \pmb { U } } ^ { ( \bar { \tau } ) } \in \mathbb { R } ^ { \bar { \pmb { N } } \times { p } }$ , and $\tilde { { \cal X } } = \tilde { { \cal X } } ^ { ( \bar { \tau } ) } \in \mathbb { R } ^ { \bar { N } \times ( n + p ) } a s$ + +$$ +\bar { \pmb { H } } = [ \bar { \pmb { h } } ^ { ( 1 ) } \dots \bar { \pmb { h } } ^ { ( \bar { N } ) } ] ^ { T } , \ \tilde { \pmb { H } } = [ \pmb { h } ^ { ( 1 ) } \dots \pmb { h } ^ { ( \bar { N } ) } ] ^ { T } , \ \tilde { \pmb { U } } = [ \pmb { u } ^ { ( 1 ) } \dots \pmb { u } ^ { ( \bar { N } ) } ] ^ { T } , \ \tilde { \pmb { X } } = [ \mu \tilde { \pmb { H } } \tilde { \pmb { U } } ] . +$$ + +Lemma D.2 (Handling perturbation). Consider the nonlinear state equation (1.1). Given sampling rate $L > 0$ and offset $\bar { \tau }$ , consider the matrices $\bar { H } , \tilde { H } , \tilde { X }$ of Definition $D . I$ and let ${ \pmb Q } = [ \gamma _ { + } ^ { - 1 / 2 } \bar { \pmb H } \tilde { \pmb U } ] \in \mathbb { R } ^ { \bar { N } \times ( n + p ) }$ e Assumptiosuch that if √ $^ { l }$ $\phi$ $\beta$ creasing, and , with probabil ${ \mathbf { \mathscr { u } } _ { t } } \overset { i . i . d . } { \sim } \mathcal { N } ( 0 , { \cal I } _ { p } )$ e exists an absol, for all matrices constantobeying $C > 0$ $\begin{array} { r } { \bar { N } \ge C \frac { \gamma _ { + } ^ { 2 } } { \gamma _ { - } ^ { 2 } } ( n + p ) } \end{array}$ $1 - 8 \exp ( - c \frac { \gamma _ { - } ^ { 2 } } { \gamma _ { + } ^ { 2 } } \bar { N } )$ $M$ $\begin{array} { r } { \| M - \bar { H } \| \le \frac { \sqrt { \gamma _ { - } \bar { N } } } { 1 0 } } \end{array}$ , the perturbed $\ b { Q }$ matrices given by, + +$$ +\tilde { Q } = [ \gamma _ { + } ^ { - 1 / 2 } M \tilde { U } ] , +$$ + +satisfy + +$$ +( \Theta + \sqrt { 2 } ) ^ { 2 } \succeq \frac { \tilde { Q } ^ { T } \tilde { Q } } { \bar { N } } \succeq \frac { \gamma _ { - } } { 2 \gamma _ { + } } . +$$ + +Proof. This result is a direct application of Theorem F.1 after determining minimum/maximum eigenvalues of population covariance. The cross covariance obeys $\mathbb { E } [ \bar { \pmb { H } } ^ { T } \tilde { \pmb { U } } ] = 0$ due to independence. Also, for $i > 1$ , the truncated state vector $\bar { \pmb { h } } ^ { ( i ) }$ is statistically identical to $\pmb { h } _ { L - 1 }$ hence $\pmb { \Sigma } [ \bar { \pmb { h } } ^ { ( i ) } ] \succeq \gamma _ { - } \bar { \pmb { I _ { n } } }$ . Consequently, $\Sigma [ { \pmb u } ^ { ( i ) } ] = I _ { p }$ , $\begin{array} { r } { \frac { 1 } { \gamma _ { + } } \pmb { \Sigma } [ \bar { \pmb { h } } ^ { ( i ) } ] \preceq { \pmb { I } } _ { n } } \end{array}$ for all $_ { i }$ and $\begin{array} { r } { \frac { \gamma _ { - } } { \gamma _ { + } } I _ { n } \preceq \frac { 1 } { \gamma _ { + } } \Sigma [ \bar { \pmb { h } } ^ { ( i ) } ] } \end{array}$ for all $i > 1$ . Hence, setting $\pmb { q } _ { i } = \left[ \frac { 1 } { \sqrt { \gamma _ { + } } } \bar { \pmb { h } } ^ { ( i ) } \right] ,$ , for all $i > 1$ + +$$ +\frac { \gamma - } { \gamma _ { + } } I _ { n } \preceq \Sigma [ { \pmb q } _ { i } ] \preceq I _ { n } . +$$ + +Set the matrix $\bar { \pmb { Q } } = [ \pmb { q } _ { 2 } \dotsm \pmb { q } _ { \bar { N } } ] ^ { T }$ and note that $Q = [ \pmb { q } _ { 1 } \bar { Q } ^ { T } ] ^ { T }$ . Applying Theorem F.1 on $\bar { Q }$ and Corollary F.2 on $\ b { Q }$ , we find that, with the desired probability, + +$$ +\theta + \sqrt { 3 / 2 } \geq \frac { 1 } { \sqrt { N } } \| Q \| \geq \frac { 1 } { \sqrt { N } } s _ { \operatorname* { m i n } } ( Q ) \geq \frac { 1 } { \sqrt { N } } s _ { \operatorname* { m i n } } ( \bar { Q } ) \geq \sqrt { \frac { N - 1 } { N } } \sqrt { \frac { 2 \gamma _ { - } } { 3 \gamma _ { + } } } \geq 0 . 9 9 \times \sqrt { \frac { 2 \gamma _ { - } } { 3 \gamma _ { + } } } . +$$ + +Setting $E = M - { \bar { H } }$ and observing $\tilde { Q } = Q + [ \gamma _ { + } ^ { - 1 / 2 } E \mathrm { ~ 0 } ]$ , the impact of the perturbation $\pmb { \cal E }$ can be bounded naively via $s _ { \operatorname* { m i n } } ( \pmb { Q } ) - \gamma _ { + } ^ { - 1 / 2 } \lVert \pmb { E } \rVert \leq s _ { \operatorname* { m i n } } ( \tilde { \pmb { Q } } ) \leq \lVert \tilde { \pmb { Q } } \rVert \leq \lVert \pmb { Q } \rVert + \gamma _ { + } ^ { - 1 / 2 } \lVert \pmb { E } \rVert$ . Using the assumed bound on $\| \pmb { E } \|$ , this yields + +$$ +\theta + \sqrt { 2 } \geq \frac { 1 } { \sqrt { \bar { N } } } \| \tilde { \pmb { Q } } \| \geq \frac { 1 } { \sqrt { \bar { N } } } s _ { \operatorname* { m i n } } ( \tilde { \pmb { Q } } ) \geq \sqrt { \frac { \gamma - \underline { { \mathbf { \Lambda } } } } { 2 \gamma + } } . +$$ + +This final inequality is identical to the desired bound (D.3). + +Theorem D.3 (Data matrixdefine the condition number n). Consider the nonlinear s. For some absolute constants tion (1.1). Given , pick a trajectory $\gamma _ { + } \geq \gamma _ { - } > 0$ $\begin{array} { r } { \rho = \frac { \gamma _ { + } } { \gamma _ { - } } } \end{array}$ $c , C > 0$ $N$ + +$$ +L = \lceil 1 - \frac { \log { ( c n \rho ) } } { \log { \| A \| } } \rceil \quad , \quad N _ { 0 } = \lfloor \frac { N } { L } \rfloor \ge C \rho ^ { 2 } ( n + p ) , +$$ + +and pick scaling $\begin{array} { r } { \mu = \frac { 1 } { \sqrt { \gamma _ { + } } } } \end{array}$ Suppose $\| A \| ~ < ~ 1$ , $\phi$ is $\beta$ -increasing, ${ \textbf { \em u } } _ { t } \stackrel { i . i . d . } { \sim } { \mathcal { N } } ( 0 , I _ { p } )$ , and Assumption $^ { l }$ holds with $\gamma _ { + } , \gamma _ { - } , \theta , L$ . Matrix $\mathbf { X } \ = \ [ \pmb { x } _ { 1 } \dots \pmb { x } _ { N } ] ^ { T }$ of (D.1) satisfies the following with probability $1 - 4 N \exp ( - 1 0 0 n ) - 8 L \exp ( - \mathcal { O } ( N _ { 0 } / \rho ^ { 2 } ) )$ . + +• Each row of $\boldsymbol { x }$ has $\ell _ { 2 }$ norm at most $c _ { 0 } { \sqrt { p + n } }$ where $c _ { 0 }$ is an absolute constant. + +• $X ^ { T } X$ obeys the bound + +$$ +( \Theta + \sqrt { 2 } ) ^ { 2 } I _ { n + p } \succeq \frac { { \bf { X } } ^ { T } { \bf { X } } } { N } \succeq \rho ^ { - 1 } I _ { n + p } / 2 . +$$ + +Proof. The first statement on $\ell _ { 2 }$ -norm bound can be concluded from Lemma D.4 and holds with probability $1 - 2 N \exp ( - 1 0 0 ( n + p ) )$ . To show the second statement, for a fixed offset $1 \le \bar { \tau } \le L$ , consider Definition D.1 and the matrices $\tilde { \pmb { H } } ^ { ( \bar { \tau } ) } , \tilde { \pmb { U } } ^ { ( \bar { \tau } ) } , \tilde { \pmb { X } } ^ { ( \bar { \tau } ) }$ . Observe that $\boldsymbol { x }$ is obtained by merging multiple sub-trajectory matrices $\{ \tilde { X } ^ { ( \bar { \tau } ) } \} _ { \bar { \tau } = 1 } ^ { L }$ . We will first show the advertised bound for an individual $\tilde { X } ^ { ( \bar { \tau } ) }$ by applying Lemma D.2 and then apply Lemma A.1 to obtain the bound on the combined matrix $\boldsymbol { x }$ . + +Recall that $\bar { N } _ { \bar { \tau } }$ is the length of the $\bar { \tau }$ th sub-trajectory i.e. number of rows of $\tilde { X } ^ { ( \bar { \tau } ) }$ . By construction $2 N _ { 0 } \geq$ $\bar { N } _ { \bar { \tau } } \geq N _ { 0 }$ for all $1 \le \bar { \tau } \le L$ . Given $1 \le \bar { \tau } \le L$ and triple $\bar { \pmb { H } } ^ { ( \bar { \tau } ) } , \tilde { \pmb { H } } ^ { ( \bar { \tau } ) } , \tilde { \pmb { U } } ^ { ( \bar { \tau } ) }$ , set $Q = [ \mu \bar { H } ^ { ( \bar { \tau } ) } \tilde { U } ^ { ( \bar { \tau } ) } ]$ . Since $N _ { 0 }$ is chosen to be large enough, applying Theorem D.2 with $\mu = 1 / \sqrt { \gamma _ { + } }$ choice, and noting $\rho = \gamma _ { + } / \gamma _ { - }$ , we find that, with probability $1 - 4 \exp ( - c _ { 1 } N _ { 0 } / \rho ^ { 2 } )$ , all matrices $M$ satisfying $\lVert M - \bar { H } ^ { ( \bar { \tau } ) } \rVert \leq \sqrt { \gamma _ { - } N _ { 0 } } / 1 0$ and $\tilde { Q }$ as in (D.2) obeys + +$$ +( \Theta + \sqrt { 2 } ) ^ { 2 } \succeq \frac { { \tilde { Q } } ^ { T } { \tilde { Q } } } { N } \succeq \rho ^ { - 1 } / 2 . +$$ + +Let us call this Event 1. To proceed, we will argue that with high probability $\| \tilde { \pmb { H } } ^ { ( \bar { \tau } ) } - \bar { \pmb { H } } ^ { ( \bar { \tau } ) } \|$ is small so that the bound above is applicable with $M = \tilde { \cal H } ^ { ( \bar { \tau } ) }$ choice; which sets $\tilde { Q } = \tilde { X } ^ { ( \bar { \tau } ) }$ in (D.5). Applying Lemma C.6, we find that, with probability $1 - 2 \bar { N } _ { \bar { \tau } } \exp ( - 1 0 0 n )$ , + +$$ +\begin{array} { r } { \| \bar { \pmb { H } } ^ { ( \bar { \tau } ) } - \tilde { \pmb { H } } ^ { ( \bar { \tau } ) } \| \leq \sqrt { 2 N _ { 0 } } \operatorname* { m a x } \{ \| \pmb { h } ^ { ( i ) } - \bar { \pmb { h } } ^ { ( i ) } \| _ { \ell _ { 2 } } \} \leq c _ { 0 } \sqrt { 2 N _ { 0 } } \sqrt { n \gamma _ { + } } \| \pmb { A } \| ^ { L - 1 } . } \end{array} +$$ + +Let us call this Event 2. We will show that our choice of $L$ ensures right hand side is small enough and guarantees $\lVert \bar { \pmb { H } } ^ { ( \bar { \tau } ) } - \tilde { \pmb { H } } ^ { ( \bar { \tau } ) } \rVert \leq \sqrt { \gamma _ { - } N _ { 0 } } / 1 0$ . Set $c = \operatorname* { m a x } \{ 2 0 0 c _ { 0 } ^ { 2 } , 1 \}$ . Desired claim follows by taking logarithms of upper/lower bounds and cancelling out $\sqrt { N _ { 0 } }$ terms as follows + +$$ +\begin{array} { r l } & { c _ { 0 } \sqrt { n } \| A \| ^ { L - 1 } \sqrt { \gamma _ { + } } \leq \sqrt { \gamma _ { - } } / 1 0 \sqrt { 2 } \iff ( L - 1 ) \log \| A \| + \log \sqrt { c n \rho } \leq 0 } \\ & { \iff - \frac { \log c n \rho } { 2 \log \| A \| } \leq L - 1 } \\ & { \iff L = \lceil 1 - \frac { \log { ( c n \rho ) } } { \log \| A \| } \rceil . } \end{array} +$$ + +Here we use the fact that $\log \| A \| < 0$ since $\| A \| < 1$ and $c n \rho \geq 0$ . Consequently, both Event 1 and Event 2 hold with probability $1 - 4 \exp ( - c _ { 1 } N _ { 0 } / \rho ^ { 2 } ) - 2 \bar { N } _ { \bar { \tau } } \exp ( - 1 0 0 n ) .$ , implying (D.5) holds with $\tilde { Q } = \tilde { X } ^ { ( \bar { \tau } ) }$ . Union bounding this over $1 \le \bar { \tau } \le L$ , (D.5) uniformly holds with $\tilde { Q } = \tilde { X } ^ { ( \bar { \tau } ) }$ and all rows of $\boldsymbol { x }$ are $\ell _ { 2 }$ -bounded with probability $1 - 4 N \exp ( - 1 0 0 n ) - 8 L \exp ( - c _ { 1 } N _ { 0 } / \rho ^ { 2 } )$ . Applying Lemma A.1 on $( \tilde { X } ^ { ( \bar { \tau } ) } ) _ { \bar { \tau } = 1 } ^ { L }$ , we conclude with the bound (D.4) on the merged matrix $\pmb { X }$ . $\boxed { \begin{array} { r l } \end{array} }$ + +Lemma D.4 $\ell _ { 2 }$ -bound on rows). Consider the setup of Theorem D.3. With probability √ $1 - 2 N \exp ( - 1 0 0 ( n +$ $p )$ ), each row of $\boldsymbol { x }$ has $\ell _ { 2 }$ -norm at most $c { \sqrt { p + n } }$ for some constant $c > 0$ . + +Proof. The tth row of $\pmb { X }$ is equal to $\begin{array} { r } { \pmb { x } _ { t } = [ \frac { \pmb { h } _ { t } ^ { T } } { \sqrt { \gamma _ { + } } } \pmb { u } _ { t } ^ { T } ] ^ { T } } \end{array}$ . Since $\| h _ { t } - \mathbb { E } [ h _ { t } ] \| _ { \psi _ { 2 } } \leq \mathcal { O } ( \sqrt { \gamma _ { + } } )$ and $\| u _ { t } \| _ { \psi _ { 2 } } ~ \le$ $\mathcal { O } ( 1 )$ , we have that $\| \pmb { x } _ { t } - \mathbb { E } [ \pmb { x } _ { t } ] \| _ { \psi _ { 2 } } \leq \mathcal { O } ( 1 )$ . Now, applying Lemma F.3 on all rows $\{ \pmb { x } _ { t } \} _ { t = 1 } ^ { N }$ , and using a√ union bound, with probability at least $1 - 2 \dot { N } \exp ( - 1 0 \bar { 0 } ( n + p ) )$ , we have that √ $\lVert \mathbf { x } _ { t } - \mathbb { E } [ \mathbf { x } _ { t } ] \rVert _ { \ell _ { 2 } } \leq c \sqrt { n + p }$ for all $t$ . To conclude, note that $\| \mathbb { E } [ \pmb { x } _ { t } ] \| _ { \ell _ { 2 } } = \| \mathbb { E } [ \pmb { h } _ { t } ] \| _ { \ell _ { 2 } } / \sqrt { \gamma _ { + } } \le \theta \le 3 \sqrt { n }$ via Assumption 1. □ + +# E PROOFS OF MAIN RESULTS + +# E.1 PROOF OF LEMMA 3.2 + +The statement follows from upper bound Lemma B.3 and lower bound Lemma B.5. + +# E.2 PROOF OF THEOREM 4.2 + +Proof. To prove this theorem, we combine Theorem D.3 with deterministic SGD convergence result of Theorem 4.1. Applying Theorem D.3, with the desired probability, inequality (D.4) holds and for all $i$ , input data satisfies the bound $\| \pmb { x } _ { i } \| _ { \ell _ { 2 } } \leq \sqrt { ( n + p ) / ( 2 c _ { 0 } ) }$ for a sufficiently small constant $c _ { 0 } > 0$ . As the next step, we will argue that these two events imply the convergence of SGD. + +Let $\pmb { \theta } ^ { ( i ) } , \pmb { c } ^ { ( i ) } \in \mathbb { R } ^ { n + p }$ denote the ith rows of $\Theta , C$ respectively. Observe that the square-loss is separable along the rows of $_ { C }$ via $\begin{array} { r } { \| \Theta - C \| _ { F } ^ { 2 } = \sum _ { i = 1 } ^ { n } \| \pmb { \theta } ^ { ( i ) } - \pmb { c } ^ { ( i ) } \| _ { \ell _ { 2 } } ^ { 2 } } \end{array}$ . Hence, SGD updates each row $\mathbf { c } ^ { ( i ) }$ via its own state equation + +$$ +\pmb { y } _ { t , i } = \phi ( \left. \pmb { c } ^ { ( i ) } , \pmb { x } _ { t } \right. ) , +$$ + +where $_ { { \mathbf { \mathit { y } } } _ { t , i } }$ is the ith entry of ${ \mathbf { } } _ { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { \mathbf { } } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } _ { } { \mathbf { } } _ { } { } \mathbf { } _ { } { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } { } \mathbf { } _ { } \mathbf { } _ { } { } \mathbf { } _ { } \mathbf { } _ { } { } \mathbf { } _ { } \mathbf { } _ { } { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } \mathbf { } _ { } \mathbf { } _ { } \mathbf { } \mathbf _ { } \mathbf { } _ { } \mathbf { } \mathbf _ { } \mathbf { } _ { } \mathbf { } _ \mathbf { } \mathbf { } _ { } \mathbf \mathbf { } _ { } \mathbf _ { } \mathbf { } \mathbf _ { } \mathbf { } \mathbf _ { } \mathbf _ { } \mathbf { } \mathbf _ { } \mathbf \mathbf { } _ \mathbf { } \mathbf _ { } \mathbf \mathbf { } \mathbf _ { } \mathbf \mathbf { } \mathbf _ { } \mathbf \mathbf \mathbf { } \mathbf _ { } \mathbf \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf _ { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf$ . Consequently, we can establish the convergence result for an individual row of $_ { C }$ . Convergence of all individual rows will imply the convergence of the overall matrix $\Theta _ { \tau }$ to the ground + +truth $_ { C }$ . Pick a row index $_ { i }$ $( 1 \leq i \leq n )$ , set $\mathbf { c } = \mathbf { c } ^ { ( i ) }$ and denote ith row of $\Theta _ { \tau }$ by $\pmb { \theta } _ { \tau }$ . Also denote the label corresponding to ith row by $y _ { t } = y _ { t , i }$ . With this notation, SGD over (2.3) runs SGD over the ith row with equations $y _ { t } \overset { \cdot } { = } \phi ( \langle c , \pmb { x } _ { t } \rangle )$ and with loss functions + +$$ +\mathcal { L } ( \pmb { \theta } ) = N ^ { - 1 } \sum _ { t = 1 } ^ { N } \mathcal { L } _ { t } ( \pmb { \theta } ) , \mathcal { L } _ { t } ( \pmb { \theta } ) = \frac { 1 } { 2 } \big ( y _ { t } - \phi ( \langle \pmb { \theta } , \pmb { x } _ { t } \rangle ) \big ) ^ { 2 } . +$$ + +Substituting our high-probability bounds on $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ (e.g. (D.4)) into Theorem 4.1, we can set $B = ( n + p ) / ( 2 c _ { 0 } )$ , $\gamma _ { + } = ( \theta + \sqrt { 2 } ) ^ { 2 }$ , and $\gamma _ { - } = \rho ^ { - 1 } / 2$ . Consequently, using the learning rate $\begin{array} { r } { \eta = c _ { 0 } \frac { \beta ^ { 2 } \rho ^ { - 1 } } { ( \theta + \sqrt { 2 } ) ^ { 2 } ( n + p ) } } \end{array}$ , for all $\tau \geq 0$ , the $\tau$ th SGD iteration $\pmb { \theta } _ { \tau }$ obeys + +$$ +\mathbb { E } [ \| \pmb { \theta } _ { \tau } - \pmb { c } \| _ { \ell _ { 2 } } ^ { 2 } ] \le \| \pmb { \theta } _ { 0 } - \pmb { c } \| _ { \ell _ { 2 } } ^ { 2 } ( 1 - c _ { 0 } \frac { \beta ^ { 4 } \rho ^ { - 2 } } { 2 ( \theta + \sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \tau } , +$$ + +where the expectation is over the random selection of SGD updates. This establishes the convergence for a particular row of $_ { C }$ . Summing up these inequalities (E.1) over all rows $\pmb { \theta } _ { \tau } ^ { ( 1 ) } , \dots , \pmb { \theta } _ { \tau } ^ { ( n ) }$ (which converge to $\bar { \mathbf { c } } ^ { ( 1 ) } , \ldots , \mathbf { c } ^ { ( n ) }$ respectively) yields the targeted bound (4.4). □ + +# E.3 PROOFS OF MAIN RESULTS ON STABLE SYSTEMS + +# E.3.1 PROOF OF THEOREM 3.3 + +Proof. Applying Lemmas B.3 and 3.2, independent of $L$ , Assumption 1 holds with parameters + +$$ +\gamma _ { + } = B _ { \infty } ^ { 2 } \quad , \quad \gamma _ { - } = \beta ^ { 2 } s _ { \mathrm { m i n } } ( B ) ^ { 2 } \quad , \quad \theta = \sqrt { 6 n } - \sqrt { 2 } \geq \sqrt { n } . +$$ + +This yields (θ + 2)2 = 6n. Hence, we can apply Theorem 4.2 with the learning rate η = c0 β26ρn(n+p) where + +$$ +\rho = \frac { B _ { \infty } ^ { 2 } } { \beta ^ { 2 } s _ { \mathrm { m i n } } ( B ) ^ { 2 } } = \frac { \gamma _ { + } } { \gamma _ { - } } , +$$ + +and convergence rate $\begin{array} { r } { 1 - \frac { \beta ^ { 2 } \eta } { 2 \rho } } \end{array}$ . To conclude with the stated result, we use the change of variable $c _ { 0 } / 6 \to c _ { 0 }$ . + +# E.3.2 PROOF OF THEOREM 3.4 + +Proof. The proof is similar to that of Theorem 3.3. Applying Lemmas B.3, B.4, and 3.2, independent of $L$ Assumption 1 holds with parameters + +$$ +\gamma _ { + } = B _ { \infty } ^ { 2 } \quad , \quad \gamma _ { - } = s _ { \mathrm { m i n } } ( B ) ^ { 2 } \quad , \quad \theta = 0 . +$$ + +Hence, we again apply Theorem 4.2 with the learning rate η = c0 β22ρ(n+p) where $\rho$ is given by (E.2). Use the change of variable $c _ { 0 } / 2 \to c _ { 0 }$ to conclude with the stated result. □ + +# E.4 LEARNING UNSTABLE SYSTEMS + +In a similar fashion to Section 4, we provide a more general result on unstable systems that makes a parametric assumption on the statistical properties of the state vector. + +Assumption 2 (Well-behaved state vector – single timestamp). Given timestamp √ $T _ { 0 } > 0$ , there exists positive scalars $\gamma _ { + } , \gamma _ { - } , \theta$ and an absolute constant $C > 0$ such that ${ \dot { \theta } } \leq 3 { \sqrt { n } }$ and the following holds + +$$ +\begin{array} { r } { \gamma _ { + } I _ { n } \succeq \Sigma [ h _ { T _ { 0 } } ] \succeq \gamma _ { - } I _ { n } \quad , \quad \| h _ { T _ { 0 } } - \mathbb { E } [ h _ { T _ { 0 } } ] \| _ { \psi _ { 2 } } \leq C \sqrt { \gamma _ { + } } \quad a n d \quad \| \mathbb { E } [ h _ { t } ] \| _ { \ell _ { 2 } } \leq \theta \sqrt { \gamma _ { + } } . } \end{array} +$$ + +The next theorem provides the parametrized result on unstable systems based on this assumption. + +Theorem E.1 (Unstable system - general). Suppose we are given $N$ independent trajectories $( { h _ { t } ^ { ( i ) } } , { u _ { t } ^ { ( i ) } } ) _ { t \geq 0 }$ for $1 \leq i \leq N$ . Sample each trajectory at time $T _ { 0 }$ to obtain $N$ samples $( { \pmb y } _ { i } , { \pmb h } _ { i } , { \pmb u } _ { i } ) _ { i = 1 } ^ { N }$ where ith sample is + +$$ +( { \pmb y } _ { i } , { \pmb h } _ { i } , { \pmb u } _ { i } ) = ( { \pmb h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \pmb h } _ { T _ { 0 } } ^ { ( i ) } , { \pmb u } _ { T _ { 0 } } ^ { ( i ) } ) . +$$ + +Let $C , c _ { 0 } > 0$ be absolute constants. Suppose Assumption $^ { l }$ holds with $T _ { 0 }$ and sample size satisfies $N \geq$ $C \rho ^ { 2 } ( n + p )$ where $\rho = \gamma _ { + } / \gamma _ { - }$ . Assume $\phi$ is $\beta$ -increasing, zero initial state conditions, and ${ \pmb u } _ { t } \overset { i . i . d . } { \sim } \mathcal { N } ( 0 , { \pmb I } _ { p } )$ Set scaling to be µ = 1/ γ+ and learning rate to be η = c0 β ρ(θ+√2)2(n+p) . Starting from $\Theta _ { 0 }$ , we run $S G D$ over the equations described in (2.2) and (2.3). With probability $1 - 2 \dot { N } \exp ( - 1 0 0 ( n + p ) ) - 4 \exp ( - \mathcal { O } ( \textstyle { \frac { N } { \rho ^ { 2 } } } ) )$ , all iterates satisfy + +$$ +\mathbb { E } [ \| \Theta _ { i } - C \| _ { F } ^ { 2 } ] \le ( 1 - c _ { 0 } \frac { \beta ^ { 4 } } { 2 \rho ^ { 2 } ( \theta + \sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \tau } \| \Theta _ { 0 } - C \| _ { F } ^ { 2 } , +$$ + +where the expectation is over the randomness of the SGD updates. + +Proof. Set $\pmb { x } _ { i } = [ \gamma _ { + } ^ { - 1 / 2 } \pmb { h } _ { i } ^ { T } \ \pmb { u } _ { i } ^ { T } ] ^ { T }$ and $\pmb { X } = [ \pmb { x } _ { 1 } ~ . ~ . ~ \pmb { x } _ { N } ] ^ { T }$ . Since $\boldsymbol { x }$ has i.i.d. rows, we can apply Theorem F.1 and Lemma F.3 to find with the desired probability that + +• Rows of $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ satisfy $\| \pmb { x } _ { i } - \mathbb { E } [ \pmb { x } _ { i } ] \| _ { \psi _ { 2 } } \leq \mathcal { O } ( 1 )$ and $\mathbb { E } [ \| \pmb { x } _ { i } \| _ { \ell _ { 2 } } ] \le 3 \sqrt { n }$ , hence all rows of $\boldsymbol { x }$ obeys $\| \pmb { x } _ { i } \| _ { \ell _ { 2 } } \leq \sqrt { ( n + p ) / ( 2 c _ { 0 } ) }$ , + +• $\boldsymbol { x }$ satisfies + +$$ +( \theta + \sqrt { 2 } ) ^ { 2 } \succeq \frac { X ^ { T } X } { N } \succeq \rho ^ { - 1 } / 2 . +$$ + +To proceed, using $\gamma _ { - } = \rho ^ { - 1 } / 2$ , $B = ( n + p ) / ( 2 c _ { 0 } )$ , and $\gamma _ { + } = ( \theta + \sqrt { 2 } ) ^ { 2 }$ , we apply Theorem 4.1 on the loss function (2.3); which yields the desired result. + +# E.5 PROOF OF THEOREM 5.1 + +Proof. The proof is a corollary of Theorem E.1. We need to substitute the proper values in Assumption 2.√ √ Applying Lemma B.3, we can substitute $\gamma _ { + } = B _ { T _ { 0 } } ^ { 2 }$ and $\theta = \sqrt { 6 n } - \sqrt { 2 } \geq \sqrt { n }$ . Next, we need to find a lower bound. Applying Lemma 3.2 for $n > 1$ and Lemma B.6 for $n = 1$ , we can substitute $\gamma _ { - } = \gamma _ { + } / \rho$ with the $\rho$ definition of (5.2). With these, the result follows as an immediate corollary of Theorem E.1. □ + +# F SUPPLEMENTARY STATISTICAL RESULTS + +The following theorem bounds the empirical covariance of matrices with independent subgaussian rows. Given a random vector $_ { \textbf { \em x } }$ , define the de-biasing operation as $\mathbf { z m } ( \pmb { x } ) = \pmb { x } - \mathbb { E } [ \pmb { x } ]$ . + +Theorem F.1. Let $\textbf { \textit { A } } \in \mathbb { R } ^ { n \times d }$ be a matrix with independent subgaussian rows $\{ { \pmb a } _ { i } \} _ { i = 1 } ^ { n }$ satisfying $\| \mathbf { z } \mathbf { m } ( \mathbf { { a } } _ { i } ) \| _ { \psi _ { 2 } } \leq \mathcal { O } ( K )$ and $\Sigma [ { \pmb a } _ { i } ] \preceq K ^ { 2 } { \pmb I } _ { d }$ for some $K > 0$ and $\| \mathbb { E } [ \pmb { a } _ { i } ] \| _ { \ell _ { 2 } } \le \theta$ . Suppose $\pmb { \Sigma } [ \pmb { a } _ { i } ] \succeq \lambda \pmb { I } _ { d }$ . Suppose $n \ge \bar { \mathcal { O } } ( K ^ { 4 } d / \lambda ^ { 2 } )$ . Then, each of the following happens with probability at least $1 - 2 \exp ( - c K ^ { - 4 } \lambda ^ { 2 } n )$ , + +$$ +\begin{array} { r } { \theta + \sqrt { 3 / 2 } K \geq \frac { 1 } { \sqrt { n } } \| A \| . } \end{array} +$$ + +• Suppose all rows of $\pmb { A }$ have equal expectations. Then $\textstyle { \frac { 1 } { \sqrt { n } } } s _ { \operatorname* { m i n } } ( A ) \geq { \sqrt { 2 \lambda / 3 } }$ + +Proof. Let $\pmb { E } = \mathbb { E } [ \pmb { A } ] , \ \bar { \pmb { A } } = \pmb { A } - \mathbb { E } [ \pmb { A } ] , \ \bar { \pmb { a } } _ { i } = \mathbf { z m } ( \pmb { a } _ { i } )$ . We will decompose $\pmb { A } = \bar { \pmb { A } } + \pmb { E }$ hence we will first focus on bounding the upper and lower singular values of $\bar { A }$ by studying the random processes $X _ { v } = \| \bar { A } v \| _ { \ell _ { 2 } } ^ { 2 }$ and $Y _ { v } = X _ { v } - \mathbb { E } [ X _ { v } ]$ over the unit sphere $S ^ { d - 1 }$ . First, we provide a deviation bound for the quantity ${ \operatorname* { s u p } } _ { v \in { \mathcal { S } } ^ { d - 1 } } | Y _ { v } |$ . To achieve this, we will utilize Talagrand’s mixed tail bound and show that increments of $Y _ { v }$ are subexpoential. Pick two unit vectors $\pmb { v } , \pmb { u } \in \mathbb { R } ^ { d }$ . Write $\pmb { x } = \pmb { u } + \pmb { v } , \pmb { y } = \pmb { u } - \pmb { v } ,$ . We have that + +$$ +{ \displaystyle { \cal X } _ { u } - { \cal X } _ { v } = \| \bar { \cal A } u \| _ { \ell _ { 2 } } ^ { 2 } - \| \bar { \cal A } v \| _ { \ell _ { 2 } } ^ { 2 } = \| \bar { \cal A } ( x + y ) / 2 \| _ { \ell _ { 2 } } ^ { 2 } - \| \bar { \cal A } ( x - y ) / 2 \| _ { \ell _ { 2 } } ^ { 2 } = x ^ { T } \bar { \cal A } ^ { T } \bar { \cal A } y = \sum _ { i = 1 } ^ { n } ( \bar { a } _ { i } ^ { T } x ) ( \bar { a } _ { i } ^ { T } y ) . } +$$ + +Letting $\hat { \pmb x } = { \pmb x } / \| { \pmb x } \| _ { \ell _ { 2 } } , \hat { \pmb y } = { \pmb y } / \| { \pmb y } \| _ { \ell _ { 2 } }$ , observe that, multiplication of subgaussians $\pmb { x } ^ { T } \bar { \pmb { a } } _ { i } , \pmb { y } ^ { T } \bar { \pmb { a } } _ { i }$ obey + +$$ +\| ( \boldsymbol { x } ^ { T } \bar { \boldsymbol { a } } _ { i } ) ( \boldsymbol { y } ^ { T } \bar { \boldsymbol { a } } _ { i } ) \| _ { \psi _ { 1 } } \leq \mathcal { O } ( \| \boldsymbol { x } \| _ { \ell _ { 2 } } \| \boldsymbol { y } \| _ { \ell _ { 2 } } K ^ { 2 } ) \leq \mathcal { O } ( K ^ { 2 } \| \boldsymbol { y } \| _ { \ell _ { 2 } } ) . +$$ + +Centering this subexponential variable around zero introduces a factor of 2 when bounding subexponential norm and yields $\| ( \boldsymbol { x } ^ { T } \bar { \boldsymbol { a } } _ { i } ) \dot { \langle } \boldsymbol { y } ^ { T } \bar { \boldsymbol { a } } _ { i } \rangle - \mathbb { E } [ ( \boldsymbol { x } ^ { T } \bar { \boldsymbol { a } } _ { i } ) ( \boldsymbol { y } ^ { T } \bar { \boldsymbol { a } } _ { i } ) ] \| _ { \psi _ { 1 } } \le \mathcal { O } ( K ^ { 2 } \| \boldsymbol { y } \| _ { \ell _ { 2 } } ) .$ . Now, using the fact that $Y _ { u } - Y _ { v }$ is sum of $_ n$ independent zero-mean subexponential random variables, we have the tail bound + +$$ +\mathbb { P } ( n ^ { - 1 } | Y _ { u } - Y _ { v } | \ge t ) \le 2 \exp ( - c ^ { \prime } n \operatorname* { m i n } \{ \frac { t ^ { 2 } } { K ^ { 4 } \| y \| _ { \ell _ { 2 } } ^ { 2 } } , \frac { t } { K ^ { 2 } \| y \| _ { \ell _ { 2 } } } \} ) . +$$ + +Applying Talagrand’s chaining bound for mixed tail processes with distance metrics $\begin{array} { r } { \rho _ { 2 } = \frac { K ^ { 2 } \| \cdot \| _ { \ell _ { 2 } } } { \sqrt { n } } , \rho _ { 1 } = } \end{array}$ K2k·k\`2n , (Theorem 3.5 of Dirksen (2013) or Theorem 2.2.23 of Talagrand (2014)) and using the fact that for unit sphere $S ^ { d - 1 }$ , Talagrand’s $\gamma$ functionals (see Talagrand (2014)) obey $\gamma _ { 1 } ( S ^ { d - 1 } ) , \gamma _ { 2 } ^ { 2 } ( S ^ { d - 1 } ) \leq \mathcal { O } ( d )$ , + +$$ +\begin{array} { r } { n ^ { - 1 } \underset { { \pmb v } \in { \pmb S } ^ { d - 1 } } { \operatorname* { s u p } } | Y _ { \pmb v } | \leq c K ^ { 2 } ( \sqrt { d / n } + d / n + t / \sqrt { n } ) , } \end{array} +$$ + +with probability √ $1 - 2 \exp ( - \operatorname* { m i n } \{ t ^ { 2 } , { \sqrt { n } } t \} )$ . Since $n \geq C \lambda ^ { - 2 } K ^ { 4 } d$ for sufficiently large $C > 0$ , picking $\textstyle t = { \dot { \frac { 1 } { 1 6 c } } } K ^ { - 2 } \lambda { \dot { \sqrt { n } } }$ , with probability $1 - 2 \exp ( - \mathcal { O } ( K ^ { - 4 } \lambda ^ { 2 } n ) )$ , we ensure that right hand side of (F.1) is less than $\lambda / 8$ . This leads to the following inequalities + +$$ +\begin{array} { r l } & { \displaystyle \frac 1 n | | \bar { \boldsymbol { A } } ^ { T } \bar { \boldsymbol { A } } - \mathbb { E } [ \bar { \boldsymbol { A } } ^ { T } \bar { \boldsymbol { A } } ] | | \le \frac \lambda 8 \Longrightarrow \frac { 9 K ^ { 2 } } { 8 } I _ { d } \succeq \frac 1 n \bar { \boldsymbol { A } } ^ { T } \bar { \boldsymbol { A } } \succeq \frac { 7 \lambda } { 8 } I _ { d } . } \\ & { \qquad \Longrightarrow \frac 9 8 K \ge \frac { 1 } { \sqrt n } | | \bar { \boldsymbol { A } } | | \ge s _ { \mathrm { m i n } } ( \bar { \boldsymbol { A } } ) \ge \sqrt { \frac { 7 } { 8 } } \lambda . } \end{array} +$$ + +Upper bound on spectral norm: For spectral norm of $\pmb { A }$ , we use the triangle inequality + +$$ +\frac { 1 } { \sqrt { n } } \| \pmb { A } \| \leq \frac { 1 } { \sqrt { n } } ( \| \pmb { E } \| + \| \bar { \pmb { A } } \| ) \leq \operatorname* { m a x } _ { 1 \leq i \leq n } \| \mathbb { E } [ \pmb { a } _ { i } ] \| _ { \ell _ { 2 } } + 9 K / 8 \leq \theta + \sqrt { 3 / 2 } K . +$$ + +Lowsize $n$ r bound on minim all ones vector by ${ \bf 1 } _ { n }$ singular value: This pand define the process √ $\begin{array} { r } { Z _ { \pmb { v } } = \frac { 1 } { \sqrt { n } } \pmb { 1 } _ { n } ^ { T } \bar { \pmb { A } } \pmb { v } } \end{array}$ ll row expectat. Observe that $\begin{array} { r } { \bar { A } ^ { T } \mathbf { 1 } _ { n } = \sum _ { i = 1 } ^ { n } \bar { \mathbf { a } } _ { i } \in \mathbb { R } ^ { d } } \end{array}$ is a vector satisfying $\| \bar { A } ^ { T } \mathbf { 1 } _ { n } / \sqrt { n } \| _ { \psi _ { 2 } } \leq \mathcal { O } ( K )$ . Hence, again using $n \geq C K ^ { 4 } \lambda ^ { - 2 } d$ for sufficiently large $C > 0$ , applying Lemma F.3 with $m = c _ { 0 } K ^ { - 4 } \lambda ^ { 2 } n > d$ by picking a sufficiently small constant $c _ { 0 } > 1 / \bar { C }$ , with probability at least $1 - 2 \exp ( - 1 0 0 c _ { 0 } K ^ { - 4 } \lambda ^ { 2 } n )$ + +$$ +\frac { 1 } { \sqrt { n } } \operatorname* { s u p } _ { \| v \| _ { 2 } = 1 } | Z _ { v } | = \frac { 1 } { n } \| \bar { \cal A } ^ { T } { \bf 1 } _ { n } \| _ { \ell _ { 2 } } \leq \frac { 1 } { 1 2 } { \cal K } { \cal K } ^ { - 2 } \lambda \leq \frac { \sqrt { \lambda } } { 1 2 } . +$$ + +Let $\begin{array} { r } { { \cal P } = I _ { n } - \frac { 1 } { n } { \bf 1 } _ { n } { \bf 1 } _ { n } ^ { T } } \end{array}$ be the projection onto the orthogonal complement of the all ones vector. Note that $P E v = 0$ as the rows of $\pmb { { \cal E } }$ are equal. With this observation, with desired probability, for any unit length $_ { v }$ , + +$$ +\begin{array} { r l } & { \| \pmb { A } \pmb { v } \| _ { \ell _ { 2 } } \ge \| \pmb { P } \pmb { A } \pmb { v } \| _ { \ell _ { 2 } } = \| \pmb { P } \pmb { \bar { A } } \pmb { v } \| _ { \ell _ { 2 } } \ge \| \pmb { \bar { A } } \pmb { v } \| _ { \ell _ { 2 } } - | Z _ { \pmb { v } } | } \\ & { \qquad \ge s _ { \operatorname* { m i n } } ( \pmb { \bar { A } } ) - \ \underset { \pmb { v } \in { S } ^ { d - 1 } } { \operatorname* { s u p } } | Z _ { \pmb { v } } | \ge ( \sqrt { 7 / 8 } - 1 / 1 2 ) \sqrt { \lambda n } , } \end{array} +$$ + +which implies $s _ { \mathrm { m i n } } ( A ) / \sqrt { n } \geq \sqrt { 2 \lambda / 3 }$ . + +The corollary below is obtained by slightly modifying the proof above by using $\begin{array} { r } { \frac { 1 } { n } \| \bar { \mathbfcal A } ^ { T } \bar { \mathbfcal A } - \mathbb { E } [ \bar { \mathbfcal A } ^ { T } \bar { \mathbfcal A } ] \| \leq \frac { K ^ { 2 } } { 8 } } \end{array}$ in line (F.2) and only focusing on the spectral norm bound. + +Corollary F.2. Let $\textbf { \textit { A } } \in \mathbb { R } ^ { n \times d }$ be a matrix with independent $\{ { \pmb a } _ { i } \} _ { i = 1 } ^ { n }$ subgaussian rows satisfying $\| \mathbf { z } \mathbf { m } ( \mathbf { { a } } _ { i } ) \| _ { \psi _ { 2 } } \leq \mathcal { O } ( K )$ and $\Sigma [ { \pmb a } _ { i } ] \preceq K ^ { 2 } { \pmb I } _ { d }$ for some $K > 0$ and $\| \mathbb { E } [ \mathbf { \underline { { a } } } _ { i } ] \rVert _ { \ell _ { 2 } } \leq \theta$ . Suppose $\pmb { \Sigma } [ \pmb { a } _ { i } ] \succeq \lambda \pmb { I } _ { d }$ Suppose $n \geq \mathcal { O } ( K ^ { 2 } d )$ . Then, with probability at least $1 - 4 \exp ( - c K ^ { - 2 } n )$ , + +$$ +\theta + { \sqrt { 3 / 2 } } K \geq { \frac { 1 } { \sqrt { n } } } \| A \| . +$$ + +The following lemma is fairly standard and is proved for the sake of completeness. + +Lemma F.3 (Subgaussian vector length). Let $\pmb { a } \in \mathbb { R } ^ { n }$ be a zero-mean subgaussian vector with $\| \pmb { a } \| _ { \psi _ { 2 } } \le L$ Then, for any $m \geq n ,$ there exists $C > 0$ such that + +$$ +\begin{array} { r } { \mathbb { P } ( \| \pmb { a } \| _ { \ell _ { 2 } } \le C L \sqrt { m } ) \ge 1 - 2 \exp ( - 1 0 0 m ) . } \end{array} +$$ + +Proof. We can pick a $1 / 2$ cover $\mathcal { C }$ of the unit $\ell _ { 2 }$ -sphere with size $\log | { \mathcal { C } } | \leq 2 n$ . For any ${ \pmb v } \in \mathcal { C }$ , subgaussianity implies, $\begin{array} { r } { \mathbb { P } ( | v ^ { T } \pmb { a } | \geq t ) \leq 2 \exp ( - \frac { c t ^ { 2 } } { 2 L ^ { 2 } } ) } \end{array}$ . Setting $t = C L { \sqrt { m } }$ for sufficiently large constant $C > 0$ , and union bounding over all $\pmb { v } \in \mathcal { C }$ , we find + +$$ +\mathbb { P } ( \bigcap _ { v \in \mathcal { C } } \| v \| _ { \ell _ { 2 } } \leq C L \sqrt { m } ) \geq 1 - 2 \exp ( 2 n - \frac { c C ^ { 2 } L ^ { 2 } m } { 2 L ^ { 2 } } ) \leq 1 - 2 \exp ( - 1 0 0 m ) . +$$ + +To conclude, let v(a) ∈ C be a’s neighbor satisfying kv − akak\`2 k\`2 ≤ 1/2. Hence, we have + +$$ +\begin{array} { r } { \| a \| _ { \ell _ { 2 } } \leq \| ( a - v ( a ) ) ^ { T } a \| _ { \ell _ { 2 } } + \| v ^ { T } a \| _ { \ell _ { 2 } } \leq \| a \| _ { \ell _ { 2 } } / 2 + C L \sqrt { m } \implies \| a \| _ { \ell _ { 2 } } \leq 2 C L \sqrt { m } . } \end{array} +$$ + +To conclude, use the change of variable $C C / 2$ . \ No newline at end of file diff --git a/parse/train/rkeMHjR9Ym/rkeMHjR9Ym_content_list.json b/parse/train/rkeMHjR9Ym/rkeMHjR9Ym_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b25a5f763d696b0e540a8d93b37573cd127c92f5 --- /dev/null +++ b/parse/train/rkeMHjR9Ym/rkeMHjR9Ym_content_list.json @@ -0,0 +1,4689 @@ +[ + { + "type": "text", + "text": "STOCHASTIC GRADIENT DESCENT LEARNS STATE EQUATIONS WITH NONLINEAR ACTIVATIONS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 797, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 174, + 400, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 238, + 544, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We study discrete time dynamical systems governed by the state equation $h _ { t + 1 } =$ $\\phi ( A h _ { t } + B u _ { t } )$ . Here $A , B$ are weight matrices, $\\phi$ is an activation function, and $\\mathbf { \\Delta } \\mathbf { u } _ { t }$ is the input data. This relation is the backbone of recurrent neural networks (e.g. LSTMs) which have broad applications in sequential learning tasks. We utilize stochastic gradient descent to learn the weight matrices from a finite input/state trajectory $\\mathbf { \\bar { \\{ u } } _ { t } , h _ { t } \\} _ { t = 0 } ^ { N }$ . We prove that SGD estimate linearly converges to the ground truth weights while using near-optimal sample size. Our results apply to increasing activations whose derivatives are bounded away from zero. The analysis is based on i) a novel SGD convergence result with nonlinear activations and ii) careful statistical characterization of the state vector. Numerical experiments verify the fast convergence of SGD on ReLU and leaky ReLU in consistence with our theory. ", + "bbox": [ + 233, + 270, + 766, + 436 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 463, + 336, + 478 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A wide range of problems involve sequential data with a natural temporal ordering. Examples include natural language processing, time series prediction, system identification, and control design, among others. State-of-the-art algorithms for sequential problems often stem from dynamical systems theory and are tailored to learn from temporally dependent data. Linear models and algorithms; such as Kalman filter, PID controller, and linear dynamical systems, have a long history and are utilized in control theory since 1960’s with great success (Brown et al. (1992); Ho & Kalman (1966); Åström & Hägglund (1995)). More recently, nonlinear models such as recurrent neural networks (RNN) found applications in complex tasks such as machine translation and speech recognition (Bahdanau et al. (2014); Graves et al. (2013); Hochreiter & Schmidhuber (1997)). Unlike feedforward neural networks, RNNs are dynamical systems that use their internal state to process inputs. The goal of this work is to shed light on the inner workings of RNNs from a theoretical point of view. In particular, we focus on the RNN state equation which is characterized by a nonlinear activation function $\\phi$ , state weight matrix $\\pmb { A }$ , and input weight matrix $\\textbf { { B } }$ as follows ", + "bbox": [ + 173, + 494, + 825, + 676 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/c25b751c217275605976e95df32cdb1ff4be84b98db6d9a3e26f9c027e78f581.jpg", + "text": "$$\nh _ { t + 1 } = \\phi ( A h _ { t } + B u _ { t } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 683, + 581, + 700 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Here $h _ { t }$ is the state vector and $\\mathbf { \\pmb { u } } _ { t }$ is the input data at timestamp $t$ . This equation is the source of dynamic behavior of RNNs and distinguishes RNN from feedforward networks. The weight matrices $\\pmb { A }$ and $\\textbf { { B } }$ govern the dynamics of the state equation and are inferred from data. We will explore the statistical and computational efficiency of stochastic gradient descent (SGD) for learning these weight matrices. ", + "bbox": [ + 174, + 707, + 823, + 776 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Contributions: Suppose we are given a finite trajectory of input/state pairs $( { \\mathbf { } } u _ { t } , h _ { t } ) _ { t = 0 } ^ { N }$ generated from the state equation (1.1). We consider a least-squares regression obtained from $N$ equations; with inputs $( { \\mathbf { } } u _ { t } , \\mathbf { \\dot { \\boldsymbol { h } } } _ { t } ) _ { t = 1 } ^ { N }$ and outputs $( h _ { t + 1 } ) _ { t = 1 } ^ { N }$ . For a class of activation functions including leaky ReLU and for stable systems1, we show that SGD linearly converges to the ground truth weight matrices while requiring near-optimal trajectory length $N$ . In particular, the required sample size is $\\mathcal { O } ( n + p )$ where $n$ and $p$ are the dimensions of the state and input vectors respectively. The results are extended to unstable systems when the samples are collected from multiple independent RNN trajectories rather than a single trajectory. Our theory applies to increasing activation functions whose derivatives are bounded away from zero, which includes leaky ReLU, and Gaussian input data. Numerical experiments on ReLU and leaky ReLU corroborate our theory and demonstrate that SGD converges faster as the activation slope increases. To obtain our results, we i) characterize the statistical properties of the state vector (e.g. well-conditioned covariance) and ii) derive a novel SGD convergence result with nonlinear activations; which may be of independent interest. As a whole, this paper provides a step towards foundational understanding of RNN training via SGD. ", + "bbox": [ + 174, + 784, + 825, + 896 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 204, + 331, + 218 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work is related to the recent optimization and statistics literature on linear dynamical systems (LDS) and neural networks. ", + "bbox": [ + 174, + 229, + 823, + 257 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Linear dynamical systems: The state-equation (1.1) reduces to a LDS when $\\phi$ is the linear activation $( \\phi ( x ) = { \\dot { x } } )$ ). Identifying the weight matrices is a core problem in linear system identification and is related to the optimal control problem (e.g. linear quadratic regulator) with unknown system dynamics. While these problems are studied since 1950’s (Ljung (1998; 1987); Åström & Eykhoff (1971)), our work is closer to the recent literature that provides data dependent bounds and characterize the non-asymptotic learning performance. Recht and coauthors have a series of papers exploring optimal control problem (Simchowitz et al. (2018); Tu et al. (2018; 2017)). In particular, Hardt et al. (2016) shows gradient descent learns single-input-single-output (SISO) LDS with polynomial guarantees. Oymak & Ozay (2018) and Faradonbeh et al. (2018) provide sample complexity bounds for learning LDS. Sanandaji et al. (2011b;a); Pereira et al. (2010) study the identification of sparse systems. ", + "bbox": [ + 173, + 265, + 825, + 406 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Neural networks: There is a growing literature on the theoretical aspects of deep learning and provable algorithms for training neural networks. Most of the existing results are concerned with feedforward networks. Ge et al. (2017); Li & Yuan (2017); Mei et al. (2018b); Soltanolkotabi (2017); Janzamin et al. (2015); Zhong et al. (2017b) consider learning fully-connected shallow networks with gradient descent. Mei et al. (2018a); Soltanolkotabi et al. (2017); Foster et al. (2018) analyze empirical landscape of related nonlinear learning problems. Brutzkus & Globerson (2017); Zhong et al. (2017a); Du et al. (2017); Goel et al. (2018) address convolutional neural networks; which is an efficient weight-sharing architecture. Brutzkus et al. (2017); Wang et al. (2018) studies over-parameterized networks when data is linearly separable. Janzamin et al. (2015); Oymak & Soltanolkotabi (2018) utilize tensor decomposition techniques for learning feedforward neural nets. For recurrent networks, Sedghi & Anandkumar (2016) proposed tensor algorithms with polynomial guarantees and Khrulkov et al. (2017) studied their expressive power. More recently, Miller & Hardt (2018) showed that stable RNNs can be approximated by feed-forward networks. ", + "bbox": [ + 173, + 412, + 825, + 593 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PROBLEM SETUP ", + "text_level": 1, + "bbox": [ + 176, + 613, + 348, + 630 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We first introduce the notation. $\\| \\cdot \\|$ returns the spectral norm of a matrix and $s _ { \\mathrm { m i n } } ( \\cdot )$ returns the minimum singular value. The activation $\\phi : \\mathbb { R } \\mathbb { R }$ applies entry-wise if its input is a vector. Throughout, $\\phi$ is assumed to be a 1-Lipschitz function. With proper scaling of its parameters, the system (1.1) with a Lipschitz activation can be transformed into a system with 1-Lipschitz activation. The functions $\\Sigma [ \\cdot ]$ and var $[ \\cdot ]$ return the covariance of a random vector and variance of a random variable respectively. ${ { I } _ { n } }$ is the identity matrix of size $n \\times n$ . Normal distribution with mean $\\pmb { \\mu }$ and covariance $\\pmb { \\Sigma }$ is denoted by $\\mathcal { N } ( \\boldsymbol { \\mu } , \\boldsymbol { \\Sigma } )$ . Throughout, $c , C , c _ { 0 } , c _ { 1 } , \\ldots$ denote positive absolute constants. ", + "bbox": [ + 173, + 643, + 825, + 756 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Setup: We consider the dynamical system parametrized by an activation function $\\phi ( \\cdot )$ and weight matrices $\\pmb { A } \\in \\mathbb { R } ^ { n \\times n } , \\pmb { B } \\in \\mathbf { \\bar { \\mathbb { R } } } ^ { n \\times p }$ as described in (1.1). Here, $\\boldsymbol { h } _ { t }$ is the $n$ dimensional state-vector and $\\mathbf { \\pmb { u } } _ { t }$ is the $p$ dimensional input to the system at time $t$ . As mentioned previously, (1.1) corresponds to the state equation of a recurrent neural network. For most RNNs of interest, the state $\\boldsymbol { h } _ { t }$ is hidden and we only get to interact with $\\boldsymbol { h } _ { t }$ via an additional output equation. For Elman networks Elman (1990), this equation is characterized by some output activation $\\phi _ { y }$ and output weights ${ C , D }$ as follows ", + "bbox": [ + 174, + 768, + 825, + 853 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/57d1dfc97b85fe497d5819d5301bea616cb4094040911f5de6af7768881e1a02.jpg", + "text": "$$\n\\mathbf { \\nabla } y _ { t } = \\phi _ { y } ( C h _ { t } + D \\mathbf { u } _ { t } ) .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 859, + 578, + 876 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, our attention is restricted to the state equation (1.1); which corresponds to setting $\\pmb { y } _ { t } = \\pmb { h } _ { t + 1 }$ in the output equation. To analyze (1.1) in a non-asymptotic data-dependent setup, we assume a finite input/state trajectory of $\\{ u _ { t } , h _ { t } \\} _ { t = 0 } ^ { N }$ generated by some ground truth weight matrices ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithm 1 Learning state equations with nonlinear activations ", + "text_level": 1, + "bbox": [ + 173, + 103, + 598, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1: Inputs: $( \\mathbf { \\boldsymbol { y } } _ { t } , h _ { t } , \\mathbf { \\boldsymbol { u } } _ { t } ) _ { t = 1 } ^ { N }$ sampled from a trajectory. Scaling $\\mu$ , learning rate $\\eta$ . Initialization \n$A _ { 0 } , B _ { 0 }$ . \n2: Outputs: Estimates ${ \\hat { A } } , { \\hat { B } }$ of the weight matrices $A , B$ . \n3: $\\mathbf { \\pmb { x } } _ { t } \\bar { } [ \\mu \\pmb { h } _ { t } ^ { T } \\mathbf { \\pmb { u } } _ { t } ^ { T } ] ^ { T }$ for $1 \\leq t \\leq N$ . \n4: $\\Theta _ { 0 } [ \\mu ^ { - 1 } A _ { 0 } \\ : B _ { 0 } ]$ \n5: for $\\tau$ from 1 to END do \n6: Pick $\\gamma _ { \\tau }$ from $\\{ 1 , 2 , \\ldots , N \\}$ uniformly at random. \n7: $\\Theta _ { \\tau } \\gets \\Theta _ { \\tau - 1 } - \\eta \\nabla \\mathcal { L } _ { \\gamma _ { \\tau } } ( \\Theta _ { \\tau - 1 } )$ \n8: end for \n9: return $[ \\hat { A } \\hat { B } ] \\Theta _ { \\mathrm { E N D } } [ \\begin{array} { c c } { \\mu I _ { n } } & { 0 } \\\\ { 0 } & { I _ { p } } \\end{array} ] .$ ", + "bbox": [ + 179, + 122, + 825, + 284 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "$( A , B )$ . Our goal is learning the unknown weights $\\pmb { A }$ and $\\textbf { { B } }$ in a data and computationally efficient way. In essence, we will show that, if the trajectory length satisfies $N \\gtrsim n + p$ , SGD can quickly and provably accomplish this goal using a constant step size. ", + "bbox": [ + 173, + 311, + 825, + 356 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Appoach: Our approach is described in Algorithm 1. It takes two hyperparameters; the scaling factor $\\mu$ and learning rate $\\eta$ . Using the RNN trajectory, we construct $N$ triples of the form $\\{ u _ { t } , h _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }$ We formulate a regression problem by defining the output vector ${ \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\mathbf { } } _ { \\mathbf { } } \\mathbf { \\Xi } _ { \\mathbf { } } \\mathbf { \\Lambda } _ { \\mathbf { } } \\mathbf { \\Lambda } _ { \\mathbf { } } \\textbf { } _ { \\mathbf { } } \\textbf { } \\textbf { } _ { \\mathrm { } }$ , input vector $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ , and the target parameter $C$ as follows ", + "bbox": [ + 173, + 361, + 826, + 417 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/99d71ba632a76cf152f474dbdb5eaee9b5132a80eefba5dcc24c959ec71e6d66.jpg", + "text": "$$\n\\begin{array} { r } { y _ { t } = h _ { t + 1 } \\quad , \\quad x _ { t } = \\left[ \\begin{array} { l } { \\mu h _ { t } } \\\\ { u _ { t } } \\end{array} \\right] \\in \\mathbb { R } ^ { n + p } \\quad , \\quad C = [ \\mu ^ { - 1 } A B ] \\in \\mathbb { R } ^ { n \\times ( n + p ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 425, + 743, + 460 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "With this reparameterization, we find the input/output identity ${ \\pmb y } _ { t } = \\phi ( { \\pmb C } { \\pmb x } _ { t } )$ . We will consider the least-squares regression given by ", + "bbox": [ + 173, + 467, + 825, + 497 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/5586b94ca53904fae9e65fc5cd3901819921f5f775d7c09d7121a03790d0c656.jpg", + "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\Theta } ) = \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\mathcal { L } _ { t } ( \\boldsymbol { \\Theta } ) \\quad \\mathrm { w h e r e } \\quad \\mathcal { L } _ { t } ( \\boldsymbol { \\Theta } ) = \\frac { 1 } { 2 } \\| y _ { t } - \\phi ( \\boldsymbol { \\Theta } x _ { t } ) \\| _ { \\ell _ { 2 } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 503, + 714, + 547 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For learning the ground truth parameter $C$ , we utilize SGD on the loss function (2.3) with a constant learning rate $\\eta$ . Starting from an initial point $\\Theta _ { 0 }$ , after END SGD iterations, Algrorithm 1 returns an estimate $\\hat { C } = \\Theta _ { \\mathrm { E N D } }$ . Estimates of $\\pmb { A }$ and $\\textbf { { B } }$ are decoded from the left and right submatrices of $\\hat { C }$ respectively. ", + "bbox": [ + 173, + 553, + 825, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 627, + 336, + 643 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 659, + 326, + 674 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The analysis of the state equation naturally depends on the choice of the activation function; which is the source of nonlinearity. We first define a class of Lipschitz and increasing activation functions. ", + "bbox": [ + 174, + 685, + 823, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 3.1 ( $\\beta$ -increasing activation). Given $1 \\geq \\beta \\geq 0$ , the activation function $\\phi$ satisfies $\\phi ( 0 ) = 0$ and $1 \\geq \\phi ^ { \\prime } ( x ) \\geq \\beta$ for all $x \\in \\mathbb { R }$ . ", + "bbox": [ + 174, + 718, + 821, + 748 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our results will apply to strictly increasing activations where $\\phi$ is $\\beta$ -increasing for some $\\beta > 0$ . Observe that, this excludes ReLU activation which has zero derivative for negative values. However, it includes Leaky ReLU which is a generalization of ReLU. Parameterized by $1 \\ge \\beta \\ge 0$ , Leaky ReLU is a $\\beta$ -increasing function given by ", + "bbox": [ + 173, + 758, + 826, + 815 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b1f001507ec9aab062e6ac49909aae227f7d54a81c863551d3f22dd776bbfb35.jpg", + "text": "$$\n{ \\mathrm { L R e L U } } ( x ) = \\operatorname* { m a x } ( \\beta x , x ) .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 821, + 589, + 839 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In general, given an increasing and 1-Lipschitz activation $\\phi$ , a $\\beta$ -increasing function $\\phi _ { \\beta }$ can be obtained by blending $\\phi$ with the linear activation, i.e. $\\phi _ { \\beta } ( x ) = ( 1 - \\beta ) \\phi ( x ) + \\beta x$ . ", + "bbox": [ + 173, + 845, + 825, + 876 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A critical property that enables SGD is that the state-vector covariance $\\pmb { \\Sigma } [ h _ { t } ]$ is well-conditioned under proper assumptions. The lemma below provides upper and lower bounds on this covariance matrix in terms of problem variables. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lemma 3.2 (State vector covariance). Consider the state equation (1.1) where $h _ { 0 } = 0$ and ${ \\mathbf { } } u _ { t } \\stackrel { i . i . d . } { \\sim }$ $\\mathcal { N } ( 0 , I _ { p } )$ . Define the upper bound term $B _ { t }$ as ", + "bbox": [ + 171, + 103, + 823, + 133 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/91b94ddd31eb131359e43000d636d2e2ea942b9558be740f218085adb857b73c.jpg", + "text": "$$\nB _ { t } = \\| B \\| \\sqrt { \\frac { 1 - \\| A \\| ^ { 2 t } } { 1 - \\| A \\| ^ { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 133, + 584, + 176 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Suppose $\\phi$ is 1-Lipschitz and $\\phi ( 0 ) = 0$ . Then, for all $t \\ge 0 , \\Sigma [ h _ { t } ] \\preceq B _ { t } ^ { 2 } I _ { n }$ . ", + "bbox": [ + 215, + 183, + 727, + 200 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Suppose $\\phi$ is a $\\beta$ -increasing function and $p \\geq n$ . Then, $\\Sigma [ h _ { t } ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } I _ { n }$ ", + "bbox": [ + 215, + 207, + 759, + 223 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As a natural extension from linear dynamical systems, we will say the system is stable if $\\| A \\| < 1$ and unstable otherwise. For activations we consider, stability implies that if the input is set to 0, state vector $\\boldsymbol { h } _ { t }$ will exponentially converge to 0 i.e. the system forgets the past states quickly. This is also the reason $( B _ { t } ) _ { t \\geq 0 }$ sequence converges for stable systems and diverges otherwise. The condition number of the covariance will play a critical role in our analysis. Using Lemma 3.2, this number can be upper bounded by $\\rho$ defined as ", + "bbox": [ + 173, + 231, + 826, + 315 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fc6b86ca3e0ac4d58e5ea587582e79573c9ab14616cffd0c7d0ea3f1d6b8f189.jpg", + "text": "$$\n\\rho = \\left( \\frac { B _ { \\infty } } { \\beta s _ { \\operatorname* { m i n } } ( B ) } \\right) ^ { 2 } = \\left( \\frac { \\| B \\| } { s _ { \\operatorname* { m i n } } ( B ) } \\right) ^ { 2 } \\frac { 1 } { \\beta ^ { 2 } ( 1 - \\| A \\| ^ { 2 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 315, + 678, + 353 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Observe that, the condition number of $\\textbf { { B } }$ appears inside the $\\rho$ term. ", + "bbox": [ + 178, + 353, + 614, + 367 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 LEARNING FROM SINGLE TRAJECTORY", + "text_level": 1, + "bbox": [ + 176, + 382, + 486, + 397 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our main result applies to stable systems $( \\left. A \\right. < 1 )$ and provides a non-asymptotic convergence guarantee for SGD in terms of the upper bound on the state vector covariance. This result characterizes the sample complexity and the rate of convergence of SGD; and also provides insights into the role of activation function and the spectral norm of $\\pmb { A }$ . ", + "bbox": [ + 173, + 409, + 825, + 465 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3.3 (Main result). Let $\\{ u _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }$ be a finite trajectory generated from the state equation (1.1). Suppose $\\| A \\| < 1$ , $\\phi$ is $\\beta$ -increasing, $h _ { 0 } = 0$ , $p \\geq n$ , and ${ \\mathbf { \\boldsymbol { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } )$ . Let $\\rho$ be same as (3.3) and $c , C , c _ { 0 }$ be properly chosen absolute constants. Pick the trajectory length $N$ to satisfy ", + "bbox": [ + 173, + 467, + 825, + 515 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/68c76a24868db4df9165ba0f5b6775760b02782628d15dba356966f0dbeac817.jpg", + "text": "$$\nN \\geq C L \\rho ^ { 2 } ( n + p ) ,\n$$", + "text_format": "latex", + "bbox": [ + 431, + 513, + 563, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where loss fu $\\begin{array} { r } { L = 1 - \\frac { \\log ( c n \\rho ) } { \\log \\| A \\| } } \\end{array}$ ) g µ = 1/B∞, learning rate η = c0 β2ρn(n+p) . Pick scalinh probability , and consider the, starting from an $1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) )$ initial point $\\Theta _ { 0 }$ , for all $\\tau \\geq 0$ , the SGD iterations described in Algorithm $^ { l }$ satisfies ", + "bbox": [ + 174, + 534, + 825, + 585 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0b5e7497e75a254dd9a481029a5296f59c29076d3da79a4a5f7d8d3daa757b36.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } n ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 587, + 684, + 621 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here the expectation is over the randomness of the SGD updates. ", + "bbox": [ + 174, + 622, + 599, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Sample complexity: Theorem 3.3 essentially requires $N \\gtrsim ( n + p ) / \\beta ^ { 4 }$ samples for learning. This can be seen by unpacking (3.3) and ignoring the logarithmic $L$ term and the condition number of $\\textbf { { B } }$ . Observe that $\\bar { \\mathcal { O } } ( n + \\bar { p } )$ growth achieves near-optimal sample size for our problem. Each state equation (1.1) consists of $n$ sub-equations (one for each entry of $h _ { t + 1 }$ ). We collect $N$ state equations to obtain a system of $N n$ equations. On the other hand, the total number of unknown parameters in $\\pmb { A }$ and $\\textbf { { B } }$ are $n ( n + p )$ . This implies Theorem 3.3 is applicable as soon as the problem is mildly overdetermined i.e. $N n \\gtrsim n ( n + p )$ . ", + "bbox": [ + 173, + 643, + 825, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Computational complexity: Theorem 3.3 requires $\\begin{array} { r } { \\mathcal { O } ( n ( n + p ) \\log \\frac { 1 } { \\varepsilon } ) } \\end{array}$ iterations to reach $\\varepsilon$ - neighborhood of the ground truth. Our analysis reveals that, this rate can be accelerated if the state vector is zero-mean. This happens for odd activation functions satisfying $\\phi ( - x ) = - \\phi ( x )$ (e.g. linear activation). The result below is a corollary and requires $\\times n$ less iterations. ", + "bbox": [ + 173, + 748, + 826, + 805 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3.4 (Faster learning for odd activations). Consider the same setup provided in Theorem 3.3. Additionally, assume that φ is an odd function. Pick scaling µ = 1/B∞, learning rate η = c0 β2ρ(n+p) , and consider the loss function (2.3). With probability $\\begin{array} { r } { 1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) ) } \\end{array}$ , starting from an initial point $\\Theta _ { 0 }$ , for all $\\tau \\geq 0$ , the SGD iterations described in Algorithm $^ { l }$ satisfies ", + "bbox": [ + 173, + 808, + 826, + 875 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8c93b654c2d8a76d1171132a4f20b7c6cb2176e5e837d3da0de397d22ecec9bf.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 875, + 676, + 910 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the expectation is over the randomness of the SGD updates. ", + "bbox": [ + 173, + 910, + 606, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Another aspect of the convergence rate is the dependence on $\\beta$ . In terms of $\\beta$ , the SGD error (3.4) decays as $( \\bar { 1 } - \\mathcal { O } ( \\beta ^ { 8 } ) ) ^ { \\tau }$ . While it is not clear how optimal is the exponent 8, numerical experiments in Section 6 demonstrate that larger $\\beta$ indeed results in drastically faster convergence. ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 MAIN IDEAS AND PROOF STRATEGY ", + "text_level": 1, + "bbox": [ + 174, + 166, + 509, + 184 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We first outline our high-level proof strategy for Theorem 3.3; which brings together ideas from statistics and optimization. ", + "bbox": [ + 176, + 199, + 823, + 228 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. We first show that input data is well-behaved by proving that state-vector $h _ { t }$ has a wellconditioned covariance as discussed in Lemma 3.2 and shown in Appendix B. The key idea is if $\\phi$ is $\\beta$ -increasing, then the random input data $\\mathbf { \\Delta } \\mathbf { u } _ { t }$ provides sufficient excitation for the output state $\\boldsymbol { h } _ { t + 1 }$ . \n2. Even if individual samples are well-behaved, analyzing (2.3) is still challenging due to temporal dependencies between the samples. These dependencies prevent us from directly using statistical learning results that typically assume i.i.d. samples. We show that the dependency between samples at time $t$ and $t + T$ decay exponentially fast in separation $T$ (for stable systems). This is outlined in Appendix C. \n3. This observation allows us to obtain nearly independent data by subsampling the original trajectory to get $( h _ { i T } , \\pmb { u } _ { i T } ) _ { i \\geq 0 }$ . Thanks to exponential decay, a logarithmically small $T$ can be chosen to generate large subtrajectories of size $N / T$ . Appendix D uses additional perturbation arguments to establish the well-behavedness of the overall data matrix. \n4. To conclude, we obtain a deterministic result which establishes fast convergence result for $\\beta$ -increasing activations and well-behaved dataset. This is provided in Theorem 4.1 and proved in Appendix A. ", + "bbox": [ + 210, + 239, + 825, + 481 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first three steps are related to the statistical nature of the problem which can be decoupled from the last step. Specifically, the last step derives a deterministic result that establishes the linear convergence of SGD for $\\beta$ -increasing functions. For linear convergence proofs, a typical strategy is showing the strong convexity of the loss function i.e. showing that, for some $\\alpha > 0$ and all points $\\mathbf { \\nabla } _ { v } , \\mathbf { \\nabla } _ { u }$ , the gradient satisfies ", + "bbox": [ + 173, + 492, + 825, + 563 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f842deeb3e5fe3ab48cb04fdd542e5f14fbeb042f1ef01f0dd961b255e711343.jpg", + "text": "$$\n\\begin{array} { r } { \\langle \\nabla \\mathcal { L } ( \\pmb { v } ) - \\nabla \\mathcal { L } ( \\pmb { u } ) , \\pmb { v } - \\pmb { u } \\rangle \\geq \\alpha \\| \\pmb { v } - \\pmb { u } \\| _ { \\ell _ { 2 } } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 569, + 642, + 589 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The core idea of our convergence result is that the strong convexity parameter of the loss function with $\\beta$ -increasing activations can be connected to the loss function with linear activations. In particular, recalling (2.3), set $\\pmb { y } _ { t } ^ { \\mathrm { l i n } } = \\pmb { C } \\pmb { x } _ { t }$ and define the linear loss to be ", + "bbox": [ + 174, + 595, + 826, + 638 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b9edfca0507553cdbfe81c766c92a7ca29c224388cfaa71278628d829283bab6.jpg", + "text": "$$\n\\mathcal { L } ^ { \\mathrm { l i n } } ( \\Theta ) = \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } \\Vert \\pmb { y } _ { t } ^ { \\mathrm { l i n } } - \\Theta \\pmb { x } _ { t } \\Vert _ { \\ell _ { 2 } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 646, + 619, + 690 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Denoting the strong convexity parameter of the original loss by $\\alpha _ { \\phi }$ and that of linear loss by $\\alpha _ { \\mathrm { l i n } }$ , we argue that $\\alpha _ { \\phi } \\geq \\beta ^ { 2 } \\alpha _ { \\mathrm { l i n } }$ ; which allows us to establish a convergence result as soon as $\\alpha _ { \\mathrm { l i n } }$ is strictly positive. Next result is our SGD convergence theorem which follows from this discussion. ", + "bbox": [ + 174, + 695, + 825, + 739 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 4.1 (Deterministic convergence). Suppose a data set $\\{ \\pmb { x } _ { i } , \\pmb { y } _ { i } \\} _ { i = 1 } ^ { N }$ is given; where output $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ is related to input $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ via $\\pmb { y } _ { i } = \\phi ( \\langle \\pmb { x } _ { i } , \\pmb { \\theta } \\rangle )$ for some $\\pmb { \\theta } \\in \\mathbb { R } ^ { n }$ . Suppose $\\beta > 0$ and $\\phi$ is a $\\beta$ -increasing. Le $\\ : \\gamma _ { + } \\geq \\gamma _ { - } > 0 \\ :$ be scalars. Assume that input samples satisfy the bounds ", + "bbox": [ + 174, + 744, + 825, + 787 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/15a20164d7bde2f542d974953eb975ab68376331ddea2c74765b2017d342449f.jpg", + "text": "$$\n\\gamma _ { + } I _ { n } \\succeq \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ { i } x _ { i } ^ { T } \\succeq \\gamma _ { - } I _ { n } \\quad , \\quad \\| x _ { i } \\| _ { \\ell _ { 2 } } ^ { 2 } \\leq B f o r a l l i .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 795, + 691, + 838 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let $\\{ r _ { \\tau } \\} _ { \\tau = 0 } ^ { \\infty }$ be a sequence of i.i.d. integers uniformly distributed between 1 to $N$ . Then, starting from an arbitrary point $\\pmb { \\theta } _ { 0 }$ , setting learning rate $\\begin{array} { r } { \\eta = \\frac { \\beta ^ { 2 } \\gamma _ { - } } { \\gamma _ { + } B } } \\end{array}$ β2γ−γ+B , for all τ ≥ 0, the SGD iterations for quadratic loss ", + "bbox": [ + 171, + 845, + 826, + 896 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1a38bb92f2b59eccb69067a7bf852fcaec48de56d42ee58525a13cfa7a876124.jpg", + "text": "$$\n\\pmb { \\theta } _ { \\tau + 1 } = \\pmb { \\theta } _ { \\tau } - \\eta ( \\phi ( \\mathbf { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) - \\pmb { y } _ { r _ { \\tau } } ) \\phi ^ { \\prime } ( \\mathbf { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) \\pmb { x } _ { r _ { \\tau } } ,\n$$", + "text_format": "latex", + "bbox": [ + 336, + 901, + 660, + 921 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "satisfies the error bound ", + "bbox": [ + 173, + 103, + 336, + 117 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8fed64bd69effa3a3706ba7f460c9a63eb1c79a0220a3973c712de5f1c205ec7.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\pmb { \\theta } _ { \\tau } - \\pmb { \\theta } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { \\theta } _ { 0 } - \\pmb { \\theta } \\| _ { \\ell _ { 2 } } ^ { 2 } \\big ( 1 - \\frac { \\beta ^ { 4 } \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } B } \\big ) ^ { \\tau } ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 125, + 642, + 160 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the expectation is over the random selection of the SGD iterations $\\{ r _ { \\tau } \\} _ { \\tau = 0 } ^ { \\infty }$ ", + "bbox": [ + 173, + 165, + 712, + 180 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This theorem provides a clean convergence rate for SGD for $\\beta$ -increasing activations and naturally generalizes standard results on linear regression which corresponds to $\\beta = 1$ . We remark that related results appear in the literature on generalized linear models. Kakade et al. (2011); Foster et al. (2018); Mei et al. (2018a) provide learning theoretic loss/gradient/hessian convergence results for isotonic regression, robust regression, and $\\beta$ -increasing activations. Goel et al. (2018) establishes a similar result for leaky ReLU activations under the assumption of symmetric input distribution and infinitely many samples (i.e. in population limit). Compared to these, we establish a deterministic linear convergence guarantee for SGD that works whenever the data matrix is full rank. We believe extensions to proximal gradient methods might be beneficial for high-dimensional nonlinear problems (e.g. sparse/low-rank approximation, manifold constraints Cai et al. (2010); Beck & Teboulle (2009); Oymak et al. (2018); Agarwal et al. (2010); Pereira et al. (2010)) and is left as a future work. ", + "bbox": [ + 173, + 190, + 825, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To derive our main results in Section 3, we need to address the first three steps outlined earlier and determine the conditions under which Theorem 4.1 is applicable to the data obtained from RNN state equation with high probability. Below we provide desirable characteristics of the state vector; which enables our statistical results. ", + "bbox": [ + 173, + 351, + 825, + 407 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assumption 1 (Well-behaved state vector). Let $L > 1$ be an integer. There exists positive scalars√ $\\gamma _ { + } , \\gamma _ { - } , \\theta$ and an absolute constant $C > 0$ such that $\\theta \\leq 3 \\sqrt { n }$ and the following holds ", + "bbox": [ + 173, + 411, + 825, + 440 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• Lower bound: $\\Sigma [ h _ { L - 1 } ] \\succeq \\gamma _ { - } I _ { n }$ , • Upper bound: for all $t ,$ , the state vector satisfies ", + "bbox": [ + 217, + 450, + 460, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 215, + 474, + 553, + 489 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/30d8372c4c8ab6d2404f446bba519232e4fad591ccb6e30b6fd2fd300bc5f5a2.jpg", + "text": "$$\n\\Sigma [ h _ { t } ] \\preceq \\gamma _ { + } I _ { n } \\quad , \\quad \\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq C \\sqrt { \\gamma _ { + } } \\quad a n d \\quad \\| \\mathbb { E } [ h _ { t } ] \\| _ { \\ell _ { 2 } } \\leq \\theta \\sqrt { \\gamma _ { + } } .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 494, + 763, + 513 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here $\\lVert \\cdot \\rVert _ { \\psi _ { 2 } }$ returns the subgaussian norm of a vector (see Def. 5.22 of Vershynin (2010)). ", + "bbox": [ + 230, + 518, + 812, + 534 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assumption 1 ensures that covariance is well-conditioned, state vector is well-concentrated, and it has a reasonably small expectation. Our next theorem establishes statistical guarantees for learning the RNN state equation based on this assumption. ", + "bbox": [ + 174, + 544, + 823, + 587 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 4.2 (General result). Let $\\{ u _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }$ be a length $N$ trajectory of the state equation (1.1). Suppose $\\| A \\| < 1$ , $\\phi$ is $\\beta$ -increasing, $h _ { 0 } = 0$ , and ${ \\mathbf { \\boldsymbol { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } )$ . Given scalars $\\gamma _ { + } \\geq \\gamma _ { - } > 0$ , set the condition number as $\\rho = \\gamma _ { + } / \\gamma _ { - }$ . For absolute constants $C , c , c _ { 0 } > 0$ , choose trajectory length $N$ to satisfy ", + "bbox": [ + 173, + 590, + 826, + 652 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/370ce44b937b28038799ac3d5f402d7858c7dd9f98c9551d61f082d525a88b2e.jpg", + "text": "$$\nN \\geq C L \\rho ^ { 2 } ( n + p ) w h e r e L = \\lceil 1 - \\frac { \\log \\left( c n \\rho \\right) } { \\log \\| A \\| } \\rceil .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 659, + 671, + 691 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Suppose Assumption $^ { l }$ holds with $L , \\gamma _ { + } , \\gamma _ { - } , \\theta$ . Pick scaling to be $\\mu = 1 / \\sqrt { \\gamma _ { + } }$ and learning rate to be η = c0 ρ(θ+√2)2(n+p) . β2 With probability $1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) )$ , starting from $\\Theta _ { 0 }$ , for all $\\tau \\geq 0$ , the SGD iterations on loss (2.3) as described in Algorithm $^ { l }$ satisfies ", + "bbox": [ + 173, + 696, + 826, + 751 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0dbc6dc1ac52f684643056b6204e414a4533d42b5bcfb72a3a8f7709e2306c67.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 284, + 755, + 714, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the expectation is over the randomness of SGD updates. ", + "bbox": [ + 174, + 796, + 583, + 813 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The advantage of this theorem is that, it isolates the optimization problem from the statistical properties of state vector. If one can prove tighter bounds on achievable $( \\gamma _ { + } , \\gamma _ { - } , \\theta )$ , it will immediately imply improved performance for SGD. In particular, Theorems 3.3 and 3.4 are simple corollaries of Theorem 4.2 with proper choices. ", + "bbox": [ + 174, + 821, + 823, + 880 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• Theorem 3.3 follows by setting $\\gamma _ { + } = B _ { \\infty } ^ { 2 }$ , $\\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 }$ , and $\\theta = { \\sqrt { n } }$ . \n• Theorem 3.4 follows by setting $\\gamma _ { + } = B _ { \\infty } ^ { 2 }$ , $\\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 }$ , and $\\theta = 0$ . ", + "bbox": [ + 215, + 888, + 735, + 926 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 LEARNING UNSTABLE SYSTEMS", + "text_level": 1, + "bbox": [ + 176, + 102, + 475, + 118 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "So far, we considered learning from a single RNN trajectory for stable systems $( \\left. A \\right. < 1 )$ . For such systems, as the time goes on, the impact of the earlier states disappear. In our analysis, this allows us to split a single trajectory into multiple nearly-independent trajectories. This approach will not work for unstable systems $\\mathbf { A }$ is arbitrary) where the impact of older states may be amplified over time. To address this, we consider a model where the data is sampled from multiple independent trajectories. ", + "bbox": [ + 173, + 132, + 825, + 204 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Suppose $N$ independent trajectories of the state-equation (1.1) are available. Pick some integer $T _ { 0 } \\geq 1$ . Denoting the ith trajectory by the triple $( { h } _ { t + 1 } ^ { ( i ) } , { h } _ { t } ^ { ( i ) } , { u } _ { t } ^ { ( i ) } ) _ { t \\geq 0 }$ , we collect a single sample from each trajectory at time $T _ { 0 }$ r he triple $( \\boldsymbol { h } _ { T _ { 0 } + 1 } ^ { ( i ) } , \\boldsymbol { h } _ { T _ { 0 } } ^ { ( i ) } , \\boldsymbol { u } _ { T _ { 0 } } ^ { ( i ) } )$ To utilize the existing $1 \\leq i \\leq N$ ", + "bbox": [ + 173, + 209, + 825, + 275 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/878a3c240f8e1e7e22880775deeb90b87892469271c34ed7119a4b83089b65cf.jpg", + "text": "$$\n( { \\bf y } _ { i } , { \\bf h } _ { i } , { \\bf u } _ { i } ) = ( { \\bf h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \\bf h } _ { T _ { 0 } } ^ { ( i ) } , { \\bf u } _ { T _ { 0 } } ^ { ( i ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 281, + 612, + 304 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "With this setup, we can again use the SGD Algorithm 1 to learn the weights $\\pmb { A }$ and $\\textbf { { B } }$ . The crucial difference compared to Section 3 is that, the samples $( { \\bf y } _ { i } , h _ { i } , { \\bf u } _ { i } ) _ { i = 1 } ^ { N }$ are now independent of each other; hence, the analysis is simplified. As previously, having an upper bound on the condition number of the state-vector covariance is critical. This upper bound can be shown to be $\\rho$ defined as ", + "bbox": [ + 174, + 308, + 825, + 364 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/46bcebc40b73552b186b6f24418092dac5768e22d3bc9c43ada9c0202ea837d2.jpg", + "text": "$$\n\\rho = \\left\\{ \\begin{array} { l l } { \\bar { \\rho } } & { \\mathrm { i f } \\ n > 1 } \\\\ { \\bar { \\rho } \\frac { \\ 1 - \\beta ^ { 2 } \\vert A \\vert ^ { 2 } } { 1 - ( \\beta \\vert A \\vert ) ^ { 2 T _ { 0 } } } } & { \\mathrm { i f } \\ n = 1 } \\end{array} \\right. \\quad \\mathrm { w h e r e } \\quad \\bar { \\rho } = \\frac { B _ { T _ { 0 } } ^ { 2 } } { \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 371, + 704, + 414 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The mod $\\bar { \\rho }$ term is similar to the earlier decation is indeed necessary since 3); ho when nvolves . On the $B _ { T _ { 0 } }$ rather than er hand, not $B _ { \\infty }$ . t, $B _ { \\infty } = \\infty$ $\\| A \\| > 1$ $B _ { T _ { 0 } } ^ { 2 }$ grows proportional to $\\| A \\| ^ { 2 T _ { 0 } }$ ; which results in exponentially bad condition number in $T _ { 0 }$ . Our $\\rho$ definition remedies this issue for single-output systems; where $n = 1$ and $\\pmb { A }$ is a scalar. In particular, when $\\beta = 1$ (e.g. $\\phi$ is linear) $\\rho$ becomes equal to the correct value $1 ^ { 2 }$ . The next theorem provides our result on unstable systems in terms of this condition number and other model parameters. ", + "bbox": [ + 173, + 417, + 826, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 5.1 (Unstable systems). Suppose we are given $N$ independent trajectories $( h _ { t } ^ { ( i ) } , u _ { t } ^ { ( i ) } ) _ { t \\geq 0 }$ for $1 \\leq i \\leq N$ . Each trajectory is sampled at time $T _ { 0 }$ to obtain $N$ samples $( { \\bf y } _ { i } , h _ { i } , { \\bf u } _ { i } ) _ { i = 1 } ^ { N }$ where the ith sample is given by (5.1). Suppose the sample size satisfies ", + "bbox": [ + 173, + 512, + 825, + 558 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/5a86f46ab4f30f54fd4e97145e201fd0ec7e6e887bd3241d2fd297c1d7983ca2.jpg", + "text": "$$\nN \\geq C \\rho ^ { 2 } ( n + p )\n$$", + "text_format": "latex", + "bbox": [ + 439, + 563, + 558, + 580 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\rho$ is given by (5.2). Assume the initial states are 0, $\\phi$ is $\\beta$ -increasing, $p \\geq n ,$ , and ${ \\pmb u } _ { t } \\stackrel { i . i . d . } { \\sim }$ $\\mathcal { N } ( 0 , { I } _ { p } )$ . Set scaling $\\mu = 1 / \\sqrt { B _ { T _ { 0 } } }$ , learning rate $\\begin{array} { r } { \\eta = c _ { 0 } \\frac { \\beta ^ { 2 } } { \\rho n ( n + p ) } } \\end{array}$ , and run SGD over the equations described in (2.2) and (2.3). Starting from $\\Theta _ { 0 }$ , with probability $1 - 2 N \\exp ( - 1 0 0 ( n + p ) ) -$ $4 \\exp ( - \\mathcal { O } ( \\frac { N } { \\rho ^ { 2 } } ) )$ , all SGD iterations satisfy ", + "bbox": [ + 173, + 590, + 825, + 659 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/1defd79b99a32e7ce6b363146a666cfd44438e1b8ba0e5d3729b229d1879e3fd.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } n ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 313, + 661, + 681, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where the expectation is over the randomness of the SGD updates. ", + "bbox": [ + 173, + 702, + 606, + 717 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 NUMERICAL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 731, + 436, + 747 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We conducted experiments on ReLU and Leaky ReLU activations. Let us first describe the experimental setup. We pick the state dimension $n = 5 0$ and the input dimension $p = 1 0 0$ . We choose the ground truth matrix $\\pmb { A }$ to be a scaled random unitary matrix; which ensures that all singular values of $\\pmb { A }$ are equal. $\\textbf { { B } }$ is generated with i.i.d. $\\mathcal { N } ( 0 , 1 )$ entries. Instead of using the theoretical scaling choice, we determine the scaling $\\mu$ from empirical covariance matrices outlined in Algorithm 2. Similar to our proof strategy, this algorithm equalizes the spectral norms of the input and state covariances to speed up convergence. We also empirically determined the learning rate and used $\\eta = 1 / 1 0 0$ in all experiments. ", + "bbox": [ + 173, + 756, + 826, + 867 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Algorithm 2 Empirical hyperparameter selection. ", + "text_level": 1, + "bbox": [ + 176, + 103, + 498, + 118 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "1: Inputs: $( h _ { t } , u _ { t } ) _ { t = 1 } ^ { N }$ sampled from a trajectory. \n2: Outputs: Scaling 3: Form the empirica $\\mu$ .covariance matrix from \n$\\Sigma _ { h }$ $\\{ h _ { t } \\} _ { t = 1 } ^ { N }$ \n4: Form the empirical covariance matrix $\\Sigma _ { u }$ $\\{ u _ { t } \\} _ { t = 1 } ^ { N }$ . \n5: return $\\sqrt { \\| \\Sigma _ { u } \\| / \\| \\Sigma _ { h } \\| }$ . ", + "bbox": [ + 179, + 123, + 573, + 199 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0bf1b15a51fed848e839755197683e302f9a9ddcb50642893bcbe2352a4d0bde.jpg", + "image_caption": [ + "Figure 1: SGD convergence behavior for Leaky ReLUs with varying minimum slope $\\beta$ . Figures a) and b) repeat the same experiments. The difference is the spectral norm of the ground truth state matrix $\\pmb { A }$ . " + ], + "image_footnote": [], + "bbox": [ + 236, + 217, + 823, + 366 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Evaluation: We consider two performance measures in the experiments. Let $\\hat { C }$ be an estimate of the ground truth parameter $\\boldsymbol { C } \\dot { = } [ \\mu ^ { - 1 } \\boldsymbol { A } \\boldsymbol { B } ]$ . The first measure is the normalized error defined as $\\| \\hat { C } - C \\| _ { F } ^ { 2 } / \\| C \\| _ { F } ^ { 2 }$ . The second measure is the normalized loss defined as ", + "bbox": [ + 174, + 410, + 825, + 455 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/3a97e193b5d9a5cf2ecb2bc97f48ab8a68fccd0bea8ba3320d05c4d0961e7e6a.jpg", + "text": "$$\n\\frac { \\sum _ { i = 1 } ^ { N } | | \\pmb { y } _ { t } - \\phi ( \\hat { C } \\pmb { x } _ { t } ) | | _ { \\ell _ { 2 } } ^ { 2 } } { \\sum _ { i = 1 } ^ { N } | | \\pmb { y } _ { t } | | _ { \\ell _ { 2 } } ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 462, + 584, + 505 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In all experiments, we run Algorithm 1 for 50000 SGD iterations and plot these measures as a function of $\\tau$ ; by using the estimate available at the end of the $\\tau$ th SGD iteration for $0 \\leq \\tau \\leq 5 0 0 0 0$ . Each curve is obtained by averaging the outcomes of 20 independent realizations.Our first experiments use $N = 5 0 0$ ; which is mildly larger than the total dimension $n + p = 1 5 0$ . In Figure 1, we plot the Leaky ReLU errors with varying slopes as described in (3.1). Here $\\beta = 0$ corresponds to ReLU and $\\beta = 1$ is the linear model. In consistence with our theory, SGD achieves linear convergence and as $\\beta$ increases, the rate of convergence drastically improves3. The improvement is more visible for less stable systems driven by $\\pmb { A }$ with a larger spectral norm. In particular, while ReLU converges for small $\\| A \\|$ , SGD gets stuck before reaching the ground truth when $\\| A \\| = 0 . 8$ . ", + "bbox": [ + 173, + 510, + 825, + 636 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To understand, how well SGD fits the training data, in Figure 2a, we plotted the normalized loss for ReLU activation. For more unstable system $\\left. \\left. A \\right. \\right. = \\bar { 0 } . 9 )$ , training loss stagnates in a similar fashion to the parameter error. We also verified that the norm of the overall gradient $\\lVert \\nabla \\mathcal { L } ( \\Theta _ { \\tau } ) \\rVert _ { F }$ continues to decay (where $\\Theta _ { \\tau }$ is the $\\tau$ th SGD iterate); which implies that SGD converges before reaching a global minima. As $\\pmb { A }$ becomes more stable, rate of convergence improves and linear rate is visible. Finally, to better understand the population landscape of the quadratic loss with ReLU activations, Figure 2b repeats the same ReLU experiments while increasing the sample size five times to $N = 2 5 0 0$ . For this more overdetermined problem, SGD converges even for $\\mathbf { | } \\mathbf { \\boldsymbol { A } } | \\mathbf { | } = 0 . 9$ ; indicating that ", + "bbox": [ + 173, + 642, + 825, + 767 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "• population landscape of loss with ReLU activation is well-behaved, • however ReLU problem requires more data compared to the Leaky ReLU for finding global minima. ", + "bbox": [ + 210, + 772, + 825, + 820 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Overall, as predicted by our theory, experiments verify that SGD indeed quickly finds the optimal weight matrices of the state equation (1.1) and as the activation slope $\\beta$ increases, the convergence rate improves. ", + "bbox": [ + 176, + 825, + 821, + 869 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/49e542a82fdc668eda12f703902eb01c8a18f213a3ab21f6cde72a6a3d9f2842.jpg", + "image_caption": [ + "Figure 2: SGD convergence behavior for ReLU with varying spectral norm of the state matrix $\\pmb { A }$ . Figures a) and b) repeats the same experiments. The difference is that a) uses $N = 5 0 0$ trajectory length whereas b) uses $N = 2 5 0 0$ (i.e. $\\times 5$ more data). Shaded regions highlight the one standard deviation around the mean. " + ], + "image_footnote": [], + "bbox": [ + 238, + 102, + 825, + 251 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 306, + 328, + 321 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work showed that SGD can learn the nonlinear dynamical system (1.1); which is characterized by weight matrices and an activation function. This problem is of interest for recurrent neural networks as well as nonlinear system identification. We showed that efficient learning is possible with optimal sample complexity and good computational performance. Our results apply to strictly increasing activations such as Leaky ReLU. We empirically showed that Leaky ReLU converges faster than ReLU and requires less samples; in consistence with our theory. We list a few unanswered problems that would provide further insights into recurrent neural networks. ", + "bbox": [ + 173, + 332, + 825, + 429 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• Covariance of the state-vector: Our results depend on the covariance of the state-vector and requires it to be positive definite. One might be able to improve the current bounds on the condition number and relax the assumptions on the activation function. Deriving similar performance bounds for ReLU is particularly interesting. ", + "bbox": [ + 174, + 429, + 825, + 484 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• Hidden state: For RNNs, the state vector is hidden and is observed through an additional equation (2.1); which further complicates the optimization landscape. Even for linear dynamical systems, learning the $( A , B , C , D { \\bar { ) } }$ system ((1.1), (2.1)) is a non-trivial task Ho & Kalman (1966); Hardt et al. (2016). What can be said when we add the nonlinear activations? ", + "bbox": [ + 173, + 484, + 825, + 539 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "• Classification task: In this work, we used normally distributed input and least-squares regression for our theoretical guarantees. More realistic input distributions might provide better insight into contemporary problems, such as natural language processing; where the goal is closer to classification (e.g. finding the best translation from another language). 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", + "bbox": [ + 173, + 853, + 823, + 881 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Kai Zhong, Zhao Song, Prateek Jain, Peter L Bartlett, and Inderjit S Dhillon. Recovery guarantees for onehidden-layer neural networks. arXiv preprint arXiv:1706.03175, 2017b. ", + "bbox": [ + 171, + 888, + 823, + 916 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A DETERMINISTIC CONVERGENCE RESULT FOR SGD ", + "text_level": 1, + "bbox": [ + 174, + 101, + 637, + 119 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof of Theorem 4.1. Given two distinct scalars $a , b$ ; define $\\begin{array} { r } { \\phi ^ { \\prime } ( a , b ) = \\frac { \\phi ( a ) - \\phi ( b ) } { a - b } } \\end{array}$ . $\\phi ^ { \\prime } ( a , b ) \\ge \\beta$ since $\\phi$ is $\\beta$ -increasing. Define $\\pmb { w } _ { \\tau }$ to be the residual ${ \\pmb w } _ { \\tau } = { \\pmb \\theta } _ { \\tau } - { \\pmb \\theta }$ . Observing ", + "bbox": [ + 173, + 128, + 825, + 160 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c97cde9e7118aa05be64d1ec4447c99c96deb339b56321a603339dc1b867bea8.jpg", + "text": "$$\n\\phi ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) - \\pmb { y } _ { r _ { \\tau } } = \\phi ^ { \\prime } ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } , \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } ) \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { w } _ { \\tau } ,\n$$", + "text_format": "latex", + "bbox": [ + 359, + 162, + 635, + 180 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "the SGD recursion obeys ", + "bbox": [ + 173, + 183, + 323, + 195 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/b5e2155e30cd4ad43ad9c440d3fcdac4ad3aeb67486db7946cae3cc549f574a9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\pmb { w } _ { \\tau + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } = \\| \\pmb { w } _ { \\tau } - \\eta \\big ( \\phi ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) - \\pmb { y } _ { r _ { \\tau } } \\big ) \\phi ^ { \\prime } \\big ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } \\big ) \\pmb { x } _ { r _ { \\tau } } \\| _ { \\ell _ { 2 } } ^ { 2 } . } \\\\ & { \\qquad = \\| \\pmb { w } _ { \\tau } - \\eta \\pmb { x } _ { r _ { \\tau } } \\phi ^ { \\prime } \\big ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } \\big ) \\phi ^ { \\prime } \\big ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } , \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } \\big ) \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } } \\\\ & { \\qquad = \\| \\big ( \\pmb { I } - \\eta \\pmb { G } _ { r _ { \\tau } } \\big ) \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 198, + 697, + 257 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where $G _ { r _ { \\tau } } = \\pmb { x } _ { r _ { \\tau } } \\phi ^ { \\prime } ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) \\phi ^ { \\prime } ( \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } , \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } ) \\pmb { x } _ { r _ { \\tau } } ^ { T }$ . Since $\\phi$ is 1-Lipschitz and $\\beta$ -increasing, $\\mathbf { { G } } _ { r _ { \\tau } }$ is a positivesemidefinite matrix satisfying ", + "bbox": [ + 173, + 260, + 821, + 287 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/1ddf5674893b7c4bb65e62b1866a5f7bf76e53a7e137cf1e2fa9f89f00795bc2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\pmb { x } _ { r _ { \\tau } } \\pmb { x } _ { r _ { \\tau } } ^ { T } \\succeq \\pmb { G } _ { r _ { \\tau } } \\succeq \\beta ^ { 2 } \\pmb { x } _ { r _ { \\tau } } \\pmb { x } _ { r _ { \\tau } } ^ { T } , } \\\\ & { \\pmb { G } _ { r _ { \\tau } } ^ { T } \\pmb { G } _ { r _ { \\tau } } \\preceq \\pmb { x } _ { r _ { \\tau } } \\pmb { x } _ { r _ { \\tau } } ^ { T } \\pmb { x } _ { r _ { \\tau } } \\pmb { x } _ { r _ { \\tau } } ^ { T } \\preceq B \\pmb { x } _ { r _ { \\tau } } \\pmb { x } _ { r _ { \\tau } } ^ { T } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 290, + 627, + 329 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Consequently, we find the following bounds in expectation ", + "bbox": [ + 178, + 329, + 521, + 343 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/b159667adb61d1632d32becb45683975ceb93c38e1460fec22806deb791a90b0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\gamma _ { + } I _ { n } \\succeq \\mathbb { E } [ G _ { r _ { \\tau } } ] \\succeq \\beta ^ { 2 } \\gamma _ { - } I _ { n } , } \\\\ & { \\mathbb { E } [ G _ { r _ { \\tau } } ^ { T } G _ { r _ { \\tau } } ] \\preceq B \\gamma _ { + } I _ { n } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 410, + 345, + 586, + 386 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Observe that (A.1) essentially lower bounds the strong convexity parameter of the problem with $\\beta ^ { 2 } \\gamma _ { - }$ ; which is the strong convexity of the identical problem with the linear activation (i.e. $\\beta = 1$ ). However, we only consider strong convexity around the ground truth parameter $\\pmb \\theta$ i.e. we restricted our attention to $( \\pmb \\theta , \\pmb \\theta _ { \\tau } )$ pairs. With this, ${ \\pmb w } _ { \\tau + 1 }$ can be controlled as, ", + "bbox": [ + 174, + 388, + 826, + 439 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/6e4326ef251b2ea42c7c3e47dbbb8e77042e19141d3742696f76469f639dc496.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\| \\pmb { w } _ { \\tau + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] = \\mathbb { E } [ \\| ( \\pmb { I } - \\eta \\pmb { G } _ { r _ { \\tau } } ) \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ] } \\\\ & { \\quad \\quad \\quad = \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } - 2 \\eta \\mathbb { E } [ \\pmb { w } _ { \\tau } ^ { T } \\pmb { G } _ { r _ { \\tau } } \\pmb { w } _ { \\tau } ] + \\eta ^ { 2 } \\mathbb { E } [ \\pmb { w } _ { \\tau } ^ { T } \\pmb { G } _ { r _ { \\tau } } ^ { T } \\pmb { G } _ { r _ { \\tau } } \\pmb { w } _ { \\tau } ] } \\\\ & { \\quad \\quad \\quad \\le \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ( 1 - 2 \\eta \\beta ^ { 2 } \\gamma _ { - } + \\eta ^ { 2 } B \\gamma _ { + } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 441, + 714, + 501 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Setting η = $\\begin{array} { r } { \\eta = \\frac { \\beta ^ { 2 } \\gamma _ { - } } { \\gamma _ { + } B } } \\end{array}$ , we find the advertised bound ", + "bbox": [ + 173, + 502, + 462, + 523 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/434bec7af88fbe0f437bdcaef0f20a55e29b7ebd1a7f5ef1e554bac731fbc73b.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\pmb { w } _ { \\tau + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\mathbb { E } [ \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ] ( 1 - \\frac { \\beta ^ { 4 } \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } B } ) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 527, + 624, + 559 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Applying induction over the iterations $\\tau$ , we find the advertised bound (4.2) ", + "bbox": [ + 171, + 560, + 620, + 575 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/f4b702b7fa9eb13b0aabcf0f0238a96fc6c60920840d1360565bc40db9b1d10f.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { w } _ { 0 } \\| _ { \\ell _ { 2 } } ^ { 2 } ( 1 - \\frac { \\beta ^ { 4 } \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } B } ) ^ { \\tau } .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 578, + 609, + 609 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Lemma A.1 (Merging $L$ splits). Assume matrices √ $\\pmb { X } ^ { ( i ) } \\in \\mathbb { R } ^ { N _ { i } \\times q }$ are given for $1 \\leq i \\leq L$ . Suppose for all $1 \\leq i \\leq L$ , rows of $\\pmb { X } ^ { ( i ) }$ has $\\ell _ { 2 }$ norm at most $\\sqrt { B }$ and each $\\pmb { X } ^ { ( i ) }$ satisfies ", + "bbox": [ + 171, + 631, + 825, + 661 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/42f23f49b4a5e0a0bf634202c11360775a0f842b04d7b287c1edb65ba8c1c494.jpg", + "text": "$$\n\\gamma _ { + } I _ { n } \\succeq \\frac { { X ^ { ( i ) } } ^ { T } X ^ { ( i ) } } { N _ { i } } \\succeq \\gamma _ { - } I _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 662, + 593, + 695 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Set $\\begin{array} { r } { N = \\sum _ { i = 1 } ^ { L } N _ { i } } \\end{array}$ and form the concatenated matrix $\\begin{array} { r } { \\pmb { X } = \\left[ \\pmb { X } ^ { ( 1 ) } \\right] . } \\\\ { \\quad } \\\\ { \\quad } \\\\ { \\pmb { X } ^ { ( L ) } } \\end{array}$ Denote ith row of $\\pmb { X }$ by $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ . Then, for ", + "bbox": [ + 173, + 700, + 826, + 765 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "each i, $\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } ^ { 2 } \\leq B$ and ", + "bbox": [ + 173, + 765, + 321, + 781 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/ac19f3abaf5cae008127a3515e31ad08263cc1be7742cb90c86f7762d1674cec.jpg", + "text": "$$\n\\gamma _ { + } I _ { n } \\varrho \\overset { \\mathbf { \\substack { \\textstyle X } } } { = } \\frac { \\mathbf { { X } } ^ { T } \\mathbf { { X } } } { N } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ { i } \\mathbf { { x } } _ { i } ^ { T } \\succeq \\gamma _ { - } I _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 780, + 629, + 819 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Proof. The bound on the rows $\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } }$ directly follows by assumption. For the remaining result, first observe that $\\begin{array} { r } { { \\pmb X } ^ { T } { \\pmb X } = \\sum _ { i = 1 } ^ { L } { \\pmb X } ^ { ( i ) ^ { T } } { \\pmb X } ^ { ( i ) } } \\end{array}$ . Next, we have ", + "bbox": [ + 173, + 832, + 825, + 864 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/5382f1915d595ae984f9c049ccea6243c8aacb77ee5f0967b53c4d21343df3e9.jpg", + "text": "$$\nN \\gamma _ { + } I _ { n } = \\sum _ { i = 1 } ^ { L } N _ { i } \\gamma _ { + } I _ { n } \\succeq \\sum _ { i = 1 } ^ { L } { X ^ { ( i ) } } ^ { T } X ^ { ( i ) } \\succeq \\sum _ { i = 1 } ^ { L } N _ { i } \\gamma _ { - } I _ { n } = N \\gamma _ { - } I _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 867, + 714, + 906 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Combining these two yields the desired upper/lower bounds on $X ^ { T } X / N$ . ", + "bbox": [ + 173, + 910, + 614, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B PROPERTIES OF THE NONLINEAR STATE EQUATIONS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 637, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This section characterizes the properties of the state vector $\\pmb { h } _ { t }$ when input sequence is normally distributed. These bounds will be crucial for obtaining upper and lower bounds for the singular values of the data matrix $\\pmb { X } = [ \\pmb { x } _ { 1 } ~ \\ldots ~ \\pmb { x } _ { N } ] ^ { T }$ described in (2.2). For probabilistic arguments, we will use the properties of subgaussian random variables. Orlicz norm provides a general framework that subsumes subgaussianity. ", + "bbox": [ + 173, + 132, + 826, + 184 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Definition B.1 (Orlicz norms). For a scalar random variable Orlicz-a norm is defined as ", + "bbox": [ + 171, + 186, + 702, + 200 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ed7529e7054e73a050b1c8e4e46687a67c4421182524773d869d9bdcfd372d55.jpg", + "text": "$$\n\\| X \\| _ { \\psi _ { a } } = \\operatorname* { s u p } _ { k \\geq 1 } k ^ { - 1 / a } ( \\mathbb { E } [ | X | ^ { k } ] ) ^ { 1 / k }\n$$", + "text_format": "latex", + "bbox": [ + 393, + 207, + 602, + 233 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Orlicz-a norm of a vector $\\pmb { x } \\in \\mathbb { R } ^ { p }$ is defined as $\\begin{array} { r } { \\| \\pmb { x } \\| _ { \\psi _ { a } } = \\operatorname* { s u p } _ { \\pmb { v } \\in B ^ { p } } \\| \\pmb { v } ^ { T } \\pmb { x } \\| _ { \\psi _ { a } } } \\end{array}$ where $B ^ { p }$ is the unit $\\ell _ { 2 }$ ball. The subexponential norm is the Orlicz-1 norm $\\lVert \\cdot \\rVert _ { \\psi _ { 1 } }$ and the subgaussian norm is the Orlicz-2 norm $\\lVert \\cdot \\rVert _ { \\psi _ { 2 } }$ . ", + "bbox": [ + 173, + 239, + 825, + 268 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma B.2 (Lipschitz properties of the state vector). Consider the state equation (1.1). Suppose activation $\\phi$ is 1-Lipschitz. Observe that $\\boldsymbol { h } _ { t + 1 }$ is a deterministic function of the input sequence $\\{ { \\pmb u } _ { \\tau } \\} _ { \\tau = 0 } ^ { t }$ . Fixing all vectors $\\{ { \\pmb u } _ { i } \\} _ { i \\neq \\tau }$ (i.e. all except ${ \\bf { u } } _ { \\tau }$ ), $\\boldsymbol { h } _ { t + 1 }$ is $\\| A \\| ^ { t - \\tau } \\| B \\|$ Lipschitz function of ${ \\pmb u } _ { \\tau }$ for $0 \\leq \\tau \\leq t$ . ", + "bbox": [ + 173, + 271, + 825, + 310 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. Fixing $\\{ { \\pmb u } _ { i } \\} _ { i \\neq \\tau }$ , denote $\\boldsymbol { h } _ { t + 1 }$ as a function of ${ \\pmb u } _ { \\tau }$ by $\\pmb { h } _ { t + 1 } ( \\pmb { u } _ { \\tau } )$ . Given a pair of vectors ${ \\bf { \\boldsymbol { u } } } _ { \\tau } , { \\bf { \\boldsymbol { u } } } _ { \\tau } ^ { \\prime }$ using 1-Lipschitzness of $\\phi$ , for any $t > \\tau$ , we have ", + "bbox": [ + 173, + 324, + 825, + 352 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/f1cf1bd772e11139bef4000ed3d11e0301248cdb11e51c18b267cc0b2712f1b0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } \\leq \\| \\phi ( A h _ { t } ( \\boldsymbol { u } _ { \\tau } ) + B \\boldsymbol { u } _ { t } ) - \\phi ( A h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) + B \\boldsymbol { u } _ { t } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| A ( h _ { t } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| A \\| \\| h _ { t } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 356, + 743, + 412 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proceeding with this recursion until $t = \\tau$ , we find ", + "bbox": [ + 174, + 416, + 475, + 429 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/46df81ca8e517a0d8dcb1945b197fd60a5721b6b4885ec0ccb9323cb3fab25c4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } \\leq \\| \\boldsymbol { A } \\| ^ { t - \\tau } \\| h _ { \\tau + 1 } ( \\boldsymbol { u } _ { \\tau } ) - h _ { \\tau + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\boldsymbol { A } \\| ^ { t - \\tau } \\| \\phi ( \\boldsymbol { A } h _ { \\tau } + \\boldsymbol { B } \\boldsymbol { u } _ { \\tau } ) - \\phi ( \\boldsymbol { A } h _ { \\tau } + \\boldsymbol { B } \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\boldsymbol { A } \\| ^ { t - \\tau } \\| \\boldsymbol { B } \\| \\| \\boldsymbol { u } _ { \\tau } - \\boldsymbol { u } _ { \\tau } ^ { \\prime } \\| _ { \\ell _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 434, + 743, + 493 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This bound implies $\\pmb { h } _ { t + 1 } ( \\pmb { u } _ { \\tau } )$ is $\\| A \\| ^ { t - \\tau } \\| B \\|$ Lipschitz function of ${ \\pmb u } _ { \\tau }$ ", + "bbox": [ + 174, + 497, + 599, + 513 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma B.3 (Upper bound). Consider the state equation governed by equation (1.1). Suppose ${ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )$ , $\\phi$ is 1-Lipschitz, $\\bar { \\phi } ( 0 ) = 0$ and $h _ { 0 } = 0$ . Recall the definition (3.2) of $B _ { t }$ . We have the following properties ", + "bbox": [ + 174, + 523, + 831, + 551 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• $\\pmb { h } _ { t }$ is a $B _ { t }$ -Lipschitz function of the vector $\\begin{array} { r } { \\pmb q _ { t } = [ \\pmb { u } _ { 0 } ^ { T } \\ . . . \\ \\pmb { u } _ { t - 1 } ^ { T } ] ^ { T } \\in \\mathbb { R } ^ { t p } . } \\end{array}$ . ", + "bbox": [ + 215, + 560, + 665, + 578 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• There exists an absolute constant $c > 0$ such that $\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq c B _ { t }$ and $\\Sigma [ h _ { t } ] \\preceq B _ { t } ^ { 2 } { \\cal I } _ { n }$ . ", + "bbox": [ + 215, + 585, + 792, + 602 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• $\\pmb { h } _ { t }$ satisfies ", + "bbox": [ + 217, + 609, + 302, + 623 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/d0934f5f47d595a3c66350bb6323db52666b2ff4f17b4a071420b06632056ff7.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\pmb { h } _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le t r ( \\pmb { B } \\pmb { B } ^ { T } ) \\frac { 1 - \\| \\pmb { A } \\| ^ { 2 t } } { 1 - \\| \\pmb { A } \\| ^ { 2 } } \\le \\operatorname* { m i n } \\{ n , p \\} B _ { t } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 621, + 686, + 652 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Also, there exists an absolute constant √ $c > 0$ such that for any $m \\geq n$ , with probability $1 -$ $2 \\exp ( - 1 0 0 m )$ , $\\| h _ { t } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { m } B _ { t }$ . ", + "bbox": [ + 232, + 655, + 820, + 681 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. i) Bounding Lipschitz constant: Observe that $\\pmb { h } _ { t }$ is a deterministic function of $\\pmb q _ { t }$ i.e. $h _ { t } = f ( q _ { t } )$ for some function $f$ . To bound Lipschitz constant of $f$ , for all (deterministic) vector pairs $\\pmb q _ { t }$ and $\\hat { \\pmb q } _ { t }$ , we find a scalar $L _ { f }$ satisfying, ", + "bbox": [ + 174, + 695, + 825, + 734 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ba4d2f007cd5e40dab104e2a9d7f6e5ce6e9a1ada8b542b6239033e1445d5191.jpg", + "text": "$$\n\\| f ( \\pmb q _ { t } ) - f ( \\hat { \\pmb q } _ { t } ) \\| _ { \\ell _ { 2 } } \\leq L _ { f } \\| \\pmb q _ { t } - \\hat { \\pmb q } _ { t } \\| _ { \\ell _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 741, + 616, + 757 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Define the vectors, $\\{ a _ { i } \\} _ { i = 0 } ^ { t }$ , as follows ", + "bbox": [ + 174, + 763, + 408, + 779 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/4216be785ac6a48836162ac81ce56b504f2ed0c71aeb6e27255c8518546845ad.jpg", + "text": "$$\n\\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\tilde { a } } _ { i } = [ \\hat { \\mathbf { u } } _ { 0 } ^ { T } \\dots \\hat { \\mathbf { u } } _ { i - 1 } ^ { T } \\mathbf { \\delta } \\mathbf { u } _ { i } ^ { T } \\dots \\mathbf { \\delta } \\mathbf { u } _ { t - 1 } ^ { T } ] ^ { T } .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 784, + 611, + 801 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Observing that ${ \\pmb a } _ { 0 } = { \\pmb q } _ { t } , { \\pmb a } _ { t } = \\hat { { \\pmb q } } _ { t }$ , we write the telescopic sum, ", + "bbox": [ + 173, + 808, + 547, + 821 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/0bae46f9c6eb884af96477d0b3ff019553d8668737a188370fac13d0d2b3dea6.jpg", + "text": "$$\n\\lVert f ( \\pmb { q } _ { t } ) - f ( \\pmb { \\hat { q } } _ { t } ) \\rVert _ { \\ell _ { 2 } } \\leq \\sum _ { i = 0 } ^ { t - 1 } \\lVert f ( \\pmb { a } _ { i + 1 } ) - f ( \\pmb { a } _ { i } ) \\rVert _ { \\ell _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 827, + 648, + 866 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Focusing on the individual terms $f ( \\pmb { a } _ { i + 1 } ) - f ( \\pmb { a } _ { i } )$ , observe that the only difference is the ${ \\mathbf { } } { \\mathbf { } } u _ { i } , \\hat { { \\mathbf { } } } { \\mathbf { } } \\bar { { \\mathbf { } } } u _ { i }$ terms. Viewing $\\pmb { h } _ { t }$ as a function of ${ \\bf { u } } _ { i }$ and applying Lemma B.2, ", + "bbox": [ + 173, + 871, + 823, + 898 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/557da1e218aa84013050b8ca5275be129b349117e30d93f73edb928a89dd72dc.jpg", + "text": "$$\n\\| f ( \\pmb { a } _ { i + 1 } ) - f ( \\pmb { a } _ { i } ) \\| _ { \\ell _ { 2 } } \\leq \\| \\pmb { A } \\| ^ { t - 1 - i } \\| \\pmb { B } \\| \\| \\pmb { u } _ { i } - \\hat { \\pmb { u } } _ { i } \\| _ { \\ell _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 904, + 663, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To bound the sum, we apply the Cauchy-Schwarz inequality; which yields ", + "bbox": [ + 173, + 103, + 612, + 119 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7b6e576cc81ea2acb2d4f9e2a91ab153430655c70db2c1bad8b1de9b9c7517c0.jpg", + "text": "$$\n\\begin{array} { r l } { | f ( q _ { t } ) - f ( \\hat { q } _ { t } ) | \\le \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\| A \\| ^ { t - 1 - i } \\| B \\| \\| u _ { i } - \\hat { u } _ { t } \\| _ { \\ell _ { 2 } } } & { } \\\\ { \\quad \\quad \\le ( \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\| A \\| ^ { 2 ( t - 1 - i ) } \\| B \\| ^ { 2 } ) ^ { 1 / 2 } \\displaystyle \\underbrace { ( \\displaystyle \\sum _ { i = 0 } ^ { t - 1 } \\| u _ { i } - \\hat { u } _ { i } \\| _ { \\ell _ { 2 } } ^ { 2 } ) ^ { 1 / 2 } } _ { \\| q _ { t } - \\hat { q } _ { t } \\| _ { \\ell _ { 2 } } } } \\\\ { \\quad \\quad \\le \\sqrt { \\displaystyle \\frac { \\| B \\| ^ { 2 } ( 1 - \\| A \\| ^ { 2 / t } ) } { 1 - \\| A \\| ^ { 2 } } } \\| q _ { t } - \\hat { q } _ { t } \\| _ { \\ell _ { 2 } } } & { } \\\\ { \\quad \\quad = B _ { t } \\| q _ { t } - \\hat { q } _ { t } \\| _ { \\ell _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 123, + 714, + 280 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The final line achieves the inequality (B.1) with $L _ { f } = B _ { t }$ hence $\\pmb { h } _ { t }$ is $B _ { t }$ Lipschitz function of $\\mathbf { \\nabla } _ { \\mathbf { \\eta } } \\mathbf { q } _ { t }$ ", + "bbox": [ + 169, + 282, + 751, + 297 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "ii) Bounding subgaussian norm: When ${ \\pmb u } _ { t } \\overset { \\mathrm { i . i . d . } } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )$ , the vector $\\pmb q _ { t }$ is distributed as $\\mathcal { N } ( 0 , { I } _ { t p } )$ . Since $\\pmb { h } _ { t }$ a $B _ { t }$ Lipschitz function of $\\pmb q _ { t }$ , for any fixed unit length vector $_ { v }$ , $\\alpha _ { \\pmb { v } } : = \\pmb { v } ^ { T } h _ { t } = \\pmb { v } ^ { T } f ( \\pmb { q } _ { t } )$ is still $B _ { t }$ -Lipschitz function of $\\pmb q _ { t }$ . Hence, using Gaussian concentration of Lipschitz functions, $\\alpha _ { v }$ satisfies ", + "bbox": [ + 174, + 305, + 825, + 348 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0134ebcafdd004d07260ebb1644e0b500a8a640e060404e92c5db2f5eecd1799.jpg", + "text": "$$\n\\mathbb { P } ( | \\alpha _ { v } - \\mathbb { E } [ \\alpha _ { v } ] | \\geq t ) \\leq 2 \\exp ( - \\frac { t ^ { 2 } } { 2 B _ { t } ^ { 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 353, + 619, + 385 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This implies that for any $\\pmb { v }$ , $\\alpha _ { v } \\mathrm { ~ - ~ } \\mathbb { E } [ \\alpha _ { v } ]$ is $\\mathcal { O } ( B _ { t } )$ subgaussian Vershynin (2010). This is true for all unit $_ { v }$ , hence using Definition B.1, the vector $\\pmb { h } _ { t }$ satisfies $\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( B _ { t } )$ as well. Secondly, $B _ { t }$ -Lipschitz function of a Gaussian vector obeys the variance inequality v $\\mathbf { a r } [ \\alpha _ { v } ] \\leq B _ { t } ^ { 2 }$ (page 49 of Ledoux (2001)), which implies the covariance bound ", + "bbox": [ + 173, + 388, + 826, + 440 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/98ce66a56af9e42f1165d8f74ace68f27082e7cf8f4ce4d21f1f4c67707798d3.jpg", + "text": "$$\n\\Sigma [ h _ { t } ] \\preceq B _ { t } ^ { 2 } I _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 439, + 549, + 457 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "iii) Bounding $\\ell _ { 2 }$ -norm: To obtain this result, we first bound $\\mathbb { E } [ \\| h _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ]$ . Since $\\phi$ is 1-Lipschitz and $\\phi ( 0 ) = 0$ we have the deterministic relation ", + "bbox": [ + 173, + 458, + 825, + 483 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/a0e52b0cab60fdc777b786ccffeda460a160862c15ea573c5c609fd83c31aa60.jpg", + "text": "$$\n\\lVert h _ { t + 1 } \\rVert _ { \\ell _ { 2 } } \\leq \\lVert A h _ { t } + B u _ { t } \\rVert _ { \\ell _ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 482, + 593, + 498 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Taking squares of both sides, expanding the right hand side, and using the independence of $\\mathbf { } h _ { t } , \\mathbf { } u _ { t }$ and the covariance information of $\\mathbf { \\Delta } \\mathbf { u } _ { t }$ , we obtain ", + "bbox": [ + 173, + 501, + 823, + 526 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d6023fa5130713f0ef269258d0c63584e7da52796c3931457b158461bbf4fdaf.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\leq \\mathbb { E } [ \\| A h _ { t } + B u _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] = \\mathbb { E } [ \\| A h _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] + \\mathbb { E } [ \\| B u _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] } \\\\ & { \\qquad \\leq \\| A \\| ^ { 2 } \\mathbb { E } [ \\| h _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] + \\mathrm { t r } ( B B ^ { T } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 530, + 704, + 569 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now that the recursion is established, expanding $\\pmb { h } _ { t }$ on the right hand side until $h _ { 0 } = 0$ , we obtain ", + "bbox": [ + 169, + 571, + 754, + 587 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/fd32011a4c51f6f7f5d6523e32a4fa2d4a0ce9f19442d42a701903cb3ff86690.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\pmb { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\sum _ { i = 0 } ^ { t } \\| \\pmb { A } \\| ^ { 2 i } \\mathrm { t r } ( \\pmb { B } \\pmb { B } ^ { T } ) \\le \\mathrm { t r } ( \\pmb { B } \\pmb { B } ^ { T } ) \\frac { 1 - \\| \\pmb { A } \\| ^ { 2 ( t + 1 ) } } { 1 - \\| \\pmb { A } \\| ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 300, + 590, + 697, + 630 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now using the fact that $\\mathrm { t r } ( B B ^ { T } ) \\leq \\mathrm { r a n k } ( B ) \\| B \\| ^ { 2 } \\leq \\mathrm { m i n } \\{ n , p \\} \\| B \\| ^ { 2 }$ , we find ", + "bbox": [ + 174, + 635, + 653, + 651 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/8ef052fae966b0e93225616dc91f3f74a2b0199689e42357f3f5f5bda572fa13.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ \\| \\boldsymbol { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ] ^ { 2 } \\le \\mathbb { E } [ \\| \\boldsymbol { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\operatorname* { m i n } \\{ n , p \\} B _ { t + 1 } ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 656, + 650, + 674 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Finally, using the fact that $\\pmb { h } _ { t }$ is $B _ { t }$ -Lipschitz function and utilizing Gaussian concentration of $\\mathbf { \\boldsymbol { q } } _ { t } \\sim \\mathcal { N } ( 0 , I _ { t p } )$ we find ", + "bbox": [ + 171, + 678, + 823, + 703 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f5cba27dd8c4380c6b6c443ff76d39a39170fcc07ad127ea728125ad3a8f8910.jpg", + "text": "$$\n\\mathbb { P } ( \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } - \\mathbb { E } [ \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } ] \\ge t ) \\le \\exp ( - \\frac { t ^ { 2 } } { 2 B _ { t } ^ { 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 700, + 650, + 732 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Setting $t = ( c - 1 ) \\sqrt { m } B _ { t }$ for sufficiently large $c > 0$ , we find $\\mathbb { P } ( \\| h _ { t } \\| _ { \\ell _ { 2 } } \\geq \\sqrt { n } B _ { t } + ( c - 1 ) \\sqrt { m } B _ { t } ) \\leq$ $\\exp ( - 1 0 0 m )$ . ", + "bbox": [ + 173, + 734, + 821, + 761 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma B.4 (Odd activations). Suppose $\\phi$ is strictly increasing and obeys $\\phi ( x ) = - \\phi ( - x )$ for all $_ x$ and $h _ { 0 } = 0$ . Consider the state equation (1.1) driven ${ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )$ . We have that $\\mathbb { E } [ h _ { t } ] = 0$ . ", + "bbox": [ + 173, + 767, + 821, + 800 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. We will inductively show that $\\{ h _ { t } \\} _ { t \\ge 0 }$ has a symmetric distribution around 0. Suppose the vector $\\pmb { h } _ { t }$ satisfies this assumption. Let $S \\subset \\mathbb { R } ^ { n }$ be a set. We will argue that $\\mathbb { P } ( h _ { t + 1 } \\subset S ) = \\mathbb { P } ( h _ { t + 1 } \\subset - S )$ . Since $\\phi$ is strictly increasing, it is bijective on vectors, and we can define the unique inverse set $S ^ { \\prime } = \\phi ^ { - 1 } ( S )$ . Also since $\\phi$ is odd, $\\phi ( - S ^ { \\prime } ) = - S$ . Since $\\mathbf { } h _ { t } , \\mathbf { } u _ { t }$ are independent and symmetric, we reach the desired conclusion as follows ", + "bbox": [ + 174, + 813, + 825, + 866 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/04b1452aad1b80c9578a4fab94dfb9a5c9e3edc98205a77b64047d3057155114.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } ( h _ { t + 1 } \\subset S ) = \\mathbb { P } ( A h _ { t } + B u _ { t } \\subset S ^ { \\prime } ) = \\mathbb { P } ( A ( - h _ { t } ) + B ( - u _ { t } ) \\subset S ^ { \\prime } ) } \\\\ & { \\qquad = \\mathbb { P } ( A h _ { t } + B u _ { t } \\subset - S ^ { \\prime } ) = \\mathbb { P } ( \\phi ( A h _ { t } + B u _ { t } ) \\subset \\phi ( - S ^ { \\prime } ) ) = \\mathbb { P } ( h _ { t + 1 } \\subset - S ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 869, + 769, + 906 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem B.5 (State-vector lower bound). Consider the nonlinear state equation (1.1) with $\\{ { \\pmb u } _ { t } \\} _ { t \\ge 0 }$ i.i.d. ∼ $\\mathcal { N } ( 0 , { \\pmb I } _ { p } )$ . Suppose $\\phi$ is a $\\beta$ -increasing function for some constant $\\beta > 0$ . For any $t \\geq 1$ , the state vector obeys ", + "bbox": [ + 171, + 103, + 823, + 131 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c0020b9459ed9c728ec193009573a65ddca6e99eb0404fba1e9b573aeb7d0647.jpg", + "text": "$$\n\\begin{array} { r } { \\Sigma [ h _ { t } ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B B ^ { T } ) I _ { n } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 413, + 137, + 584, + 155 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. The proof is an application of Lemma B.7. The main idea is to write $\\pmb { h } _ { t }$ as sum of two independent vectors, one of which has independent entries. Consider a multivariate Gaussian vector $\\mathbf { \\sigma } _ { \\mathbf { \\mathcal { g } } } \\sim \\mathcal { N } ( \\bar { 0 , } \\Sigma )$ . $\\pmb { g }$ is statistically identical to ${ \\pmb g } _ { 1 } + { \\pmb g } _ { 2 }$ where ${ \\pmb g } _ { 1 } \\sim \\mathcal { N } ( 0 , s _ { \\mathrm { m i n } } ( { \\pmb \\Sigma } ) { \\pmb I } _ { d } )$ and $g _ { 2 } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } - s _ { \\mathrm { m i n } } ( \\pmb { \\Sigma } ) \\pmb { I } _ { d } )$ are independent multivariate Gaussians. ", + "bbox": [ + 173, + 167, + 826, + 220 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since $B { \\boldsymbol { u } } _ { t } \\sim \\mathcal { N } ( 0 , B B ^ { T } )$ , setting $\\pmb { \\Sigma } = \\pmb { B } \\pmb { B } ^ { T }$ and $s _ { \\mathrm { m i n } } = s _ { \\mathrm { m i n } } ( \\Sigma )$ , we have that $B u _ { t } \\sim g _ { 1 } + g _ { 2 }$ where ${ \\bf { \\mathit { g } } } _ { 1 } , { \\bf { \\mathit { g } } } _ { 2 }$ are independent and ${ \\pmb g } _ { 1 } \\sim \\mathcal { N } ( 0 , s _ { \\mathrm { m i n } } { \\pmb I } _ { n } )$ and $\\pmb { g } _ { 2 } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } - s _ { \\mathrm { m i n } } \\pmb { I } _ { n } )$ . Consequently, we may write ", + "bbox": [ + 173, + 226, + 825, + 253 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/3e308247203f6acd9824321cbae0181bc44ed86a7383abeaa717e1069498cb00.jpg", + "text": "$$\nB u _ { t } + A h _ { t } \\sim g _ { 1 } + g _ { 2 } + A h _ { t } .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 260, + 598, + 275 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For lower bound, the crucial component will be the $\\pmb { g } _ { 1 }$ term; which has i.i.d. entries. Applying Lemma B.7 by setting ${ \\pmb x } = { \\pmb g } _ { 1 }$ and $\\pmb { y } = \\pmb { g } _ { 2 } + \\pmb { A } \\hat { h } _ { t }$ , and using the fact that $h _ { t } , g _ { 1 } , g _ { 2 }$ are all independent of each other, we find the advertised bound, for all $t \\geq 0$ , via ", + "bbox": [ + 174, + 280, + 825, + 320 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/0005d34fc28006e4db2441493c0c34e267513d7bf3abd84d80c4f4177a93b2ae.jpg", + "text": "$$\n\\Sigma [ h _ { t + 1 } ] = \\Sigma [ \\phi ( g _ { 1 } + g _ { 2 } + A h _ { t } ) ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } I _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 324, + 647, + 342 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The next theorem applies to multiple-input-single-output (MISO) systems where $\\pmb { A }$ is a scalar and $_ B$ is a row vector. The goal is refining the lower bound of Theorem B.5. ", + "bbox": [ + 173, + 376, + 825, + 402 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Theorem B.6 (MISO lower bound). Consider the setup of Theorem B.5 with single output i.e. $n = 1$ . For any $t \\geq 1$ , the state vector obeys ", + "bbox": [ + 171, + 405, + 823, + 431 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/44abcb6d72d25dbd8fdd2dfddaf0193181e028550cd028b31632dddd3d835111.jpg", + "text": "$$\n\\mathbf { v a r } [ \\pmb { h } _ { t } ] \\geq \\beta ^ { 2 } \\| \\pmb { B } \\| _ { \\ell _ { 2 } } ^ { 2 } \\frac { 1 - ( \\beta | \\pmb { A } | ) ^ { 2 t } } { 1 - \\beta ^ { 2 } | \\pmb { A } | ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 429, + 606, + 462 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. For any random variable $X$ , applying Lemma B.7, we have va ${ \\mathfrak { r } } [ \\phi ( X ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ X ]$ . Recursively, this yields ", + "bbox": [ + 173, + 476, + 826, + 503 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6633a7ff85f49fee6a9ec18e775d5263ccd2d49a8e76ae5a546b7efdd0a23caf.jpg", + "text": "$$\n\\mathbf { v a r } [ h _ { t + 1 } ] = \\mathbf { v a r } [ \\phi ( A h _ { t } + B u _ { t } ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ A h _ { t } + B u _ { t } ] = \\beta ^ { 2 } ( | A | ^ { 2 } \\mathbf { v a r } [ h _ { t } ] + \\| B \\| _ { \\ell _ { 2 } } ^ { 2 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 228, + 508, + 766, + 526 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Expanding these inequalities till $\\scriptstyle h _ { 0 }$ , we obtain the desired bound ", + "bbox": [ + 173, + 531, + 560, + 546 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/8c144c4eb62532a3c0625658554e3fae1da1c6c7a3ab9e7c59eca69b3fc6a5c8.jpg", + "text": "$$\n\\mathbf { v a r } [ \\pmb { h } _ { t } ] \\geq \\sum _ { i = 1 } ^ { t } ( \\beta ^ { i } | \\pmb { A } | ^ { i - 1 } \\| \\pmb { B } \\| _ { \\ell _ { 2 } } ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 551, + 604, + 590 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Lemma B.7 (Vector lower bound). Suppose $\\phi$ is a $\\beta$ -increasing function. Let ${ \\pmb x } = [ { \\pmb x } _ { 1 } ~ . ~ . ~ { \\pmb x } _ { n } ] ^ { T }$ be a vector with i.i.d. entries distributed as $x _ { i } \\sim X$ . Let $\\textbf { { y } }$ be a random vector independent of $_ { \\textbf { \\em x } }$ . Then, ", + "bbox": [ + 169, + 616, + 823, + 643 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/213d762ed80863dc48729ddbb319e2c4bcc6340c7ddb9f0e650fd1c34f13944a.jpg", + "text": "$$\n\\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) ] \\succeq \\beta ^ { 2 } { \\mathbf { v a r } } [ X ] { \\cal I } _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 647, + 589, + 666 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. We first apply law of total covariance (e.g. Lemma B.8) to simplify the problem using the following lower bound based on the independence of $_ { \\textbf { \\em x } }$ and $\\pmb { y }$ , ", + "bbox": [ + 173, + 679, + 823, + 707 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/5f584816276ff3a576ef62768cc88df1104585866e70470c1b160e6a3e30c5bb.jpg", + "text": "$$\n\\begin{array} { r } { \\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) ] \\succeq \\mathbb { E } _ { { \\pmb y } } [ \\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) | { \\pmb y } ] ] } \\\\ { = \\mathbb { E } _ { { \\pmb y } } [ \\Sigma _ { \\pmb x } [ \\phi ( { \\pmb x } + { \\pmb y } ) ] ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 710, + 614, + 747 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now, focusing on the covariance $\\Sigma _ { x } [ \\phi ( { \\pmb x } + { \\pmb y } ) ]$ , fixing a realization of $\\textbf { { y } }$ , and using the fact that $_ { \\textbf { \\em x } }$ has i.i.d. entries; $\\phi ( { \\pmb x } + { \\pmb y } )$ has independent entries as $\\phi$ applies entry-wise. This implies that $\\Sigma _ { \\pmb { x } } [ \\phi ( \\pmb { x } + \\pmb { y } ) ]$ is a diagonal matrix. Consequently, its lowest eigenvalue is the minimum variance over all entries, ", + "bbox": [ + 174, + 751, + 825, + 790 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/26d2978b2dd2794600d62a5d4ab004125b839f14e7d5879deb1a0553ba151b7f.jpg", + "text": "$$\n\\pmb { \\Sigma } _ { \\pmb { x } } [ \\phi ( \\pmb { x } + \\pmb { y } ) ] \\succeq \\operatorname* { m i n } _ { 1 \\leq i \\leq n } \\mathbf { v a r } [ \\phi ( \\pmb { x } _ { i } + \\pmb { y } _ { i } ) ] \\pmb { I } _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 795, + 632, + 819 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Fortunately, Lemma B.9 provides the lower bound $\\mathbf { v a r } [ \\phi ( \\pmb { x } _ { i } + \\pmb { y } _ { i } ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ X ]$ . Since this lower bound holds for any fixed realization of $\\textbf { { y } }$ , it still holds after taking expectation over $\\textbf { { y } }$ ; which concludes the proof. □ ", + "bbox": [ + 173, + 825, + 823, + 853 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The next two lemmas are helper results for Lemma B.7 and are provided for the sake of completeness. ", + "bbox": [ + 168, + 867, + 790, + 882 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Lemma B.8 (Law of total covariance). Let $\\mathbf { \\nabla } _ { \\mathbf { x } , \\mathbf { y } }$ be two random vectors and assume $\\textbf { { y } }$ has finite covariance. Then ", + "bbox": [ + 169, + 885, + 821, + 909 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/44d55fbe10b1d640569c7c17cc2cc9aef0b7ce68a2a43581bc9b22c8c3a059d0.jpg", + "text": "$$\n\\pmb { \\Sigma } [ \\pmb { y } ] = \\mathbb { E } [ \\pmb { \\Sigma } [ \\pmb { y } \\mid \\pmb { x } ] ] + \\pmb { \\Sigma } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 907, + 607, + 926 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. First, write $\\Sigma [ { \\pmb y } ] = \\mathbb { E } [ { \\pmb y } { \\pmb y } ^ { T } ] - \\mathbb { E } [ { \\pmb y } ] \\mathbb { E } [ { \\pmb y } ^ { T } ]$ . Then, applying the law of total expectation to each term, ", + "bbox": [ + 171, + 102, + 812, + 119 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/ae3d5e164ef16e7b936f71d53a104985a9d5229e1585e41ca785e85df5460e82.jpg", + "text": "$$\n\\pmb { \\Sigma } [ \\pmb { y } ] = \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] - \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } ^ { T } \\mid \\pmb { x } ] ]\n$$", + "text_format": "latex", + "bbox": [ + 343, + 121, + 648, + 138 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Next, we can write the conditional expectation as $\\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] = \\mathbb { E } [ \\pmb { \\Sigma } [ \\pmb { y } \\mid \\pmb { x } ] ] + \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] ^ { T } .$ . To conclude, we obtain the covariance of $\\mathbb { E } [ { \\pmb y } \\mid { \\pmb x } ]$ via the difference, ", + "bbox": [ + 174, + 138, + 823, + 169 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/23772338c41be9a58f0e01f27b4886b0c960b4747eec6e9dcbfeca8a85e1ba99.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] ^ { T } - \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] = \\pmb { \\Sigma } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 171, + 696, + 189 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "which yields the desired bound. ", + "bbox": [ + 173, + 189, + 361, + 203 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Lemma B.9 (Scalar lower bound). Suppose $\\phi$ is a $\\beta$ -increasing function with $\\beta > 0$ as defined in Definition 3.1. Given a random variable $X$ and a scalar $_ y$ , we have ", + "bbox": [ + 171, + 209, + 823, + 236 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d91a8d679f115481ec73193dc053402121b809b0fe0f607b2d2702a944d7d2ca.jpg", + "text": "$$\n\\mathbf { v a r } [ { \\phi } ( X + y ) ] \\geq { \\beta } ^ { 2 } \\mathbf { v a r } [ X ] .\n$$", + "text_format": "latex", + "bbox": [ + 410, + 236, + 586, + 253 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Since $\\phi$ is $\\beta$ -increasing, it is invertible and $\\phi ^ { - 1 }$ is strictly increasing. Additionally, $\\phi ^ { - 1 }$ is $1 / \\beta$ Lipschitz since, ", + "bbox": [ + 173, + 265, + 823, + 291 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Using this observation and the fact that $\\mathbb { E } [ X ]$ minimizes $\\mathbb { E } ( X - \\alpha ) ^ { 2 }$ over $\\alpha$ , $\\mathbf { v a r } [ \\phi ( X + y ) ]$ can be lower bounded as follows ", + "bbox": [ + 171, + 306, + 823, + 332 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/59d943e38de5e0ffe9fb584eafd9376cd2fefc44425f91dc32a27433d99ec06c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { v a r } [ \\phi ( X + y ) ] = \\mathbb { E } ( \\phi ( X + y ) - \\mathbb { E } [ \\phi ( X + y ) ] ) ^ { 2 } } \\\\ & { \\qquad \\geq \\beta ^ { 2 } \\mathbb { E } ( ( X + y ) - \\phi ^ { - 1 } ( \\mathbb { E } [ \\phi ( X + y ) ] ) ) ^ { 2 } } \\\\ & { \\qquad \\geq \\beta ^ { 2 } \\mathbb { E } ( X + y - \\mathbb { E } [ X + y ] ) ^ { 2 } } \\\\ & { \\qquad = \\beta ^ { 2 } \\mathbb { E } ( X - \\mathbb { E } X ) ^ { 2 } = \\beta ^ { 2 } \\mathbf { v a r } [ X ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 330, + 673, + 407 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note that, the final line is the desired conclusion. ", + "bbox": [ + 176, + 406, + 462, + 420 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C TRUNCATING STABLE SYSTEMS ", + "text_level": 1, + "bbox": [ + 176, + 439, + 475, + 455 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "One of the challenges in analyzing dynamical systems is the fact that samples from the same trajectory have temporal dependence. This section shows that, for stable systems, the impact of the past states decay exponentially fast and the system can be approximated by using the recent inputs only. We first define the truncation of the state vector. ", + "bbox": [ + 173, + 468, + 825, + 518 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Definition C.1 (Truncated state vector). Suppose $\\phi ( 0 ) = 0$ , initial condition $h _ { 0 } = 0$ , and consider the state equation (1.1). Given a timestamp $t$ , $L$ -truncation of the state vector $\\pmb { h } _ { t }$ is denoted by $\\bar { h } _ { t , L }$ and is equal to $\\mathbf { \\nabla } _ { \\mathbf { \\eta } } \\mathbf { q } _ { t }$ where ", + "bbox": [ + 173, + 520, + 823, + 559 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/446a96b3504994399aa5ffecbf626bffb6669f479f0405ec924e2a4bf13b05b4.jpg", + "text": "$$\n\\pmb { q } \\tau + 1 = \\phi ( \\pmb { A } \\pmb { q } _ { \\tau } + \\pmb { B } \\pmb { u } _ { \\tau } ^ { \\prime } ) \\quad , \\quad q _ { 0 } = 0\n$$", + "text_format": "latex", + "bbox": [ + 382, + 560, + 616, + 577 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "is the state vector generated by the inputs ${ \\pmb u } _ { \\tau } ^ { \\prime }$ satisfying ", + "bbox": [ + 176, + 579, + 501, + 592 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/85b7b37191a5fc384df5b34ce1bf6e9203b7a99510a176013e8bcac592b30617.jpg", + "text": "$$\n{ \\pmb u } _ { \\tau } ^ { \\prime } = \\left\\{ \\begin{array} { l l } { 0 i f \\tau < t - L } \\\\ { { \\pmb u } _ { \\tau } e l s e } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 416, + 593, + 580, + 631 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In words, $L$ truncated state vector $\\bar { h } _ { t , L }$ is obtained by unrolling $\\mathbf { \\delta } _ { h _ { t } }$ until time $t - L$ and setting the contribution of the state vector $\\boldsymbol { h } _ { t - L }$ to 0. This way, $\\bar { h } _ { t , L }$ depends only on the variables $\\{ u _ { \\tau } \\} _ { \\tau = t - L } ^ { t - 1 }$ . ", + "bbox": [ + 166, + 640, + 823, + 667 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The following lemma states that impact of truncation can be made fairly small for stable systems $( \\left. A \\right. < 1 )$ . Lemma C.2 (Truncation impact – deterministic). Consider the state vector $\\pmb { h } _ { t }$ and its $L$ -truncation $\\bar { h } _ { t , L }$ from Definition C.1. Suppose $\\phi$ is 1-Lipschitz. We have that ", + "bbox": [ + 173, + 674, + 825, + 715 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/9d8edc1566a5e0b82315ed1e75bc76aafb951a6daf4e2ed43431f0c8c9f0601e.jpg", + "text": "$$\n\\| h _ { t } - \\bar { h } _ { t , L } \\| _ { \\ell _ { 2 } } \\leq \\left\\{ \\begin{array} { l l } { 0 i f t \\leq L } \\\\ { \\| A \\| ^ { L } \\| h _ { t - L } \\| _ { \\ell _ { 2 } } e l s e } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 715, + 633, + 755 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. When $t \\leq L$ , Definition C.1 implies ${ \\pmb u } _ { \\tau } ^ { \\prime } = { \\pmb u } _ { \\tau }$ hence ${ \\pmb h } _ { t } = { \\pmb q } _ { t } = \\bar { { \\pmb h } } _ { t , L }$ . When $t > L$ , we again use Definition C.1 and recall that ${ \\pmb u } _ { \\tau } ^ { \\prime } = 0$ until time $\\tau = t - L - 1$ . For all $t - L < \\tau \\leq t$ , using 1-Lipschitzness of $\\phi$ , we have that ", + "bbox": [ + 173, + 767, + 825, + 808 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/237f124d4e461a8981e182c3c3d4c9af98e84254ca898c20dce707d0ef9ffdf5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| h _ { \\tau } - { \\pmb q } _ { \\tau } \\| _ { \\ell _ { 2 } } = \\| \\phi ( { \\pmb A } h _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) - \\phi ( { \\pmb A } { \\pmb q } _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\le \\| ( { \\pmb A } h _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) - ( { \\pmb A } { \\pmb q } _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\le \\| { \\pmb A } ( h _ { \\tau - 1 } - { \\pmb q } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } \\le \\| { \\pmb A } \\| \\| h _ { \\tau - 1 } - { \\pmb q } _ { \\tau - 1 } \\| _ { \\ell _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 808, + 702, + 858 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Applying this recursion between $t - L < \\tau \\leq t$ and using the fact that $\\pmb q _ { t - L } = 0$ implies the advertised result ", + "bbox": [ + 179, + 858, + 821, + 872 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/94e0305accc9c22f5eaf86897b61391a47c90a0fa63e2ad21af6875fb2667f92.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\pmb { h } _ { t } - \\pmb { q } _ { t } \\| _ { \\ell _ { 2 } } \\leq \\| \\pmb { A } \\| ^ { L } \\| \\pmb { h } _ { t - L } - \\pmb { q } _ { t - L } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\pmb { A } \\| ^ { L } \\| \\pmb { h } _ { t - L } \\| _ { \\ell _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 872, + 622, + 911 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We will now argue that, for stable systems, a single trajectory can be split into multiple nearly independent trajectories. First, we describe how the sub-trajectories are constructed. ", + "bbox": [ + 171, + 127, + 823, + 155 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition C.3 (Sub-trajectory). Let sampling rate $L \\geq 1$ and offset $1 \\le \\bar { \\tau } \\le L$ be two integers. Let $\\bar { N } = \\bar { N } _ { \\bar { \\tau } }$ be the largest integer obeying $( { \\bar { N } } - 1 ) { \\bar { L } } + { \\bar { \\tau } } \\leq N$ . We sample the trajectory $\\{ h _ { t } , \\boldsymbol { u } _ { t } \\} _ { t = 0 } ^ { N }$ at the points $\\bar { \\tau } , \\bar { \\tau } + L , \\dots , \\bar { \\tau } + ( \\bar { N } - 1 ) L + \\bar { \\tau }$ and define the τ¯th sub-trajectory as ", + "bbox": [ + 174, + 159, + 823, + 196 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/cfb605345930fd8d087ff1239d81c2fb3dd9aec4377a75146c63ceb6f04c964f.jpg", + "text": "$$\n( { \\pmb h } ^ { ( i ) } , { \\pmb u } ^ { ( i ) } ) : = ( { \\pmb h } ^ { ( i , \\bar { \\tau } ) } , { \\pmb u } ^ { ( i , \\bar { \\tau } ) } ) = ( { \\pmb h } _ { ( i - 1 ) L + \\bar { \\tau } } , { \\pmb u } _ { ( i - 1 ) L + \\bar { \\tau } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 204, + 676, + 222 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition C.4 (Truncated sub-trajectory). Consider the state equation (1.1) and recall Definition C.1. Given offset $\\bar { \\tau }$ and sampling rate $L$ , for $1 \\leq i \\leq \\bar { N }$ , the ith truncated sub-trajectory states are $\\{ \\bar { h } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }$ where the ith state is defined as ", + "bbox": [ + 174, + 227, + 823, + 267 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/696e48086acd07d35240914316a784acf6191a950542cb24728e28c4b4156742.jpg", + "text": "$$\n\\bar { \\pmb { h } } ^ { ( i ) } = \\bar { \\pmb { h } } _ { L ( i - 1 ) + \\bar { \\tau } , L - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 426, + 265, + 570, + 285 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The truncated samples are independent of each other as shown in the next lemma. ", + "bbox": [ + 173, + 294, + 655, + 308 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma C.5. Consider the truncated states of Definition C.4. If (1.1) is generated by independent vectors {ut}t≥0, for any offset τ¯ and sampling rate L, the vectors {h¯ (i)}N¯i=1 $\\{ \\bar { \\pmb { h } } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } } , \\{ \\pmb { u } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }$ are all independent of each other. ", + "bbox": [ + 173, + 310, + 826, + 352 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. By construction $\\bar { \\pmb { h } } ^ { ( i ) }$ only depends on the vectors $\\left\\{ \\pmb { u } _ { \\tau } \\right\\} _ { \\tau = L ( i - 2 ) + \\bar { \\tau } + 1 } ^ { L ( i - 1 ) + \\bar { \\tau } - 1 }$ . Note that the dependence ranges $[ L ( i - 2 ) + \\bar { \\tau } + 1 , L ( i - 1 ) + \\bar { \\tau } - 1 ]$ are disjoint intervals for different $_ { i }$ ’s; hence $( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }$ are independent of each other. To show the independence of $\\mathbf { \\pmb { u } } ^ { ( i ) }$ and $\\bar { \\pmb { h } } ^ { ( i ) }$ ; observe that inputs $\\pmb { u } ^ { ( i ) } = \\pmb { u } _ { L ( i - 1 ) + \\hat { \\tau } }$ have timestamp $\\bar { \\tau }$ modulo $L$ ; which is not covered by the dependence range of $( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }$ . □ ", + "bbox": [ + 173, + 364, + 825, + 430 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "If the input is randomly generated, Lemma C.2 can be combined with a probabilistic bound on $\\pmb { h } _ { t }$ , to show that truncated states $\\bar { \\pmb { h } } ^ { ( i ) }$ are fairly close to the actual states $\\mathbf { \\delta } _ { h } ( i )$ . ", + "bbox": [ + 171, + 444, + 823, + 472 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Lemma C.6 (Truncation impact – random). Given offset $\\bar { \\tau }$ and sampling rate $L$ , consider the state vectors of the sub-trajectory $\\{ h ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }$ and $L - 1$ -truncations $( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }$ . Suppose $\\{ \\pmb { u } _ { t } \\} _ { t \\geq 0 } \\stackrel { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } ) , \\lVert \\pmb { A } \\rVert < 1$ $h _ { 0 } = 0$ , $\\phi$ is 1-Lipschitz, and $\\phi ( 0 ) = 0$ . Also suppose upper bound (4.3) of Assumption $^ { l }$ holds for some $\\theta \\leq$ ${ \\sqrt { n } } , \\gamma _ { + } > 0$ . There exists an absolute constant $c > 0$ such that with probability at least $1 - 2 \\bar { N } \\mathrm { \\bar { e x p } } ( - 1 0 0 n )$ , for all $1 \\leq i \\leq \\bar { N }$ , the following bound holds ", + "bbox": [ + 173, + 476, + 826, + 544 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8e187455e36ffdd751ac435d8796a1fc724da5d51e0b78956a59b374e08d727f.jpg", + "text": "$$\n\\| h ^ { ( i ) } - \\bar { h } ^ { ( i ) } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { n } \\| A \\| ^ { L - 1 } \\sqrt { \\gamma _ { + } } .\n$$", + "text_format": "latex", + "bbox": [ + 382, + 549, + 614, + 568 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In particular, we can always pick $\\gamma _ { + } = B _ { \\infty } ^ { 2 }$ (via Lemma $B . 3$ ). ", + "bbox": [ + 173, + 574, + 540, + 589 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. Using Assumption 1, we can apply Lemma F.3 on vectors $\\{ h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\} _ { i = 1 } ^ { \\bar { N } }$ . Using a union bound, with desired probability, all vectors obey ", + "bbox": [ + 173, + 603, + 825, + 632 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/1102ab5eb959e5e9a6081c3a6a9df0efabf7791f01273a9dbb06cd61fe65c2c9.jpg", + "text": "$$\n\\begin{array} { r } { \\| h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } - \\mathbb { E } \\big [ h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\big ] \\| _ { \\ell _ { 2 } } \\leq ( c - 1 ) \\sqrt { n \\gamma _ { + } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 637, + 665, + 654 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "for sufficiently large $c$ . Since $\\theta \\leq \\sqrt { n }$ , triangle inequality implies $\\| h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { n \\gamma _ { + } }$ . Now, applying Lemma C.2, for all $1 \\leq i \\leq \\bar { N }$ , we find ", + "bbox": [ + 171, + 660, + 823, + 689 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5bf4df24229dac93eaa49840a340d125c6e63576b608103a3bd7da010fb011b6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\pmb { h } ^ { ( i ) } - \\bar { \\pmb { h } } ^ { ( i ) } \\| _ { \\ell _ { 2 } } = \\| \\pmb { h } _ { ( i - 1 ) L + \\bar { \\tau } } - \\bar { \\pmb { h } } _ { ( i - 1 ) L + \\bar { \\tau } , L - 1 } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\pmb { A } \\| ^ { L - 1 } \\| \\pmb { h } _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq c \\| \\pmb { A } \\| ^ { L - 1 } \\sqrt { n \\gamma + 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 693, + 658, + 753 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "D PROPERTIES OF THE DATA MATRIX", + "text_level": 1, + "bbox": [ + 173, + 789, + 498, + 805 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "This section utilizes the probabilistic estimates from Section B to provide bounds on the condition number of data matrices obtained from the RNN trajectory (1.1). Following (2.2), these matrices $_ { H , U }$ and $\\boldsymbol { x }$ are defined as ", + "bbox": [ + 174, + 819, + 825, + 858 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5d887163c25f83c4d904cc186b06cd4e8d76a784333b168f2b713d44414471d1.jpg", + "text": "$$\n\\mathbf { { \\cal H } } = [ h _ { 1 } \\ \\dots \\ h _ { N } ] ^ { T } \\quad , \\quad \\mathbf { { \\cal U } } = \\mathbf { { \\cal H } } = [ u _ { 1 } \\ \\dots \\ u _ { N } ] ^ { T } \\quad , \\quad \\mathbf { { \\cal X } } = [ x _ { 1 } \\ \\dots \\ x _ { N } ] ^ { T } .\n$$", + "text_format": "latex", + "bbox": [ + 259, + 861, + 740, + 880 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The challenge is that, the state matrix $_ H$ has dependent rows; which will be addressed by carefully splitting the trajectory $\\{ u _ { t } , h _ { t } \\} _ { t = 0 } ^ { N }$ into multiple sub-trajectories which are internally weakly dependent as discussed in Section C. We first define the matrices obtained from these sub-trajectories. ", + "bbox": [ + 174, + 885, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition D.1. Given sampling rate $L$ and offset $\\bar { \\tau }$ , consider the $L$ -subsampled trajectory $\\{ \\boldsymbol { h } ^ { ( i ) } , \\boldsymbol { u } ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ i) }Ni=1 as described in Definitions C.3 and C.4. Define the matrices $\\bar { \\pmb { H } } = \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { \\pmb { N } } \\times { n } }$ , $\\tilde { \\pmb { H } } = \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { \\cal N } \\times n }$ , $\\tilde { \\pmb { U } } =$ $\\tilde { \\pmb { U } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { \\pmb { N } } \\times { p } }$ , and $\\tilde { { \\cal X } } = \\tilde { { \\cal X } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { N } \\times ( n + p ) } a s$ ", + "bbox": [ + 173, + 103, + 826, + 147 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/79c3804269bf0100a77d8e21ea068947f21f4a2c6eefa3b110dc58dc617804aa.jpg", + "text": "$$\n\\bar { \\pmb { H } } = [ \\bar { \\pmb { h } } ^ { ( 1 ) } \\dots \\bar { \\pmb { h } } ^ { ( \\bar { N } ) } ] ^ { T } , \\ \\tilde { \\pmb { H } } = [ \\pmb { h } ^ { ( 1 ) } \\dots \\pmb { h } ^ { ( \\bar { N } ) } ] ^ { T } , \\ \\tilde { \\pmb { U } } = [ \\pmb { u } ^ { ( 1 ) } \\dots \\pmb { u } ^ { ( \\bar { N } ) } ] ^ { T } , \\ \\tilde { \\pmb { X } } = [ \\mu \\tilde { \\pmb { H } } \\tilde { \\pmb { U } } ] .\n$$", + "text_format": "latex", + "bbox": [ + 225, + 154, + 769, + 172 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma D.2 (Handling perturbation). Consider the nonlinear state equation (1.1). Given sampling rate $L > 0$ and offset $\\bar { \\tau }$ , consider the matrices $\\bar { H } , \\tilde { H } , \\tilde { X }$ of Definition $D . I$ and let ${ \\pmb Q } = [ \\gamma _ { + } ^ { - 1 / 2 } \\bar { \\pmb H } \\tilde { \\pmb U } ] \\in \\mathbb { R } ^ { \\bar { N } \\times ( n + p ) }$ e Assumptiosuch that if √ $^ { l }$ $\\phi$ $\\beta$ creasing, and , with probabil ${ \\mathbf { \\mathscr { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\cal I } _ { p } )$ e exists an absol, for all matrices constantobeying $C > 0$ $\\begin{array} { r } { \\bar { N } \\ge C \\frac { \\gamma _ { + } ^ { 2 } } { \\gamma _ { - } ^ { 2 } } ( n + p ) } \\end{array}$ $1 - 8 \\exp ( - c \\frac { \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } ^ { 2 } } \\bar { N } )$ $M$ $\\begin{array} { r } { \\| M - \\bar { H } \\| \\le \\frac { \\sqrt { \\gamma _ { - } \\bar { N } } } { 1 0 } } \\end{array}$ , the perturbed $\\ b { Q }$ matrices given by, ", + "bbox": [ + 173, + 178, + 826, + 272 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/3fdf1e5e331b950134e5e9a15c945d2c42688e7ebafaea0254cbd9a35b61e1f8.jpg", + "text": "$$\n\\tilde { Q } = [ \\gamma _ { + } ^ { - 1 / 2 } M \\tilde { U } ] ,\n$$", + "text_format": "latex", + "bbox": [ + 437, + 277, + 560, + 296 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "satisfy ", + "bbox": [ + 173, + 303, + 215, + 316 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/31e590a2fd52e062ccef23b7723ae20f4145016590654297f00df4592938d594.jpg", + "text": "$$\n( \\Theta + \\sqrt { 2 } ) ^ { 2 } \\succeq \\frac { \\tilde { Q } ^ { T } \\tilde { Q } } { \\bar { N } } \\succeq \\frac { \\gamma _ { - } } { 2 \\gamma _ { + } } .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 321, + 591, + 356 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. This result is a direct application of Theorem F.1 after determining minimum/maximum eigenvalues of population covariance. The cross covariance obeys $\\mathbb { E } [ \\bar { \\pmb { H } } ^ { T } \\tilde { \\pmb { U } } ] = 0$ due to independence. Also, for $i > 1$ , the truncated state vector $\\bar { \\pmb { h } } ^ { ( i ) }$ is statistically identical to $\\pmb { h } _ { L - 1 }$ hence $\\pmb { \\Sigma } [ \\bar { \\pmb { h } } ^ { ( i ) } ] \\succeq \\gamma _ { - } \\bar { \\pmb { I _ { n } } }$ . Consequently, $\\Sigma [ { \\pmb u } ^ { ( i ) } ] = I _ { p }$ , $\\begin{array} { r } { \\frac { 1 } { \\gamma _ { + } } \\pmb { \\Sigma } [ \\bar { \\pmb { h } } ^ { ( i ) } ] \\preceq { \\pmb { I } } _ { n } } \\end{array}$ for all $_ { i }$ and $\\begin{array} { r } { \\frac { \\gamma _ { - } } { \\gamma _ { + } } I _ { n } \\preceq \\frac { 1 } { \\gamma _ { + } } \\Sigma [ \\bar { \\pmb { h } } ^ { ( i ) } ] } \\end{array}$ for all $i > 1$ . Hence, setting $\\pmb { q } _ { i } = \\left[ \\frac { 1 } { \\sqrt { \\gamma _ { + } } } \\bar { \\pmb { h } } ^ { ( i ) } \\right] ,$ , for all $i > 1$ ", + "bbox": [ + 173, + 367, + 826, + 448 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7eaf3abc4e7b1cb90fa84636ed15ad747d18ad248e3888a3639a3b9aec9a7732.jpg", + "text": "$$\n\\frac { \\gamma - } { \\gamma _ { + } } I _ { n } \\preceq \\Sigma [ { \\pmb q } _ { i } ] \\preceq I _ { n } .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 453, + 566, + 481 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Set the matrix $\\bar { \\pmb { Q } } = [ \\pmb { q } _ { 2 } \\dotsm \\pmb { q } _ { \\bar { N } } ] ^ { T }$ and note that $Q = [ \\pmb { q } _ { 1 } \\bar { Q } ^ { T } ] ^ { T }$ . Applying Theorem F.1 on $\\bar { Q }$ and Corollary F.2 on $\\ b { Q }$ , we find that, with the desired probability, ", + "bbox": [ + 173, + 486, + 825, + 515 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/afe580357b7156b889c09a393065f108ef780236beb64055ccd1dce53b577697.jpg", + "text": "$$\n\\theta + \\sqrt { 3 / 2 } \\geq \\frac { 1 } { \\sqrt { N } } \\| Q \\| \\geq \\frac { 1 } { \\sqrt { N } } s _ { \\operatorname* { m i n } } ( Q ) \\geq \\frac { 1 } { \\sqrt { N } } s _ { \\operatorname* { m i n } } ( \\bar { Q } ) \\geq \\sqrt { \\frac { N - 1 } { N } } \\sqrt { \\frac { 2 \\gamma _ { - } } { 3 \\gamma _ { + } } } \\geq 0 . 9 9 \\times \\sqrt { \\frac { 2 \\gamma _ { - } } { 3 \\gamma _ { + } } } .\n$$", + "text_format": "latex", + "bbox": [ + 202, + 521, + 792, + 555 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Setting $E = M - { \\bar { H } }$ and observing $\\tilde { Q } = Q + [ \\gamma _ { + } ^ { - 1 / 2 } E \\mathrm { ~ 0 } ]$ , the impact of the perturbation $\\pmb { \\cal E }$ can be bounded naively via $s _ { \\operatorname* { m i n } } ( \\pmb { Q } ) - \\gamma _ { + } ^ { - 1 / 2 } \\lVert \\pmb { E } \\rVert \\leq s _ { \\operatorname* { m i n } } ( \\tilde { \\pmb { Q } } ) \\leq \\lVert \\tilde { \\pmb { Q } } \\rVert \\leq \\lVert \\pmb { Q } \\rVert + \\gamma _ { + } ^ { - 1 / 2 } \\lVert \\pmb { E } \\rVert$ . Using the assumed bound on $\\| \\pmb { E } \\|$ , this yields ", + "bbox": [ + 173, + 561, + 826, + 607 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/049cd6f4010c0350d142863ffbc3675008577aa10ed78380a97400da0e691af5.jpg", + "text": "$$\n\\theta + \\sqrt { 2 } \\geq \\frac { 1 } { \\sqrt { \\bar { N } } } \\| \\tilde { \\pmb { Q } } \\| \\geq \\frac { 1 } { \\sqrt { \\bar { N } } } s _ { \\operatorname* { m i n } } ( \\tilde { \\pmb { Q } } ) \\geq \\sqrt { \\frac { \\gamma - \\underline { { \\mathbf { \\Lambda } } } } { 2 \\gamma + } } .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 606, + 648, + 637 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "This final inequality is identical to the desired bound (D.3). ", + "bbox": [ + 173, + 638, + 524, + 654 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem D.3 (Data matrixdefine the condition number n). Consider the nonlinear s. For some absolute constants tion (1.1). Given , pick a trajectory $\\gamma _ { + } \\geq \\gamma _ { - } > 0$ $\\begin{array} { r } { \\rho = \\frac { \\gamma _ { + } } { \\gamma _ { - } } } \\end{array}$ $c , C > 0$ $N$ ", + "bbox": [ + 173, + 661, + 826, + 690 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/dbb4d709294ae8ab6863d574912596a40eed551b99fec5bc7606fda76508cbd2.jpg", + "text": "$$\nL = \\lceil 1 - \\frac { \\log { ( c n \\rho ) } } { \\log { \\| A \\| } } \\rceil \\quad , \\quad N _ { 0 } = \\lfloor \\frac { N } { L } \\rfloor \\ge C \\rho ^ { 2 } ( n + p ) ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 698, + 666, + 728 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "and pick scaling $\\begin{array} { r } { \\mu = \\frac { 1 } { \\sqrt { \\gamma _ { + } } } } \\end{array}$ Suppose $\\| A \\| ~ < ~ 1$ , $\\phi$ is $\\beta$ -increasing, ${ \\textbf { \\em u } } _ { t } \\stackrel { i . i . d . } { \\sim } { \\mathcal { N } } ( 0 , I _ { p } )$ , and Assumption $^ { l }$ holds with $\\gamma _ { + } , \\gamma _ { - } , \\theta , L$ . Matrix $\\mathbf { X } \\ = \\ [ \\pmb { x } _ { 1 } \\dots \\pmb { x } _ { N } ] ^ { T }$ of (D.1) satisfies the following with probability $1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( N _ { 0 } / \\rho ^ { 2 } ) )$ . ", + "bbox": [ + 173, + 736, + 825, + 785 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• Each row of $\\boldsymbol { x }$ has $\\ell _ { 2 }$ norm at most $c _ { 0 } { \\sqrt { p + n } }$ where $c _ { 0 }$ is an absolute constant. ", + "bbox": [ + 215, + 795, + 709, + 810 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• $X ^ { T } X$ obeys the bound ", + "bbox": [ + 215, + 818, + 375, + 833 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/adb62a33f910556490a4a46f9a66ac4a1107349fa57e25d7c4e297a878fe775a.jpg", + "text": "$$\n( \\Theta + \\sqrt { 2 } ) ^ { 2 } I _ { n + p } \\succeq \\frac { { \\bf { X } } ^ { T } { \\bf { X } } } { N } \\succeq \\rho ^ { - 1 } I _ { n + p } / 2 .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 839, + 660, + 869 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. The first statement on $\\ell _ { 2 }$ -norm bound can be concluded from Lemma D.4 and holds with probability $1 - 2 N \\exp ( - 1 0 0 ( n + p ) )$ . To show the second statement, for a fixed offset $1 \\le \\bar { \\tau } \\le L$ , consider Definition D.1 and the matrices $\\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } , \\tilde { \\pmb { U } } ^ { ( \\bar { \\tau } ) } , \\tilde { \\pmb { X } } ^ { ( \\bar { \\tau } ) }$ . Observe that $\\boldsymbol { x }$ is obtained by merging multiple sub-trajectory matrices $\\{ \\tilde { X } ^ { ( \\bar { \\tau } ) } \\} _ { \\bar { \\tau } = 1 } ^ { L }$ . We will first show the advertised bound for an individual $\\tilde { X } ^ { ( \\bar { \\tau } ) }$ by applying Lemma D.2 and then apply Lemma A.1 to obtain the bound on the combined matrix $\\boldsymbol { x }$ . ", + "bbox": [ + 173, + 882, + 825, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 102, + 825, + 131 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Recall that $\\bar { N } _ { \\bar { \\tau } }$ is the length of the $\\bar { \\tau }$ th sub-trajectory i.e. number of rows of $\\tilde { X } ^ { ( \\bar { \\tau } ) }$ . By construction $2 N _ { 0 } \\geq$ $\\bar { N } _ { \\bar { \\tau } } \\geq N _ { 0 }$ for all $1 \\le \\bar { \\tau } \\le L$ . Given $1 \\le \\bar { \\tau } \\le L$ and triple $\\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } , \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } , \\tilde { \\pmb { U } } ^ { ( \\bar { \\tau } ) }$ , set $Q = [ \\mu \\bar { H } ^ { ( \\bar { \\tau } ) } \\tilde { U } ^ { ( \\bar { \\tau } ) } ]$ . Since $N _ { 0 }$ is chosen to be large enough, applying Theorem D.2 with $\\mu = 1 / \\sqrt { \\gamma _ { + } }$ choice, and noting $\\rho = \\gamma _ { + } / \\gamma _ { - }$ , we find that, with probability $1 - 4 \\exp ( - c _ { 1 } N _ { 0 } / \\rho ^ { 2 } )$ , all matrices $M$ satisfying $\\lVert M - \\bar { H } ^ { ( \\bar { \\tau } ) } \\rVert \\leq \\sqrt { \\gamma _ { - } N _ { 0 } } / 1 0$ and $\\tilde { Q }$ as in (D.2) obeys ", + "bbox": [ + 173, + 137, + 825, + 210 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/64e7640c75ef8b7bfc2e9f8aa0c337486a47b771142f3c39503e0406cc7b5624.jpg", + "text": "$$\n( \\Theta + \\sqrt { 2 } ) ^ { 2 } \\succeq \\frac { { \\tilde { Q } } ^ { T } { \\tilde { Q } } } { N } \\succeq \\rho ^ { - 1 } / 2 .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 215, + 598, + 246 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let us call this Event 1. To proceed, we will argue that with high probability $\\| \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } - \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\|$ is small so that the bound above is applicable with $M = \\tilde { \\cal H } ^ { ( \\bar { \\tau } ) }$ choice; which sets $\\tilde { Q } = \\tilde { X } ^ { ( \\bar { \\tau } ) }$ in (D.5). Applying Lemma C.6, we find that, with probability $1 - 2 \\bar { N } _ { \\bar { \\tau } } \\exp ( - 1 0 0 n )$ , ", + "bbox": [ + 174, + 252, + 826, + 295 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/026f93854a61c2e7b5bc6bbc66d6fe9ed499bf3e12ea2a389e17b7994cc645f0.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } - \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\| \\leq \\sqrt { 2 N _ { 0 } } \\operatorname* { m a x } \\{ \\| \\pmb { h } ^ { ( i ) } - \\bar { \\pmb { h } } ^ { ( i ) } \\| _ { \\ell _ { 2 } } \\} \\leq c _ { 0 } \\sqrt { 2 N _ { 0 } } \\sqrt { n \\gamma _ { + } } \\| \\pmb { A } \\| ^ { L - 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 301, + 733, + 320 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let us call this Event 2. We will show that our choice of $L$ ensures right hand side is small enough and guarantees $\\lVert \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } - \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\rVert \\leq \\sqrt { \\gamma _ { - } N _ { 0 } } / 1 0$ . Set $c = \\operatorname* { m a x } \\{ 2 0 0 c _ { 0 } ^ { 2 } , 1 \\}$ . Desired claim follows by taking logarithms of upper/lower bounds and cancelling out $\\sqrt { N _ { 0 } }$ terms as follows ", + "bbox": [ + 176, + 325, + 823, + 367 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/2d649d4022ac6d43344413ab73349f4d80f20ea5385ff0cf9acb1543c2d92e86.jpg", + "text": "$$\n\\begin{array} { r l } & { c _ { 0 } \\sqrt { n } \\| A \\| ^ { L - 1 } \\sqrt { \\gamma _ { + } } \\leq \\sqrt { \\gamma _ { - } } / 1 0 \\sqrt { 2 } \\iff ( L - 1 ) \\log \\| A \\| + \\log \\sqrt { c n \\rho } \\leq 0 } \\\\ & { \\iff - \\frac { \\log c n \\rho } { 2 \\log \\| A \\| } \\leq L - 1 } \\\\ & { \\iff L = \\lceil 1 - \\frac { \\log { ( c n \\rho ) } } { \\log \\| A \\| } \\rceil . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 372, + 732, + 458 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Here we use the fact that $\\log \\| A \\| < 0$ since $\\| A \\| < 1$ and $c n \\rho \\geq 0$ . Consequently, both Event 1 and Event 2 hold with probability $1 - 4 \\exp ( - c _ { 1 } N _ { 0 } / \\rho ^ { 2 } ) - 2 \\bar { N } _ { \\bar { \\tau } } \\exp ( - 1 0 0 n ) .$ , implying (D.5) holds with $\\tilde { Q } = \\tilde { X } ^ { ( \\bar { \\tau } ) }$ . Union bounding this over $1 \\le \\bar { \\tau } \\le L$ , (D.5) uniformly holds with $\\tilde { Q } = \\tilde { X } ^ { ( \\bar { \\tau } ) }$ and all rows of $\\boldsymbol { x }$ are $\\ell _ { 2 }$ -bounded with probability $1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - c _ { 1 } N _ { 0 } / \\rho ^ { 2 } )$ . Applying Lemma A.1 on $( \\tilde { X } ^ { ( \\bar { \\tau } ) } ) _ { \\bar { \\tau } = 1 } ^ { L }$ , we conclude with the bound (D.4) on the merged matrix $\\pmb { X }$ . $\\boxed { \\begin{array} { r l } \\end{array} }$ ", + "bbox": [ + 173, + 460, + 825, + 531 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma D.4 $\\ell _ { 2 }$ -bound on rows). Consider the setup of Theorem D.3. With probability √ $1 - 2 N \\exp ( - 1 0 0 ( n +$ $p )$ ), each row of $\\boldsymbol { x }$ has $\\ell _ { 2 }$ -norm at most $c { \\sqrt { p + n } }$ for some constant $c > 0$ . ", + "bbox": [ + 171, + 537, + 826, + 565 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. The tth row of $\\pmb { X }$ is equal to $\\begin{array} { r } { \\pmb { x } _ { t } = [ \\frac { \\pmb { h } _ { t } ^ { T } } { \\sqrt { \\gamma _ { + } } } \\pmb { u } _ { t } ^ { T } ] ^ { T } } \\end{array}$ . Since $\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( \\sqrt { \\gamma _ { + } } )$ and $\\| u _ { t } \\| _ { \\psi _ { 2 } } ~ \\le$ $\\mathcal { O } ( 1 )$ , we have that $\\| \\pmb { x } _ { t } - \\mathbb { E } [ \\pmb { x } _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( 1 )$ . Now, applying Lemma F.3 on all rows $\\{ \\pmb { x } _ { t } \\} _ { t = 1 } ^ { N }$ , and using a√ union bound, with probability at least $1 - 2 \\dot { N } \\exp ( - 1 0 \\bar { 0 } ( n + p ) )$ , we have that √ $\\lVert \\mathbf { x } _ { t } - \\mathbb { E } [ \\mathbf { x } _ { t } ] \\rVert _ { \\ell _ { 2 } } \\leq c \\sqrt { n + p }$ for all $t$ . To conclude, note that $\\| \\mathbb { E } [ \\pmb { x } _ { t } ] \\| _ { \\ell _ { 2 } } = \\| \\mathbb { E } [ \\pmb { h } _ { t } ] \\| _ { \\ell _ { 2 } } / \\sqrt { \\gamma _ { + } } \\le \\theta \\le 3 \\sqrt { n }$ via Assumption 1. □ ", + "bbox": [ + 173, + 580, + 826, + 643 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E PROOFS OF MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 661, + 436, + 678 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.1 PROOF OF LEMMA 3.2 ", + "text_level": 1, + "bbox": [ + 174, + 691, + 372, + 705 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The statement follows from upper bound Lemma B.3 and lower bound Lemma B.5. ", + "bbox": [ + 174, + 717, + 665, + 731 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.2 PROOF OF THEOREM 4.2 ", + "text_level": 1, + "bbox": [ + 174, + 746, + 388, + 761 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. To prove this theorem, we combine Theorem D.3 with deterministic SGD convergence result of Theorem 4.1. Applying Theorem D.3, with the desired probability, inequality (D.4) holds and for all $i$ , input data satisfies the bound $\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } \\leq \\sqrt { ( n + p ) / ( 2 c _ { 0 } ) }$ for a sufficiently small constant $c _ { 0 } > 0$ . As the next step, we will argue that these two events imply the convergence of SGD. ", + "bbox": [ + 173, + 771, + 825, + 825 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Let $\\pmb { \\theta } ^ { ( i ) } , \\pmb { c } ^ { ( i ) } \\in \\mathbb { R } ^ { n + p }$ denote the ith rows of $\\Theta , C$ respectively. Observe that the square-loss is separable along the rows of $_ { C }$ via $\\begin{array} { r } { \\| \\Theta - C \\| _ { F } ^ { 2 } = \\sum _ { i = 1 } ^ { n } \\| \\pmb { \\theta } ^ { ( i ) } - \\pmb { c } ^ { ( i ) } \\| _ { \\ell _ { 2 } } ^ { 2 } } \\end{array}$ . Hence, SGD updates each row $\\mathbf { c } ^ { ( i ) }$ via its own state equation ", + "bbox": [ + 173, + 832, + 825, + 872 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/e471507f348577c59117ce026533ce95c06cccc9e0f3efcf37057d3c0ec295cf.jpg", + "text": "$$\n\\pmb { y } _ { t , i } = \\phi ( \\left. \\pmb { c } ^ { ( i ) } , \\pmb { x } _ { t } \\right. ) ,\n$$", + "text_format": "latex", + "bbox": [ + 431, + 871, + 563, + 896 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $_ { { \\mathbf { \\mathit { y } } } _ { t , i } }$ is the ith entry of ${ \\mathbf { } } _ { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { \\mathbf { } } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } _ { } { \\mathbf { } } _ { } { } \\mathbf { } _ { } { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } { } \\mathbf { } _ { } \\mathbf { } _ { } { } \\mathbf { } _ { } \\mathbf { } _ { } { } \\mathbf { } _ { } \\mathbf { } _ { } { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } \\mathbf { } _ { } \\mathbf { } _ { } \\mathbf { } \\mathbf _ { } \\mathbf { } _ { } \\mathbf { } \\mathbf _ { } \\mathbf { } _ { } \\mathbf { } _ \\mathbf { } \\mathbf { } _ { } \\mathbf \\mathbf { } _ { } \\mathbf _ { } \\mathbf { } \\mathbf _ { } \\mathbf { } \\mathbf _ { } \\mathbf _ { } \\mathbf { } \\mathbf _ { } \\mathbf \\mathbf { } _ \\mathbf { } \\mathbf _ { } \\mathbf \\mathbf { } \\mathbf _ { } \\mathbf \\mathbf { } \\mathbf _ { } \\mathbf \\mathbf \\mathbf { } \\mathbf _ { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf _ { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf$ . Consequently, we can establish the convergence result for an individual row of $_ { C }$ . Convergence of all individual rows will imply the convergence of the overall matrix $\\Theta _ { \\tau }$ to the ground ", + "bbox": [ + 169, + 897, + 828, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "truth $_ { C }$ . Pick a row index $_ { i }$ $( 1 \\leq i \\leq n )$ , set $\\mathbf { c } = \\mathbf { c } ^ { ( i ) }$ and denote ith row of $\\Theta _ { \\tau }$ by $\\pmb { \\theta } _ { \\tau }$ . Also denote the label corresponding to ith row by $y _ { t } = y _ { t , i }$ . With this notation, SGD over (2.3) runs SGD over the ith row with equations $y _ { t } \\overset { \\cdot } { = } \\phi ( \\langle c , \\pmb { x } _ { t } \\rangle )$ and with loss functions ", + "bbox": [ + 173, + 103, + 825, + 143 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/81edf3d5401aa679cfd7b1a43b894284bc107a49259a5f91279b7a26002bbec8.jpg", + "text": "$$\n\\mathcal { L } ( \\pmb { \\theta } ) = N ^ { - 1 } \\sum _ { t = 1 } ^ { N } \\mathcal { L } _ { t } ( \\pmb { \\theta } ) , \\mathcal { L } _ { t } ( \\pmb { \\theta } ) = \\frac { 1 } { 2 } \\big ( y _ { t } - \\phi ( \\langle \\pmb { \\theta } , \\pmb { x } _ { t } \\rangle ) \\big ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 145, + 671, + 184 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Substituting our high-probability bounds on $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ (e.g. (D.4)) into Theorem 4.1, we can set $B = ( n + p ) / ( 2 c _ { 0 } )$ , $\\gamma _ { + } = ( \\theta + \\sqrt { 2 } ) ^ { 2 }$ , and $\\gamma _ { - } = \\rho ^ { - 1 } / 2$ . Consequently, using the learning rate $\\begin{array} { r } { \\eta = c _ { 0 } \\frac { \\beta ^ { 2 } \\rho ^ { - 1 } } { ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } } \\end{array}$ , for all $\\tau \\geq 0$ , the $\\tau$ th SGD iteration $\\pmb { \\theta } _ { \\tau }$ obeys ", + "bbox": [ + 173, + 186, + 826, + 233 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a2a34de338e81f6f5bfb551642d33031ebb1a1cbff91529d11c41478fa0cfbbd.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\pmb { \\theta } _ { \\tau } - \\pmb { c } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { \\theta } _ { 0 } - \\pmb { c } \\| _ { \\ell _ { 2 } } ^ { 2 } ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } \\rho ^ { - 2 } } { 2 ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } ,\n$$", + "text_format": "latex", + "bbox": [ + 315, + 234, + 679, + 267 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where the expectation is over the random selection of SGD updates. This establishes the convergence for a particular row of $_ { C }$ . Summing up these inequalities (E.1) over all rows $\\pmb { \\theta } _ { \\tau } ^ { ( 1 ) } , \\dots , \\pmb { \\theta } _ { \\tau } ^ { ( n ) }$ (which converge to $\\bar { \\mathbf { c } } ^ { ( 1 ) } , \\ldots , \\mathbf { c } ^ { ( n ) }$ respectively) yields the targeted bound (4.4). □ ", + "bbox": [ + 174, + 268, + 826, + 313 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.3 PROOFS OF MAIN RESULTS ON STABLE SYSTEMS ", + "text_level": 1, + "bbox": [ + 173, + 328, + 552, + 342 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.3.1 PROOF OF THEOREM 3.3 ", + "text_level": 1, + "bbox": [ + 174, + 352, + 401, + 367 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. Applying Lemmas B.3 and 3.2, independent of $L$ , Assumption 1 holds with parameters ", + "bbox": [ + 173, + 376, + 733, + 390 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/bf81b85bc5403bd484b6f8d4f718375aec9603d21852fc0ff9a982dc793e4df6.jpg", + "text": "$$\n\\gamma _ { + } = B _ { \\infty } ^ { 2 } \\quad , \\quad \\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } \\quad , \\quad \\theta = \\sqrt { 6 n } - \\sqrt { 2 } \\geq \\sqrt { n } .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 392, + 692, + 410 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "This yields (θ + 2)2 = 6n. Hence, we can apply Theorem 4.2 with the learning rate η = c0 β26ρn(n+p) where ", + "bbox": [ + 173, + 415, + 826, + 433 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/df580270ea63b2cf7c717a642d6652bf23e3cc5c5b6752bd2654c8fc9574fdd6.jpg", + "text": "$$\n\\rho = \\frac { B _ { \\infty } ^ { 2 } } { \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } } = \\frac { \\gamma _ { + } } { \\gamma _ { - } } ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 434, + 576, + 465 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and convergence rate $\\begin{array} { r } { 1 - \\frac { \\beta ^ { 2 } \\eta } { 2 \\rho } } \\end{array}$ . To conclude with the stated result, we use the change of variable $c _ { 0 } / 6 \\to c _ { 0 }$ . ", + "bbox": [ + 169, + 470, + 812, + 487 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.3.2 PROOF OF THEOREM 3.4 ", + "text_level": 1, + "bbox": [ + 174, + 500, + 401, + 515 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. The proof is similar to that of Theorem 3.3. Applying Lemmas B.3, B.4, and 3.2, independent of $L$ Assumption 1 holds with parameters ", + "bbox": [ + 174, + 523, + 825, + 550 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/73f78ed1f120e6b20278d967d2729c7d15ff90e1628a632a598d54e5c2af9a47.jpg", + "text": "$$\n\\gamma _ { + } = B _ { \\infty } ^ { 2 } \\quad , \\quad \\gamma _ { - } = s _ { \\mathrm { m i n } } ( B ) ^ { 2 } \\quad , \\quad \\theta = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 551, + 637, + 569 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Hence, we again apply Theorem 4.2 with the learning rate η = c0 β22ρ(n+p) where $\\rho$ is given by (E.2). Use the change of variable $c _ { 0 } / 2 \\to c _ { 0 }$ to conclude with the stated result. □ ", + "bbox": [ + 173, + 574, + 825, + 604 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.4 LEARNING UNSTABLE SYSTEMS ", + "text_level": 1, + "bbox": [ + 176, + 619, + 437, + 633 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In a similar fashion to Section 4, we provide a more general result on unstable systems that makes a parametric assumption on the statistical properties of the state vector. ", + "bbox": [ + 171, + 643, + 823, + 671 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Assumption 2 (Well-behaved state vector – single timestamp). Given timestamp √ $T _ { 0 } > 0$ , there exists positive scalars $\\gamma _ { + } , \\gamma _ { - } , \\theta$ and an absolute constant $C > 0$ such that ${ \\dot { \\theta } } \\leq 3 { \\sqrt { n } }$ and the following holds ", + "bbox": [ + 174, + 672, + 823, + 699 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/9438b91dda38899e0a2d67a2c2af2ddd25b8dff5fc8f27ecc69daf059185b4cc.jpg", + "text": "$$\n\\begin{array} { r } { \\gamma _ { + } I _ { n } \\succeq \\Sigma [ h _ { T _ { 0 } } ] \\succeq \\gamma _ { - } I _ { n } \\quad , \\quad \\| h _ { T _ { 0 } } - \\mathbb { E } [ h _ { T _ { 0 } } ] \\| _ { \\psi _ { 2 } } \\leq C \\sqrt { \\gamma _ { + } } \\quad a n d \\quad \\| \\mathbb { E } [ h _ { t } ] \\| _ { \\ell _ { 2 } } \\leq \\theta \\sqrt { \\gamma _ { + } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 702, + 754, + 717 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The next theorem provides the parametrized result on unstable systems based on this assumption. ", + "bbox": [ + 171, + 726, + 750, + 739 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Theorem E.1 (Unstable system - general). Suppose we are given $N$ independent trajectories $( { h _ { t } ^ { ( i ) } } , { u _ { t } ^ { ( i ) } } ) _ { t \\geq 0 }$ for $1 \\leq i \\leq N$ . Sample each trajectory at time $T _ { 0 }$ to obtain $N$ samples $( { \\pmb y } _ { i } , { \\pmb h } _ { i } , { \\pmb u } _ { i } ) _ { i = 1 } ^ { N }$ where ith sample is ", + "bbox": [ + 171, + 744, + 823, + 773 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a9a3926ae266e800ef88f4df5fedb737db10ee409ea1235fc9b294e04220741a.jpg", + "text": "$$\n( { \\pmb y } _ { i } , { \\pmb h } _ { i } , { \\pmb u } _ { i } ) = ( { \\pmb h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \\pmb h } _ { T _ { 0 } } ^ { ( i ) } , { \\pmb u } _ { T _ { 0 } } ^ { ( i ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 775, + 601, + 795 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Let $C , c _ { 0 } > 0$ be absolute constants. Suppose Assumption $^ { l }$ holds with $T _ { 0 }$ and sample size satisfies $N \\geq$ $C \\rho ^ { 2 } ( n + p )$ where $\\rho = \\gamma _ { + } / \\gamma _ { - }$ . Assume $\\phi$ is $\\beta$ -increasing, zero initial state conditions, and ${ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )$ Set scaling to be µ = 1/ γ+ and learning rate to be η = c0 β ρ(θ+√2)2(n+p) . Starting from $\\Theta _ { 0 }$ , we run $S G D$ over the equations described in (2.2) and (2.3). With probability $1 - 2 \\dot { N } \\exp ( - 1 0 0 ( n + p ) ) - 4 \\exp ( - \\mathcal { O } ( \\textstyle { \\frac { N } { \\rho ^ { 2 } } } ) )$ , all iterates satisfy ", + "bbox": [ + 171, + 795, + 826, + 875 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/40a187810ccb779821ff164c2bc79e99bdffd289d5c17f904282562bb74864e7.jpg", + "text": "$$\n\\mathbb { E } [ \\| \\Theta _ { i } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 877, + 692, + 909 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where the expectation is over the randomness of the SGD updates. ", + "bbox": [ + 173, + 910, + 563, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. Set $\\pmb { x } _ { i } = [ \\gamma _ { + } ^ { - 1 / 2 } \\pmb { h } _ { i } ^ { T } \\ \\pmb { u } _ { i } ^ { T } ] ^ { T }$ and $\\pmb { X } = [ \\pmb { x } _ { 1 } ~ . ~ . ~ \\pmb { x } _ { N } ] ^ { T }$ . Since $\\boldsymbol { x }$ has i.i.d. rows, we can apply Theorem F.1 and Lemma F.3 to find with the desired probability that ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Rows of $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ satisfy $\\| \\pmb { x } _ { i } - \\mathbb { E } [ \\pmb { x } _ { i } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( 1 )$ and $\\mathbb { E } [ \\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } ] \\le 3 \\sqrt { n }$ , hence all rows of $\\boldsymbol { x }$ obeys $\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } \\leq \\sqrt { ( n + p ) / ( 2 c _ { 0 } ) }$ , ", + "bbox": [ + 217, + 147, + 823, + 179 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• $\\boldsymbol { x }$ satisfies ", + "bbox": [ + 217, + 186, + 302, + 200 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/3d224d945f8ed522394feb5bf26a9b05f69d8d27c01ea6e360fb5f732a91d988.jpg", + "text": "$$\n( \\theta + \\sqrt { 2 } ) ^ { 2 } \\succeq \\frac { X ^ { T } X } { N } \\succeq \\rho ^ { - 1 } / 2 .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 198, + 625, + 227 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "To proceed, using $\\gamma _ { - } = \\rho ^ { - 1 } / 2$ , $B = ( n + p ) / ( 2 c _ { 0 } )$ , and $\\gamma _ { + } = ( \\theta + \\sqrt { 2 } ) ^ { 2 }$ , we apply Theorem 4.1 on the loss function (2.3); which yields the desired result. ", + "bbox": [ + 173, + 244, + 825, + 272 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.5 PROOF OF THEOREM 5.1 ", + "text_level": 1, + "bbox": [ + 174, + 289, + 387, + 304 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. The proof is a corollary of Theorem E.1. We need to substitute the proper values in Assumption 2.√ √ Applying Lemma B.3, we can substitute $\\gamma _ { + } = B _ { T _ { 0 } } ^ { 2 }$ and $\\theta = \\sqrt { 6 n } - \\sqrt { 2 } \\geq \\sqrt { n }$ . Next, we need to find a lower bound. Applying Lemma 3.2 for $n > 1$ and Lemma B.6 for $n = 1$ , we can substitute $\\gamma _ { - } = \\gamma _ { + } / \\rho$ with the $\\rho$ definition of (5.2). With these, the result follows as an immediate corollary of Theorem E.1. □ ", + "bbox": [ + 173, + 314, + 826, + 368 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F SUPPLEMENTARY STATISTICAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 388, + 549, + 405 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The following theorem bounds the empirical covariance of matrices with independent subgaussian rows. Given a random vector $_ { \\textbf { \\em x } }$ , define the de-biasing operation as $\\mathbf { z m } ( \\pmb { x } ) = \\pmb { x } - \\mathbb { E } [ \\pmb { x } ]$ . ", + "bbox": [ + 171, + 419, + 823, + 446 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Theorem F.1. Let $\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { n \\times d }$ be a matrix with independent subgaussian rows $\\{ { \\pmb a } _ { i } \\} _ { i = 1 } ^ { n }$ satisfying $\\| \\mathbf { z } \\mathbf { m } ( \\mathbf { { a } } _ { i } ) \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )$ and $\\Sigma [ { \\pmb a } _ { i } ] \\preceq K ^ { 2 } { \\pmb I } _ { d }$ for some $K > 0$ and $\\| \\mathbb { E } [ \\pmb { a } _ { i } ] \\| _ { \\ell _ { 2 } } \\le \\theta$ . Suppose $\\pmb { \\Sigma } [ \\pmb { a } _ { i } ] \\succeq \\lambda \\pmb { I } _ { d }$ . Suppose $n \\ge \\bar { \\mathcal { O } } ( K ^ { 4 } d / \\lambda ^ { 2 } )$ . Then, each of the following happens with probability at least $1 - 2 \\exp ( - c K ^ { - 4 } \\lambda ^ { 2 } n )$ , ", + "bbox": [ + 171, + 450, + 825, + 492 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/1d6c6892247eec94579b048e95bc254ffa5b0901d3e00f09933ceef579b9d2da.jpg", + "text": "$$\n\\begin{array} { r } { \\theta + \\sqrt { 3 / 2 } K \\geq \\frac { 1 } { \\sqrt { n } } \\| A \\| . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 502, + 385, + 522 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Suppose all rows of $\\pmb { A }$ have equal expectations. Then $\\textstyle { \\frac { 1 } { \\sqrt { n } } } s _ { \\operatorname* { m i n } } ( A ) \\geq { \\sqrt { 2 \\lambda / 3 } }$ ", + "bbox": [ + 217, + 531, + 694, + 550 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. Let $\\pmb { E } = \\mathbb { E } [ \\pmb { A } ] , \\ \\bar { \\pmb { A } } = \\pmb { A } - \\mathbb { E } [ \\pmb { A } ] , \\ \\bar { \\pmb { a } } _ { i } = \\mathbf { z m } ( \\pmb { a } _ { i } )$ . We will decompose $\\pmb { A } = \\bar { \\pmb { A } } + \\pmb { E }$ hence we will first focus on bounding the upper and lower singular values of $\\bar { A }$ by studying the random processes $X _ { v } = \\| \\bar { A } v \\| _ { \\ell _ { 2 } } ^ { 2 }$ and $Y _ { v } = X _ { v } - \\mathbb { E } [ X _ { v } ]$ over the unit sphere $S ^ { d - 1 }$ . First, we provide a deviation bound for the quantity ${ \\operatorname* { s u p } } _ { v \\in { \\mathcal { S } } ^ { d - 1 } } | Y _ { v } |$ . To achieve this, we will utilize Talagrand’s mixed tail bound and show that increments of $Y _ { v }$ are subexpoential. Pick two unit vectors $\\pmb { v } , \\pmb { u } \\in \\mathbb { R } ^ { d }$ . Write $\\pmb { x } = \\pmb { u } + \\pmb { v } , \\pmb { y } = \\pmb { u } - \\pmb { v } ,$ . We have that ", + "bbox": [ + 173, + 569, + 826, + 640 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/7ea56592793109b32bd96fd4d8a6dbb46b2cd41baf9db46d71e52ec2cc025a6d.jpg", + "text": "$$\n{ \\displaystyle { \\cal X } _ { u } - { \\cal X } _ { v } = \\| \\bar { \\cal A } u \\| _ { \\ell _ { 2 } } ^ { 2 } - \\| \\bar { \\cal A } v \\| _ { \\ell _ { 2 } } ^ { 2 } = \\| \\bar { \\cal A } ( x + y ) / 2 \\| _ { \\ell _ { 2 } } ^ { 2 } - \\| \\bar { \\cal A } ( x - y ) / 2 \\| _ { \\ell _ { 2 } } ^ { 2 } = x ^ { T } \\bar { \\cal A } ^ { T } \\bar { \\cal A } y = \\sum _ { i = 1 } ^ { n } ( \\bar { a } _ { i } ^ { T } x ) ( \\bar { a } _ { i } ^ { T } y ) . }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 647, + 825, + 684 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Letting $\\hat { \\pmb x } = { \\pmb x } / \\| { \\pmb x } \\| _ { \\ell _ { 2 } } , \\hat { \\pmb y } = { \\pmb y } / \\| { \\pmb y } \\| _ { \\ell _ { 2 } }$ , observe that, multiplication of subgaussians $\\pmb { x } ^ { T } \\bar { \\pmb { a } } _ { i } , \\pmb { y } ^ { T } \\bar { \\pmb { a } } _ { i }$ obey ", + "bbox": [ + 169, + 693, + 771, + 708 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/901e3c5ba18531c25f17d53ac587a6b5dcf583757c69fee376a1a60fe94034ca.jpg", + "text": "$$\n\\| ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ( \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) \\| _ { \\psi _ { 1 } } \\leq \\mathcal { O } ( \\| \\boldsymbol { x } \\| _ { \\ell _ { 2 } } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } K ^ { 2 } ) \\leq \\mathcal { O } ( K ^ { 2 } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 715, + 684, + 733 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Centering this subexponential variable around zero introduces a factor of 2 when bounding subexponential norm and yields $\\| ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) \\dot { \\langle } \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } \\rangle - \\mathbb { E } [ ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ( \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ] \\| _ { \\psi _ { 1 } } \\le \\mathcal { O } ( K ^ { 2 } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } ) .$ . Now, using the fact that $Y _ { u } - Y _ { v }$ is sum of $_ n$ independent zero-mean subexponential random variables, we have the tail bound ", + "bbox": [ + 174, + 741, + 825, + 781 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6a9fd07645c919ab83931ca3a4780b10630053296d4d36d623dba49056424056.jpg", + "text": "$$\n\\mathbb { P } ( n ^ { - 1 } | Y _ { u } - Y _ { v } | \\ge t ) \\le 2 \\exp ( - c ^ { \\prime } n \\operatorname* { m i n } \\{ \\frac { t ^ { 2 } } { K ^ { 4 } \\| y \\| _ { \\ell _ { 2 } } ^ { 2 } } , \\frac { t } { K ^ { 2 } \\| y \\| _ { \\ell _ { 2 } } } \\} ) .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 787, + 702, + 820 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Applying Talagrand’s chaining bound for mixed tail processes with distance metrics $\\begin{array} { r } { \\rho _ { 2 } = \\frac { K ^ { 2 } \\| \\cdot \\| _ { \\ell _ { 2 } } } { \\sqrt { n } } , \\rho _ { 1 } = } \\end{array}$ K2k·k\\`2n , (Theorem 3.5 of Dirksen (2013) or Theorem 2.2.23 of Talagrand (2014)) and using the fact that for unit sphere $S ^ { d - 1 }$ , Talagrand’s $\\gamma$ functionals (see Talagrand (2014)) obey $\\gamma _ { 1 } ( S ^ { d - 1 } ) , \\gamma _ { 2 } ^ { 2 } ( S ^ { d - 1 } ) \\leq \\mathcal { O } ( d )$ , ", + "bbox": [ + 173, + 832, + 825, + 886 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/38badad730e8cf1d9e9950c4138decc8a57a64d27d8b40caa42fa9007cd96965.jpg", + "text": "$$\n\\begin{array} { r } { n ^ { - 1 } \\underset { { \\pmb v } \\in { \\pmb S } ^ { d - 1 } } { \\operatorname* { s u p } } | Y _ { \\pmb v } | \\leq c K ^ { 2 } ( \\sqrt { d / n } + d / n + t / \\sqrt { n } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 893, + 648, + 921 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "with probability √ $1 - 2 \\exp ( - \\operatorname* { m i n } \\{ t ^ { 2 } , { \\sqrt { n } } t \\} )$ . Since $n \\geq C \\lambda ^ { - 2 } K ^ { 4 } d$ for sufficiently large $C > 0$ , picking $\\textstyle t = { \\dot { \\frac { 1 } { 1 6 c } } } K ^ { - 2 } \\lambda { \\dot { \\sqrt { n } } }$ , with probability $1 - 2 \\exp ( - \\mathcal { O } ( K ^ { - 4 } \\lambda ^ { 2 } n ) )$ , we ensure that right hand side of (F.1) is less than $\\lambda / 8$ . This leads to the following inequalities ", + "bbox": [ + 173, + 102, + 823, + 143 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/6a31d31682b3051c04df5600fc56ac5a526cbff53fad7ede45785c1b1e94827e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac 1 n | | \\bar { \\boldsymbol { A } } ^ { T } \\bar { \\boldsymbol { A } } - \\mathbb { E } [ \\bar { \\boldsymbol { A } } ^ { T } \\bar { \\boldsymbol { A } } ] | | \\le \\frac \\lambda 8 \\Longrightarrow \\frac { 9 K ^ { 2 } } { 8 } I _ { d } \\succeq \\frac 1 n \\bar { \\boldsymbol { A } } ^ { T } \\bar { \\boldsymbol { A } } \\succeq \\frac { 7 \\lambda } { 8 } I _ { d } . } \\\\ & { \\qquad \\Longrightarrow \\frac 9 8 K \\ge \\frac { 1 } { \\sqrt n } | | \\bar { \\boldsymbol { A } } | | \\ge s _ { \\mathrm { m i n } } ( \\bar { \\boldsymbol { A } } ) \\ge \\sqrt { \\frac { 7 } { 8 } } \\lambda . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 148, + 723, + 215 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Upper bound on spectral norm: For spectral norm of $\\pmb { A }$ , we use the triangle inequality ", + "bbox": [ + 174, + 219, + 696, + 233 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/f6dde6e85e70133aede052fcf9d7ccbb6684b191d6025a067852effd0917d066.jpg", + "text": "$$\n\\frac { 1 } { \\sqrt { n } } \\| \\pmb { A } \\| \\leq \\frac { 1 } { \\sqrt { n } } ( \\| \\pmb { E } \\| + \\| \\bar { \\pmb { A } } \\| ) \\leq \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\| \\mathbb { E } [ \\pmb { a } _ { i } ] \\| _ { \\ell _ { 2 } } + 9 K / 8 \\leq \\theta + \\sqrt { 3 / 2 } K .\n$$", + "text_format": "latex", + "bbox": [ + 266, + 238, + 730, + 270 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lowsize $n$ r bound on minim all ones vector by ${ \\bf 1 } _ { n }$ singular value: This pand define the process √ $\\begin{array} { r } { Z _ { \\pmb { v } } = \\frac { 1 } { \\sqrt { n } } \\pmb { 1 } _ { n } ^ { T } \\bar { \\pmb { A } } \\pmb { v } } \\end{array}$ ll row expectat. Observe that $\\begin{array} { r } { \\bar { A } ^ { T } \\mathbf { 1 } _ { n } = \\sum _ { i = 1 } ^ { n } \\bar { \\mathbf { a } } _ { i } \\in \\mathbb { R } ^ { d } } \\end{array}$ is a vector satisfying $\\| \\bar { A } ^ { T } \\mathbf { 1 } _ { n } / \\sqrt { n } \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )$ . Hence, again using $n \\geq C K ^ { 4 } \\lambda ^ { - 2 } d$ for sufficiently large $C > 0$ , applying Lemma F.3 with $m = c _ { 0 } K ^ { - 4 } \\lambda ^ { 2 } n > d$ by picking a sufficiently small constant $c _ { 0 } > 1 / \\bar { C }$ , with probability at least $1 - 2 \\exp ( - 1 0 0 c _ { 0 } K ^ { - 4 } \\lambda ^ { 2 } n )$ ", + "bbox": [ + 173, + 281, + 826, + 352 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/ef8ed4bc3249b9fef4bb30908aff4d969b4de6464af0b98c10a45ec75772d77c.jpg", + "text": "$$\n\\frac { 1 } { \\sqrt { n } } \\operatorname* { s u p } _ { \\| v \\| _ { 2 } = 1 } | Z _ { v } | = \\frac { 1 } { n } \\| \\bar { \\cal A } ^ { T } { \\bf 1 } _ { n } \\| _ { \\ell _ { 2 } } \\leq \\frac { 1 } { 1 2 } { \\cal K } { \\cal K } ^ { - 2 } \\lambda \\leq \\frac { \\sqrt { \\lambda } } { 1 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 358, + 674, + 393 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Let $\\begin{array} { r } { { \\cal P } = I _ { n } - \\frac { 1 } { n } { \\bf 1 } _ { n } { \\bf 1 } _ { n } ^ { T } } \\end{array}$ be the projection onto the orthogonal complement of the all ones vector. Note that $P E v = 0$ as the rows of $\\pmb { { \\cal E } }$ are equal. With this observation, with desired probability, for any unit length $_ { v }$ , ", + "bbox": [ + 169, + 401, + 823, + 429 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/362cedfdc475dad89402176709671c2e32d457104a23fc049106e0daf5f007d5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\pmb { A } \\pmb { v } \\| _ { \\ell _ { 2 } } \\ge \\| \\pmb { P } \\pmb { A } \\pmb { v } \\| _ { \\ell _ { 2 } } = \\| \\pmb { P } \\pmb { \\bar { A } } \\pmb { v } \\| _ { \\ell _ { 2 } } \\ge \\| \\pmb { \\bar { A } } \\pmb { v } \\| _ { \\ell _ { 2 } } - | Z _ { \\pmb { v } } | } \\\\ & { \\qquad \\ge s _ { \\operatorname* { m i n } } ( \\pmb { \\bar { A } } ) - \\ \\underset { \\pmb { v } \\in { S } ^ { d - 1 } } { \\operatorname* { s u p } } | Z _ { \\pmb { v } } | \\ge ( \\sqrt { 7 / 8 } - 1 / 1 2 ) \\sqrt { \\lambda n } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 433, + 684, + 481 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "which implies $s _ { \\mathrm { m i n } } ( A ) / \\sqrt { n } \\geq \\sqrt { 2 \\lambda / 3 }$ . ", + "bbox": [ + 174, + 488, + 415, + 505 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The corollary below is obtained by slightly modifying the proof above by using $\\begin{array} { r } { \\frac { 1 } { n } \\| \\bar { \\mathbfcal A } ^ { T } \\bar { \\mathbfcal A } - \\mathbb { E } [ \\bar { \\mathbfcal A } ^ { T } \\bar { \\mathbfcal A } ] \\| \\leq \\frac { K ^ { 2 } } { 8 } } \\end{array}$ in line (F.2) and only focusing on the spectral norm bound. ", + "bbox": [ + 174, + 521, + 821, + 549 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Corollary F.2. Let $\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { n \\times d }$ be a matrix with independent $\\{ { \\pmb a } _ { i } \\} _ { i = 1 } ^ { n }$ subgaussian rows satisfying $\\| \\mathbf { z } \\mathbf { m } ( \\mathbf { { a } } _ { i } ) \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )$ and $\\Sigma [ { \\pmb a } _ { i } ] \\preceq K ^ { 2 } { \\pmb I } _ { d }$ for some $K > 0$ and $\\| \\mathbb { E } [ \\mathbf { \\underline { { a } } } _ { i } ] \\rVert _ { \\ell _ { 2 } } \\leq \\theta$ . Suppose $\\pmb { \\Sigma } [ \\pmb { a } _ { i } ] \\succeq \\lambda \\pmb { I } _ { d }$ Suppose $n \\geq \\mathcal { O } ( K ^ { 2 } d )$ . Then, with probability at least $1 - 4 \\exp ( - c K ^ { - 2 } n )$ , ", + "bbox": [ + 174, + 553, + 823, + 592 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/f1f28689a063fcd9425a36acbd0539e8f550978cfda9084a87afeb7030e5a357.jpg", + "text": "$$\n\\theta + { \\sqrt { 3 / 2 } } K \\geq { \\frac { 1 } { \\sqrt { n } } } \\| A \\| .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 598, + 576, + 628 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The following lemma is fairly standard and is proved for the sake of completeness. ", + "bbox": [ + 173, + 641, + 663, + 656 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma F.3 (Subgaussian vector length). Let $\\pmb { a } \\in \\mathbb { R } ^ { n }$ be a zero-mean subgaussian vector with $\\| \\pmb { a } \\| _ { \\psi _ { 2 } } \\le L$ Then, for any $m \\geq n ,$ there exists $C > 0$ such that ", + "bbox": [ + 173, + 659, + 820, + 685 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/87a698569083f817b8ddfe1cebd33738e9ba8f8d4f8cd25e08ad33cc93fccb48.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } ( \\| \\pmb { a } \\| _ { \\ell _ { 2 } } \\le C L \\sqrt { m } ) \\ge 1 - 2 \\exp ( - 1 0 0 m ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 691, + 633, + 708 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof. We can pick a $1 / 2$ cover $\\mathcal { C }$ of the unit $\\ell _ { 2 }$ -sphere with size $\\log | { \\mathcal { C } } | \\leq 2 n$ . For any ${ \\pmb v } \\in \\mathcal { C }$ , subgaussianity implies, $\\begin{array} { r } { \\mathbb { P } ( | v ^ { T } \\pmb { a } | \\geq t ) \\leq 2 \\exp ( - \\frac { c t ^ { 2 } } { 2 L ^ { 2 } } ) } \\end{array}$ . Setting $t = C L { \\sqrt { m } }$ for sufficiently large constant $C > 0$ , and union bounding over all $\\pmb { v } \\in \\mathcal { C }$ , we find ", + "bbox": [ + 173, + 722, + 826, + 765 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/905ac3d7de230ea2bfd0b2e10b7518950215c6b624da0400760313bcceb7d4cf.jpg", + "text": "$$\n\\mathbb { P } ( \\bigcap _ { v \\in \\mathcal { C } } \\| v \\| _ { \\ell _ { 2 } } \\leq C L \\sqrt { m } ) \\geq 1 - 2 \\exp ( 2 n - \\frac { c C ^ { 2 } L ^ { 2 } m } { 2 L ^ { 2 } } ) \\leq 1 - 2 \\exp ( - 1 0 0 m ) .\n$$", + "text_format": "latex", + "bbox": [ + 256, + 770, + 740, + 806 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To conclude, let v(a) ∈ C be a’s neighbor satisfying kv − akak\\`2 k\\`2 ≤ 1/2. Hence, we have ", + "bbox": [ + 173, + 813, + 728, + 832 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/4b3f1cd8afcc305b2e98880ecf87a213468ff4e7b8aa7c70ea3ed937bf4985d7.jpg", + "text": "$$\n\\begin{array} { r } { \\| a \\| _ { \\ell _ { 2 } } \\leq \\| ( a - v ( a ) ) ^ { T } a \\| _ { \\ell _ { 2 } } + \\| v ^ { T } a \\| _ { \\ell _ { 2 } } \\leq \\| a \\| _ { \\ell _ { 2 } } / 2 + C L \\sqrt { m } \\implies \\| a \\| _ { \\ell _ { 2 } } \\leq 2 C L \\sqrt { m } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 839, + 772, + 857 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To conclude, use the change of variable $C C / 2$ . ", + "bbox": [ + 173, + 863, + 477, + 877 + ], + "page_idx": 21 + } +] \ No newline at end of file diff --git a/parse/train/rkeMHjR9Ym/rkeMHjR9Ym_middle.json b/parse/train/rkeMHjR9Ym/rkeMHjR9Ym_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..f28bbb50046ef1b61fc85f17c698e3c6cc3c66ce --- /dev/null +++ b/parse/train/rkeMHjR9Ym/rkeMHjR9Ym_middle.json @@ -0,0 +1,92578 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 488, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 77, + 428, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 77, + 428, + 97 + ], + "score": 1.0, + "content": "STOCHASTIC GRADIENT DESCENT LEARNS", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 490, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 490, + 118 + ], + "score": 1.0, + "content": "STATE EQUATIONS WITH NONLINEAR ACTIVATIONS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 138, + 245, + 159 + ], + "lines": [ + { + "bbox": [ + 113, + 138, + 201, + 150 + ], + "spans": [ + { + "bbox": [ + 113, + 138, + 201, + 150 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 148, + 245, + 160 + ], + "spans": [ + { + "bbox": [ + 111, + 148, + 245, + 160 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 189, + 333, + 201 + ], + "lines": [ + { + "bbox": [ + 276, + 187, + 336, + 203 + ], + "spans": [ + { + "bbox": [ + 276, + 187, + 336, + 203 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 214, + 469, + 346 + ], + "lines": [ + { + "bbox": [ + 141, + 213, + 470, + 227 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 435, + 227 + ], + "score": 1.0, + "content": "We study discrete time dynamical systems governed by the state equation", + "type": "text" + }, + { + "bbox": [ + 436, + 214, + 470, + 226 + ], + "score": 0.9, + "content": "h _ { t + 1 } =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 143, + 224, + 470, + 238 + ], + "spans": [ + { + "bbox": [ + 143, + 225, + 207, + 237 + ], + "score": 0.9, + "content": "\\phi ( A h _ { t } + B u _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 224, + 234, + 238 + ], + "score": 1.0, + "content": ". 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Numerical experiments verify", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 324, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 324, + 470, + 335 + ], + "score": 1.0, + "content": "the fast convergence of SGD on ReLU and leaky ReLU in consistence with our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 173, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 173, + 348 + ], + "score": 1.0, + "content": "theory.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 367, + 206, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 208, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 208, + 383 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "A wide range of problems involve sequential data with a natural temporal ordering. 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We prove that SGD estimate linearly converges to the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 280, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 470, + 292 + ], + "score": 1.0, + "content": "ground truth weights while using near-optimal sample size. Our results apply to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 470, + 303 + ], + "score": 1.0, + "content": "increasing activations whose derivatives are bounded away from zero. The analysis", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 302, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 302, + 470, + 313 + ], + "score": 1.0, + "content": "is based on i) a novel SGD convergence result with nonlinear activations and ii)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 312, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 325 + ], + "score": 1.0, + "content": "careful statistical characterization of the state vector. Numerical experiments verify", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 324, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 324, + 470, + 335 + ], + "score": 1.0, + "content": "the fast convergence of SGD on ReLU and leaky ReLU in consistence with our", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 173, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 173, + 348 + ], + "score": 1.0, + "content": "theory.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10.5, + "bbox_fs": [ + 139, + 213, + 472, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 367, + 206, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 208, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 208, + 383 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "A wide range of problems involve sequential data with a natural temporal ordering. Examples include", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 401, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 506, + 417 + ], + "score": 1.0, + "content": "natural language processing, time series prediction, system identification, and control design, among", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 428 + ], + "score": 1.0, + "content": "others. State-of-the-art algorithms for sequential problems often stem from dynamical systems theory", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "and are tailored to learn from temporally dependent data. Linear models and algorithms; such as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "Kalman filter, PID controller, and linear dynamical systems, have a long history and are utilized in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "control theory since 1960’s with great success (Brown et al. (1992); Ho & Kalman (1966); Åström", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "& Hägglund (1995)). More recently, nonlinear models such as recurrent neural networks (RNN)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "found applications in complex tasks such as machine translation and speech recognition (Bahdanau", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "et al. (2014); Graves et al. (2013); Hochreiter & Schmidhuber (1997)). Unlike feedforward neural", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 505 + ], + "score": 1.0, + "content": "networks, RNNs are dynamical systems that use their internal state to process inputs. The goal of this", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "work is to shed light on the inner workings of RNNs from a theoretical point of view. In particular,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 474, + 527 + ], + "score": 1.0, + "content": "we focus on the RNN state equation which is characterized by a nonlinear activation function", + "type": "text" + }, + { + "bbox": [ + 474, + 514, + 481, + 525 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 513, + 505, + 527 + ], + "score": 1.0, + "content": ", state", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 524, + 331, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 164, + 538 + ], + "score": 1.0, + "content": "weight matrix", + "type": "text" + }, + { + "bbox": [ + 165, + 525, + 174, + 534 + ], + "score": 0.7, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 524, + 276, + 538 + ], + "score": 1.0, + "content": ", and input weight matrix", + "type": "text" + }, + { + "bbox": [ + 276, + 525, + 286, + 534 + ], + "score": 0.84, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 524, + 331, + 538 + ], + "score": 1.0, + "content": "as follows", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 393, + 506, + 538 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 541, + 356, + 555 + ], + "lines": [ + { + "bbox": [ + 254, + 541, + 356, + 555 + ], + "spans": [ + { + "bbox": [ + 254, + 541, + 356, + 555 + ], + "score": 0.92, + "content": "h _ { t + 1 } = \\phi ( A h _ { t } + B u _ { t } ) ,", + "type": "interline_equation", + "image_path": "c25b751c217275605976e95df32cdb1ff4be84b98db6d9a3e26f9c027e78f581.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 254, + 541, + 356, + 555 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 504, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 129, + 573 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 129, + 561, + 140, + 572 + ], + "score": 0.88, + "content": "h _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 560, + 234, + 573 + ], + "score": 1.0, + "content": "is the state vector and", + "type": "text" + }, + { + "bbox": [ + 234, + 562, + 245, + 572 + ], + "score": 0.86, + "content": "\\mathbf { \\pmb { u } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 560, + 370, + 573 + ], + "score": 1.0, + "content": "is the input data at timestamp", + "type": "text" + }, + { + "bbox": [ + 371, + 562, + 376, + 570 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 560, + 505, + 573 + ], + "score": 1.0, + "content": ". This equation is the source of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "dynamic behavior of RNNs and distinguishes RNN from feedforward networks. The weight matrices", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 582, + 117, + 593 + ], + "score": 0.76, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 582, + 136, + 595 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 136, + 583, + 146, + 593 + ], + "score": 0.78, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "govern the dynamics of the state equation and are inferred from data. We will explore", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "the statistical and computational efficiency of stochastic gradient descent (SGD) for learning these", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 174, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 174, + 617 + ], + "score": 1.0, + "content": "weight matrices.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 560, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 104, + 617, + 509, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 617, + 415, + 637 + ], + "score": 1.0, + "content": "Contributions: Suppose we are given a finite trajectory of input/state pairs", + "type": "text" + }, + { + "bbox": [ + 415, + 621, + 461, + 633 + ], + "score": 0.93, + "content": "( { \\mathbf { } } u _ { t } , h _ { t } ) _ { t = 0 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 617, + 509, + 637 + ], + "score": 1.0, + "content": "generated", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 450, + 645 + ], + "score": 1.0, + "content": "from the state equation (1.1). 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For a class of activation functions including leaky", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "ReLU and for stable systems1, we show that SGD linearly converges to the ground truth weight", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 328, + 678 + ], + "score": 1.0, + "content": "matrices while requiring near-optimal trajectory length", + "type": "text" + }, + { + "bbox": [ + 329, + 666, + 338, + 675 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ". 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The results", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "are extended to unstable systems when the samples are collected from multiple independent RNN", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "trajectories rather than a single trajectory. Our theory applies to increasing activation functions", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "whose derivatives are bounded away from zero, which includes leaky ReLU, and Gaussian input", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "data. Numerical experiments on ReLU and leaky ReLU corroborate our theory and demonstrate that", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "SGD converges faster as the activation slope increases. To obtain our results, we i) characterize the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "statistical properties of the state vector (e.g. well-conditioned covariance) and ii) derive a novel SGD", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "convergence result with nonlinear activations; which may be of independent interest. As a whole, this", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 446, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 446, + 150 + ], + "score": 1.0, + "content": "paper provides a step towards foundational understanding of RNN training via SGD.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 617, + 509, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "whose derivatives are bounded away from zero, which includes leaky ReLU, and Gaussian input", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "data. Numerical experiments on ReLU and leaky ReLU corroborate our theory and demonstrate that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "SGD converges faster as the activation slope increases. To obtain our results, we i) characterize the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "statistical properties of the state vector (e.g. well-conditioned covariance) and ii) derive a novel SGD", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "convergence result with nonlinear activations; which may be of independent interest. As a whole, this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 446, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 446, + 150 + ], + "score": 1.0, + "content": "paper provides a step towards foundational understanding of RNN training via SGD.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 162, + 203, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 205, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 205, + 175 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 182, + 504, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "score": 1.0, + "content": "Our work is related to the recent optimization and statistics literature on linear dynamical systems", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 219, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 219, + 205 + ], + "score": 1.0, + "content": "(LDS) and neural networks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 409, + 223 + ], + "score": 1.0, + "content": "Linear dynamical systems: The state-equation (1.1) reduces to a LDS when", + "type": "text" + }, + { + "bbox": [ + 410, + 211, + 417, + 222 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "is the linear activation", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 108, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 108, + 221, + 150, + 234 + ], + "score": 0.87, + "content": "( \\phi ( x ) = { \\dot { x } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "). Identifying the weight matrices is a core problem in linear system identification and is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "related to the optimal control problem (e.g. linear quadratic regulator) with unknown system dynamics.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "While these problems are studied since 1950’s (Ljung (1998; 1987); Åström & Eykhoff (1971)),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 268 + ], + "score": 1.0, + "content": "our work is closer to the recent literature that provides data dependent bounds and characterize the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "non-asymptotic learning performance. Recht and coauthors have a series of papers exploring optimal", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "control problem (Simchowitz et al. (2018); Tu et al. (2018; 2017)). In particular, Hardt et al. (2016)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "shows gradient descent learns single-input-single-output (SISO) LDS with polynomial guarantees.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 298, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 314 + ], + "score": 1.0, + "content": "Oymak & Ozay (2018) and Faradonbeh et al. (2018) provide sample complexity bounds for learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 488, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 488, + 324 + ], + "score": 1.0, + "content": "LDS. Sanandaji et al. (2011b;a); Pereira et al. (2010) study the identification of sparse systems.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "Neural networks: There is a growing literature on the theoretical aspects of deep learning and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "provable algorithms for training neural networks. Most of the existing results are concerned with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "feedforward networks. Ge et al. (2017); Li & Yuan (2017); Mei et al. (2018b); Soltanolkotabi", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "(2017); Janzamin et al. (2015); Zhong et al. (2017b) consider learning fully-connected shallow", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "networks with gradient descent. Mei et al. (2018a); Soltanolkotabi et al. (2017); Foster et al. 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(2018) studies", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "over-parameterized networks when data is linearly separable. Janzamin et al. (2015); Oymak &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "Soltanolkotabi (2018) utilize tensor decomposition techniques for learning feedforward neural nets.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "For recurrent networks, Sedghi & Anandkumar (2016) proposed tensor algorithms with polynomial", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "guarantees and Khrulkov et al. (2017) studied their expressive power. More recently, Miller & Hardt", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 459, + 432, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 432, + 472 + ], + "score": 1.0, + "content": "(2018) showed that stable RNNs can be approximated by feed-forward networks.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 486, + 213, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 214, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 214, + 501 + ], + "score": 1.0, + "content": "2 PROBLEM SETUP", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 237, + 524 + ], + "score": 1.0, + "content": "We first introduce the notation.", + "type": "text" + }, + { + "bbox": [ + 237, + 511, + 256, + 523 + ], + "score": 0.9, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 510, + 427, + 524 + ], + "score": 1.0, + "content": "returns the spectral norm of a matrix and", + "type": "text" + }, + { + "bbox": [ + 428, + 511, + 457, + 523 + ], + "score": 0.91, + "content": "s _ { \\mathrm { m i n } } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "returns the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 278, + 534 + ], + "score": 1.0, + "content": "minimum singular value. 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However,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 427, + 637 + ], + "score": 1.0, + "content": "it includes Leaky ReLU which is a generalization of ReLU. Parameterized by", + "type": "text" + }, + { + "bbox": [ + 428, + 624, + 473, + 636 + ], + "score": 0.93, + "content": "1 \\ge \\beta \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 624, + 506, + 637 + ], + "score": 1.0, + "content": ", Leaky", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 274, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 149, + 649 + ], + "score": 1.0, + "content": "ReLU is a", + "type": "text" + }, + { + "bbox": [ + 149, + 636, + 157, + 646 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 635, + 274, + 649 + ], + "score": 1.0, + "content": "-increasing function given by", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 602, + 507, + 649 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 651, + 361, + 665 + ], + "lines": [ + { + "bbox": [ + 249, + 651, + 361, + 665 + ], + "spans": [ + { + "bbox": [ + 249, + 651, + 361, + 665 + ], + "score": 0.91, + "content": "{ \\mathrm { L R e L U } } ( x ) = \\operatorname* { m a x } ( \\beta x , x ) .", + "type": "interline_equation", + "image_path": "b1f001507ec9aab062e6ac49909aae227f7d54a81c863551d3f22dd776bbfb35.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 249, + 651, + 361, + 665 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 670, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 348, + 684 + ], + "score": 1.0, + "content": "In general, given an increasing and 1-Lipschitz activation", + "type": "text" + }, + { + "bbox": [ + 349, + 671, + 356, + 682 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 671, + 369, + 684 + ], + "score": 1.0, + "content": ", a", + "type": "text" + }, + { + "bbox": [ + 369, + 671, + 376, + 682 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 671, + 461, + 684 + ], + "score": 1.0, + "content": "-increasing function", + "type": "text" + }, + { + "bbox": [ + 461, + 671, + 473, + 683 + ], + "score": 0.89, + "content": "\\phi _ { \\beta }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 681, + 438, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 193, + 694 + ], + "score": 1.0, + "content": "obtained by blending", + "type": "text" + }, + { + "bbox": [ + 193, + 682, + 200, + 693 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 681, + 320, + 694 + ], + "score": 1.0, + "content": "with the linear activation, i.e.", + "type": "text" + }, + { + "bbox": [ + 321, + 682, + 434, + 694 + ], + "score": 0.91, + "content": "\\phi _ { \\beta } ( x ) = ( 1 - \\beta ) \\phi ( x ) + \\beta x", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 681, + 438, + 694 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 671, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 397, + 711 + ], + "score": 1.0, + "content": "A critical property that enables SGD is that the state-vector covariance", + "type": "text" + }, + { + "bbox": [ + 398, + 699, + 423, + 711 + ], + "score": 0.92, + "content": "\\pmb { \\Sigma } [ h _ { t } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "is well-conditioned", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "under proper assumptions. The lemma below provides upper and lower bounds on this covariance", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 257, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 257, + 733 + ], + "score": 1.0, + "content": "matrix in terms of problem variables.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 699, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 503, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 427, + 97 + ], + "score": 1.0, + "content": "Lemma 3.2 (State vector covariance). Consider the state equation (1.1) where", + "type": "text" + }, + { + "bbox": [ + 427, + 84, + 458, + 95 + ], + "score": 0.91, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 80, + 477, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 82, + 503, + 95 + ], + "score": 0.58, + "content": "{ \\mathbf { } } u _ { t } \\stackrel { i . i . d . } { \\sim }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 291, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 144, + 108 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , I _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 93, + 266, + 109 + ], + "score": 1.0, + "content": ". Define the upper bound term", + "type": "text" + }, + { + "bbox": [ + 266, + 95, + 277, + 106 + ], + "score": 0.88, + "content": "B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 93, + 291, + 109 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 106, + 358, + 140 + ], + "lines": [ + { + "bbox": [ + 253, + 106, + 358, + 140 + ], + "spans": [ + { + "bbox": [ + 253, + 106, + 358, + 140 + ], + "score": 0.94, + "content": "B _ { t } = \\| B \\| \\sqrt { \\frac { 1 - \\| A \\| ^ { 2 t } } { 1 - \\| A \\| ^ { 2 } } } .", + "type": "interline_equation", + "image_path": "91b94ddd31eb131359e43000d636d2e2ea942b9558be740f218085adb857b73c.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 253, + 106, + 358, + 123.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 253, + 123.0, + 358, + 140.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 132, + 145, + 445, + 159 + ], + "lines": [ + { + "bbox": [ + 132, + 145, + 446, + 160 + ], + "spans": [ + { + "bbox": [ + 132, + 145, + 178, + 160 + ], + "score": 1.0, + "content": "• Suppose", + "type": "text" + }, + { + "bbox": [ + 178, + 147, + 185, + 158 + ], + "score": 0.8, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 145, + 259, + 160 + ], + "score": 1.0, + "content": "is 1-Lipschitz and", + "type": "text" + }, + { + "bbox": [ + 260, + 146, + 298, + 159 + ], + "score": 0.93, + "content": "\\phi ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 145, + 354, + 160 + ], + "score": 1.0, + "content": ". Then, for all", + "type": "text" + }, + { + "bbox": [ + 355, + 145, + 443, + 159 + ], + "score": 0.6, + "content": "t \\ge 0 , \\Sigma [ h _ { t } ] \\preceq B _ { t } ^ { 2 } I _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 145, + 446, + 160 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 132, + 164, + 465, + 177 + ], + "lines": [ + { + "bbox": [ + 131, + 162, + 463, + 180 + ], + "spans": [ + { + "bbox": [ + 131, + 162, + 178, + 180 + ], + "score": 1.0, + "content": "• Suppose", + "type": "text" + }, + { + "bbox": [ + 178, + 166, + 185, + 176 + ], + "score": 0.79, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 162, + 203, + 180 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 203, + 165, + 210, + 177 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 162, + 309, + 180 + ], + "score": 1.0, + "content": "-increasing function and", + "type": "text" + }, + { + "bbox": [ + 309, + 166, + 335, + 177 + ], + "score": 0.9, + "content": "p \\geq n", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 162, + 365, + 180 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 365, + 164, + 463, + 177 + ], + "score": 0.91, + "content": "\\Sigma [ h _ { t } ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } I _ { n }", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 506, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 467, + 196 + ], + "score": 1.0, + "content": "As a natural extension from linear dynamical systems, we will say the system is stable if", + "type": "text" + }, + { + "bbox": [ + 467, + 183, + 505, + 196 + ], + "score": 0.92, + "content": "\\| A \\| < 1", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "and unstable otherwise. For activations we consider, stability implies that if the input is set to 0, state", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 133, + 218 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 133, + 206, + 145, + 217 + ], + "score": 0.88, + "content": "\\boldsymbol { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "will exponentially converge to 0 i.e. the system forgets the past states quickly. This is also", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 151, + 231 + ], + "score": 1.0, + "content": "the reason", + "type": "text" + }, + { + "bbox": [ + 151, + 217, + 185, + 229 + ], + "score": 0.92, + "content": "( B _ { t } ) _ { t \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "sequence converges for stable systems and diverges otherwise. The condition", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "number of the covariance will play a critical role in our analysis. Using Lemma 3.2, this number can", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 239, + 244, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 192, + 251 + ], + "score": 1.0, + "content": "be upper bounded by", + "type": "text" + }, + { + "bbox": [ + 193, + 240, + 200, + 250 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 239, + 244, + 251 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 250, + 415, + 280 + ], + "lines": [ + { + "bbox": [ + 196, + 250, + 415, + 280 + ], + "spans": [ + { + "bbox": [ + 196, + 250, + 415, + 280 + ], + "score": 0.93, + "content": "\\rho = \\left( \\frac { B _ { \\infty } } { \\beta s _ { \\operatorname* { m i n } } ( B ) } \\right) ^ { 2 } = \\left( \\frac { \\| B \\| } { s _ { \\operatorname* { m i n } } ( B ) } \\right) ^ { 2 } \\frac { 1 } { \\beta ^ { 2 } ( 1 - \\| A \\| ^ { 2 } ) } .", + "type": "interline_equation", + "image_path": "fc6b86ca3e0ac4d58e5ea587582e79573c9ab14616cffd0c7d0ea3f1d6b8f189.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 250, + 415, + 265.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 196, + 265.0, + 415, + 280.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 280, + 376, + 291 + ], + "lines": [ + { + "bbox": [ + 107, + 278, + 376, + 293 + ], + "spans": [ + { + "bbox": [ + 107, + 278, + 261, + 293 + ], + "score": 1.0, + "content": "Observe that, the condition number of", + "type": "text" + }, + { + "bbox": [ + 261, + 281, + 271, + 290 + ], + "score": 0.83, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 278, + 345, + 293 + ], + "score": 1.0, + "content": "appears inside the", + "type": "text" + }, + { + "bbox": [ + 346, + 282, + 353, + 291 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 278, + 376, + 293 + ], + "score": 1.0, + "content": "term.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 303, + 298, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 299, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 299, + 316 + ], + "score": 1.0, + "content": "3.2 LEARNING FROM SINGLE TRAJECTORY", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 279, + 338 + ], + "score": 1.0, + "content": "Our main result applies to stable systems", + "type": "text" + }, + { + "bbox": [ + 279, + 324, + 322, + 336 + ], + "score": 0.9, + "content": "( \\left. A \\right. < 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "and provides a non-asymptotic convergence", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "guarantee for SGD in terms of the upper bound on the state vector covariance. This result characterizes", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "the sample complexity and the rate of convergence of SGD; and also provides insights into the role of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 358, + 297, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 283, + 370 + ], + "score": 1.0, + "content": "activation function and the spectral norm of", + "type": "text" + }, + { + "bbox": [ + 283, + 358, + 293, + 367 + ], + "score": 0.76, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 358, + 297, + 370 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 233, + 385 + ], + "score": 1.0, + "content": "Theorem 3.3 (Main result). Let", + "type": "text" + }, + { + "bbox": [ + 234, + 369, + 294, + 383 + ], + "score": 0.89, + "content": "\\{ u _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 369, + 506, + 385 + ], + "score": 1.0, + "content": "be a finite trajectory generated from the state equation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 102, + 379, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 102, + 379, + 167, + 401 + ], + "score": 1.0, + "content": "(1.1). Suppose", + "type": "text" + }, + { + "bbox": [ + 168, + 385, + 205, + 397 + ], + "score": 0.87, + "content": "\\| A \\| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 379, + 209, + 401 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 209, + 385, + 216, + 397 + ], + "score": 0.75, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 379, + 226, + 401 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 227, + 385, + 234, + 397 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 379, + 283, + 401 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 283, + 385, + 314, + 397 + ], + "score": 0.86, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 379, + 317, + 401 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 385, + 343, + 397 + ], + "score": 0.72, + "content": "p \\geq n", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 379, + 365, + 401 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 365, + 383, + 430, + 398 + ], + "score": 0.91, + "content": "{ \\mathbf { \\boldsymbol { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 379, + 451, + 401 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 451, + 388, + 457, + 397 + ], + "score": 0.55, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 379, + 507, + 401 + ], + "score": 1.0, + "content": "be same as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 488, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 145, + 410 + ], + "score": 1.0, + "content": "(3.3) and", + "type": "text" + }, + { + "bbox": [ + 146, + 397, + 175, + 408 + ], + "score": 0.91, + "content": "c , C , c _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 395, + 438, + 410 + ], + "score": 1.0, + "content": "be properly chosen absolute constants. Pick the trajectory length", + "type": "text" + }, + { + "bbox": [ + 439, + 397, + 448, + 406 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 395, + 488, + 410 + ], + "score": 1.0, + "content": "to satisfy", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 407, + 345, + 422 + ], + "lines": [ + { + "bbox": [ + 264, + 407, + 345, + 422 + ], + "spans": [ + { + "bbox": [ + 264, + 407, + 345, + 422 + ], + "score": 0.89, + "content": "N \\geq C L \\rho ^ { 2 } ( n + p ) ,", + "type": "interline_equation", + "image_path": "68c76a24868db4df9165ba0f5b6775760b02782628d15dba356966f0dbeac817.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 264, + 407, + 345, + 422 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 509, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 133, + 459 + ], + "score": 1.0, + "content": "where loss fu", + "type": "text" + }, + { + "bbox": [ + 134, + 423, + 204, + 441 + ], + "score": 0.92, + "content": "\\begin{array} { r } { L = 1 - \\frac { \\log ( c n \\rho ) } { \\log \\| A \\| } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 423, + 435, + 441 + ], + "score": 1.0, + "content": ") g µ = 1/B∞, learning rate η = c0 β2ρn(n+p)", + "type": "text" + }, + { + "bbox": [ + 204, + 420, + 257, + 459 + ], + "score": 1.0, + "content": ". Pick scalinh probability", + "type": "text" + }, + { + "bbox": [ + 432, + 420, + 509, + 459 + ], + "score": 1.0, + "content": ", and consider the, starting from an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 257, + 440, + 431, + 454 + ], + "spans": [ + { + "bbox": [ + 257, + 440, + 431, + 454 + ], + "score": 0.87, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 453, + 443, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 156, + 465 + ], + "score": 1.0, + "content": "initial point", + "type": "text" + }, + { + "bbox": [ + 156, + 453, + 170, + 464 + ], + "score": 0.88, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 453, + 201, + 465 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 201, + 453, + 226, + 464 + ], + "score": 0.89, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 453, + 402, + 465 + ], + "score": 1.0, + "content": ", the SGD iterations described in Algorithm", + "type": "text" + }, + { + "bbox": [ + 402, + 455, + 407, + 463 + ], + "score": 0.57, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 453, + 443, + 465 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 465, + 419, + 492 + ], + "lines": [ + { + "bbox": [ + 192, + 465, + 419, + 492 + ], + "spans": [ + { + "bbox": [ + 192, + 465, + 419, + 492 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } n ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "0b5e7497e75a254dd9a481029a5296f59c29076d3da79a4a5f7d8d3daa757b36.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 192, + 465, + 419, + 492 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 367, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 368, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 368, + 506 + ], + "score": 1.0, + "content": "Here the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 104, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 326, + 525 + ], + "score": 1.0, + "content": "Sample complexity: Theorem 3.3 essentially requires", + "type": "text" + }, + { + "bbox": [ + 326, + 510, + 396, + 523 + ], + "score": 0.92, + "content": "N \\gtrsim ( n + p ) / \\beta ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "samples for learning. This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 355, + 535 + ], + "score": 1.0, + "content": "can be seen by unpacking (3.3) and ignoring the logarithmic", + "type": "text" + }, + { + "bbox": [ + 356, + 523, + 364, + 532 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "term and the condition number of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 117, + 543 + ], + "score": 0.73, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 532, + 176, + 546 + ], + "score": 1.0, + "content": ". Observe that", + "type": "text" + }, + { + "bbox": [ + 176, + 533, + 216, + 545 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { O } } ( n + \\bar { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "growth achieves near-optimal sample size for our problem. Each state", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 208, + 558 + ], + "score": 1.0, + "content": "equation (1.1) consists of", + "type": "text" + }, + { + "bbox": [ + 208, + 546, + 216, + 554 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 544, + 360, + 558 + ], + "score": 1.0, + "content": "sub-equations (one for each entry of", + "type": "text" + }, + { + "bbox": [ + 361, + 545, + 382, + 556 + ], + "score": 0.9, + "content": "h _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 544, + 433, + 558 + ], + "score": 1.0, + "content": "). We collect", + "type": "text" + }, + { + "bbox": [ + 433, + 545, + 443, + 554 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "state equations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 195, + 567 + ], + "score": 1.0, + "content": "to obtain a system of", + "type": "text" + }, + { + "bbox": [ + 195, + 556, + 211, + 565 + ], + "score": 0.78, + "content": "N n", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "equations. On the other hand, the total number of unknown parameters", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 117, + 579 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 567, + 127, + 577 + ], + "score": 0.73, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 566, + 145, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 145, + 567, + 155, + 577 + ], + "score": 0.81, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 566, + 171, + 579 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 172, + 566, + 209, + 578 + ], + "score": 0.92, + "content": "n ( n + p )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 566, + 505, + 579 + ], + "score": 1.0, + "content": ". This implies Theorem 3.3 is applicable as soon as the problem is mildly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 256, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 186, + 590 + ], + "score": 1.0, + "content": "overdetermined i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 577, + 252, + 589 + ], + "score": 0.93, + "content": "N n \\gtrsim n ( n + p )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 576, + 256, + 590 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 506, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 330, + 607 + ], + "score": 1.0, + "content": "Computational complexity: Theorem 3.3 requires", + "type": "text" + }, + { + "bbox": [ + 330, + 593, + 410, + 607 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { O } ( n ( n + p ) \\log \\frac { 1 } { \\varepsilon } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 593, + 496, + 607 + ], + "score": 1.0, + "content": "iterations to reach", + "type": "text" + }, + { + "bbox": [ + 496, + 596, + 502, + 604 + ], + "score": 0.55, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "neighborhood of the ground truth. Our analysis reveals that, this rate can be accelerated if the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 434, + 629 + ], + "score": 1.0, + "content": "state vector is zero-mean. This happens for odd activation functions satisfying", + "type": "text" + }, + { + "bbox": [ + 434, + 616, + 505, + 628 + ], + "score": 0.92, + "content": "\\phi ( - x ) = - \\phi ( x )", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 450, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 374, + 639 + ], + "score": 1.0, + "content": "(e.g. linear activation). The result below is a corollary and requires", + "type": "text" + }, + { + "bbox": [ + 374, + 628, + 390, + 637 + ], + "score": 0.86, + "content": "\\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 627, + 450, + 639 + ], + "score": 1.0, + "content": "less iterations.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 507, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 507, + 653 + ], + "score": 1.0, + "content": "Theorem 3.4 (Faster learning for odd activations). Consider the same setup provided in Theorem 3.3.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 651, + 508, + 669 + ], + "spans": [ + { + "bbox": [ + 103, + 651, + 508, + 669 + ], + "score": 1.0, + "content": "Additionally, assume that φ is an odd function. Pick scaling µ = 1/B∞, learning rate η = c0 β2ρ(n+p) ,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 102, + 663, + 509, + 686 + ], + "spans": [ + { + "bbox": [ + 102, + 663, + 328, + 686 + ], + "score": 1.0, + "content": "and consider the loss function (2.3). 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Consider the state equation (1.1) where", + "type": "text" + }, + { + "bbox": [ + 427, + 84, + 458, + 95 + ], + "score": 0.91, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 80, + 477, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 82, + 503, + 95 + ], + "score": 0.58, + "content": "{ \\mathbf { } } u _ { t } \\stackrel { i . i . d . } { \\sim }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 291, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 144, + 108 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , I _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 93, + 266, + 109 + ], + "score": 1.0, + "content": ". 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Then,", + "type": "text" + }, + { + "bbox": [ + 365, + 164, + 463, + 177 + ], + "score": 0.91, + "content": "\\Sigma [ h _ { t } ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } I _ { n }", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 131, + 162, + 463, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 506, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 467, + 196 + ], + "score": 1.0, + "content": "As a natural extension from linear dynamical systems, we will say the system is stable if", + "type": "text" + }, + { + "bbox": [ + 467, + 183, + 505, + 196 + ], + "score": 0.92, + "content": "\\| A \\| < 1", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "and unstable otherwise. For activations we consider, stability implies that if the input is set to 0, state", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 133, + 218 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 133, + 206, + 145, + 217 + ], + "score": 0.88, + "content": "\\boldsymbol { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "will exponentially converge to 0 i.e. the system forgets the past states quickly. This is also", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 151, + 231 + ], + "score": 1.0, + "content": "the reason", + "type": "text" + }, + { + "bbox": [ + 151, + 217, + 185, + 229 + ], + "score": 0.92, + "content": "( B _ { t } ) _ { t \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "sequence converges for stable systems and diverges otherwise. The condition", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "number of the covariance will play a critical role in our analysis. Using Lemma 3.2, this number can", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 239, + 244, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 192, + 251 + ], + "score": 1.0, + "content": "be upper bounded by", + "type": "text" + }, + { + "bbox": [ + 193, + 240, + 200, + 250 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 239, + 244, + 251 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 183, + 506, + 251 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 250, + 415, + 280 + ], + "lines": [ + { + "bbox": [ + 196, + 250, + 415, + 280 + ], + "spans": [ + { + "bbox": [ + 196, + 250, + 415, + 280 + ], + "score": 0.93, + "content": "\\rho = \\left( \\frac { B _ { \\infty } } { \\beta s _ { \\operatorname* { m i n } } ( B ) } \\right) ^ { 2 } = \\left( \\frac { \\| B \\| } { s _ { \\operatorname* { m i n } } ( B ) } \\right) ^ { 2 } \\frac { 1 } { \\beta ^ { 2 } ( 1 - \\| A \\| ^ { 2 } ) } .", + "type": "interline_equation", + "image_path": "fc6b86ca3e0ac4d58e5ea587582e79573c9ab14616cffd0c7d0ea3f1d6b8f189.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 250, + 415, + 265.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 196, + 265.0, + 415, + 280.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 280, + 376, + 291 + ], + "lines": [ + { + "bbox": [ + 107, + 278, + 376, + 293 + ], + "spans": [ + { + "bbox": [ + 107, + 278, + 261, + 293 + ], + "score": 1.0, + "content": "Observe that, the condition number of", + "type": "text" + }, + { + "bbox": [ + 261, + 281, + 271, + 290 + ], + "score": 0.83, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 278, + 345, + 293 + ], + "score": 1.0, + "content": "appears inside the", + "type": "text" + }, + { + "bbox": [ + 346, + 282, + 353, + 291 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 278, + 376, + 293 + ], + "score": 1.0, + "content": "term.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 107, + 278, + 376, + 293 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 303, + 298, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 299, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 299, + 316 + ], + "score": 1.0, + "content": "3.2 LEARNING FROM SINGLE TRAJECTORY", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 324, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 279, + 338 + ], + "score": 1.0, + "content": "Our main result applies to stable systems", + "type": "text" + }, + { + "bbox": [ + 279, + 324, + 322, + 336 + ], + "score": 0.9, + "content": "( \\left. 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This result characterizes", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "the sample complexity and the rate of convergence of SGD; and also provides insights into the role of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 358, + 297, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 283, + 370 + ], + "score": 1.0, + "content": "activation function and the spectral norm of", + "type": "text" + }, + { + "bbox": [ + 283, + 358, + 293, + 367 + ], + "score": 0.76, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 358, + 297, + 370 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 324, + 506, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 233, + 385 + ], + "score": 1.0, + "content": "Theorem 3.3 (Main result). Let", + "type": "text" + }, + { + "bbox": [ + 234, + 369, + 294, + 383 + ], + "score": 0.89, + "content": "\\{ u _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 369, + 506, + 385 + ], + "score": 1.0, + "content": "be a finite trajectory generated from the state equation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 102, + 379, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 102, + 379, + 167, + 401 + ], + "score": 1.0, + "content": "(1.1). Suppose", + "type": "text" + }, + { + "bbox": [ + 168, + 385, + 205, + 397 + ], + "score": 0.87, + "content": "\\| A \\| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 379, + 209, + 401 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 209, + 385, + 216, + 397 + ], + "score": 0.75, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 379, + 226, + 401 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 227, + 385, + 234, + 397 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 379, + 283, + 401 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 283, + 385, + 314, + 397 + ], + "score": 0.86, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 379, + 317, + 401 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 385, + 343, + 397 + ], + "score": 0.72, + "content": "p \\geq n", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 379, + 365, + 401 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 365, + 383, + 430, + 398 + ], + "score": 0.91, + "content": "{ \\mathbf { \\boldsymbol { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 379, + 451, + 401 + ], + "score": 1.0, + "content": ". 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Pick the trajectory length", + "type": "text" + }, + { + "bbox": [ + 439, + 397, + 448, + 406 + ], + "score": 0.78, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 395, + 488, + 410 + ], + "score": 1.0, + "content": "to satisfy", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 102, + 369, + 507, + 410 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 407, + 345, + 422 + ], + "lines": [ + { + "bbox": [ + 264, + 407, + 345, + 422 + ], + "spans": [ + { + "bbox": [ + 264, + 407, + 345, + 422 + ], + "score": 0.89, + "content": "N \\geq C L \\rho ^ { 2 } ( n + p ) ,", + "type": "interline_equation", + "image_path": "68c76a24868db4df9165ba0f5b6775760b02782628d15dba356966f0dbeac817.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 264, + 407, + 345, + 422 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 509, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 133, + 459 + ], + "score": 1.0, + "content": "where loss fu", + "type": "text" + }, + { + "bbox": [ + 134, + 423, + 204, + 441 + ], + "score": 0.92, + "content": "\\begin{array} { r } { L = 1 - \\frac { \\log ( c n \\rho ) } { \\log \\| A \\| } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 423, + 435, + 441 + ], + "score": 1.0, + "content": ") g µ = 1/B∞, learning rate η = c0 β2ρn(n+p)", + "type": "text" + }, + { + "bbox": [ + 204, + 420, + 257, + 459 + ], + "score": 1.0, + "content": ". Pick scalinh probability", + "type": "text" + }, + { + "bbox": [ + 432, + 420, + 509, + 459 + ], + "score": 1.0, + "content": ", and consider the, starting from an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 257, + 440, + 431, + 454 + ], + "spans": [ + { + "bbox": [ + 257, + 440, + 431, + 454 + ], + "score": 0.87, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 453, + 443, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 156, + 465 + ], + "score": 1.0, + "content": "initial point", + "type": "text" + }, + { + "bbox": [ + 156, + 453, + 170, + 464 + ], + "score": 0.88, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 453, + 201, + 465 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 201, + 453, + 226, + 464 + ], + "score": 0.89, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 453, + 402, + 465 + ], + "score": 1.0, + "content": ", the SGD iterations described in Algorithm", + "type": "text" + }, + { + "bbox": [ + 402, + 455, + 407, + 463 + ], + "score": 0.57, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 453, + 443, + 465 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 420, + 509, + 465 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 465, + 419, + 492 + ], + "lines": [ + { + "bbox": [ + 192, + 465, + 419, + 492 + ], + "spans": [ + { + "bbox": [ + 192, + 465, + 419, + 492 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } n ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } .", + "type": "interline_equation", + "image_path": "0b5e7497e75a254dd9a481029a5296f59c29076d3da79a4a5f7d8d3daa757b36.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 192, + 465, + 419, + 492 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 367, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 368, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 368, + 506 + ], + "score": 1.0, + "content": "Here the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 491, + 368, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 104, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 326, + 525 + ], + "score": 1.0, + "content": "Sample complexity: Theorem 3.3 essentially requires", + "type": "text" + }, + { + "bbox": [ + 326, + 510, + 396, + 523 + ], + "score": 0.92, + "content": "N \\gtrsim ( n + p ) / \\beta ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "samples for learning. This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 355, + 535 + ], + "score": 1.0, + "content": "can be seen by unpacking (3.3) and ignoring the logarithmic", + "type": "text" + }, + { + "bbox": [ + 356, + 523, + 364, + 532 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "term and the condition number of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 117, + 543 + ], + "score": 0.73, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 532, + 176, + 546 + ], + "score": 1.0, + "content": ". Observe that", + "type": "text" + }, + { + "bbox": [ + 176, + 533, + 216, + 545 + ], + "score": 0.93, + "content": "\\bar { \\mathcal { O } } ( n + \\bar { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "growth achieves near-optimal sample size for our problem. Each state", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 208, + 558 + ], + "score": 1.0, + "content": "equation (1.1) consists of", + "type": "text" + }, + { + "bbox": [ + 208, + 546, + 216, + 554 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 544, + 360, + 558 + ], + "score": 1.0, + "content": "sub-equations (one for each entry of", + "type": "text" + }, + { + "bbox": [ + 361, + 545, + 382, + 556 + ], + "score": 0.9, + "content": "h _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 544, + 433, + 558 + ], + "score": 1.0, + "content": "). We collect", + "type": "text" + }, + { + "bbox": [ + 433, + 545, + 443, + 554 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "state equations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 195, + 567 + ], + "score": 1.0, + "content": "to obtain a system of", + "type": "text" + }, + { + "bbox": [ + 195, + 556, + 211, + 565 + ], + "score": 0.78, + "content": "N n", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "equations. On the other hand, the total number of unknown parameters", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 117, + 579 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 567, + 127, + 577 + ], + "score": 0.73, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 566, + 145, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 145, + 567, + 155, + 577 + ], + "score": 0.81, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 566, + 171, + 579 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 172, + 566, + 209, + 578 + ], + "score": 0.92, + "content": "n ( n + p )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 566, + 505, + 579 + ], + "score": 1.0, + "content": ". This implies Theorem 3.3 is applicable as soon as the problem is mildly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 256, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 186, + 590 + ], + "score": 1.0, + "content": "overdetermined i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 577, + 252, + 589 + ], + "score": 0.93, + "content": "N n \\gtrsim n ( n + p )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 576, + 256, + 590 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 510, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 506, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 330, + 607 + ], + "score": 1.0, + "content": "Computational complexity: Theorem 3.3 requires", + "type": "text" + }, + { + "bbox": [ + 330, + 593, + 410, + 607 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { O } ( n ( n + p ) \\log \\frac { 1 } { \\varepsilon } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 593, + 496, + 607 + ], + "score": 1.0, + "content": "iterations to reach", + "type": "text" + }, + { + "bbox": [ + 496, + 596, + 502, + 604 + ], + "score": 0.55, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "neighborhood of the ground truth. Our analysis reveals that, this rate can be accelerated if the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 434, + 629 + ], + "score": 1.0, + "content": "state vector is zero-mean. This happens for odd activation functions satisfying", + "type": "text" + }, + { + "bbox": [ + 434, + 616, + 505, + 628 + ], + "score": 0.92, + "content": "\\phi ( - x ) = - \\phi ( x )", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 450, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 374, + 639 + ], + "score": 1.0, + "content": "(e.g. linear activation). The result below is a corollary and requires", + "type": "text" + }, + { + "bbox": [ + 374, + 628, + 390, + 637 + ], + "score": 0.86, + "content": "\\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 627, + 450, + 639 + ], + "score": 1.0, + "content": "less iterations.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 593, + 507, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 640, + 507, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 507, + 653 + ], + "score": 1.0, + "content": "Theorem 3.4 (Faster learning for odd activations). Consider the same setup provided in Theorem 3.3.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 651, + 508, + 669 + ], + "spans": [ + { + "bbox": [ + 103, + 651, + 508, + 669 + ], + "score": 1.0, + "content": "Additionally, assume that φ is an odd function. Pick scaling µ = 1/B∞, learning rate η = c0 β2ρ(n+p) ,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 102, + 663, + 509, + 686 + ], + "spans": [ + { + "bbox": [ + 102, + 663, + 328, + 686 + ], + "score": 1.0, + "content": "and consider the loss function (2.3). With probability", + "type": "text" + }, + { + "bbox": [ + 328, + 668, + 502, + 683 + ], + "score": 0.86, + "content": "\\begin{array} { r } { 1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 663, + 509, + 686 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 221, + 694 + ], + "score": 1.0, + "content": "starting from an initial point", + "type": "text" + }, + { + "bbox": [ + 221, + 681, + 235, + 692 + ], + "score": 0.87, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 681, + 266, + 694 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 266, + 681, + 291, + 692 + ], + "score": 0.9, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 681, + 464, + 694 + ], + "score": 1.0, + "content": ", the SGD iterations described in Algorithm", + "type": "text" + }, + { + "bbox": [ + 464, + 683, + 470, + 691 + ], + "score": 0.49, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5, + "bbox_fs": [ + 102, + 640, + 509, + 694 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 693, + 414, + 721 + ], + "lines": [ + { + "bbox": [ + 196, + 693, + 414, + 721 + ], + "spans": [ + { + "bbox": [ + 196, + 693, + 414, + 721 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "8c93b654c2d8a76d1171132a4f20b7c6cb2176e5e837d3da0de397d22ecec9bf.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 196, + 693, + 414, + 721 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 721, + 371, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 372, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 372, + 734 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45, + "bbox_fs": [ + 106, + 720, + 372, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 354, + 96 + ], + "score": 1.0, + "content": "Another aspect of the convergence rate is the dependence on", + "type": "text" + }, + { + "bbox": [ + 354, + 83, + 361, + 94 + ], + "score": 0.8, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 81, + 412, + 96 + ], + "score": 1.0, + "content": ". In terms of", + "type": "text" + }, + { + "bbox": [ + 413, + 83, + 420, + 94 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 81, + 506, + 96 + ], + "score": 1.0, + "content": ", the SGD error (3.4)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 147, + 105 + ], + "score": 1.0, + "content": "decays as", + "type": "text" + }, + { + "bbox": [ + 147, + 93, + 204, + 106 + ], + "score": 0.93, + "content": "( \\bar { 1 } - \\mathcal { O } ( \\beta ^ { 8 } ) ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ". While it is not clear how optimal is the exponent 8, numerical experiments", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 450, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 250, + 117 + ], + "score": 1.0, + "content": "in Section 6 demonstrate that larger", + "type": "text" + }, + { + "bbox": [ + 251, + 105, + 258, + 116 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 104, + 450, + 117 + ], + "score": 1.0, + "content": "indeed results in drastically faster convergence.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 312, + 146 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 313, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 313, + 147 + ], + "score": 1.0, + "content": "4 MAIN IDEAS AND PROOF STRATEGY", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 158, + 504, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 172 + ], + "score": 1.0, + "content": "We first outline our high-level proof strategy for Theorem 3.3; which brings together ideas from", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 169, + 216, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 216, + 182 + ], + "score": 1.0, + "content": "statistics and optimization.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 129, + 190, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 129, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 129, + 191, + 446, + 204 + ], + "score": 1.0, + "content": "1. We first show that input data is well-behaved by proving that state-vector", + "type": "text" + }, + { + "bbox": [ + 446, + 191, + 457, + 202 + ], + "score": 0.88, + "content": "h _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "has a well-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 202, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 142, + 202, + 505, + 214 + ], + "score": 1.0, + "content": "conditioned covariance as discussed in Lemma 3.2 and shown in Appendix B. The key idea", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 160, + 225 + ], + "score": 1.0, + "content": "is if", + "type": "text" + }, + { + "bbox": [ + 160, + 213, + 168, + 225 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 213, + 178, + 225 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 178, + 213, + 185, + 224 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 213, + 344, + 225 + ], + "score": 1.0, + "content": "-increasing, then the random input data", + "type": "text" + }, + { + "bbox": [ + 344, + 214, + 356, + 224 + ], + "score": 0.85, + "content": "\\mathbf { \\Delta } \\mathbf { u } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "provides sufficient excitation for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 222, + 217, + 237 + ], + "spans": [ + { + "bbox": [ + 142, + 222, + 191, + 237 + ], + "score": 1.0, + "content": "output state", + "type": "text" + }, + { + "bbox": [ + 191, + 224, + 212, + 236 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 222, + 217, + 237 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 130, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 130, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "2. Even if individual samples are well-behaved, analyzing (2.3) is still challenging due to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 251, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 142, + 251, + 505, + 262 + ], + "score": 1.0, + "content": "temporal dependencies between the samples. These dependencies prevent us from directly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "using statistical learning results that typically assume i.i.d. samples. We show that the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 272, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 293, + 285 + ], + "score": 1.0, + "content": "dependency between samples at time", + "type": "text" + }, + { + "bbox": [ + 293, + 273, + 298, + 282 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 272, + 316, + 285 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 273, + 341, + 283 + ], + "score": 0.91, + "content": "t + T", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 272, + 495, + 285 + ], + "score": 1.0, + "content": "decay exponentially fast in separation", + "type": "text" + }, + { + "bbox": [ + 495, + 273, + 504, + 282 + ], + "score": 0.78, + "content": "T", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 283, + 353, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 283, + 353, + 296 + ], + "score": 1.0, + "content": "(for stable systems). This is outlined in Appendix C.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 128, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 128, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "3. This observation allows us to obtain nearly independent data by subsampling the original", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 310, + 504, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 211, + 322 + ], + "score": 1.0, + "content": "trajectory to get", + "type": "text" + }, + { + "bbox": [ + 211, + 310, + 269, + 322 + ], + "score": 0.93, + "content": "( h _ { i T } , \\pmb { u } _ { i T } ) _ { i \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 310, + 495, + 322 + ], + "score": 1.0, + "content": ". Thanks to exponential decay, a logarithmically small", + "type": "text" + }, + { + "bbox": [ + 495, + 311, + 504, + 320 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 363, + 333 + ], + "score": 1.0, + "content": "can be chosen to generate large subtrajectories of size", + "type": "text" + }, + { + "bbox": [ + 363, + 321, + 385, + 333 + ], + "score": 0.9, + "content": "N / T", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 321, + 505, + 333 + ], + "score": 1.0, + "content": ". Appendix D uses additional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 333, + 478, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 478, + 344 + ], + "score": 1.0, + "content": "perturbation arguments to establish the well-behavedness of the overall data matrix.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 129, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 129, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "4. To conclude, we obtain a deterministic result which establishes fast convergence result for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 142, + 360, + 150, + 370 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "-increasing activations and well-behaved dataset. This is provided in Theorem 4.1 and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 370, + 237, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 237, + 382 + ], + "score": 1.0, + "content": "proved in Appendix A.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "The first three steps are related to the statistical nature of the problem which can be decoupled", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "from the last step. Specifically, the last step derives a deterministic result that establishes the linear", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 206, + 426 + ], + "score": 1.0, + "content": "convergence of SGD for", + "type": "text" + }, + { + "bbox": [ + 206, + 414, + 213, + 425 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "-increasing functions. For linear convergence proofs, a typical strategy is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 419, + 437 + ], + "score": 1.0, + "content": "showing the strong convexity of the loss function i.e. showing that, for some", + "type": "text" + }, + { + "bbox": [ + 420, + 425, + 446, + 434 + ], + "score": 0.9, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "and all points", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 435, + 213, + 447 + ], + "spans": [ + { + "bbox": [ + 107, + 436, + 125, + 446 + ], + "score": 0.84, + "content": "\\mathbf { \\nabla } _ { v } , \\mathbf { \\nabla } _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 435, + 213, + 447 + ], + "score": 1.0, + "content": ", the gradient satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 451, + 393, + 467 + ], + "lines": [ + { + "bbox": [ + 217, + 451, + 393, + 467 + ], + "spans": [ + { + "bbox": [ + 217, + 451, + 393, + 467 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\langle \\nabla \\mathcal { L } ( \\pmb { v } ) - \\nabla \\mathcal { L } ( \\pmb { u } ) , \\pmb { v } - \\pmb { u } \\rangle \\geq \\alpha \\| \\pmb { v } - \\pmb { u } \\| _ { \\ell _ { 2 } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "f842deeb3e5fe3ab48cb04fdd542e5f14fbeb042f1ef01f0dd961b255e711343.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 217, + 451, + 393, + 467 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "The core idea of our convergence result is that the strong convexity parameter of the loss function with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 107, + 484, + 114, + 495 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "-increasing activations can be connected to the loss function with linear activations. In particular,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 493, + 355, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 181, + 508 + ], + "score": 1.0, + "content": "recalling (2.3), set", + "type": "text" + }, + { + "bbox": [ + 182, + 494, + 229, + 506 + ], + "score": 0.93, + "content": "\\pmb { y } _ { t } ^ { \\mathrm { l i n } } = \\pmb { C } \\pmb { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 493, + 355, + 508 + ], + "score": 1.0, + "content": "and define the linear loss to be", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 512, + 379, + 547 + ], + "lines": [ + { + "bbox": [ + 232, + 512, + 379, + 547 + ], + "spans": [ + { + "bbox": [ + 232, + 512, + 379, + 547 + ], + "score": 0.95, + "content": "\\mathcal { L } ^ { \\mathrm { l i n } } ( \\Theta ) = \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } \\Vert \\pmb { y } _ { t } ^ { \\mathrm { l i n } } - \\Theta \\pmb { x } _ { t } \\Vert _ { \\ell _ { 2 } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "b9edfca0507553cdbfe81c766c92a7ca29c224388cfaa71278628d829283bab6.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 512, + 379, + 529.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 232, + 529.5, + 379, + 547.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 358, + 565 + ], + "score": 1.0, + "content": "Denoting the strong convexity parameter of the original loss by", + "type": "text" + }, + { + "bbox": [ + 358, + 554, + 371, + 565 + ], + "score": 0.87, + "content": "\\alpha _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 551, + 471, + 565 + ], + "score": 1.0, + "content": "and that of linear loss by", + "type": "text" + }, + { + "bbox": [ + 472, + 554, + 487, + 563 + ], + "score": 0.86, + "content": "\\alpha _ { \\mathrm { l i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 551, + 505, + 565 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 149, + 577 + ], + "score": 1.0, + "content": "argue that", + "type": "text" + }, + { + "bbox": [ + 149, + 563, + 200, + 576 + ], + "score": 0.92, + "content": "\\alpha _ { \\phi } \\geq \\beta ^ { 2 } \\alpha _ { \\mathrm { l i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 564, + 448, + 577 + ], + "score": 1.0, + "content": "; which allows us to establish a convergence result as soon as", + "type": "text" + }, + { + "bbox": [ + 449, + 566, + 464, + 575 + ], + "score": 0.86, + "content": "\\alpha _ { \\mathrm { l i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "is strictly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 575, + 469, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 469, + 587 + ], + "score": 1.0, + "content": "positive. Next result is our SGD convergence theorem which follows from this discussion.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 355, + 605 + ], + "score": 1.0, + "content": "Theorem 4.1 (Deterministic convergence). Suppose a data set", + "type": "text" + }, + { + "bbox": [ + 356, + 590, + 403, + 603 + ], + "score": 0.92, + "content": "\\{ \\pmb { x } _ { i } , \\pmb { y } _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 588, + 493, + 605 + ], + "score": 1.0, + "content": "is given; where output", + "type": "text" + }, + { + "bbox": [ + 493, + 593, + 504, + 602 + ], + "score": 0.76, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 179, + 615 + ], + "score": 1.0, + "content": "is related to input", + "type": "text" + }, + { + "bbox": [ + 179, + 603, + 190, + 612 + ], + "score": 0.85, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 600, + 206, + 615 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 206, + 601, + 270, + 613 + ], + "score": 0.91, + "content": "\\pmb { y } _ { i } = \\phi ( \\langle \\pmb { x } _ { i } , \\pmb { \\theta } \\rangle )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 600, + 309, + 615 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 310, + 601, + 341, + 612 + ], + "score": 0.9, + "content": "\\pmb { \\theta } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 600, + 381, + 615 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 382, + 603, + 407, + 613 + ], + "score": 0.92, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 600, + 426, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 602, + 433, + 613 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 600, + 451, + 615 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 451, + 602, + 458, + 613 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "-increasing.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 613, + 408, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 119, + 625 + ], + "score": 1.0, + "content": "Le", + "type": "text" + }, + { + "bbox": [ + 119, + 613, + 178, + 624 + ], + "score": 0.89, + "content": "\\ : \\gamma _ { + } \\geq \\gamma _ { - } > 0 \\ :", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 613, + 408, + 625 + ], + "score": 1.0, + "content": "be scalars. Assume that input samples satisfy the bounds", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 630, + 423, + 664 + ], + "lines": [ + { + "bbox": [ + 187, + 630, + 423, + 664 + ], + "spans": [ + { + "bbox": [ + 187, + 630, + 423, + 664 + ], + "score": 0.94, + "content": "\\gamma _ { + } I _ { n } \\succeq \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ { i } x _ { i } ^ { T } \\succeq \\gamma _ { - } I _ { n } \\quad , \\quad \\| x _ { i } \\| _ { \\ell _ { 2 } } ^ { 2 } \\leq B f o r a l l i .", + "type": "interline_equation", + "image_path": "15a20164d7bde2f542d974953eb975ab68376331ddea2c74765b2017d342449f.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 630, + 423, + 647.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 187, + 647.0, + 423, + 664.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 670, + 506, + 710 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 122, + 684 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 671, + 158, + 683 + ], + "score": 0.92, + "content": "\\{ r _ { \\tau } \\} _ { \\tau = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 669, + 429, + 684 + ], + "score": 1.0, + "content": "be a sequence of i.i.d. integers uniformly distributed between 1 to", + "type": "text" + }, + { + "bbox": [ + 430, + 671, + 440, + 681 + ], + "score": 0.71, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 669, + 505, + 684 + ], + "score": 1.0, + "content": ". Then, starting", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 679, + 508, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 204, + 698 + ], + "score": 1.0, + "content": "from an arbitrary point", + "type": "text" + }, + { + "bbox": [ + 205, + 685, + 216, + 696 + ], + "score": 0.87, + "content": "\\pmb { \\theta } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 679, + 306, + 698 + ], + "score": 1.0, + "content": ", setting learning rate", + "type": "text" + }, + { + "bbox": [ + 307, + 682, + 348, + 700 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\eta = \\frac { \\beta ^ { 2 } \\gamma _ { - } } { \\gamma _ { + } B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 681, + 508, + 702 + ], + "score": 1.0, + "content": "β2γ−γ+B , for all τ ≥ 0, the SGD iterations for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 166, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 166, + 711 + ], + "score": 1.0, + "content": "quadratic loss", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 714, + 404, + 730 + ], + "lines": [ + { + "bbox": [ + 206, + 714, + 404, + 730 + ], + "spans": [ + { + "bbox": [ + 206, + 714, + 404, + 730 + ], + "score": 0.9, + "content": "\\pmb { \\theta } _ { \\tau + 1 } = \\pmb { \\theta } _ { \\tau } - \\eta ( \\phi ( \\mathbf { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) - \\pmb { y } _ { r _ { \\tau } } ) \\phi ^ { \\prime } ( \\mathbf { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) \\pmb { x } _ { r _ { \\tau } } ,", + "type": "interline_equation", + "image_path": "1a38bb92f2b59eccb69067a7bf852fcaec48de56d42ee58525a13cfa7a876124.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 206, + 714, + 404, + 730 + ], + "spans": [], + "index": 44 + } + ] + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 354, + 96 + ], + "score": 1.0, + "content": "Another aspect of the convergence rate is the dependence on", + "type": "text" + }, + { + "bbox": [ + 354, + 83, + 361, + 94 + ], + "score": 0.8, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 81, + 412, + 96 + ], + "score": 1.0, + "content": ". In terms of", + "type": "text" + }, + { + "bbox": [ + 413, + 83, + 420, + 94 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 81, + 506, + 96 + ], + "score": 1.0, + "content": ", the SGD error (3.4)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 147, + 105 + ], + "score": 1.0, + "content": "decays as", + "type": "text" + }, + { + "bbox": [ + 147, + 93, + 204, + 106 + ], + "score": 0.93, + "content": "( \\bar { 1 } - \\mathcal { O } ( \\beta ^ { 8 } ) ) ^ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ". While it is not clear how optimal is the exponent 8, numerical experiments", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 450, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 250, + 117 + ], + "score": 1.0, + "content": "in Section 6 demonstrate that larger", + "type": "text" + }, + { + "bbox": [ + 251, + 105, + 258, + 116 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 104, + 450, + 117 + ], + "score": 1.0, + "content": "indeed results in drastically faster convergence.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 117 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 312, + 146 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 313, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 313, + 147 + ], + "score": 1.0, + "content": "4 MAIN IDEAS AND PROOF STRATEGY", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 108, + 158, + 504, + 181 + ], + "lines": [ + { + "bbox": [ + 105, + 156, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 172 + ], + "score": 1.0, + "content": "We first outline our high-level proof strategy for Theorem 3.3; which brings together ideas from", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 169, + 216, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 216, + 182 + ], + "score": 1.0, + "content": "statistics and optimization.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 156, + 506, + 182 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 190, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 129, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 129, + 191, + 446, + 204 + ], + "score": 1.0, + "content": "1. We first show that input data is well-behaved by proving that state-vector", + "type": "text" + }, + { + "bbox": [ + 446, + 191, + 457, + 202 + ], + "score": 0.88, + "content": "h _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "has a well-", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 202, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 142, + 202, + 505, + 214 + ], + "score": 1.0, + "content": "conditioned covariance as discussed in Lemma 3.2 and shown in Appendix B. The key idea", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 141, + 213, + 160, + 225 + ], + "score": 1.0, + "content": "is if", + "type": "text" + }, + { + "bbox": [ + 160, + 213, + 168, + 225 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 213, + 178, + 225 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 178, + 213, + 185, + 224 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 213, + 344, + 225 + ], + "score": 1.0, + "content": "-increasing, then the random input data", + "type": "text" + }, + { + "bbox": [ + 344, + 214, + 356, + 224 + ], + "score": 0.85, + "content": "\\mathbf { \\Delta } \\mathbf { u } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "provides sufficient excitation for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 222, + 217, + 237 + ], + "spans": [ + { + "bbox": [ + 142, + 222, + 191, + 237 + ], + "score": 1.0, + "content": "output state", + "type": "text" + }, + { + "bbox": [ + 191, + 224, + 212, + 236 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 222, + 217, + 237 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + }, + { + "bbox": [ + 130, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 130, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "2. Even if individual samples are well-behaved, analyzing (2.3) is still challenging due to", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 251, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 142, + 251, + 505, + 262 + ], + "score": 1.0, + "content": "temporal dependencies between the samples. These dependencies prevent us from directly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "using statistical learning results that typically assume i.i.d. samples. 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Appendix D uses additional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 333, + 478, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 478, + 344 + ], + "score": 1.0, + "content": "perturbation arguments to establish the well-behavedness of the overall data matrix.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 129, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "4. To conclude, we obtain a deterministic result which establishes fast convergence result for", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 142, + 360, + 150, + 370 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "-increasing activations and well-behaved dataset. This is provided in Theorem 4.1 and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 370, + 237, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 237, + 382 + ], + "score": 1.0, + "content": "proved in Appendix A.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 128, + 191, + 506, + 382 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "The first three steps are related to the statistical nature of the problem which can be decoupled", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "from the last step. Specifically, the last step derives a deterministic result that establishes the linear", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 206, + 426 + ], + "score": 1.0, + "content": "convergence of SGD for", + "type": "text" + }, + { + "bbox": [ + 206, + 414, + 213, + 425 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "-increasing functions. For linear convergence proofs, a typical strategy is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 419, + 437 + ], + "score": 1.0, + "content": "showing the strong convexity of the loss function i.e. showing that, for some", + "type": "text" + }, + { + "bbox": [ + 420, + 425, + 446, + 434 + ], + "score": 0.9, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "and all points", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 435, + 213, + 447 + ], + "spans": [ + { + "bbox": [ + 107, + 436, + 125, + 446 + ], + "score": 0.84, + "content": "\\mathbf { \\nabla } _ { v } , \\mathbf { \\nabla } _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 435, + 213, + 447 + ], + "score": 1.0, + "content": ", the gradient satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 390, + 505, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 451, + 393, + 467 + ], + "lines": [ + { + "bbox": [ + 217, + 451, + 393, + 467 + ], + "spans": [ + { + "bbox": [ + 217, + 451, + 393, + 467 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\langle \\nabla \\mathcal { L } ( \\pmb { v } ) - \\nabla \\mathcal { L } ( \\pmb { u } ) , \\pmb { v } - \\pmb { u } \\rangle \\geq \\alpha \\| \\pmb { v } - \\pmb { u } \\| _ { \\ell _ { 2 } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "f842deeb3e5fe3ab48cb04fdd542e5f14fbeb042f1ef01f0dd961b255e711343.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 217, + 451, + 393, + 467 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "The core idea of our convergence result is that the strong convexity parameter of the loss function with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 107, + 484, + 114, + 495 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "-increasing activations can be connected to the loss function with linear activations. In particular,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 493, + 355, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 181, + 508 + ], + "score": 1.0, + "content": "recalling (2.3), set", + "type": "text" + }, + { + "bbox": [ + 182, + 494, + 229, + 506 + ], + "score": 0.93, + "content": "\\pmb { y } _ { t } ^ { \\mathrm { l i n } } = \\pmb { C } \\pmb { x } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 493, + 355, + 508 + ], + "score": 1.0, + "content": "and define the linear loss to be", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 471, + 506, + 508 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 512, + 379, + 547 + ], + "lines": [ + { + "bbox": [ + 232, + 512, + 379, + 547 + ], + "spans": [ + { + "bbox": [ + 232, + 512, + 379, + 547 + ], + "score": 0.95, + "content": "\\mathcal { L } ^ { \\mathrm { l i n } } ( \\Theta ) = \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } \\Vert \\pmb { y } _ { t } ^ { \\mathrm { l i n } } - \\Theta \\pmb { x } _ { t } \\Vert _ { \\ell _ { 2 } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "b9edfca0507553cdbfe81c766c92a7ca29c224388cfaa71278628d829283bab6.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 512, + 379, + 529.5 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 232, + 529.5, + 379, + 547.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 551, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 358, + 565 + ], + "score": 1.0, + "content": "Denoting the strong convexity parameter of the original loss by", + "type": "text" + }, + { + "bbox": [ + 358, + 554, + 371, + 565 + ], + "score": 0.87, + "content": "\\alpha _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 551, + 471, + 565 + ], + "score": 1.0, + "content": "and that of linear loss by", + "type": "text" + }, + { + "bbox": [ + 472, + 554, + 487, + 563 + ], + "score": 0.86, + "content": "\\alpha _ { \\mathrm { l i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 551, + 505, + 565 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 149, + 577 + ], + "score": 1.0, + "content": "argue that", + "type": "text" + }, + { + "bbox": [ + 149, + 563, + 200, + 576 + ], + "score": 0.92, + "content": "\\alpha _ { \\phi } \\geq \\beta ^ { 2 } \\alpha _ { \\mathrm { l i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 564, + 448, + 577 + ], + "score": 1.0, + "content": "; which allows us to establish a convergence result as soon as", + "type": "text" + }, + { + "bbox": [ + 449, + 566, + 464, + 575 + ], + "score": 0.86, + "content": "\\alpha _ { \\mathrm { l i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "is strictly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 575, + 469, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 469, + 587 + ], + "score": 1.0, + "content": "positive. Next result is our SGD convergence theorem which follows from this discussion.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 551, + 505, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 355, + 605 + ], + "score": 1.0, + "content": "Theorem 4.1 (Deterministic convergence). Suppose a data set", + "type": "text" + }, + { + "bbox": [ + 356, + 590, + 403, + 603 + ], + "score": 0.92, + "content": "\\{ \\pmb { x } _ { i } , \\pmb { y } _ { i } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 588, + 493, + 605 + ], + "score": 1.0, + "content": "is given; where output", + "type": "text" + }, + { + "bbox": [ + 493, + 593, + 504, + 602 + ], + "score": 0.76, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 179, + 615 + ], + "score": 1.0, + "content": "is related to input", + "type": "text" + }, + { + "bbox": [ + 179, + 603, + 190, + 612 + ], + "score": 0.85, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 600, + 206, + 615 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 206, + 601, + 270, + 613 + ], + "score": 0.91, + "content": "\\pmb { y } _ { i } = \\phi ( \\langle \\pmb { x } _ { i } , \\pmb { \\theta } \\rangle )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 600, + 309, + 615 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 310, + 601, + 341, + 612 + ], + "score": 0.9, + "content": "\\pmb { \\theta } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 600, + 381, + 615 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 382, + 603, + 407, + 613 + ], + "score": 0.92, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 600, + 426, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 602, + 433, + 613 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 600, + 451, + 615 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 451, + 602, + 458, + 613 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "-increasing.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 613, + 408, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 119, + 625 + ], + "score": 1.0, + "content": "Le", + "type": "text" + }, + { + "bbox": [ + 119, + 613, + 178, + 624 + ], + "score": 0.89, + "content": "\\ : \\gamma _ { + } \\geq \\gamma _ { - } > 0 \\ :", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 613, + 408, + 625 + ], + "score": 1.0, + "content": "be scalars. Assume that input samples satisfy the bounds", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 588, + 506, + 625 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 630, + 423, + 664 + ], + "lines": [ + { + "bbox": [ + 187, + 630, + 423, + 664 + ], + "spans": [ + { + "bbox": [ + 187, + 630, + 423, + 664 + ], + "score": 0.94, + "content": "\\gamma _ { + } I _ { n } \\succeq \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ { i } x _ { i } ^ { T } \\succeq \\gamma _ { - } I _ { n } \\quad , \\quad \\| x _ { i } \\| _ { \\ell _ { 2 } } ^ { 2 } \\leq B f o r a l l i .", + "type": "interline_equation", + "image_path": "15a20164d7bde2f542d974953eb975ab68376331ddea2c74765b2017d342449f.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 187, + 630, + 423, + 647.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 187, + 647.0, + 423, + 664.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 670, + 506, + 710 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 122, + 684 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 671, + 158, + 683 + ], + "score": 0.92, + "content": "\\{ r _ { \\tau } \\} _ { \\tau = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 669, + 429, + 684 + ], + "score": 1.0, + "content": "be a sequence of i.i.d. integers uniformly distributed between 1 to", + "type": "text" + }, + { + "bbox": [ + 430, + 671, + 440, + 681 + ], + "score": 0.71, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 669, + 505, + 684 + ], + "score": 1.0, + "content": ". Then, starting", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 679, + 508, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 204, + 698 + ], + "score": 1.0, + "content": "from an arbitrary point", + "type": "text" + }, + { + "bbox": [ + 205, + 685, + 216, + 696 + ], + "score": 0.87, + "content": "\\pmb { \\theta } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 679, + 306, + 698 + ], + "score": 1.0, + "content": ", setting learning rate", + "type": "text" + }, + { + "bbox": [ + 307, + 682, + 348, + 700 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\eta = \\frac { \\beta ^ { 2 } \\gamma _ { - } } { \\gamma _ { + } B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 681, + 508, + 702 + ], + "score": 1.0, + "content": "β2γ−γ+B , for all τ ≥ 0, the SGD iterations for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 166, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 166, + 711 + ], + "score": 1.0, + "content": "quadratic loss", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 669, + 508, + 711 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 714, + 404, + 730 + ], + "lines": [ + { + "bbox": [ + 206, + 714, + 404, + 730 + ], + "spans": [ + { + "bbox": [ + 206, + 714, + 404, + 730 + ], + "score": 0.9, + "content": "\\pmb { \\theta } _ { \\tau + 1 } = \\pmb { \\theta } _ { \\tau } - \\eta ( \\phi ( \\mathbf { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) - \\pmb { y } _ { r _ { \\tau } } ) \\phi ^ { \\prime } ( \\mathbf { x } _ { r _ { \\tau } } ^ { T } \\pmb { \\theta } _ { \\tau } ) \\pmb { x } _ { r _ { \\tau } } ,", + "type": "interline_equation", + "image_path": "1a38bb92f2b59eccb69067a7bf852fcaec48de56d42ee58525a13cfa7a876124.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 206, + 714, + 404, + 730 + ], + "spans": [], + "index": 44 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 206, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 207, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 207, + 95 + ], + "score": 1.0, + "content": "satisfies the error bound", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 99, + 393, + 127 + ], + "lines": [ + { + "bbox": [ + 216, + 99, + 393, + 127 + ], + "spans": [ + { + "bbox": [ + 216, + 99, + 393, + 127 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\pmb { \\theta } _ { \\tau } - \\pmb { \\theta } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { \\theta } _ { 0 } - \\pmb { \\theta } \\| _ { \\ell _ { 2 } } ^ { 2 } \\big ( 1 - \\frac { \\beta ^ { 4 } \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } B } \\big ) ^ { \\tau } ,", + "type": "interline_equation", + "image_path": "8fed64bd69effa3a3706ba7f460c9a63eb1c79a0220a3973c712de5f1c205ec7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 216, + 99, + 393, + 127 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 436, + 143 + ], + "lines": [ + { + "bbox": [ + 104, + 127, + 434, + 147 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 398, + 147 + ], + "score": 1.0, + "content": "where the expectation is over the random selection of the SGD iterations", + "type": "text" + }, + { + "bbox": [ + 399, + 131, + 434, + 144 + ], + "score": 0.92, + "content": "\\{ r _ { \\tau } \\} _ { \\tau = 0 } ^ { \\infty }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 151, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 351, + 165 + ], + "score": 1.0, + "content": "This theorem provides a clean convergence rate for SGD for", + "type": "text" + }, + { + "bbox": [ + 351, + 153, + 358, + 164 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 151, + 505, + 165 + ], + "score": 1.0, + "content": "-increasing activations and naturally", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 163, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 383, + 174 + ], + "score": 1.0, + "content": "generalizes standard results on linear regression which corresponds to", + "type": "text" + }, + { + "bbox": [ + 383, + 163, + 409, + 174 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 163, + 506, + 174 + ], + "score": 1.0, + "content": ". We remark that related", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "results appear in the literature on generalized linear models. Kakade et al. (2011); Foster et al.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 197 + ], + "score": 1.0, + "content": "(2018); Mei et al. (2018a) provide learning theoretic loss/gradient/hessian convergence results for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 277, + 208 + ], + "score": 1.0, + "content": "isotonic regression, robust regression, and", + "type": "text" + }, + { + "bbox": [ + 277, + 196, + 285, + 207 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "-increasing activations. Goel et al. (2018) establishes a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 206, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 506, + 220 + ], + "score": 1.0, + "content": "similar result for leaky ReLU activations under the assumption of symmetric input distribution and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "infinitely many samples (i.e. in population limit). Compared to these, we establish a deterministic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "linear convergence guarantee for SGD that works whenever the data matrix is full rank. We believe", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "extensions to proximal gradient methods might be beneficial for high-dimensional nonlinear problems", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "score": 1.0, + "content": "(e.g. sparse/low-rank approximation, manifold constraints Cai et al. (2010); Beck & Teboulle (2009);", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 478, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 478, + 273 + ], + "score": 1.0, + "content": "Oymak et al. (2018); Agarwal et al. (2010); Pereira et al. (2010)) and is left as a future work.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "To derive our main results in Section 3, we need to address the first three steps outlined earlier and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "determine the conditions under which Theorem 4.1 is applicable to the data obtained from RNN state", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "equation with high probability. Below we provide desirable characteristics of the state vector; which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 225, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 225, + 323 + ], + "score": 1.0, + "content": "enables our statistical results.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 302, + 339 + ], + "score": 1.0, + "content": "Assumption 1 (Well-behaved state vector). Let", + "type": "text" + }, + { + "bbox": [ + 302, + 327, + 329, + 336 + ], + "score": 0.89, + "content": "L > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "be an integer. There exists positive scalars√", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 336, + 453, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 145, + 349 + ], + "score": 0.91, + "content": "\\gamma _ { + } , \\gamma _ { - } , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 336, + 249, + 350 + ], + "score": 1.0, + "content": "and an absolute constant", + "type": "text" + }, + { + "bbox": [ + 249, + 337, + 276, + 347 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 336, + 316, + 350 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 316, + 337, + 355, + 349 + ], + "score": 0.93, + "content": "\\theta \\leq 3 \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 336, + 453, + 350 + ], + "score": 1.0, + "content": "and the following holds", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 133, + 357, + 282, + 370 + ], + "lines": [ + { + "bbox": [ + 131, + 355, + 283, + 372 + ], + "spans": [ + { + "bbox": [ + 131, + 355, + 206, + 372 + ], + "score": 1.0, + "content": "• Lower bound:", + "type": "text" + }, + { + "bbox": [ + 206, + 357, + 279, + 370 + ], + "score": 0.9, + "content": "\\Sigma [ h _ { L - 1 } ] \\succeq \\gamma _ { - } I _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 355, + 283, + 372 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 132, + 376, + 339, + 388 + ], + "lines": [ + { + "bbox": [ + 132, + 376, + 339, + 390 + ], + "spans": [ + { + "bbox": [ + 132, + 376, + 232, + 390 + ], + "score": 1.0, + "content": "• Upper bound: for all", + "type": "text" + }, + { + "bbox": [ + 233, + 378, + 238, + 387 + ], + "score": 0.5, + "content": "t ,", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 376, + 339, + 390 + ], + "score": 1.0, + "content": ", the state vector satisfies", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 392, + 467, + 407 + ], + "lines": [ + { + "bbox": [ + 160, + 392, + 467, + 407 + ], + "spans": [ + { + "bbox": [ + 160, + 392, + 467, + 407 + ], + "score": 0.87, + "content": "\\Sigma [ h _ { t } ] \\preceq \\gamma _ { + } I _ { n } \\quad , \\quad \\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq C \\sqrt { \\gamma _ { + } } \\quad a n d \\quad \\| \\mathbb { E } [ h _ { t } ] \\| _ { \\ell _ { 2 } } \\leq \\theta \\sqrt { \\gamma _ { + } } .", + "type": "interline_equation", + "image_path": "30d8372c4c8ab6d2404f446bba519232e4fad591ccb6e30b6fd2fd300bc5f5a2.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 160, + 392, + 467, + 407 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 411, + 497, + 423 + ], + "lines": [ + { + "bbox": [ + 140, + 409, + 499, + 425 + ], + "spans": [ + { + "bbox": [ + 140, + 409, + 164, + 425 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 164, + 411, + 187, + 424 + ], + "score": 0.91, + "content": "\\lVert \\cdot \\rVert _ { \\psi _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 409, + 499, + 425 + ], + "score": 1.0, + "content": "returns the subgaussian norm of a vector (see Def. 5.22 of Vershynin (2010)).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 504, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 443 + ], + "score": 1.0, + "content": "Assumption 1 ensures that covariance is well-conditioned, state vector is well-concentrated, and it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 442, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 506, + 457 + ], + "score": 1.0, + "content": "has a reasonably small expectation. Our next theorem establishes statistical guarantees for learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 455, + 307, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 307, + 467 + ], + "score": 1.0, + "content": "the RNN state equation based on this assumption.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 506, + 517 + ], + "lines": [ + { + "bbox": [ + 104, + 466, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 245, + 484 + ], + "score": 1.0, + "content": "Theorem 4.2 (General result). Let", + "type": "text" + }, + { + "bbox": [ + 245, + 468, + 304, + 481 + ], + "score": 0.93, + "content": "\\{ u _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 466, + 350, + 484 + ], + "score": 1.0, + "content": "be a length", + "type": "text" + }, + { + "bbox": [ + 350, + 469, + 360, + 479 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 466, + 507, + 484 + ], + "score": 1.0, + "content": "trajectory of the state equation (1.1).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 143, + 497 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 143, + 483, + 182, + 496 + ], + "score": 0.82, + "content": "\\| A \\| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 482, + 185, + 497 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 186, + 484, + 193, + 495 + ], + "score": 0.63, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 482, + 204, + 497 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 204, + 484, + 211, + 495 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 482, + 261, + 497 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 261, + 483, + 292, + 495 + ], + "score": 0.89, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 482, + 314, + 497 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 315, + 481, + 381, + 496 + ], + "score": 0.89, + "content": "{ \\mathbf { \\boldsymbol { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 482, + 445, + 497 + ], + "score": 1.0, + "content": ". Given scalars", + "type": "text" + }, + { + "bbox": [ + 445, + 484, + 503, + 496 + ], + "score": 0.9, + "content": "\\gamma _ { + } \\geq \\gamma _ { - } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 482, + 507, + 497 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 493, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 224, + 508 + ], + "score": 1.0, + "content": "set the condition number as", + "type": "text" + }, + { + "bbox": [ + 224, + 495, + 273, + 506 + ], + "score": 0.92, + "content": "\\rho = \\gamma _ { + } / \\gamma _ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 493, + 376, + 508 + ], + "score": 1.0, + "content": ". 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Pick scaling to be", + "type": "text" + }, + { + "bbox": [ + 371, + 553, + 421, + 567 + ], + "score": 0.94, + "content": "\\mu = 1 / \\sqrt { \\gamma _ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "and learning rate to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 563, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 207, + 586 + ], + "score": 1.0, + "content": "be η = c0 ρ(θ+√2)2(n+p) .", + "type": "text" + }, + { + "bbox": [ + 168, + 563, + 184, + 575 + ], + "score": 1.0, + "content": "β2", + "type": "text" + }, + { + "bbox": [ + 208, + 567, + 275, + 583 + ], + "score": 1.0, + "content": "With probability", + "type": "text" + }, + { + "bbox": [ + 276, + 568, + 447, + 583 + ], + "score": 0.86, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 567, + 506, + 583 + ], + "score": 1.0, + "content": ", starting from", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 456, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 120, + 594 + ], + "score": 0.81, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 582, + 151, + 595 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 152, + 584, + 176, + 594 + ], + "score": 0.86, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 582, + 415, + 595 + ], + "score": 1.0, + "content": ", the SGD iterations on loss (2.3) as described in Algorithm", + "type": "text" + }, + { + "bbox": [ + 416, + 584, + 421, + 592 + ], + "score": 0.59, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 582, + 456, + 595 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 598, + 437, + 627 + ], + "lines": [ + { + "bbox": [ + 174, + 598, + 437, + 627 + ], + "spans": [ + { + "bbox": [ + 174, + 598, + 437, + 627 + ], + "score": 0.94, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "0dbc6dc1ac52f684643056b6204e414a4533d42b5bcfb72a3a8f7709e2306c67.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 174, + 598, + 437, + 607.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 174, + 607.6666666666666, + 437, + 617.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 174, + 617.3333333333333, + 437, + 626.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 357, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 358, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 358, + 646 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of SGD updates.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 651, + 504, + 697 + ], + "lines": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "The advantage of this theorem is that, it isolates the optimization problem from the statistical properties", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 352, + 676 + ], + "score": 1.0, + "content": "of state vector. If one can prove tighter bounds on achievable", + "type": "text" + }, + { + "bbox": [ + 352, + 663, + 397, + 675 + ], + "score": 0.91, + "content": "( \\gamma _ { + } , \\gamma _ { - } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 662, + 505, + 676 + ], + "score": 1.0, + "content": ", it will immediately imply", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 673, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 687 + ], + "score": 1.0, + "content": "improved performance for SGD. In particular, Theorems 3.3 and 3.4 are simple corollaries of Theorem", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 685, + 205, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 205, + 698 + ], + "score": 1.0, + "content": "4.2 with proper choices.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 132, + 704, + 450, + 734 + ], + "lines": [ + { + "bbox": [ + 131, + 703, + 451, + 719 + ], + "spans": [ + { + "bbox": [ + 131, + 703, + 269, + 719 + ], + "score": 1.0, + "content": "• Theorem 3.3 follows by setting", + "type": "text" + }, + { + "bbox": [ + 270, + 705, + 311, + 718 + ], + "score": 0.9, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 703, + 316, + 719 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 705, + 392, + 718 + ], + "score": 0.91, + "content": "\\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 703, + 412, + 719 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 413, + 705, + 446, + 718 + ], + "score": 0.91, + "content": "\\theta = { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 703, + 451, + 719 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 131, + 718, + 442, + 734 + ], + "spans": [ + { + "bbox": [ + 131, + 718, + 269, + 734 + ], + "score": 1.0, + "content": "• Theorem 3.4 follows by setting", + "type": "text" + }, + { + "bbox": [ + 270, + 720, + 311, + 733 + ], + "score": 0.75, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 718, + 315, + 734 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 720, + 392, + 732 + ], + "score": 0.89, + "content": "\\gamma _ { - 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\\pmb { \\theta } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { \\theta } _ { 0 } - \\pmb { \\theta } \\| _ { \\ell _ { 2 } } ^ { 2 } \\big ( 1 - \\frac { \\beta ^ { 4 } \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } B } \\big ) ^ { \\tau } ,", + "type": "interline_equation", + "image_path": "8fed64bd69effa3a3706ba7f460c9a63eb1c79a0220a3973c712de5f1c205ec7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 216, + 99, + 393, + 127 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 436, + 143 + ], + "lines": [ + { + "bbox": [ + 104, + 127, + 434, + 147 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 398, + 147 + ], + "score": 1.0, + "content": "where the expectation is over the random selection of the SGD iterations", + "type": "text" + }, + { + "bbox": [ + 399, + 131, + 434, + 144 + ], + "score": 0.92, + "content": "\\{ r _ { \\tau } \\} _ { \\tau = 0 } ^ { \\infty }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 127, + 434, + 147 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 151, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 351, + 165 + ], + "score": 1.0, + "content": "This theorem provides a clean convergence rate for SGD for", + "type": "text" + }, + { + "bbox": [ + 351, + 153, + 358, + 164 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 151, + 505, + 165 + ], + "score": 1.0, + "content": "-increasing activations and naturally", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 163, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 383, + 174 + ], + "score": 1.0, + "content": "generalizes standard results on linear regression which corresponds to", + "type": "text" + }, + { + "bbox": [ + 383, + 163, + 409, + 174 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 163, + 506, + 174 + ], + "score": 1.0, + "content": ". We remark that related", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 186 + ], + "score": 1.0, + "content": "results appear in the literature on generalized linear models. Kakade et al. (2011); Foster et al.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 197 + ], + "score": 1.0, + "content": "(2018); Mei et al. (2018a) provide learning theoretic loss/gradient/hessian convergence results for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 277, + 208 + ], + "score": 1.0, + "content": "isotonic regression, robust regression, and", + "type": "text" + }, + { + "bbox": [ + 277, + 196, + 285, + 207 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "-increasing activations. Goel et al. (2018) establishes a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 206, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 506, + 220 + ], + "score": 1.0, + "content": "similar result for leaky ReLU activations under the assumption of symmetric input distribution and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "infinitely many samples (i.e. in population limit). Compared to these, we establish a deterministic", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "linear convergence guarantee for SGD that works whenever the data matrix is full rank. We believe", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "extensions to proximal gradient methods might be beneficial for high-dimensional nonlinear problems", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "score": 1.0, + "content": "(e.g. sparse/low-rank approximation, manifold constraints Cai et al. (2010); Beck & Teboulle (2009);", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 478, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 478, + 273 + ], + "score": 1.0, + "content": "Oymak et al. (2018); Agarwal et al. (2010); Pereira et al. (2010)) and is left as a future work.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 151, + 507, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "To derive our main results in Section 3, we need to address the first three steps outlined earlier and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "determine the conditions under which Theorem 4.1 is applicable to the data obtained from RNN state", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "equation with high probability. Below we provide desirable characteristics of the state vector; which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 225, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 225, + 323 + ], + "score": 1.0, + "content": "enables our statistical results.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 277, + 505, + 323 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 302, + 339 + ], + "score": 1.0, + "content": "Assumption 1 (Well-behaved state vector). Let", + "type": "text" + }, + { + "bbox": [ + 302, + 327, + 329, + 336 + ], + "score": 0.89, + "content": "L > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "be an integer. There exists positive scalars√", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 336, + 453, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 145, + 349 + ], + "score": 0.91, + "content": "\\gamma _ { + } , \\gamma _ { - } , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 336, + 249, + 350 + ], + "score": 1.0, + "content": "and an absolute constant", + "type": "text" + }, + { + "bbox": [ + 249, + 337, + 276, + 347 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 336, + 316, + 350 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 316, + 337, + 355, + 349 + ], + "score": 0.93, + "content": "\\theta \\leq 3 \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 336, + 453, + 350 + ], + "score": 1.0, + "content": "and the following holds", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 326, + 505, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 357, + 282, + 370 + ], + "lines": [ + { + "bbox": [ + 131, + 355, + 283, + 372 + ], + "spans": [ + { + "bbox": [ + 131, + 355, + 206, + 372 + ], + "score": 1.0, + "content": "• Lower bound:", + "type": "text" + }, + { + "bbox": [ + 206, + 357, + 279, + 370 + ], + "score": 0.9, + "content": "\\Sigma [ h _ { L - 1 } ] \\succeq \\gamma _ { - } I _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 355, + 283, + 372 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 132, + 376, + 339, + 390 + ], + "spans": [ + { + "bbox": [ + 132, + 376, + 232, + 390 + ], + "score": 1.0, + "content": "• Upper bound: for all", + "type": "text" + }, + { + "bbox": [ + 233, + 378, + 238, + 387 + ], + "score": 0.5, + "content": "t ,", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 376, + 339, + 390 + ], + "score": 1.0, + "content": ", the state vector satisfies", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 131, + 355, + 283, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 376, + 339, + 388 + ], + "lines": [], + "index": 21, + "bbox_fs": [ + 132, + 376, + 339, + 390 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 392, + 467, + 407 + ], + "lines": [ + { + "bbox": [ + 160, + 392, + 467, + 407 + ], + "spans": [ + { + "bbox": [ + 160, + 392, + 467, + 407 + ], + "score": 0.87, + "content": "\\Sigma [ h _ { t } ] \\preceq \\gamma _ { + } I _ { n } \\quad , \\quad \\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq C \\sqrt { \\gamma _ { + } } \\quad a n d \\quad \\| \\mathbb { E } [ h _ { t } ] \\| _ { \\ell _ { 2 } } \\leq \\theta \\sqrt { \\gamma _ { + } } .", + "type": "interline_equation", + "image_path": "30d8372c4c8ab6d2404f446bba519232e4fad591ccb6e30b6fd2fd300bc5f5a2.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 160, + 392, + 467, + 407 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 411, + 497, + 423 + ], + "lines": [ + { + "bbox": [ + 140, + 409, + 499, + 425 + ], + "spans": [ + { + "bbox": [ + 140, + 409, + 164, + 425 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 164, + 411, + 187, + 424 + ], + "score": 0.91, + "content": "\\lVert \\cdot \\rVert _ { \\psi _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 409, + 499, + 425 + ], + "score": 1.0, + "content": "returns the subgaussian norm of a vector (see Def. 5.22 of Vershynin (2010)).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 140, + 409, + 499, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 431, + 504, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 443 + ], + "score": 1.0, + "content": "Assumption 1 ensures that covariance is well-conditioned, state vector is well-concentrated, and it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 442, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 506, + 457 + ], + "score": 1.0, + "content": "has a reasonably small expectation. Our next theorem establishes statistical guarantees for learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 455, + 307, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 307, + 467 + ], + "score": 1.0, + "content": "the RNN state equation based on this assumption.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 433, + 506, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 506, + 517 + ], + "lines": [ + { + "bbox": [ + 104, + 466, + 507, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 245, + 484 + ], + "score": 1.0, + "content": "Theorem 4.2 (General result). Let", + "type": "text" + }, + { + "bbox": [ + 245, + 468, + 304, + 481 + ], + "score": 0.93, + "content": "\\{ u _ { t } , h _ { t + 1 } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 466, + 350, + 484 + ], + "score": 1.0, + "content": "be a length", + "type": "text" + }, + { + "bbox": [ + 350, + 469, + 360, + 479 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 466, + 507, + 484 + ], + "score": 1.0, + "content": "trajectory of the state equation (1.1).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 143, + 497 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 143, + 483, + 182, + 496 + ], + "score": 0.82, + "content": "\\| A \\| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 482, + 185, + 497 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 186, + 484, + 193, + 495 + ], + "score": 0.63, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 482, + 204, + 497 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 204, + 484, + 211, + 495 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 482, + 261, + 497 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 261, + 483, + 292, + 495 + ], + "score": 0.89, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 482, + 314, + 497 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 315, + 481, + 381, + 496 + ], + "score": 0.89, + "content": "{ \\mathbf { \\boldsymbol { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 482, + 445, + 497 + ], + "score": 1.0, + "content": ". Given scalars", + "type": "text" + }, + { + "bbox": [ + 445, + 484, + 503, + 496 + ], + "score": 0.9, + "content": "\\gamma _ { + } \\geq \\gamma _ { - } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 482, + 507, + 497 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 493, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 224, + 508 + ], + "score": 1.0, + "content": "set the condition number as", + "type": "text" + }, + { + "bbox": [ + 224, + 495, + 273, + 506 + ], + "score": 0.92, + "content": "\\rho = \\gamma _ { + } / \\gamma _ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 493, + 376, + 508 + ], + "score": 1.0, + "content": ". For absolute constants", + "type": "text" + }, + { + "bbox": [ + 377, + 495, + 427, + 506 + ], + "score": 0.91, + "content": "C , c , c _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 493, + 506, + 508 + ], + "score": 1.0, + "content": ", choose trajectory", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 504, + 184, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 134, + 519 + ], + "score": 1.0, + "content": "length", + "type": "text" + }, + { + "bbox": [ + 135, + 506, + 144, + 515 + ], + "score": 0.7, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 504, + 184, + 519 + ], + "score": 1.0, + "content": "to satisfy", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 466, + 507, + 519 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 522, + 411, + 548 + ], + "lines": [ + { + "bbox": [ + 199, + 522, + 411, + 548 + ], + "spans": [ + { + "bbox": [ + 199, + 522, + 411, + 548 + ], + "score": 0.9, + "content": "N \\geq C L \\rho ^ { 2 } ( n + p ) w h e r e L = \\lceil 1 - \\frac { \\log \\left( c n \\rho \\right) } { \\log \\| A \\| } \\rceil .", + "type": "interline_equation", + "image_path": "370ce44b937b28038799ac3d5f402d7858c7dd9f98c9551d61f082d525a88b2e.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 199, + 522, + 411, + 548 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 506, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 191, + 566 + ], + "score": 1.0, + "content": "Suppose Assumption", + "type": "text" + }, + { + "bbox": [ + 192, + 554, + 197, + 563 + ], + "score": 0.49, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 552, + 243, + 566 + ], + "score": 1.0, + "content": "holds with", + "type": "text" + }, + { + "bbox": [ + 243, + 554, + 292, + 565 + ], + "score": 0.9, + "content": "L , \\gamma _ { + } , \\gamma _ { - } , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 552, + 370, + 566 + ], + "score": 1.0, + "content": ". Pick scaling to be", + "type": "text" + }, + { + "bbox": [ + 371, + 553, + 421, + 567 + ], + "score": 0.94, + "content": "\\mu = 1 / \\sqrt { \\gamma _ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "and learning rate to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 563, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 207, + 586 + ], + "score": 1.0, + "content": "be η = c0 ρ(θ+√2)2(n+p) .", + "type": "text" + }, + { + "bbox": [ + 168, + 563, + 184, + 575 + ], + "score": 1.0, + "content": "β2", + "type": "text" + }, + { + "bbox": [ + 208, + 567, + 275, + 583 + ], + "score": 1.0, + "content": "With probability", + "type": "text" + }, + { + "bbox": [ + 276, + 568, + 447, + 583 + ], + "score": 0.86, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( \\frac { N } { L \\rho ^ { 2 } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 567, + 506, + 583 + ], + "score": 1.0, + "content": ", starting from", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 456, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 120, + 594 + ], + "score": 0.81, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 582, + 151, + 595 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 152, + 584, + 176, + 594 + ], + "score": 0.86, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 582, + 415, + 595 + ], + "score": 1.0, + "content": ", the SGD iterations on loss (2.3) as described in Algorithm", + "type": "text" + }, + { + "bbox": [ + 416, + 584, + 421, + 592 + ], + "score": 0.59, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 582, + 456, + 595 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 552, + 506, + 595 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 598, + 437, + 627 + ], + "lines": [ + { + "bbox": [ + 174, + 598, + 437, + 627 + ], + "spans": [ + { + "bbox": [ + 174, + 598, + 437, + 627 + ], + "score": 0.94, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "0dbc6dc1ac52f684643056b6204e414a4533d42b5bcfb72a3a8f7709e2306c67.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 174, + 598, + 437, + 607.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 174, + 607.6666666666666, + 437, + 617.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 174, + 617.3333333333333, + 437, + 626.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 357, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 358, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 358, + 646 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of SGD updates.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 630, + 358, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 651, + 504, + 697 + ], + "lines": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "The advantage of this theorem is that, it isolates the optimization problem from the statistical properties", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 352, + 676 + ], + "score": 1.0, + "content": "of state vector. If one can prove tighter bounds on achievable", + "type": "text" + }, + { + "bbox": [ + 352, + 663, + 397, + 675 + ], + "score": 0.91, + "content": "( \\gamma _ { + } , \\gamma _ { - } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 662, + 505, + 676 + ], + "score": 1.0, + "content": ", it will immediately imply", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 673, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 687 + ], + "score": 1.0, + "content": "improved performance for SGD. In particular, Theorems 3.3 and 3.4 are simple corollaries of Theorem", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 685, + 205, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 205, + 698 + ], + "score": 1.0, + "content": "4.2 with proper choices.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 652, + 505, + 698 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 704, + 450, + 734 + ], + "lines": [ + { + "bbox": [ + 131, + 703, + 451, + 719 + ], + "spans": [ + { + "bbox": [ + 131, + 703, + 269, + 719 + ], + "score": 1.0, + "content": "• Theorem 3.3 follows by setting", + "type": "text" + }, + { + "bbox": [ + 270, + 705, + 311, + 718 + ], + "score": 0.9, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 703, + 316, + 719 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 705, + 392, + 718 + ], + "score": 0.91, + "content": "\\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 703, + 412, + 719 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 413, + 705, + 446, + 718 + ], + "score": 0.91, + "content": "\\theta = { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 703, + 451, + 719 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43, + "is_list_end_line": true + }, + { + "bbox": [ + 131, + 718, + 442, + 734 + ], + "spans": [ + { + "bbox": [ + 131, + 718, + 269, + 734 + ], + "score": 1.0, + "content": "• Theorem 3.4 follows by setting", + "type": "text" + }, + { + "bbox": [ + 270, + 720, + 311, + 733 + ], + "score": 0.75, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 718, + 315, + 734 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 720, + 392, + 732 + ], + "score": 0.89, + "content": "\\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 718, + 412, + 734 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 413, + 721, + 437, + 731 + ], + "score": 0.89, + "content": "\\theta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 718, + 442, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 43.5, + "bbox_fs": [ + 131, + 703, + 451, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 291, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "score": 1.0, + "content": "5 LEARNING UNSTABLE SYSTEMS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 420, + 119 + ], + "score": 1.0, + "content": "So far, we considered learning from a single RNN trajectory for stable systems", + "type": "text" + }, + { + "bbox": [ + 420, + 106, + 463, + 118 + ], + "score": 0.89, + "content": "( \\left. A \\right. < 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 106, + 505, + 119 + ], + "score": 1.0, + "content": ". For such", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "systems, as the time goes on, the impact of the earlier states disappear. In our analysis, this allows us", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "to split a single trajectory into multiple nearly-independent trajectories. This approach will not work", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 192, + 152 + ], + "score": 1.0, + "content": "for unstable systems", + "type": "text" + }, + { + "bbox": [ + 192, + 140, + 202, + 149 + ], + "score": 0.57, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "is arbitrary) where the impact of older states may be amplified over time. To", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "address this, we consider a model where the data is sampled from multiple independent trajectories.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 144, + 180 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 144, + 167, + 155, + 177 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 165, + 506, + 180 + ], + "score": 1.0, + "content": "independent trajectories of the state-equation (1.1) are available. Pick some integer", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 137, + 192 + ], + "score": 0.89, + "content": "T _ { 0 } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 171, + 307, + 200 + ], + "score": 1.0, + "content": ". 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The crucial", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 254, + 507, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 324, + 271 + ], + "score": 1.0, + "content": "difference compared to Section 3 is that, the samples", + "type": "text" + }, + { + "bbox": [ + 325, + 255, + 384, + 268 + ], + "score": 0.93, + "content": "( { \\bf y } _ { i } , h _ { i } , { \\bf u } _ { i } ) _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 254, + 507, + 271 + ], + "score": 1.0, + "content": "are now independent of each", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "other; hence, the analysis is simplified. As previously, having an upper bound on the condition", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 278, + 503, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 452, + 290 + ], + "score": 1.0, + "content": "number of the state-vector covariance is critical. This upper bound can be shown to be", + "type": "text" + }, + { + "bbox": [ + 453, + 279, + 459, + 289 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 278, + 503, + 290 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 294, + 431, + 328 + ], + "lines": [ + { + "bbox": [ + 180, + 294, + 431, + 328 + ], + "spans": [ + { + "bbox": [ + 180, + 294, + 431, + 328 + ], + "score": 0.94, + "content": "\\rho = \\left\\{ \\begin{array} { l l } { \\bar { \\rho } } & { \\mathrm { i f } \\ n > 1 } \\\\ { \\bar { \\rho } \\frac { \\ 1 - \\beta ^ { 2 } \\vert A \\vert ^ { 2 } } { 1 - ( \\beta \\vert A \\vert ) ^ { 2 T _ { 0 } } } } & { \\mathrm { i f } \\ n = 1 } \\end{array} \\right. \\quad \\mathrm { w h e r e } \\quad \\bar { \\rho } = \\frac { B _ { T _ { 0 } } ^ { 2 } } { \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "46bcebc40b73552b186b6f24418092dac5768e22d3bc9c43ada9c0202ea837d2.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 180, + 294, + 431, + 305.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 180, + 305.3333333333333, + 431, + 316.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 180, + 316.66666666666663, + 431, + 327.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 103, + 333, + 486, + 361 + ], + "spans": [ + { + "bbox": [ + 103, + 333, + 124, + 361 + ], + "score": 1.0, + "content": "The mod", + "type": "text" + }, + { + "bbox": [ + 125, + 334, + 131, + 344 + ], + "score": 0.82, + "content": "\\bar { \\rho }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 333, + 262, + 361 + ], + "score": 1.0, + "content": "term is similar to the earlier decation is indeed necessary since", + "type": "text" + }, + { + "bbox": [ + 302, + 333, + 327, + 361 + ], + "score": 1.0, + "content": "3); ho when", + "type": "text" + }, + { + "bbox": [ + 366, + 333, + 396, + 361 + ], + "score": 1.0, + "content": "nvolves . On the", + "type": "text" + }, + { + "bbox": [ + 397, + 333, + 414, + 344 + ], + "score": 0.91, + "content": "B _ { T _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 333, + 462, + 361 + ], + "score": 1.0, + "content": "rather than er hand, not", + "type": "text" + }, + { + "bbox": [ + 462, + 333, + 479, + 344 + ], + "score": 0.9, + "content": "B _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 333, + 486, + 361 + ], + "score": 1.0, + "content": ". t,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 262, + 343, + 503, + 357 + ], + "spans": [ + { + "bbox": [ + 262, + 344, + 302, + 355 + ], + "score": 0.91, + "content": "B _ { \\infty } = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 343, + 365, + 356 + ], + "score": 0.93, + "content": "\\| A \\| > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 343, + 503, + 357 + ], + "score": 0.9, + "content": "B _ { T _ { 0 } } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 354, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 104, + 354, + 197, + 371 + ], + "score": 1.0, + "content": "grows proportional to", + "type": "text" + }, + { + "bbox": [ + 198, + 355, + 230, + 369 + ], + "score": 0.93, + "content": "\\| A \\| ^ { 2 T _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 354, + 462, + 371 + ], + "score": 1.0, + "content": "; which results in exponentially bad condition number in", + "type": "text" + }, + { + "bbox": [ + 462, + 357, + 474, + 367 + ], + "score": 0.85, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 354, + 497, + 371 + ], + "score": 1.0, + "content": ". Our", + "type": "text" + }, + { + "bbox": [ + 497, + 358, + 504, + 368 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 354, + 380 + ], + "score": 1.0, + "content": "definition remedies this issue for single-output systems; where", + "type": "text" + }, + { + "bbox": [ + 355, + 368, + 380, + 378 + ], + "score": 0.9, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 367, + 398, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 368, + 408, + 378 + ], + "score": 0.74, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "is a scalar. In particular,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 129, + 392 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 379, + 156, + 390 + ], + "score": 0.9, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 378, + 176, + 392 + ], + "score": 1.0, + "content": "(e.g.", + "type": "text" + }, + { + "bbox": [ + 177, + 379, + 184, + 390 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 378, + 222, + 392 + ], + "score": 1.0, + "content": "is linear)", + "type": "text" + }, + { + "bbox": [ + 222, + 380, + 229, + 390 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 378, + 367, + 392 + ], + "score": 1.0, + "content": "becomes equal to the correct value", + "type": "text" + }, + { + "bbox": [ + 368, + 378, + 378, + 388 + ], + "score": 0.69, + "content": "1 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 378, + 506, + 392 + ], + "score": 1.0, + "content": ". The next theorem provides our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 465, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 465, + 402 + ], + "score": 1.0, + "content": "result on unstable systems in terms of this condition number and other model parameters.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 103, + 400, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 103, + 400, + 334, + 423 + ], + "score": 1.0, + "content": "Theorem 5.1 (Unstable systems). Suppose we are given", + "type": "text" + }, + { + "bbox": [ + 334, + 408, + 344, + 417 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 400, + 444, + 423 + ], + "score": 1.0, + "content": "independent trajectories", + "type": "text" + }, + { + "bbox": [ + 445, + 404, + 504, + 420 + ], + "score": 0.92, + "content": "( h _ { t } ^ { ( i ) } , u _ { t } ^ { ( i ) } ) _ { t \\geq 0 }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 417, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 120, + 433 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 419, + 165, + 430 + ], + "score": 0.9, + "content": "1 \\leq i \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 417, + 307, + 433 + ], + "score": 1.0, + "content": ". Each trajectory is sampled at time", + "type": "text" + }, + { + "bbox": [ + 307, + 419, + 318, + 430 + ], + "score": 0.87, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 417, + 357, + 433 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 357, + 419, + 367, + 429 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 417, + 403, + 433 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 403, + 419, + 463, + 431 + ], + "score": 0.92, + "content": "( { \\bf y } _ { i } , h _ { i } , { \\bf u } _ { i } ) _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 417, + 506, + 433 + ], + "score": 1.0, + "content": "where the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 431, + 353, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 353, + 442 + ], + "score": 1.0, + "content": "ith sample is given by (5.1). Suppose the sample size satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 446, + 342, + 460 + ], + "lines": [ + { + "bbox": [ + 269, + 446, + 342, + 460 + ], + "spans": [ + { + "bbox": [ + 269, + 446, + 342, + 460 + ], + "score": 0.91, + "content": "N \\geq C \\rho ^ { 2 } ( n + p )", + "type": "interline_equation", + "image_path": "5a86f46ab4f30f54fd4e97145e201fd0ec7e6e887bd3241d2fd297c1d7983ca2.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 269, + 446, + 342, + 460 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 134, + 483 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 472, + 141, + 482 + ], + "score": 0.73, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 466, + 347, + 483 + ], + "score": 1.0, + "content": "is given by (5.2). Assume the initial states are 0,", + "type": "text" + }, + { + "bbox": [ + 348, + 471, + 355, + 481 + ], + "score": 0.74, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 466, + 367, + 483 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 367, + 470, + 374, + 481 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 466, + 424, + 483 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 425, + 471, + 453, + 482 + ], + "score": 0.88, + "content": "p \\geq n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 466, + 475, + 483 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 476, + 468, + 504, + 481 + ], + "score": 0.62, + "content": "{ \\pmb u } _ { t } \\stackrel { i . i . d . } { \\sim }", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 479, + 507, + 502 + ], + "spans": [ + { + "bbox": [ + 107, + 483, + 144, + 496 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , { I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 479, + 194, + 502 + ], + "score": 1.0, + "content": ". Set scaling", + "type": "text" + }, + { + "bbox": [ + 195, + 482, + 251, + 497 + ], + "score": 0.94, + "content": "\\mu = 1 / \\sqrt { B _ { T _ { 0 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 479, + 309, + 502 + ], + "score": 1.0, + "content": ", learning rate", + "type": "text" + }, + { + "bbox": [ + 309, + 482, + 370, + 498 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\eta = c _ { 0 } \\frac { \\beta ^ { 2 } } { \\rho n ( n + p ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 479, + 507, + 502 + ], + "score": 1.0, + "content": ", and run SGD over the equations", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 290, + 511 + ], + "score": 1.0, + "content": "described in (2.2) and (2.3). Starting from", + "type": "text" + }, + { + "bbox": [ + 290, + 497, + 304, + 509 + ], + "score": 0.87, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 495, + 379, + 511 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 379, + 497, + 505, + 510 + ], + "score": 0.9, + "content": "1 - 2 N \\exp ( - 1 0 0 ( n + p ) ) -", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 507, + 279, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 509, + 171, + 524 + ], + "score": 0.92, + "content": "4 \\exp ( - \\mathcal { O } ( \\frac { N } { \\rho ^ { 2 } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 507, + 279, + 523 + ], + "score": 1.0, + "content": ", all SGD iterations satisfy", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 524, + 417, + 551 + ], + "lines": [ + { + "bbox": [ + 192, + 524, + 417, + 551 + ], + "spans": [ + { + "bbox": [ + 192, + 524, + 417, + 551 + ], + "score": 0.91, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } n ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "1defd79b99a32e7ce6b363146a666cfd44438e1b8ba0e5d3729b229d1879e3fd.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 192, + 524, + 417, + 551 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 371, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 373, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 373, + 570 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 107, + 579, + 267, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 268, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 268, + 594 + ], + "score": 1.0, + "content": "6 NUMERICAL EXPERIMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 506, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "We conducted experiments on ReLU and Leaky ReLU activations. Let us first describe the experi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 276, + 623 + ], + "score": 1.0, + "content": "mental setup. We pick the state dimension", + "type": "text" + }, + { + "bbox": [ + 276, + 611, + 307, + 621 + ], + "score": 0.89, + "content": "n = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 611, + 406, + 623 + ], + "score": 1.0, + "content": "and the input dimension", + "type": "text" + }, + { + "bbox": [ + 406, + 611, + 441, + 622 + ], + "score": 0.9, + "content": "p = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 611, + 506, + 623 + ], + "score": 1.0, + "content": ". We choose the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 186, + 634 + ], + "score": 1.0, + "content": "ground truth matrix", + "type": "text" + }, + { + "bbox": [ + 186, + 622, + 196, + 631 + ], + "score": 0.8, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "to be a scaled random unitary matrix; which ensures that all singular values of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 632, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 117, + 643 + ], + "score": 0.63, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 632, + 158, + 646 + ], + "score": 1.0, + "content": "are equal.", + "type": "text" + }, + { + "bbox": [ + 158, + 633, + 169, + 643 + ], + "score": 0.74, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 632, + 258, + 646 + ], + "score": 1.0, + "content": "is generated with i.i.d.", + "type": "text" + }, + { + "bbox": [ + 258, + 632, + 291, + 644 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 632, + 507, + 646 + ], + "score": 1.0, + "content": "entries. Instead of using the theoretical scaling choice,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 208, + 655 + ], + "score": 1.0, + "content": "we determine the scaling", + "type": "text" + }, + { + "bbox": [ + 209, + 645, + 216, + 655 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 644, + 506, + 655 + ], + "score": 1.0, + "content": "from empirical covariance matrices outlined in Algorithm 2. Similar to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "our proof strategy, this algorithm equalizes the spectral norms of the input and state covariances to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 436, + 678 + ], + "score": 1.0, + "content": "speed up convergence. We also empirically determined the learning rate and used", + "type": "text" + }, + { + "bbox": [ + 436, + 665, + 481, + 677 + ], + "score": 0.95, + "content": "\\eta = 1 / 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "in all", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 160, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 160, + 689 + ], + "score": 1.0, + "content": "experiments.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 701, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 186, + 713 + ], + "score": 1.0, + "content": "2Clearly, any nonzero", + "type": "text" + }, + { + "bbox": [ + 186, + 702, + 208, + 711 + ], + "score": 0.86, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "covariance matrix has condition number 1. However, due to subtleties in the proof", + "type": "text" + } + ] + }, + { + "bbox": [ + 109, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 109, + 711, + 190, + 723 + ], + "score": 1.0, + "content": "strategy, we don’t use", + "type": "text" + }, + { + "bbox": [ + 190, + 712, + 213, + 722 + ], + "score": 0.9, + "content": "\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 711, + 227, + 723 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 227, + 712, + 250, + 722 + ], + "score": 0.87, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 711, + 506, + 723 + ], + "score": 1.0, + "content": ". Obtaining tighter bounds on the subgaussian norm of the state-vector", + "type": "text" + } + ] + }, + { + "bbox": [ + 110, + 722, + 217, + 732 + ], + "spans": [ + { + "bbox": [ + 110, + 722, + 217, + 732 + ], + "score": 1.0, + "content": "would help resolve this issue.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 291, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 293, + 96 + ], + "score": 1.0, + "content": "5 LEARNING UNSTABLE SYSTEMS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 420, + 119 + ], + "score": 1.0, + "content": "So far, we considered learning from a single RNN trajectory for stable systems", + "type": "text" + }, + { + "bbox": [ + 420, + 106, + 463, + 118 + ], + "score": 0.89, + "content": "( \\left. A \\right. < 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 106, + 505, + 119 + ], + "score": 1.0, + "content": ". For such", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "systems, as the time goes on, the impact of the earlier states disappear. In our analysis, this allows us", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "to split a single trajectory into multiple nearly-independent trajectories. This approach will not work", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 192, + 152 + ], + "score": 1.0, + "content": "for unstable systems", + "type": "text" + }, + { + "bbox": [ + 192, + 140, + 202, + 149 + ], + "score": 0.57, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "is arbitrary) where the impact of older states may be amplified over time. To", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "address this, we consider a model where the data is sampled from multiple independent trajectories.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 106, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 144, + 180 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 144, + 167, + 155, + 177 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 165, + 506, + 180 + ], + "score": 1.0, + "content": "independent trajectories of the state-equation (1.1) are available. Pick some integer", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 137, + 192 + ], + "score": 0.89, + "content": "T _ { 0 } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 171, + 307, + 200 + ], + "score": 1.0, + "content": ". Denoting the ith trajectory by the triple", + "type": "text" + }, + { + "bbox": [ + 307, + 178, + 390, + 193 + ], + "score": 0.92, + "content": "( { h } _ { t + 1 } ^ { ( i ) } , { h } _ { t } ^ { ( i ) } , { u } _ { t } ^ { ( i ) } ) _ { t \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 171, + 506, + 200 + ], + "score": 1.0, + "content": ", we collect a single sample", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 100, + 186, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 100, + 186, + 228, + 217 + ], + "score": 1.0, + "content": "from each trajectory at time", + "type": "text" + }, + { + "bbox": [ + 229, + 196, + 240, + 207 + ], + "score": 0.87, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 186, + 244, + 217 + ], + "score": 0.786, + "content": "r", + "type": "text" + }, + { + "bbox": [ + 289, + 186, + 326, + 217 + ], + "score": 1.0, + "content": "he triple", + "type": "text" + }, + { + "bbox": [ + 326, + 193, + 401, + 209 + ], + "score": 0.93, + "content": "( \\boldsymbol { h } _ { T _ { 0 } + 1 } ^ { ( i ) } , \\boldsymbol { h } _ { T _ { 0 } } ^ { ( i ) } , \\boldsymbol { u } _ { T _ { 0 } } ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "To utilize the existing", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 244, + 207, + 289, + 218 + ], + "spans": [ + { + "bbox": [ + 244, + 207, + 289, + 218 + ], + "score": 0.91, + "content": "1 \\leq i \\leq N", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 100, + 165, + 506, + 218 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 223, + 375, + 241 + ], + "lines": [ + { + "bbox": [ + 235, + 223, + 375, + 241 + ], + "spans": [ + { + "bbox": [ + 235, + 223, + 375, + 241 + ], + "score": 0.93, + "content": "( { \\bf y } _ { i } , { \\bf h } _ { i } , { \\bf u } _ { i } ) = ( { \\bf h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \\bf h } _ { T _ { 0 } } ^ { ( i ) } , { \\bf u } _ { T _ { 0 } } ^ { ( i ) } ) .", + "type": "interline_equation", + "image_path": "878a3c240f8e1e7e22880775deeb90b87892469271c34ed7119a4b83089b65cf.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 235, + 223, + 375, + 241 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 414, + 257 + ], + "score": 1.0, + "content": "With this setup, we can again use the SGD Algorithm 1 to learn the weights", + "type": "text" + }, + { + "bbox": [ + 414, + 245, + 424, + 255 + ], + "score": 0.81, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 244, + 442, + 257 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 442, + 245, + 452, + 255 + ], + "score": 0.8, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 244, + 505, + 257 + ], + "score": 1.0, + "content": ". The crucial", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 254, + 507, + 271 + ], + "spans": [ + { + "bbox": [ + 104, + 254, + 324, + 271 + ], + "score": 1.0, + "content": "difference compared to Section 3 is that, the samples", + "type": "text" + }, + { + "bbox": [ + 325, + 255, + 384, + 268 + ], + "score": 0.93, + "content": "( { \\bf y } _ { i } , h _ { i } , { \\bf u } _ { i } ) _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 254, + 507, + 271 + ], + "score": 1.0, + "content": "are now independent of each", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "other; hence, the analysis is simplified. As previously, having an upper bound on the condition", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 278, + 503, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 452, + 290 + ], + "score": 1.0, + "content": "number of the state-vector covariance is critical. This upper bound can be shown to be", + "type": "text" + }, + { + "bbox": [ + 453, + 279, + 459, + 289 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 278, + 503, + 290 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 244, + 507, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 294, + 431, + 328 + ], + "lines": [ + { + "bbox": [ + 180, + 294, + 431, + 328 + ], + "spans": [ + { + "bbox": [ + 180, + 294, + 431, + 328 + ], + "score": 0.94, + "content": "\\rho = \\left\\{ \\begin{array} { l l } { \\bar { \\rho } } & { \\mathrm { i f } \\ n > 1 } \\\\ { \\bar { \\rho } \\frac { \\ 1 - \\beta ^ { 2 } \\vert A \\vert ^ { 2 } } { 1 - ( \\beta \\vert A \\vert ) ^ { 2 T _ { 0 } } } } & { \\mathrm { i f } \\ n = 1 } \\end{array} \\right. \\quad \\mathrm { w h e r e } \\quad \\bar { \\rho } = \\frac { B _ { T _ { 0 } } ^ { 2 } } { \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "46bcebc40b73552b186b6f24418092dac5768e22d3bc9c43ada9c0202ea837d2.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 180, + 294, + 431, + 305.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 180, + 305.3333333333333, + 431, + 316.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 180, + 316.66666666666663, + 431, + 327.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 103, + 333, + 486, + 361 + ], + "spans": [ + { + "bbox": [ + 103, + 333, + 124, + 361 + ], + "score": 1.0, + "content": "The mod", + "type": "text" + }, + { + "bbox": [ + 125, + 334, + 131, + 344 + ], + "score": 0.82, + "content": "\\bar { \\rho }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 333, + 262, + 361 + ], + "score": 1.0, + "content": "term is similar to the earlier decation is indeed necessary since", + "type": "text" + }, + { + "bbox": [ + 302, + 333, + 327, + 361 + ], + "score": 1.0, + "content": "3); ho when", + "type": "text" + }, + { + "bbox": [ + 366, + 333, + 396, + 361 + ], + "score": 1.0, + "content": "nvolves . On the", + "type": "text" + }, + { + "bbox": [ + 397, + 333, + 414, + 344 + ], + "score": 0.91, + "content": "B _ { T _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 333, + 462, + 361 + ], + "score": 1.0, + "content": "rather than er hand, not", + "type": "text" + }, + { + "bbox": [ + 462, + 333, + 479, + 344 + ], + "score": 0.9, + "content": "B _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 333, + 486, + 361 + ], + "score": 1.0, + "content": ". t,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 262, + 343, + 503, + 357 + ], + "spans": [ + { + "bbox": [ + 262, + 344, + 302, + 355 + ], + "score": 0.91, + "content": "B _ { \\infty } = \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 343, + 365, + 356 + ], + "score": 0.93, + "content": "\\| A \\| > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 343, + 503, + 357 + ], + "score": 0.9, + "content": "B _ { T _ { 0 } } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 354, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 104, + 354, + 197, + 371 + ], + "score": 1.0, + "content": "grows proportional to", + "type": "text" + }, + { + "bbox": [ + 198, + 355, + 230, + 369 + ], + "score": 0.93, + "content": "\\| A \\| ^ { 2 T _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 354, + 462, + 371 + ], + "score": 1.0, + "content": "; which results in exponentially bad condition number in", + "type": "text" + }, + { + "bbox": [ + 462, + 357, + 474, + 367 + ], + "score": 0.85, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 354, + 497, + 371 + ], + "score": 1.0, + "content": ". Our", + "type": "text" + }, + { + "bbox": [ + 497, + 358, + 504, + 368 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 354, + 380 + ], + "score": 1.0, + "content": "definition remedies this issue for single-output systems; where", + "type": "text" + }, + { + "bbox": [ + 355, + 368, + 380, + 378 + ], + "score": 0.9, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 367, + 398, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 368, + 408, + 378 + ], + "score": 0.74, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "is a scalar. In particular,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 129, + 392 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 379, + 156, + 390 + ], + "score": 0.9, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 378, + 176, + 392 + ], + "score": 1.0, + "content": "(e.g.", + "type": "text" + }, + { + "bbox": [ + 177, + 379, + 184, + 390 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 378, + 222, + 392 + ], + "score": 1.0, + "content": "is linear)", + "type": "text" + }, + { + "bbox": [ + 222, + 380, + 229, + 390 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 378, + 367, + 392 + ], + "score": 1.0, + "content": "becomes equal to the correct value", + "type": "text" + }, + { + "bbox": [ + 368, + 378, + 378, + 388 + ], + "score": 0.69, + "content": "1 ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 378, + 506, + 392 + ], + "score": 1.0, + "content": ". The next theorem provides our", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 465, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 465, + 402 + ], + "score": 1.0, + "content": "result on unstable systems in terms of this condition number and other model parameters.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 103, + 333, + 506, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 103, + 400, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 103, + 400, + 334, + 423 + ], + "score": 1.0, + "content": "Theorem 5.1 (Unstable systems). Suppose we are given", + "type": "text" + }, + { + "bbox": [ + 334, + 408, + 344, + 417 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 400, + 444, + 423 + ], + "score": 1.0, + "content": "independent trajectories", + "type": "text" + }, + { + "bbox": [ + 445, + 404, + 504, + 420 + ], + "score": 0.92, + "content": "( h _ { t } ^ { ( i ) } , u _ { t } ^ { ( i ) } ) _ { t \\geq 0 }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 417, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 120, + 433 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 419, + 165, + 430 + ], + "score": 0.9, + "content": "1 \\leq i \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 417, + 307, + 433 + ], + "score": 1.0, + "content": ". Each trajectory is sampled at time", + "type": "text" + }, + { + "bbox": [ + 307, + 419, + 318, + 430 + ], + "score": 0.87, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 417, + 357, + 433 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 357, + 419, + 367, + 429 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 417, + 403, + 433 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 403, + 419, + 463, + 431 + ], + "score": 0.92, + "content": "( { \\bf y } _ { i } , h _ { i } , { \\bf u } _ { i } ) _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 417, + 506, + 433 + ], + "score": 1.0, + "content": "where the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 431, + 353, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 353, + 442 + ], + "score": 1.0, + "content": "ith sample is given by (5.1). Suppose the sample size satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 103, + 400, + 506, + 442 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 269, + 446, + 342, + 460 + ], + "lines": [ + { + "bbox": [ + 269, + 446, + 342, + 460 + ], + "spans": [ + { + "bbox": [ + 269, + 446, + 342, + 460 + ], + "score": 0.91, + "content": "N \\geq C \\rho ^ { 2 } ( n + p )", + "type": "interline_equation", + "image_path": "5a86f46ab4f30f54fd4e97145e201fd0ec7e6e887bd3241d2fd297c1d7983ca2.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 269, + 446, + 342, + 460 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 134, + 483 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 472, + 141, + 482 + ], + "score": 0.73, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 466, + 347, + 483 + ], + "score": 1.0, + "content": "is given by (5.2). 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Set scaling", + "type": "text" + }, + { + "bbox": [ + 195, + 482, + 251, + 497 + ], + "score": 0.94, + "content": "\\mu = 1 / \\sqrt { B _ { T _ { 0 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 479, + 309, + 502 + ], + "score": 1.0, + "content": ", learning rate", + "type": "text" + }, + { + "bbox": [ + 309, + 482, + 370, + 498 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\eta = c _ { 0 } \\frac { \\beta ^ { 2 } } { \\rho n ( n + p ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 479, + 507, + 502 + ], + "score": 1.0, + "content": ", and run SGD over the equations", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 495, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 290, + 511 + ], + "score": 1.0, + "content": "described in (2.2) and (2.3). Starting from", + "type": "text" + }, + { + "bbox": [ + 290, + 497, + 304, + 509 + ], + "score": 0.87, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 495, + 379, + 511 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 379, + 497, + 505, + 510 + ], + "score": 0.9, + "content": "1 - 2 N \\exp ( - 1 0 0 ( n + p ) ) -", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 507, + 279, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 509, + 171, + 524 + ], + "score": 0.92, + "content": "4 \\exp ( - \\mathcal { O } ( \\frac { N } { \\rho ^ { 2 } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 507, + 279, + 523 + ], + "score": 1.0, + "content": ", all SGD iterations satisfy", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 466, + 507, + 524 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 524, + 417, + 551 + ], + "lines": [ + { + "bbox": [ + 192, + 524, + 417, + 551 + ], + "spans": [ + { + "bbox": [ + 192, + 524, + 417, + 551 + ], + "score": 0.91, + "content": "\\mathbb { E } [ \\| \\Theta _ { \\tau } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } n ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "1defd79b99a32e7ce6b363146a666cfd44438e1b8ba0e5d3729b229d1879e3fd.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 192, + 524, + 417, + 551 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 371, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 373, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 373, + 570 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 555, + 373, + 570 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 579, + 267, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 268, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 268, + 594 + ], + "score": 1.0, + "content": "6 NUMERICAL EXPERIMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 506, + 687 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "We conducted experiments on ReLU and Leaky ReLU activations. Let us first describe the experi-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 276, + 623 + ], + "score": 1.0, + "content": "mental setup. We pick the state dimension", + "type": "text" + }, + { + "bbox": [ + 276, + 611, + 307, + 621 + ], + "score": 0.89, + "content": "n = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 611, + 406, + 623 + ], + "score": 1.0, + "content": "and the input dimension", + "type": "text" + }, + { + "bbox": [ + 406, + 611, + 441, + 622 + ], + "score": 0.9, + "content": "p = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 611, + 506, + 623 + ], + "score": 1.0, + "content": ". We choose the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 186, + 634 + ], + "score": 1.0, + "content": "ground truth matrix", + "type": "text" + }, + { + "bbox": [ + 186, + 622, + 196, + 631 + ], + "score": 0.8, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "to be a scaled random unitary matrix; which ensures that all singular values of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 632, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 117, + 643 + ], + "score": 0.63, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 632, + 158, + 646 + ], + "score": 1.0, + "content": "are equal.", + "type": "text" + }, + { + "bbox": [ + 158, + 633, + 169, + 643 + ], + "score": 0.74, + "content": "\\textbf { { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 632, + 258, + 646 + ], + "score": 1.0, + "content": "is generated with i.i.d.", + "type": "text" + }, + { + "bbox": [ + 258, + 632, + 291, + 644 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 632, + 507, + 646 + ], + "score": 1.0, + "content": "entries. Instead of using the theoretical scaling choice,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 208, + 655 + ], + "score": 1.0, + "content": "we determine the scaling", + "type": "text" + }, + { + "bbox": [ + 209, + 645, + 216, + 655 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 644, + 506, + 655 + ], + "score": 1.0, + "content": "from empirical covariance matrices outlined in Algorithm 2. Similar to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "our proof strategy, this algorithm equalizes the spectral norms of the input and state covariances to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 436, + 678 + ], + "score": 1.0, + "content": "speed up convergence. We also empirically determined the learning rate and used", + "type": "text" + }, + { + "bbox": [ + 436, + 665, + 481, + 677 + ], + "score": 0.95, + "content": "\\eta = 1 / 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "in all", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 160, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 160, + 689 + ], + "score": 1.0, + "content": "experiments.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 600, + 507, + 689 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 305, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 307, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 307, + 96 + ], + "score": 1.0, + "content": "Algorithm 2 Empirical hyperparameter selection.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 110, + 98, + 351, + 158 + ], + "lines": [ + { + "bbox": [ + 109, + 95, + 313, + 115 + ], + "spans": [ + { + "bbox": [ + 109, + 95, + 158, + 115 + ], + "score": 1.0, + "content": "1: Inputs:", + "type": "text" + }, + { + "bbox": [ + 158, + 98, + 205, + 111 + ], + "score": 0.92, + "content": "( h _ { t } , u _ { t } ) _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 95, + 313, + 115 + ], + "score": 1.0, + "content": "sampled from a trajectory.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 109, + 313, + 142 + ], + "spans": [ + { + "bbox": [ + 111, + 109, + 197, + 142 + ], + "score": 1.0, + "content": "2: Outputs: Scaling 3: Form the empirica", + "type": "text" + }, + { + "bbox": [ + 197, + 113, + 204, + 122 + ], + "score": 0.71, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 109, + 275, + 142 + ], + "score": 1.0, + "content": ".covariance matrix", + "type": "text" + }, + { + "bbox": [ + 290, + 109, + 313, + 142 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 276, + 120, + 348, + 132 + ], + "spans": [ + { + "bbox": [ + 276, + 121, + 289, + 132 + ], + "score": 0.88, + "content": "\\Sigma _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 120, + 348, + 132 + ], + "score": 0.78, + "content": "\\{ h _ { t } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 108, + 128, + 353, + 147 + ], + "spans": [ + { + "bbox": [ + 108, + 128, + 276, + 147 + ], + "score": 1.0, + "content": "4: Form the empirical covariance matrix", + "type": "text" + }, + { + "bbox": [ + 276, + 133, + 289, + 143 + ], + "score": 0.88, + "content": "\\Sigma _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 133, + 347, + 144 + ], + "score": 0.67, + "content": "\\{ u _ { t } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 128, + 353, + 147 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 110, + 143, + 218, + 158 + ], + "spans": [ + { + "bbox": [ + 110, + 144, + 153, + 158 + ], + "score": 1.0, + "content": "5: return", + "type": "text" + }, + { + "bbox": [ + 154, + 143, + 214, + 157 + ], + "score": 0.93, + "content": "\\sqrt { \\| \\Sigma _ { u } \\| / \\| \\Sigma _ { h } \\| }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 144, + 218, + 158 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 145, + 172, + 504, + 290 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 172, + 504, + 290 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 172, + 504, + 290 + ], + "spans": [ + { + "bbox": [ + 145, + 172, + 504, + 290 + ], + "score": 0.955, + "type": "image", + "image_path": "0bf1b15a51fed848e839755197683e302f9a9ddcb50642893bcbe2352a4d0bde.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 145, + 172, + 504, + 211.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 145, + 211.33333333333334, + 504, + 250.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 145, + 250.66666666666669, + 504, + 290.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 291, + 505, + 313 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 428, + 304 + ], + "score": 1.0, + "content": "Figure 1: SGD convergence behavior for Leaky ReLUs with varying minimum slope", + "type": "text" + }, + { + "bbox": [ + 428, + 292, + 435, + 302 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 290, + 506, + 304 + ], + "score": 1.0, + "content": ". Figures a) and b)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 302, + 467, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 455, + 314 + ], + "score": 1.0, + "content": "repeat the same experiments. The difference is the spectral norm of the ground truth state matrix", + "type": "text" + }, + { + "bbox": [ + 455, + 303, + 464, + 312 + ], + "score": 0.73, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 302, + 467, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 421, + 338 + ], + "score": 1.0, + "content": "Evaluation: We consider two performance measures in the experiments. Let", + "type": "text" + }, + { + "bbox": [ + 422, + 324, + 432, + 335 + ], + "score": 0.86, + "content": "\\hat { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "be an estimate of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 219, + 348 + ], + "score": 1.0, + "content": "the ground truth parameter", + "type": "text" + }, + { + "bbox": [ + 219, + 335, + 286, + 348 + ], + "score": 0.93, + "content": "\\boldsymbol { C } \\dot { = } [ \\mu ^ { - 1 } \\boldsymbol { A } \\boldsymbol { B } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 336, + 505, + 348 + ], + "score": 1.0, + "content": ". The first measure is the normalized error defined as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 347, + 405, + 362 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 185, + 361 + ], + "score": 0.93, + "content": "\\| \\hat { C } - C \\| _ { F } ^ { 2 } / \\| C \\| _ { F } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 348, + 405, + 362 + ], + "score": 1.0, + "content": ". The second measure is the normalized loss defined as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 366, + 358, + 400 + ], + "lines": [ + { + "bbox": [ + 252, + 366, + 358, + 400 + ], + "spans": [ + { + "bbox": [ + 252, + 366, + 358, + 400 + ], + "score": 0.94, + "content": "\\frac { \\sum _ { i = 1 } ^ { N } | | \\pmb { y } _ { t } - \\phi ( \\hat { C } \\pmb { x } _ { t } ) | | _ { \\ell _ { 2 } } ^ { 2 } } { \\sum _ { i = 1 } ^ { N } | | \\pmb { y } _ { t } | | _ { \\ell _ { 2 } } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "3a97e193b5d9a5cf2ecb2bc97f48ab8a68fccd0bea8ba3320d05c4d0961e7e6a.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 366, + 358, + 383.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 252, + 383.0, + 358, + 400.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "In all experiments, we run Algorithm 1 for 50000 SGD iterations and plot these measures as a function", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 117, + 428 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 417, + 124, + 425 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 415, + 324, + 428 + ], + "score": 1.0, + "content": "; by using the estimate available at the end of the", + "type": "text" + }, + { + "bbox": [ + 324, + 417, + 330, + 425 + ], + "score": 0.69, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 415, + 414, + 428 + ], + "score": 1.0, + "content": "th SGD iteration for", + "type": "text" + }, + { + "bbox": [ + 414, + 416, + 477, + 426 + ], + "score": 0.91, + "content": "0 \\leq \\tau \\leq 5 0 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 415, + 505, + 428 + ], + "score": 1.0, + "content": ". Each", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "curve is obtained by averaging the outcomes of 20 independent realizations.Our first experiments", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 123, + 450 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 123, + 438, + 162, + 448 + ], + "score": 0.89, + "content": "N = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 437, + 361, + 450 + ], + "score": 1.0, + "content": "; which is mildly larger than the total dimension", + "type": "text" + }, + { + "bbox": [ + 362, + 438, + 416, + 448 + ], + "score": 0.9, + "content": "n + p = 1 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ". In Figure 1, we plot", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 390, + 461 + ], + "score": 1.0, + "content": "the Leaky ReLU errors with varying slopes as described in (3.1). Here", + "type": "text" + }, + { + "bbox": [ + 390, + 449, + 416, + 460 + ], + "score": 0.89, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "corresponds to ReLU", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 123, + 472 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 460, + 149, + 471 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "is the linear model. In consistence with our theory, SGD achieves linear convergence and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 117, + 483 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 117, + 471, + 125, + 482 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "increases, the rate of convergence drastically improves3. The improvement is more visible for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 223, + 494 + ], + "score": 1.0, + "content": "less stable systems driven by", + "type": "text" + }, + { + "bbox": [ + 223, + 482, + 232, + 492 + ], + "score": 0.61, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "with a larger spectral norm. In particular, while ReLU converges for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 492, + 424, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 131, + 505 + ], + "score": 1.0, + "content": "small", + "type": "text" + }, + { + "bbox": [ + 131, + 492, + 150, + 504 + ], + "score": 0.91, + "content": "\\| A \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 492, + 374, + 505 + ], + "score": 1.0, + "content": ", SGD gets stuck before reaching the ground truth when", + "type": "text" + }, + { + "bbox": [ + 375, + 492, + 420, + 504 + ], + "score": 0.93, + "content": "\\| A \\| = 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 492, + 424, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "To understand, how well SGD fits the training data, in Figure 2a, we plotted the normalized loss", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 307, + 533 + ], + "score": 1.0, + "content": "for ReLU activation. For more unstable system", + "type": "text" + }, + { + "bbox": [ + 307, + 520, + 357, + 532 + ], + "score": 0.9, + "content": "\\left. \\left. A \\right. \\right. = \\bar { 0 } . 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 519, + 506, + 533 + ], + "score": 1.0, + "content": ", training loss stagnates in a similar", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 450, + 546 + ], + "score": 1.0, + "content": "fashion to the parameter error. We also verified that the norm of the overall gradient", + "type": "text" + }, + { + "bbox": [ + 451, + 531, + 504, + 543 + ], + "score": 0.92, + "content": "\\lVert \\nabla \\mathcal { L } ( \\Theta _ { \\tau } ) \\rVert _ { F }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 216, + 555 + ], + "score": 1.0, + "content": "continues to decay (where", + "type": "text" + }, + { + "bbox": [ + 217, + 542, + 231, + 553 + ], + "score": 0.89, + "content": "\\Theta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 541, + 257, + 555 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 257, + 543, + 263, + 552 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "th SGD iterate); which implies that SGD converges before", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 227, + 566 + ], + "score": 1.0, + "content": "reaching a global minima. As", + "type": "text" + }, + { + "bbox": [ + 227, + 554, + 237, + 563 + ], + "score": 0.68, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "becomes more stable, rate of convergence improves and linear rate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "is visible. Finally, to better understand the population landscape of the quadratic loss with ReLU", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 575, + 504, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 504, + 587 + ], + "score": 1.0, + "content": "activations, Figure 2b repeats the same ReLU experiments while increasing the sample size five", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 141, + 599 + ], + "score": 1.0, + "content": "times to", + "type": "text" + }, + { + "bbox": [ + 142, + 586, + 186, + 596 + ], + "score": 0.9, + "content": "N = 2 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 585, + 456, + 599 + ], + "score": 1.0, + "content": ". For this more overdetermined problem, SGD converges even for", + "type": "text" + }, + { + "bbox": [ + 456, + 586, + 502, + 598 + ], + "score": 0.95, + "content": "\\mathbf { | } \\mathbf { \\boldsymbol { A } } | \\mathbf { | } = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 585, + 506, + 599 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 597, + 167, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 167, + 609 + ], + "score": 1.0, + "content": "indicating that", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 129, + 612, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 131, + 612, + 414, + 625 + ], + "spans": [ + { + "bbox": [ + 131, + 612, + 414, + 625 + ], + "score": 1.0, + "content": "• population landscape of loss with ReLU activation is well-behaved,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 131, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 131, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "• however ReLU problem requires more data compared to the Leaky ReLU for finding global", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 639, + 178, + 651 + ], + "spans": [ + { + "bbox": [ + 141, + 639, + 178, + 651 + ], + "score": 1.0, + "content": "minima.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 108, + 654, + 503, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Overall, as predicted by our theory, experiments verify that SGD indeed quickly finds the optimal", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 387, + 678 + ], + "score": 1.0, + "content": "weight matrices of the state equation (1.1) and as the activation slope", + "type": "text" + }, + { + "bbox": [ + 387, + 666, + 394, + 677 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "increases, the convergence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 678, + 165, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 165, + 689 + ], + "score": 1.0, + "content": "rate improves.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 691, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 287, + 704 + ], + "score": 1.0, + "content": "3Note that convergence becomes faster for larger", + "type": "text" + }, + { + "bbox": [ + 287, + 693, + 293, + 702 + ], + "score": 0.89, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 689, + 505, + 704 + ], + "score": 1.0, + "content": "under the realizable model i.e. there exists a ground truth", + "type": "text" + } + ] + }, + { + "bbox": [ + 109, + 702, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 109, + 702, + 239, + 712 + ], + "score": 1.0, + "content": "state equation with activation slope", + "type": "text" + }, + { + "bbox": [ + 239, + 703, + 245, + 712 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 702, + 505, + 712 + ], + "score": 1.0, + "content": "that can fit the observed trajectory. 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Figures a) and b)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 302, + 467, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 455, + 314 + ], + "score": 1.0, + "content": "repeat the same experiments. The difference is the spectral norm of the ground truth state matrix", + "type": "text" + }, + { + "bbox": [ + 455, + 303, + 464, + 312 + ], + "score": 0.73, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 302, + 467, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 421, + 338 + ], + "score": 1.0, + "content": "Evaluation: We consider two performance measures in the experiments. Let", + "type": "text" + }, + { + "bbox": [ + 422, + 324, + 432, + 335 + ], + "score": 0.86, + "content": "\\hat { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "be an estimate of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 219, + 348 + ], + "score": 1.0, + "content": "the ground truth parameter", + "type": "text" + }, + { + "bbox": [ + 219, + 335, + 286, + 348 + ], + "score": 0.93, + "content": "\\boldsymbol { C } \\dot { = } [ \\mu ^ { - 1 } \\boldsymbol { A } \\boldsymbol { B } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 336, + 505, + 348 + ], + "score": 1.0, + "content": ". The first measure is the normalized error defined as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 347, + 405, + 362 + ], + "spans": [ + { + "bbox": [ + 107, + 347, + 185, + 361 + ], + "score": 0.93, + "content": "\\| \\hat { C } - C \\| _ { F } ^ { 2 } / \\| C \\| _ { F } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 348, + 405, + 362 + ], + "score": 1.0, + "content": ". The second measure is the normalized loss defined as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 324, + 506, + 362 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 366, + 358, + 400 + ], + "lines": [ + { + "bbox": [ + 252, + 366, + 358, + 400 + ], + "spans": [ + { + "bbox": [ + 252, + 366, + 358, + 400 + ], + "score": 0.94, + "content": "\\frac { \\sum _ { i = 1 } ^ { N } | | \\pmb { y } _ { t } - \\phi ( \\hat { C } \\pmb { x } _ { t } ) | | _ { \\ell _ { 2 } } ^ { 2 } } { \\sum _ { i = 1 } ^ { N } | | \\pmb { y } _ { t } | | _ { \\ell _ { 2 } } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "3a97e193b5d9a5cf2ecb2bc97f48ab8a68fccd0bea8ba3320d05c4d0961e7e6a.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 252, + 366, + 358, + 383.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 252, + 383.0, + 358, + 400.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "In all experiments, we run Algorithm 1 for 50000 SGD iterations and plot these measures as a function", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 117, + 428 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 417, + 124, + 425 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 415, + 324, + 428 + ], + "score": 1.0, + "content": "; by using the estimate available at the end of the", + "type": "text" + }, + { + "bbox": [ + 324, + 417, + 330, + 425 + ], + "score": 0.69, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 415, + 414, + 428 + ], + "score": 1.0, + "content": "th SGD iteration for", + "type": "text" + }, + { + "bbox": [ + 414, + 416, + 477, + 426 + ], + "score": 0.91, + "content": "0 \\leq \\tau \\leq 5 0 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 415, + 505, + 428 + ], + "score": 1.0, + "content": ". Each", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "curve is obtained by averaging the outcomes of 20 independent realizations.Our first experiments", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 123, + 450 + ], + "score": 1.0, + "content": "use", + "type": "text" + }, + { + "bbox": [ + 123, + 438, + 162, + 448 + ], + "score": 0.89, + "content": "N = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 437, + 361, + 450 + ], + "score": 1.0, + "content": "; which is mildly larger than the total dimension", + "type": "text" + }, + { + "bbox": [ + 362, + 438, + 416, + 448 + ], + "score": 0.9, + "content": "n + p = 1 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ". In Figure 1, we plot", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 390, + 461 + ], + "score": 1.0, + "content": "the Leaky ReLU errors with varying slopes as described in (3.1). Here", + "type": "text" + }, + { + "bbox": [ + 390, + 449, + 416, + 460 + ], + "score": 0.89, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "corresponds to ReLU", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 123, + 472 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 460, + 149, + 471 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "is the linear model. In consistence with our theory, SGD achieves linear convergence and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 117, + 483 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 117, + 471, + 125, + 482 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "increases, the rate of convergence drastically improves3. The improvement is more visible for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 223, + 494 + ], + "score": 1.0, + "content": "less stable systems driven by", + "type": "text" + }, + { + "bbox": [ + 223, + 482, + 232, + 492 + ], + "score": 0.61, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "with a larger spectral norm. In particular, while ReLU converges for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 492, + 424, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 131, + 505 + ], + "score": 1.0, + "content": "small", + "type": "text" + }, + { + "bbox": [ + 131, + 492, + 150, + 504 + ], + "score": 0.91, + "content": "\\| A \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 492, + 374, + 505 + ], + "score": 1.0, + "content": ", SGD gets stuck before reaching the ground truth when", + "type": "text" + }, + { + "bbox": [ + 375, + 492, + 420, + 504 + ], + "score": 0.93, + "content": "\\| A \\| = 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 492, + 424, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 404, + 506, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 522 + ], + "score": 1.0, + "content": "To understand, how well SGD fits the training data, in Figure 2a, we plotted the normalized loss", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 307, + 533 + ], + "score": 1.0, + "content": "for ReLU activation. For more unstable system", + "type": "text" + }, + { + "bbox": [ + 307, + 520, + 357, + 532 + ], + "score": 0.9, + "content": "\\left. \\left. A \\right. \\right. = \\bar { 0 } . 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 519, + 506, + 533 + ], + "score": 1.0, + "content": ", training loss stagnates in a similar", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 450, + 546 + ], + "score": 1.0, + "content": "fashion to the parameter error. We also verified that the norm of the overall gradient", + "type": "text" + }, + { + "bbox": [ + 451, + 531, + 504, + 543 + ], + "score": 0.92, + "content": "\\lVert \\nabla \\mathcal { L } ( \\Theta _ { \\tau } ) \\rVert _ { F }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 216, + 555 + ], + "score": 1.0, + "content": "continues to decay (where", + "type": "text" + }, + { + "bbox": [ + 217, + 542, + 231, + 553 + ], + "score": 0.89, + "content": "\\Theta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 541, + 257, + 555 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 257, + 543, + 263, + 552 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "th SGD iterate); which implies that SGD converges before", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 227, + 566 + ], + "score": 1.0, + "content": "reaching a global minima. As", + "type": "text" + }, + { + "bbox": [ + 227, + 554, + 237, + 563 + ], + "score": 0.68, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "becomes more stable, rate of convergence improves and linear rate", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "is visible. Finally, to better understand the population landscape of the quadratic loss with ReLU", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 575, + 504, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 504, + 587 + ], + "score": 1.0, + "content": "activations, Figure 2b repeats the same ReLU experiments while increasing the sample size five", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 141, + 599 + ], + "score": 1.0, + "content": "times to", + "type": "text" + }, + { + "bbox": [ + 142, + 586, + 186, + 596 + ], + "score": 0.9, + "content": "N = 2 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 585, + 456, + 599 + ], + "score": 1.0, + "content": ". For this more overdetermined problem, SGD converges even for", + "type": "text" + }, + { + "bbox": [ + 456, + 586, + 502, + 598 + ], + "score": 0.95, + "content": "\\mathbf { | } \\mathbf { \\boldsymbol { A } } | \\mathbf { | } = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 585, + 506, + 599 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 597, + 167, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 167, + 609 + ], + "score": 1.0, + "content": "indicating that", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 508, + 506, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 612, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 131, + 612, + 414, + 625 + ], + "spans": [ + { + "bbox": [ + 131, + 612, + 414, + 625 + ], + "score": 1.0, + "content": "• population landscape of loss with ReLU activation is well-behaved,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 131, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 131, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "• however ReLU problem requires more data compared to the Leaky ReLU for finding global", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 639, + 178, + 651 + ], + "spans": [ + { + "bbox": [ + 141, + 639, + 178, + 651 + ], + "score": 1.0, + "content": "minima.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 131, + 612, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 654, + 503, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "Overall, as predicted by our theory, experiments verify that SGD indeed quickly finds the optimal", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 387, + 678 + ], + "score": 1.0, + "content": "weight matrices of the state equation (1.1) and as the activation slope", + "type": "text" + }, + { + "bbox": [ + 387, + 666, + 394, + 677 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "increases, the convergence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 678, + 165, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 165, + 689 + ], + "score": 1.0, + "content": "rate improves.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 654, + 506, + 689 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 146, + 81, + 505, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 81, + 505, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 81, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 146, + 81, + 505, + 199 + ], + "score": 0.97, + "type": "image", + "image_path": "49e542a82fdc668eda12f703902eb01c8a18f213a3ab21f6cde72a6a3d9f2842.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 146, + 81, + 505, + 120.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 146, + 120.33333333333334, + 505, + 159.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 159.66666666666669, + 505, + 199.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 200, + 504, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 453, + 212 + ], + "score": 1.0, + "content": "Figure 2: SGD convergence behavior for ReLU with varying spectral norm of the state matrix", + "type": "text" + }, + { + "bbox": [ + 454, + 200, + 463, + 209 + ], + "score": 0.33, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 198, + 506, + 212 + ], + "score": 1.0, + "content": ". Figures a)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 211, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 347, + 222 + ], + "score": 1.0, + "content": "and b) repeats the same experiments. The difference is that a) uses", + "type": "text" + }, + { + "bbox": [ + 348, + 211, + 384, + 221 + ], + "score": 0.9, + "content": "N = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 211, + 505, + 222 + ], + "score": 1.0, + "content": "trajectory length whereas b) uses", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 221, + 474, + 234 + ], + "spans": [ + { + "bbox": [ + 107, + 222, + 147, + 231 + ], + "score": 0.89, + "content": "N = 2 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 221, + 164, + 234 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + }, + { + "bbox": [ + 164, + 222, + 177, + 232 + ], + "score": 0.83, + "content": "\\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 221, + 474, + 234 + ], + "score": 1.0, + "content": "more data). Shaded regions highlight the one standard deviation around the mean.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 243, + 201, + 255 + ], + "lines": [ + { + "bbox": [ + 104, + 241, + 203, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 203, + 259 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 263, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "This work showed that SGD can learn the nonlinear dynamical system (1.1); which is characterized by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "weight matrices and an activation function. This problem is of interest for recurrent neural networks", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "score": 1.0, + "content": "as well as nonlinear system identification. We showed that efficient learning is possible with optimal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 310 + ], + "score": 1.0, + "content": "sample complexity and good computational performance. Our results apply to strictly increasing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "activations such as Leaky ReLU. We empirically showed that Leaky ReLU converges faster than", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "ReLU and requires less samples; in consistence with our theory. We list a few unanswered problems", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 329, + 373, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 373, + 342 + ], + "score": 1.0, + "content": "that would provide further insights into recurrent neural networks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "• Covariance of the state-vector: Our results depend on the covariance of the state-vector and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "requires it to be positive definite. One might be able to improve the current bounds on the condition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "number and relax the assumptions on the activation function. 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Even for linear dynamical systems,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 155, + 417 + ], + "score": 1.0, + "content": "learning the", + "type": "text" + }, + { + "bbox": [ + 156, + 405, + 213, + 417 + ], + "score": 0.93, + "content": "( A , B , C , D { \\bar { ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "system ((1.1), (2.1)) is a non-trivial task Ho & Kalman (1966); Hardt et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 416, + 369, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 369, + 429 + ], + "score": 1.0, + "content": "(2016). 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Shaded regions highlight the one standard deviation around the mean.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 243, + 201, + 255 + ], + "lines": [ + { + "bbox": [ + 104, + 241, + 203, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 203, + 259 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 263, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "This work showed that SGD can learn the nonlinear dynamical system (1.1); which is characterized by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "weight matrices and an activation function. This problem is of interest for recurrent neural networks", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "score": 1.0, + "content": "as well as nonlinear system identification. We showed that efficient learning is possible with optimal", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 295, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 310 + ], + "score": 1.0, + "content": "sample complexity and good computational performance. Our results apply to strictly increasing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "activations such as Leaky ReLU. We empirically showed that Leaky ReLU converges faster than", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "ReLU and requires less samples; in consistence with our theory. We list a few unanswered problems", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 329, + 373, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 373, + 342 + ], + "score": 1.0, + "content": "that would provide further insights into recurrent neural networks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 263, + 505, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "• Covariance of the state-vector: Our results depend on the covariance of the state-vector and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "requires it to be positive definite. 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Deriving similar performance bounds", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 371, + 253, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 253, + 387 + ], + "score": 1.0, + "content": "for ReLU is particularly interesting.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 339, + 505, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "• Hidden state: For RNNs, the state vector is hidden and is observed through an additional equation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "(2.1); which further complicates the optimization landscape. Even for linear dynamical systems,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 155, + 417 + ], + "score": 1.0, + "content": "learning the", + "type": "text" + }, + { + "bbox": [ + 156, + 405, + 213, + 417 + ], + "score": 0.93, + "content": "( A , B , C , D { \\bar { ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "system ((1.1), (2.1)) is a non-trivial task Ho & Kalman (1966); Hardt et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 416, + 369, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 369, + 429 + ], + "score": 1.0, + "content": "(2016). What can be said when we add the nonlinear activations?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 382, + 506, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "• Classification task: In this work, we used normally distributed input and least-squares regression", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 439, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 450 + ], + "score": 1.0, + "content": "for our theoretical guarantees. More realistic input distributions might provide better insight into", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "contemporary problems, such as natural language processing; where the goal is closer to classification", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 460, + 333, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 333, + 473 + ], + "score": 1.0, + "content": "(e.g. finding the best translation from another language).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 426, + 505, + 473 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 506, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 110 + ], + "score": 1.0, + "content": "Alekh Agarwal, Sahand Negahban, and Martin J Wainwright. 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2 \\eta \\mathbb { E } [ \\pmb { w } _ { \\tau } ^ { T } \\pmb { G } _ { r _ { \\tau } } \\pmb { w } _ { \\tau } ] + \\eta ^ { 2 } \\mathbb { E } [ \\pmb { w } _ { \\tau } ^ { T } \\pmb { G } _ { r _ { \\tau } } ^ { T } \\pmb { G } _ { r _ { \\tau } } \\pmb { w } _ { \\tau } ] } \\\\ & { \\quad \\quad \\quad \\le \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ( 1 - 2 \\eta \\beta ^ { 2 } \\gamma _ { - } + \\eta ^ { 2 } B \\gamma _ { + } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "6e4326ef251b2ea42c7c3e47dbbb8e77042e19141d3742696f76469f639dc496.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 174, + 350, + 437, + 365.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 174, + 365.6666666666667, + 437, + 381.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 174, + 381.33333333333337, + 437, + 397.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 398, + 283, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 283, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 147, + 414 + ], + "score": 1.0, + "content": "Setting η =", + "type": "text" + }, + { + "bbox": [ + 134, + 398, + 173, + 416 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\eta = \\frac { \\beta ^ { 2 } \\gamma _ { - } } { \\gamma _ { + } B } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 402, + 283, + 412 + ], + "score": 1.0, + "content": ", we find the advertised bound", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 418, + 382, + 443 + ], + "lines": [ + { + "bbox": [ + 228, + 418, + 382, + 443 + ], + "spans": [ + { + "bbox": [ + 228, + 418, + 382, + 443 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\pmb { w } _ { \\tau + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\mathbb { E } [ \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ] ( 1 - \\frac { \\beta ^ { 4 } \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } B } ) .", + "type": "interline_equation", + "image_path": "434bec7af88fbe0f437bdcaef0f20a55e29b7ebd1a7f5ef1e554bac731fbc73b.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 228, + 418, + 382, + 443 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 444, + 380, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 380, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 245, + 457 + ], + "score": 1.0, + "content": "Applying induction over the iterations", + "type": "text" + }, + { + "bbox": [ + 246, + 449, + 251, + 454 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 444, + 380, + 457 + ], + "score": 1.0, + "content": ", we find the advertised bound (4.2)", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 458, + 373, + 483 + ], + "lines": [ + { + "bbox": [ + 237, + 458, + 373, + 483 + ], + "spans": [ + { + "bbox": [ + 237, + 458, + 373, + 483 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\pmb { w } _ { \\tau } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { w } _ { 0 } \\| _ { \\ell _ { 2 } } ^ { 2 } ( 1 - 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h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } \\leq \\| \\phi ( A h _ { t } ( \\boldsymbol { u } _ { \\tau } ) + B \\boldsymbol { u } _ { t } ) - \\phi ( A h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) + B \\boldsymbol { u } _ { t } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| A ( h _ { t } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| A \\| \\| h _ { t } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "f1cf1bd772e11139bef4000ed3d11e0301248cdb11e51c18b267cc0b2712f1b0.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 155, + 282, + 455, + 297.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 155, + 297.0, + 455, + 312.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 155, + 312.0, + 455, + 327.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 291, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 292, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 237, + 341 + ], + "score": 1.0, + "content": "Proceeding with this recursion until", + "type": "text" + }, + { + "bbox": [ + 237, + 331, + 258, + 339 + ], + "score": 0.89, + "content": "t = \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 329, + 292, + 341 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 344, + 455, + 391 + ], + "lines": [ + { + "bbox": [ + 155, + 344, + 455, + 391 + ], + "spans": [ + { + "bbox": [ + 155, + 344, + 455, + 391 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ) - 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1 } ^ { T } ] ^ { T } \\in \\mathbb { R } ^ { t p } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 443, + 406, + 460 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 132, + 464, + 485, + 477 + ], + "lines": [ + { + "bbox": [ + 131, + 464, + 486, + 478 + ], + "spans": [ + { + "bbox": [ + 131, + 464, + 263, + 478 + ], + "score": 1.0, + "content": "• There exists an absolute constant", + "type": "text" + }, + { + "bbox": [ + 264, + 466, + 286, + 475 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 464, + 322, + 478 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 322, + 465, + 408, + 477 + ], + "score": 0.92, + "content": "\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq c B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 464, + 426, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 465, + 482, + 477 + ], + "score": 0.92, + "content": "\\Sigma [ h _ { t } ] \\preceq B _ { t } ^ { 2 } { \\cal I } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 464, + 486, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 133, + 483, + 185, + 494 + ], + "lines": [ + { + "bbox": [ + 132, + 482, + 185, + 495 + ], + "spans": [ + { + "bbox": [ + 132, + 482, + 142, + 495 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 484, + 153, + 493 + ], + "score": 0.83, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 482, + 185, + 495 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 492, + 420, + 517 + ], + "lines": [ + { + "bbox": [ + 225, + 492, + 420, + 517 + ], + "spans": [ + { + "bbox": [ + 225, + 492, + 420, + 517 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\pmb { h } _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le t r ( \\pmb { B } \\pmb { B } ^ { T } ) \\frac { 1 - 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h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } \\leq \\| \\phi ( A h _ { t } ( \\boldsymbol { u } _ { \\tau } ) + B \\boldsymbol { u } _ { t } ) - \\phi ( A h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) + B \\boldsymbol { u } _ { t } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| A ( h _ { t } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| A \\| \\| h _ { t } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "f1cf1bd772e11139bef4000ed3d11e0301248cdb11e51c18b267cc0b2712f1b0.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 155, + 282, + 455, + 297.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 155, + 297.0, + 455, + 312.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 155, + 312.0, + 455, + 327.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 291, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 292, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 237, + 341 + ], + "score": 1.0, + "content": "Proceeding with this recursion until", + "type": "text" + }, + { + "bbox": [ + 237, + 331, + 258, + 339 + ], + "score": 0.89, + "content": "t = \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 329, + 292, + 341 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 329, + 292, + 341 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 344, + 455, + 391 + ], + "lines": [ + { + "bbox": [ + 155, + 344, + 455, + 391 + ], + "spans": [ + { + "bbox": [ + 155, + 344, + 455, + 391 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ) - h _ { t + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } \\leq \\| \\boldsymbol { A } \\| ^ { t - \\tau } \\| h _ { \\tau + 1 } ( \\boldsymbol { u } _ { \\tau } ) - h _ { \\tau + 1 } ( \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\boldsymbol { A } \\| ^ { t - \\tau } \\| \\phi ( \\boldsymbol { A } h _ { \\tau } + \\boldsymbol { B } \\boldsymbol { u } _ { \\tau } ) - \\phi ( \\boldsymbol { A } h _ { \\tau } + \\boldsymbol { B } \\boldsymbol { u } _ { \\tau } ^ { \\prime } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\boldsymbol { A } \\| ^ { t - \\tau } \\| \\boldsymbol { B } \\| \\| \\boldsymbol { u } _ { \\tau } - \\boldsymbol { u } _ { \\tau } ^ { \\prime } \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "46df81ca8e517a0d8dcb1945b197fd60a5721b6b4885ec0ccb9323cb3fab25c4.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 155, + 344, + 455, + 359.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 155, + 359.6666666666667, + 455, + 375.33333333333337 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 155, + 375.33333333333337, + 455, + 391.00000000000006 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 367, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 365, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 177, + 408 + ], + "score": 1.0, + "content": "This bound implies", + "type": "text" + }, + { + "bbox": [ + 178, + 395, + 216, + 406 + ], + "score": 0.92, + "content": "\\pmb { h } _ { t + 1 } ( \\pmb { u } _ { \\tau } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 394, + 225, + 408 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 225, + 394, + 274, + 406 + ], + "score": 0.93, + "content": "\\| A \\| ^ { t - 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Consider the state equation governed by equation (1.1). Suppose", + "type": "text" + }, + { + "bbox": [ + 443, + 413, + 503, + 427 + ], + "score": 0.91, + "content": "{ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 410, + 509, + 431 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 425, + 490, + 438 + ], + "spans": [ + { + "bbox": [ + 107, + 426, + 113, + 436 + ], + "score": 0.8, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 425, + 167, + 438 + ], + "score": 1.0, + "content": "is 1-Lipschitz,", + "type": "text" + }, + { + "bbox": [ + 167, + 426, + 202, + 437 + ], + "score": 0.93, + "content": "\\bar { \\phi } ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 425, + 219, + 438 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 426, + 247, + 435 + ], + "score": 0.9, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 425, + 354, + 438 + ], + "score": 1.0, + "content": ". 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We have the following properties", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 102, + 410, + 509, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 444, + 407, + 458 + ], + "lines": [ + { + "bbox": [ + 131, + 443, + 406, + 460 + ], + "spans": [ + { + "bbox": [ + 131, + 443, + 142, + 460 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 447, + 153, + 456 + ], + "score": 0.85, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 443, + 169, + 460 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 169, + 447, + 180, + 456 + ], + "score": 0.85, + "content": "B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 443, + 295, + 460 + ], + "score": 1.0, + "content": "-Lipschitz function of the vector", + "type": "text" + }, + { + "bbox": [ + 296, + 445, + 403, + 458 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\pmb q _ { t } = [ \\pmb { u } _ { 0 } ^ { T } \\ . . . \\ \\pmb { u } _ { t - 1 } ^ { T } ] ^ { T } \\in \\mathbb { R } ^ { t p } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 443, + 406, + 460 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 131, + 443, + 406, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 464, + 485, + 477 + ], + "lines": [ + { + "bbox": [ + 131, + 464, + 486, + 478 + ], + "spans": [ + { + "bbox": [ + 131, + 464, + 263, + 478 + ], + "score": 1.0, + "content": "• There exists an absolute constant", + "type": "text" + }, + { + "bbox": [ + 264, + 466, + 286, + 475 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 464, + 322, + 478 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 322, + 465, + 408, + 477 + ], + "score": 0.92, + "content": "\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq c B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 464, + 426, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 426, + 465, + 482, + 477 + ], + "score": 0.92, + "content": "\\Sigma [ h _ { t } ] \\preceq B _ { t } ^ { 2 } { \\cal I } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 464, + 486, + 478 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 131, + 464, + 486, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 483, + 185, + 494 + ], + "lines": [ + { + "bbox": [ + 132, + 482, + 185, + 495 + ], + "spans": [ + { + "bbox": [ + 132, + 482, + 142, + 495 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 484, + 153, + 493 + ], + "score": 0.83, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 482, + 185, + 495 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 132, + 482, + 185, + 495 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 492, + 420, + 517 + ], + "lines": [ + { + "bbox": [ + 225, + 492, + 420, + 517 + ], + "spans": [ + { + "bbox": [ + 225, + 492, + 420, + 517 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\pmb { h } _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le t r ( \\pmb { B } \\pmb { B } ^ { T } ) \\frac { 1 - 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This is true for all unit", + "type": "text" + }, + { + "bbox": [ + 496, + 312, + 502, + 318 + ], + "score": 0.73, + "content": "_ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 308, + 506, + 321 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 245, + 332 + ], + "score": 1.0, + "content": "hence using Definition B.1, the vector", + "type": "text" + }, + { + "bbox": [ + 246, + 321, + 256, + 329 + ], + "score": 0.88, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 319, + 288, + 332 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 289, + 320, + 386, + 330 + ], + "score": 0.89, + "content": "\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( B _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 319, + 456, + 332 + ], + "score": 1.0, + "content": "as well. 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Since", + "type": "text" + }, + { + "bbox": [ + 392, + 365, + 399, + 374 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 362, + 467, + 376 + ], + "score": 1.0, + "content": "is 1-Lipschitz and", + "type": "text" + }, + { + "bbox": [ + 468, + 363, + 502, + 375 + ], + "score": 0.92, + "content": "\\phi ( 0 ) = 0", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 374, + 230, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 230, + 384 + ], + "score": 1.0, + "content": "we have the deterministic relation", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 382, + 363, + 395 + ], + "lines": [ + { + "bbox": [ + 247, + 382, + 363, + 395 + ], + "spans": [ + { + "bbox": [ + 247, + 382, + 363, + 395 + ], + "score": 0.9, + "content": "\\lVert h _ { t + 1 } \\rVert _ { \\ell _ { 2 } } \\leq \\lVert A h _ { t } + B u _ { t } \\rVert _ { \\ell _ { 2 } } .", + "type": "interline_equation", + "image_path": "a0e52b0cab60fdc777b786ccffeda460a160862c15ea573c5c609fd83c31aa60.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 247, + 382, + 363, + 395 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 504, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 449, + 408 + ], + "score": 1.0, + "content": "Taking squares of both sides, expanding the right hand side, and using the independence of", + "type": "text" + }, + { + "bbox": [ + 450, + 397, + 473, + 407 + ], + "score": 0.91, + "content": "\\mathbf { } h _ { t } , \\mathbf { } u _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 254, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 202, + 418 + ], + "score": 1.0, + "content": "covariance information of", + "type": "text" + }, + { + "bbox": [ + 202, + 408, + 212, + 416 + ], + "score": 0.8, + "content": "\\mathbf { \\Delta } \\mathbf { u } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 407, + 254, + 418 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 420, + 431, + 451 + ], + "lines": [ + { + "bbox": [ + 180, + 420, + 431, + 451 + ], + "spans": [ + { + "bbox": [ + 180, + 420, + 431, + 451 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\mathbb { E } [ \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\leq \\mathbb { E } [ \\| A h _ { t } + B u _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] = \\mathbb { E } [ \\| A h _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] + \\mathbb { E } [ \\| B u _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] } \\\\ & { \\qquad \\leq \\| A \\| ^ { 2 } \\mathbb { E } [ \\| h _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] + \\mathrm { t r } ( B B ^ { T } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "d6023fa5130713f0ef269258d0c63584e7da52796c3931457b158461bbf4fdaf.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 180, + 420, + 431, + 430.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 180, + 430.3333333333333, + 431, + 440.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 180, + 440.66666666666663, + 431, + 450.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 453, + 462, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 462, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 282, + 466 + ], + "score": 1.0, + "content": "Now that the recursion is established, expanding", + "type": "text" + }, + { + "bbox": [ + 282, + 454, + 293, + 464 + ], + "score": 0.89, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 452, + 393, + 466 + ], + "score": 1.0, + "content": "on the right hand side until", + "type": "text" + }, + { + "bbox": [ + 393, + 454, + 421, + 464 + ], + "score": 0.91, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 452, + 462, + 466 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 468, + 427, + 499 + ], + "lines": [ + { + "bbox": [ + 184, + 468, + 427, + 499 + ], + "spans": [ + { + "bbox": [ + 184, + 468, + 427, + 499 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\pmb { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\sum _ { i = 0 } ^ { t } \\| \\pmb { A } \\| ^ { 2 i } \\mathrm { t r } ( \\pmb { B } \\pmb { B } ^ { T } ) \\le \\mathrm { t r } ( \\pmb { B } \\pmb { B } ^ { T } ) \\frac { 1 - \\| \\pmb { A } \\| ^ { 2 ( t + 1 ) } } { 1 - \\| \\pmb { A } \\| ^ { 2 } } .", + "type": "interline_equation", + "image_path": "fd32011a4c51f6f7f5d6523e32a4fa2d4a0ce9f19442d42a701903cb3ff86690.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 468, + 427, + 483.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 184, + 483.5, + 427, + 499.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 400, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 400, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 192, + 518 + ], + "score": 1.0, + "content": "Now using the fact that", + "type": "text" + }, + { + "bbox": [ + 192, + 503, + 367, + 516 + ], + "score": 0.92, + "content": "\\mathrm { t r } ( B B ^ { T } ) \\leq \\mathrm { r a n k } ( B ) \\| B \\| ^ { 2 } \\leq \\mathrm { m i n } \\{ n , p \\} \\| B \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 502, + 400, + 518 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 520, + 398, + 534 + ], + "lines": [ + { + "bbox": [ + 213, + 520, + 398, + 534 + ], + "spans": [ + { + "bbox": [ + 213, + 520, + 398, + 534 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } [ \\| \\boldsymbol { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ] ^ { 2 } \\le \\mathbb { E } [ \\| \\boldsymbol { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\operatorname* { m i n } \\{ n , p \\} B _ { t + 1 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8ef052fae966b0e93225616dc91f3f74a2b0199689e42357f3f5f5bda572fa13.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 213, + 520, + 398, + 534 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 537, + 504, + 557 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 502, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 201, + 550 + ], + "score": 1.0, + "content": "Finally, using the fact that", + "type": "text" + }, + { + "bbox": [ + 201, + 538, + 212, + 548 + ], + "score": 0.88, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 536, + 221, + 550 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 222, + 538, + 232, + 548 + ], + "score": 0.89, + "content": "B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 536, + 444, + 550 + ], + "score": 1.0, + "content": "-Lipschitz function and utilizing Gaussian concentration of", + "type": "text" + }, + { + "bbox": [ + 444, + 538, + 502, + 549 + ], + "score": 0.92, + "content": "\\mathbf { \\boldsymbol { q } } _ { t } \\sim \\mathcal { N } ( 0 , I _ { t p } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 547, + 136, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 136, + 559 + ], + "score": 1.0, + "content": "we find", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 555, + 398, + 580 + ], + "lines": [ + { + "bbox": [ + 212, + 555, + 398, + 580 + ], + "spans": [ + { + "bbox": [ + 212, + 555, + 398, + 580 + ], + "score": 0.91, + "content": "\\mathbb { P } ( \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } - \\mathbb { E } [ \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } ] \\ge t ) \\le \\exp ( - \\frac { t ^ { 2 } } { 2 B _ { t } ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "f5cba27dd8c4380c6b6c443ff76d39a39170fcc07ad127ea728125ad3a8f8910.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 212, + 555, + 398, + 580 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 503, + 603 + ], + "lines": [ + { + "bbox": [ + 104, + 578, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 135, + 597 + ], + "score": 1.0, + "content": "Setting", + "type": "text" + }, + { + "bbox": [ + 135, + 582, + 208, + 593 + ], + "score": 0.91, + "content": "t = ( c - 1 ) \\sqrt { m } B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 578, + 288, + 597 + ], + "score": 1.0, + "content": "for sufficiently large", + "type": "text" + }, + { + "bbox": [ + 289, + 583, + 313, + 592 + ], + "score": 0.88, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 578, + 348, + 597 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + }, + { + "bbox": [ + 348, + 582, + 505, + 594 + ], + "score": 0.87, + "content": "\\mathbb { P } ( \\| h _ { t } \\| _ { \\ell _ { 2 } } \\geq \\sqrt { n } B _ { t } + ( c - 1 ) \\sqrt { m } B _ { t } ) \\leq", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 591, + 163, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 592, + 158, + 603 + ], + "score": 0.88, + "content": "\\exp ( - 1 0 0 m )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 591, + 163, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 503, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 259, + 621 + ], + "score": 1.0, + "content": "Lemma B.4 (Odd activations). Suppose", + "type": "text" + }, + { + "bbox": [ + 260, + 610, + 267, + 619 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 607, + 387, + 621 + ], + "score": 1.0, + "content": "is strictly increasing and obeys", + "type": "text" + }, + { + "bbox": [ + 387, + 609, + 452, + 620 + ], + "score": 0.92, + "content": "\\phi ( x ) = - \\phi ( - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 607, + 480, + 621 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 481, + 611, + 487, + 618 + ], + "score": 0.73, + "content": "_ x", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 441, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 135, + 632 + ], + "score": 0.89, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 616, + 284, + 636 + ], + "score": 1.0, + "content": ". Consider the state equation (1.1) driven", + "type": "text" + }, + { + "bbox": [ + 285, + 620, + 345, + 633 + ], + "score": 0.91, + "content": "{ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 616, + 397, + 636 + ], + "score": 1.0, + "content": ". We have that", + "type": "text" + }, + { + "bbox": [ + 397, + 622, + 435, + 633 + ], + "score": 0.92, + "content": "\\mathbb { E } [ h _ { t } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 616, + 441, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 504, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 245, + 658 + ], + "score": 1.0, + "content": "Proof. We will inductively show that", + "type": "text" + }, + { + "bbox": [ + 245, + 645, + 277, + 656 + ], + "score": 0.92, + "content": "\\{ h _ { t } \\} _ { t \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 644, + 493, + 658 + ], + "score": 1.0, + "content": "has a symmetric distribution around 0. Suppose the vector", + "type": "text" + }, + { + "bbox": [ + 493, + 646, + 504, + 655 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 212, + 667 + ], + "score": 1.0, + "content": "satisfies this assumption. Let", + "type": "text" + }, + { + "bbox": [ + 212, + 655, + 242, + 664 + ], + "score": 0.91, + "content": "S \\subset \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 654, + 343, + 667 + ], + "score": 1.0, + "content": "be a set. We will argue that", + "type": "text" + }, + { + "bbox": [ + 343, + 655, + 463, + 666 + ], + "score": 0.9, + "content": "\\mathbb { P } ( h _ { t + 1 } \\subset S ) = \\mathbb { P } ( h _ { t + 1 } \\subset - S )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 654, + 489, + 667 + ], + "score": 1.0, + "content": ". 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\\leq \\lVert A h _ { t } + B u _ { t } \\rVert _ { \\ell _ { 2 } } .", + "type": "interline_equation", + "image_path": "a0e52b0cab60fdc777b786ccffeda460a160862c15ea573c5c609fd83c31aa60.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 247, + 382, + 363, + 395 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 504, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 449, + 408 + ], + "score": 1.0, + "content": "Taking squares of both sides, expanding the right hand side, and using the independence of", + "type": "text" + }, + { + "bbox": [ + 450, + 397, + 473, + 407 + ], + "score": 0.91, + "content": "\\mathbf { } h _ { t } , \\mathbf { } u _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 254, + 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+ \\mathbb { E } [ \\| B u _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] } \\\\ & { \\qquad \\leq \\| A \\| ^ { 2 } \\mathbb { E } [ \\| h _ { t } \\| _ { \\ell _ { 2 } } ^ { 2 } ] + \\mathrm { t r } ( B B ^ { T } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "d6023fa5130713f0ef269258d0c63584e7da52796c3931457b158461bbf4fdaf.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 180, + 420, + 431, + 430.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 180, + 430.3333333333333, + 431, + 440.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 180, + 440.66666666666663, + 431, + 450.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 453, + 462, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 462, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 282, + 466 + ], + "score": 1.0, + "content": "Now that the recursion is established, expanding", + "type": "text" + }, + { + "bbox": [ + 282, + 454, + 293, + 464 + ], + "score": 0.89, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 452, + 393, + 466 + ], + "score": 1.0, + "content": "on the right hand side until", + "type": "text" + }, + { + "bbox": [ + 393, + 454, + 421, + 464 + ], + "score": 0.91, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 452, + 462, + 466 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 452, + 462, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 468, + 427, + 499 + ], + "lines": [ + { + "bbox": [ + 184, + 468, + 427, + 499 + ], + "spans": [ + { + "bbox": [ + 184, + 468, + 427, + 499 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\pmb { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\sum _ { i = 0 } ^ { t } \\| \\pmb { A } \\| ^ { 2 i } \\mathrm { t r } ( \\pmb { B } \\pmb { B } ^ { T } ) \\le \\mathrm { t r } ( \\pmb { B } \\pmb { B } ^ { T } ) \\frac { 1 - \\| \\pmb { A } \\| ^ { 2 ( t + 1 ) } } { 1 - \\| \\pmb { A } \\| ^ { 2 } } .", + "type": "interline_equation", + "image_path": "fd32011a4c51f6f7f5d6523e32a4fa2d4a0ce9f19442d42a701903cb3ff86690.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 468, + 427, + 483.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 184, + 483.5, + 427, + 499.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 400, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 400, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 192, + 518 + ], + "score": 1.0, + "content": "Now using the fact that", + "type": "text" + }, + { + "bbox": [ + 192, + 503, + 367, + 516 + ], + "score": 0.92, + "content": "\\mathrm { t r } ( B B ^ { T } ) \\leq \\mathrm { r a n k } ( B ) \\| B \\| ^ { 2 } \\leq \\mathrm { m i n } \\{ n , p \\} \\| B \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 502, + 400, + 518 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 502, + 400, + 518 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 520, + 398, + 534 + ], + "lines": [ + { + "bbox": [ + 213, + 520, + 398, + 534 + ], + "spans": [ + { + "bbox": [ + 213, + 520, + 398, + 534 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } [ \\| \\boldsymbol { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ] ^ { 2 } \\le \\mathbb { E } [ \\| \\boldsymbol { h } _ { t + 1 } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\operatorname* { m i n } \\{ n , p \\} B _ { t + 1 } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "8ef052fae966b0e93225616dc91f3f74a2b0199689e42357f3f5f5bda572fa13.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 213, + 520, + 398, + 534 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 537, + 504, + 557 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 502, + 550 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 201, + 550 + ], + "score": 1.0, + "content": "Finally, using the fact that", + "type": "text" + }, + { + "bbox": [ + 201, + 538, + 212, + 548 + ], + "score": 0.88, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 536, + 221, + 550 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 222, + 538, + 232, + 548 + ], + "score": 0.89, + "content": "B _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 536, + 444, + 550 + ], + "score": 1.0, + "content": "-Lipschitz function and utilizing Gaussian concentration of", + "type": "text" + }, + { + "bbox": [ + 444, + 538, + 502, + 549 + ], + "score": 0.92, + "content": "\\mathbf { \\boldsymbol { q } } _ { t } \\sim \\mathcal { N } ( 0 , I _ { t p } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 547, + 136, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 136, + 559 + ], + "score": 1.0, + "content": "we find", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 536, + 502, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 555, + 398, + 580 + ], + "lines": [ + { + "bbox": [ + 212, + 555, + 398, + 580 + ], + "spans": [ + { + "bbox": [ + 212, + 555, + 398, + 580 + ], + "score": 0.91, + "content": "\\mathbb { P } ( \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } - \\mathbb { E } [ \\| h _ { t + 1 } \\| _ { \\ell _ { 2 } } ] \\ge t ) \\le \\exp ( - \\frac { t ^ { 2 } } { 2 B _ { t } ^ { 2 } } ) .", + "type": "interline_equation", + "image_path": "f5cba27dd8c4380c6b6c443ff76d39a39170fcc07ad127ea728125ad3a8f8910.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 212, + 555, + 398, + 580 + ], + 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\\leq", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 591, + 163, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 592, + 158, + 603 + ], + "score": 0.88, + "content": "\\exp ( - 1 0 0 m )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 591, + 163, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 578, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 503, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 259, + 621 + ], + "score": 1.0, + "content": "Lemma B.4 (Odd activations). Suppose", + "type": "text" + }, + { + "bbox": [ + 260, + 610, + 267, + 619 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 607, + 387, + 621 + ], + "score": 1.0, + "content": "is strictly increasing and obeys", + "type": "text" + }, + { + "bbox": [ + 387, + 609, + 452, + 620 + ], + "score": 0.92, + "content": "\\phi ( x ) = - \\phi ( - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 607, + 480, + 621 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 481, + 611, + 487, + 618 + ], + "score": 0.73, + "content": "_ x", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 607, + 505, + 621 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 441, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 135, + 632 + ], + "score": 0.89, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 616, + 284, + 636 + ], + "score": 1.0, + "content": ". Consider the state equation (1.1) driven", + "type": "text" + }, + { + "bbox": [ + 285, + 620, + 345, + 633 + ], + "score": 0.91, + "content": "{ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 616, + 397, + 636 + ], + "score": 1.0, + "content": ". We have that", + "type": "text" + }, + { + "bbox": [ + 397, + 622, + 435, + 633 + ], + "score": 0.92, + "content": "\\mathbb { E } [ h _ { t } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 616, + 441, + 636 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 607, + 505, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 504, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 245, + 658 + ], + "score": 1.0, + "content": "Proof. We will inductively show that", + "type": "text" + }, + { + "bbox": [ + 245, + 645, + 277, + 656 + ], + "score": 0.92, + "content": "\\{ h _ { t } \\} _ { t \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 644, + 493, + 658 + ], + "score": 1.0, + "content": "has a symmetric distribution around 0. Suppose the vector", + "type": "text" + }, + { + "bbox": [ + 493, + 646, + 504, + 655 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 212, + 667 + ], + "score": 1.0, + "content": "satisfies this assumption. Let", + "type": "text" + }, + { + "bbox": [ + 212, + 655, + 242, + 664 + ], + "score": 0.91, + "content": "S \\subset \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 654, + 343, + 667 + ], + "score": 1.0, + "content": "be a set. We will argue that", + "type": "text" + }, + { + "bbox": [ + 343, + 655, + 463, + 666 + ], + "score": 0.9, + "content": "\\mathbb { P } ( h _ { t + 1 } \\subset S ) = \\mathbb { P } ( h _ { t + 1 } \\subset - S )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 654, + 489, + 667 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 489, + 656, + 496, + 665 + ], + "score": 0.87, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 662, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 403, + 677 + ], + "score": 1.0, + "content": "strictly increasing, it is bijective on vectors, and we can define the unique inverse set", + "type": "text" + }, + { + "bbox": [ + 404, + 665, + 454, + 676 + ], + "score": 0.9, + "content": "S ^ { \\prime } = \\phi ^ { - 1 } ( S )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 662, + 497, + 677 + ], + "score": 1.0, + "content": ". Also since", + "type": "text" + }, + { + "bbox": [ + 497, + 666, + 504, + 675 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 675, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 132, + 687 + ], + "score": 1.0, + "content": "is odd,", + "type": "text" + }, + { + "bbox": [ + 133, + 675, + 187, + 686 + ], + "score": 0.93, + "content": "\\phi ( - S ^ { \\prime } ) = - S", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 675, + 213, + 687 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 213, + 676, + 237, + 685 + ], + "score": 0.88, + "content": "\\mathbf { } h _ { t } , \\mathbf { } u _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 675, + 506, + 687 + ], + "score": 1.0, + "content": "are independent and symmetric, we reach the desired conclusion as follows", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 644, + 506, + 687 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 689, + 471, + 718 + ], + "lines": [ + { + "bbox": [ + 120, + 689, + 471, + 718 + ], + "spans": [ + { + "bbox": [ + 120, + 689, + 471, + 718 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbb { P } ( h _ { t + 1 } \\subset S ) = \\mathbb { P } ( A h _ { t } + B u _ { t } \\subset S ^ { \\prime } ) = \\mathbb { P } ( A ( - h _ { t } ) + B ( - u _ { t } ) \\subset S ^ { \\prime } ) } \\\\ & { \\qquad = \\mathbb { P } ( A h _ { t } + B u _ { t } \\subset - S ^ { \\prime } ) = \\mathbb { P } ( \\phi ( A h _ { t } + B u _ { t } ) \\subset \\phi ( - S ^ { \\prime } ) ) = \\mathbb { P } ( h _ { t + 1 } \\subset - S ) . } \\end{array}", + "type": "interline_equation", + "image_path": "04b1452aad1b80c9578a4fab94dfb9a5c9e3edc98205a77b64047d3057155114.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 120, + 689, + 471, + 698.6666666666666 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 120, + 698.6666666666666, + 471, + 708.3333333333333 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 120, + 708.3333333333333, + 471, + 717.9999999999999 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 455, + 96 + ], + "score": 1.0, + "content": "Theorem B.5 (State-vector lower bound). Consider the nonlinear state equation (1.1) with", + "type": "text" + }, + { + "bbox": [ + 456, + 82, + 488, + 95 + ], + "score": 0.57, + "content": "\\{ { \\pmb u } _ { t } \\} _ { t \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 79, + 505, + 96 + ], + "score": 1.0, + "content": "i.i.d. ∼", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 141, + 104 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 93, + 177, + 105 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 177, + 94, + 184, + 104 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 93, + 199, + 105 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 200, + 94, + 207, + 104 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 93, + 344, + 105 + ], + "score": 1.0, + "content": "-increasing function for some constant", + "type": "text" + }, + { + "bbox": [ + 345, + 93, + 368, + 104 + ], + "score": 0.9, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 93, + 401, + 105 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 402, + 94, + 423, + 104 + ], + "score": 0.89, + "content": "t \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ", the state vector obeys", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 109, + 358, + 123 + ], + "lines": [ + { + "bbox": [ + 253, + 109, + 358, + 123 + ], + "spans": [ + { + "bbox": [ + 253, + 109, + 358, + 123 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\Sigma [ h _ { t } ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B B ^ { T } ) I _ { n } . } \\end{array}", + "type": "interline_equation", + "image_path": "c0020b9459ed9c728ec193009573a65ddca6e99eb0404fba1e9b573aeb7d0647.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 253, + 109, + 358, + 123 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 506, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 391, + 146 + ], + "score": 1.0, + "content": "Proof. The proof is an application of Lemma B.7. The main idea is to write", + "type": "text" + }, + { + "bbox": [ + 391, + 135, + 402, + 144 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 134, + 505, + 146 + ], + "score": 1.0, + "content": "as sum of two independent", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 142, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 437, + 158 + ], + "score": 1.0, + "content": "vectors, one of which has independent entries. Consider a multivariate Gaussian vector", + "type": "text" + }, + { + "bbox": [ + 437, + 144, + 491, + 155 + ], + "score": 0.89, + "content": "\\mathbf { \\sigma } _ { \\mathbf { \\mathcal { g } } } \\sim \\mathcal { N } ( \\bar { 0 , } \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 142, + 496, + 158 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 146, + 504, + 155 + ], + "score": 0.46, + "content": "\\pmb { g }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 205, + 167 + ], + "score": 1.0, + "content": "is statistically identical to", + "type": "text" + }, + { + "bbox": [ + 206, + 155, + 238, + 165 + ], + "score": 0.9, + "content": "{ \\pmb g } _ { 1 } + { \\pmb g } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 153, + 265, + 167 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 266, + 154, + 358, + 165 + ], + "score": 0.91, + "content": "{ \\pmb g } _ { 1 } \\sim \\mathcal { N } ( 0 , s _ { \\mathrm { m i n } } ( { \\pmb \\Sigma } ) { \\pmb I } _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 153, + 376, + 167 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 154, + 489, + 165 + ], + "score": 0.89, + "content": "g _ { 2 } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } - s _ { \\mathrm { m i n } } ( \\pmb { \\Sigma } ) \\pmb { I } _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 239, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 239, + 176 + ], + "score": 1.0, + "content": "independent multivariate Gaussians.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 129, + 192 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 130, + 179, + 209, + 191 + ], + "score": 0.91, + "content": "B { \\boldsymbol { u } } _ { t } \\sim \\mathcal { N } ( 0 , B B ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 178, + 239, + 192 + ], + "score": 1.0, + "content": ", setting", + "type": "text" + }, + { + "bbox": [ + 240, + 179, + 284, + 189 + ], + "score": 0.91, + "content": "\\pmb { \\Sigma } = \\pmb { B } \\pmb { B } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 178, + 301, + 192 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 180, + 364, + 190 + ], + "score": 0.91, + "content": "s _ { \\mathrm { m i n } } = s _ { \\mathrm { m i n } } ( \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 178, + 417, + 192 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 417, + 181, + 479, + 191 + ], + "score": 0.94, + "content": "B u _ { t } \\sim g _ { 1 } + g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 178, + 506, + 192 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 498, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 130, + 201 + ], + "score": 0.89, + "content": "{ \\bf { \\mathit { g } } } _ { 1 } , { \\bf { \\mathit { g } } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 189, + 205, + 203 + ], + "score": 1.0, + "content": "are independent and", + "type": "text" + }, + { + "bbox": [ + 206, + 190, + 279, + 201 + ], + "score": 0.9, + "content": "{ \\pmb g } _ { 1 } \\sim \\mathcal { N } ( 0 , s _ { \\mathrm { m i n } } { \\pmb I } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 189, + 295, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 190, + 389, + 201 + ], + "score": 0.86, + "content": "\\pmb { g } _ { 2 } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } - s _ { \\mathrm { m i n } } \\pmb { I } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 189, + 498, + 203 + ], + "score": 1.0, + "content": ". Consequently, we may write", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 206, + 366, + 218 + ], + "lines": [ + { + "bbox": [ + 244, + 206, + 366, + 218 + ], + "spans": [ + { + "bbox": [ + 244, + 206, + 366, + 218 + ], + "score": 0.88, + "content": "B u _ { t } + A h _ { t } \\sim g _ { 1 } + g _ { 2 } + A h _ { t } .", + "type": "interline_equation", + "image_path": "3e308247203f6acd9824321cbae0181bc44ed86a7383abeaa717e1069498cb00.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 244, + 206, + 366, + 218 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 293, + 235 + ], + "score": 1.0, + "content": "For lower bound, the crucial component will be the", + "type": "text" + }, + { + "bbox": [ + 294, + 225, + 303, + 233 + ], + "score": 0.85, + "content": "\\pmb { g } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "term; which has i.i.d. entries. Applying Lemma B.7 by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 132, + 244 + ], + "score": 1.0, + "content": "setting", + "type": "text" + }, + { + "bbox": [ + 132, + 234, + 161, + 243 + ], + "score": 0.9, + "content": "{ \\pmb x } = { \\pmb g } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 232, + 177, + 244 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 233, + 234, + 243 + ], + "score": 0.91, + "content": "\\pmb { y } = \\pmb { g } _ { 2 } + \\pmb { A } \\hat { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 232, + 318, + 244 + ], + "score": 1.0, + "content": ", and using the fact that", + "type": "text" + }, + { + "bbox": [ + 318, + 234, + 355, + 243 + ], + "score": 0.92, + "content": "h _ { t } , g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "are all independent of each other, we find", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 248, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 209, + 254 + ], + "score": 1.0, + "content": "the advertised bound, for all", + "type": "text" + }, + { + "bbox": [ + 210, + 244, + 231, + 253 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 243, + 248, + 254 + ], + "score": 1.0, + "content": ", via", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 257, + 396, + 271 + ], + "lines": [ + { + "bbox": [ + 214, + 257, + 396, + 271 + ], + "spans": [ + { + "bbox": [ + 214, + 257, + 396, + 271 + ], + "score": 0.91, + "content": "\\Sigma [ h _ { t + 1 } ] = \\Sigma [ \\phi ( g _ { 1 } + g _ { 2 } + A h _ { t } ) ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } I _ { n } .", + "type": "interline_equation", + "image_path": "0005d34fc28006e4db2441493c0c34e267513d7bf3abd84d80c4f4177a93b2ae.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 214, + 257, + 396, + 271 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 401, + 309 + ], + "score": 1.0, + "content": "The next theorem applies to multiple-input-single-output (MISO) systems where", + "type": "text" + }, + { + "bbox": [ + 401, + 299, + 410, + 307 + ], + "score": 0.81, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 298, + 464, + 309 + ], + "score": 1.0, + "content": "is a scalar and", + "type": "text" + }, + { + "bbox": [ + 464, + 299, + 473, + 307 + ], + "score": 0.82, + "content": "_ B", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "is a row", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 308, + 327, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 327, + 319 + ], + "score": 1.0, + "content": "vector. The goal is refining the lower bound of Theorem B.5.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 105, + 321, + 504, + 342 + ], + "lines": [ + { + "bbox": [ + 104, + 319, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 319, + 447, + 335 + ], + "score": 1.0, + "content": "Theorem B.6 (MISO lower bound). Consider the setup of Theorem B.5 with single output i.e.", + "type": "text" + }, + { + "bbox": [ + 447, + 323, + 471, + 331 + ], + "score": 0.87, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 319, + 506, + 335 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 331, + 211, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 128, + 342 + ], + "score": 0.88, + "content": "t \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 331, + 211, + 344 + ], + "score": 1.0, + "content": ", the state vector obeys", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 340, + 371, + 366 + ], + "lines": [ + { + "bbox": [ + 239, + 340, + 371, + 366 + ], + "spans": [ + { + "bbox": [ + 239, + 340, + 371, + 366 + ], + "score": 0.94, + "content": "\\mathbf { v a r } [ \\pmb { h } _ { t } ] \\geq \\beta ^ { 2 } \\| \\pmb { B } \\| _ { \\ell _ { 2 } } ^ { 2 } \\frac { 1 - ( \\beta | \\pmb { A } | ) ^ { 2 t } } { 1 - \\beta ^ { 2 } | \\pmb { A } | ^ { 2 } } .", + "type": "interline_equation", + "image_path": "44abcb6d72d25dbd8fdd2dfddaf0193181e028550cd028b31632dddd3d835111.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 239, + 340, + 371, + 366 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 506, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 224, + 390 + ], + "score": 1.0, + "content": "Proof. For any random variable", + "type": "text" + }, + { + "bbox": [ + 225, + 379, + 234, + 387 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 376, + 362, + 390 + ], + "score": 1.0, + "content": ", applying Lemma B.7, we have va", + "type": "text" + }, + { + "bbox": [ + 362, + 377, + 439, + 389 + ], + "score": 0.91, + "content": "{ \\mathfrak { r } } [ \\phi ( X ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 376, + 505, + 390 + ], + "score": 1.0, + "content": ". Recursively, this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 103, + 387, + 132, + 400 + ], + "spans": [ + { + "bbox": [ + 103, + 387, + 132, + 400 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 403, + 469, + 417 + ], + "lines": [ + { + "bbox": [ + 140, + 403, + 469, + 417 + ], + "spans": [ + { + "bbox": [ + 140, + 403, + 469, + 417 + ], + "score": 0.88, + "content": "\\mathbf { v a r } [ h _ { t + 1 } ] = \\mathbf { v a r } [ \\phi ( A h _ { t } + B u _ { t } ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ A h _ { t } + B u _ { t } ] = \\beta ^ { 2 } ( | A | ^ { 2 } \\mathbf { v a r } [ h _ { t } ] + \\| B \\| _ { \\ell _ { 2 } } ^ { 2 } ) .", + "type": "interline_equation", + "image_path": "6633a7ff85f49fee6a9ec18e775d5263ccd2d49a8e76ae5a546b7efdd0a23caf.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 140, + 403, + 469, + 417 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 343, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 343, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 224, + 434 + ], + "score": 1.0, + "content": "Expanding these inequalities till", + "type": "text" + }, + { + "bbox": [ + 224, + 423, + 235, + 432 + ], + "score": 0.86, + "content": "\\scriptstyle h _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 421, + 343, + 434 + ], + "score": 1.0, + "content": ", we obtain the desired bound", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 437, + 370, + 468 + ], + "lines": [ + { + "bbox": [ + 240, + 437, + 370, + 468 + ], + "spans": [ + { + "bbox": [ + 240, + 437, + 370, + 468 + ], + "score": 0.94, + "content": "\\mathbf { v a r } [ \\pmb { h } _ { t } ] \\geq \\sum _ { i = 1 } ^ { t } ( \\beta ^ { i } | \\pmb { A } | ^ { i - 1 } \\| \\pmb { B } \\| _ { \\ell _ { 2 } } ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "8c144c4eb62532a3c0625658554e3fae1da1c6c7a3ab9e7c59eca69b3fc6a5c8.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 437, + 370, + 452.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 240, + 452.5, + 370, + 468.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 488, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 270, + 501 + ], + "score": 1.0, + "content": "Lemma B.7 (Vector lower bound). Suppose", + "type": "text" + }, + { + "bbox": [ + 271, + 489, + 277, + 499 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 488, + 293, + 501 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 293, + 489, + 300, + 499 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 488, + 391, + 501 + ], + "score": 1.0, + "content": "-increasing function. Let", + "type": "text" + }, + { + "bbox": [ + 392, + 488, + 461, + 500 + ], + "score": 0.89, + "content": "{ \\pmb x } = [ { \\pmb x } _ { 1 } ~ . ~ . ~ { \\pmb x } _ { n } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "be a vector", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 438, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 221, + 511 + ], + "score": 1.0, + "content": "with i.i.d. entries distributed as", + "type": "text" + }, + { + "bbox": [ + 221, + 499, + 251, + 509 + ], + "score": 0.91, + "content": "x _ { i } \\sim X", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 497, + 269, + 511 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 269, + 501, + 276, + 509 + ], + "score": 0.73, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 497, + 404, + 511 + ], + "score": 1.0, + "content": "be a random vector independent of", + "type": "text" + }, + { + "bbox": [ + 405, + 501, + 411, + 508 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 497, + 438, + 511 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 513, + 361, + 528 + ], + "lines": [ + { + "bbox": [ + 250, + 513, + 361, + 528 + ], + "spans": [ + { + "bbox": [ + 250, + 513, + 361, + 528 + ], + "score": 0.91, + "content": "\\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) ] \\succeq \\beta ^ { 2 } { \\mathbf { v a r } } [ X ] { \\cal I } _ { n } .", + "type": "interline_equation", + "image_path": "213d762ed80863dc48729ddbb319e2c4bcc6340c7ddb9f0e650fd1c34f13944a.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 250, + 513, + 361, + 528 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "score": 1.0, + "content": "Proof. We first apply law of total covariance (e.g. Lemma B.8) to simplify the problem using the following", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 296, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 262, + 561 + ], + "score": 1.0, + "content": "lower bound based on the independence of", + "type": "text" + }, + { + "bbox": [ + 262, + 551, + 269, + 558 + ], + "score": 0.8, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 547, + 285, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 551, + 292, + 559 + ], + "score": 0.81, + "content": "\\pmb { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 547, + 296, + 561 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 563, + 376, + 592 + ], + "lines": [ + { + "bbox": [ + 234, + 563, + 376, + 592 + ], + "spans": [ + { + "bbox": [ + 234, + 563, + 376, + 592 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) ] \\succeq \\mathbb { E } _ { { \\pmb y } } [ \\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) | { \\pmb y } ] ] } \\\\ { = \\mathbb { E } _ { { \\pmb y } } [ \\Sigma _ { \\pmb x } [ \\phi ( { \\pmb x } + { \\pmb y } ) ] ] . } \\end{array}", + "type": "interline_equation", + "image_path": "5f584816276ff3a576ef62768cc88df1104585866e70470c1b160e6a3e30c5bb.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 563, + 376, + 577.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 234, + 577.5, + 376, + 592.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 234, + 608 + ], + "score": 1.0, + "content": "Now, focusing on the covariance", + "type": "text" + }, + { + "bbox": [ + 235, + 595, + 290, + 606 + ], + "score": 0.92, + "content": "\\Sigma _ { x } [ \\phi ( { \\pmb x } + { \\pmb y } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 595, + 380, + 608 + ], + "score": 1.0, + "content": ", fixing a realization of", + "type": "text" + }, + { + "bbox": [ + 381, + 597, + 388, + 606 + ], + "score": 0.79, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 595, + 481, + 608 + ], + "score": 1.0, + "content": ", and using the fact that", + "type": "text" + }, + { + "bbox": [ + 481, + 597, + 488, + 605 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 154, + 617 + ], + "score": 1.0, + "content": "i.i.d. entries;", + "type": "text" + }, + { + "bbox": [ + 155, + 606, + 191, + 617 + ], + "score": 0.92, + "content": "\\phi ( { \\pmb x } + { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 605, + 290, + 617 + ], + "score": 1.0, + "content": "has independent entries as", + "type": "text" + }, + { + "bbox": [ + 291, + 606, + 297, + 616 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 605, + 434, + 617 + ], + "score": 1.0, + "content": "applies entry-wise. This implies that", + "type": "text" + }, + { + "bbox": [ + 434, + 605, + 489, + 617 + ], + "score": 0.92, + "content": "\\Sigma _ { \\pmb { x } } [ \\phi ( \\pmb { x } + \\pmb { y } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 616, + 447, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 447, + 627 + ], + "score": 1.0, + "content": "diagonal matrix. Consequently, its lowest eigenvalue is the minimum variance over all entries,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 630, + 387, + 649 + ], + "lines": [ + { + "bbox": [ + 223, + 630, + 387, + 649 + ], + "spans": [ + { + "bbox": [ + 223, + 630, + 387, + 649 + ], + "score": 0.92, + "content": "\\pmb { \\Sigma } _ { \\pmb { x } } [ \\phi ( \\pmb { x } + \\pmb { y } ) ] \\succeq \\operatorname* { m i n } _ { 1 \\leq i \\leq n } \\mathbf { v a r } [ \\phi ( \\pmb { x } _ { i } + \\pmb { y } _ { i } ) ] \\pmb { I } _ { n } .", + "type": "interline_equation", + "image_path": "26d2978b2dd2794600d62a5d4ab004125b839f14e7d5879deb1a0553ba151b7f.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 223, + 630, + 387, + 649 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 504, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 286, + 667 + ], + "score": 1.0, + "content": "Fortunately, Lemma B.9 provides the lower bound", + "type": "text" + }, + { + "bbox": [ + 287, + 654, + 396, + 666 + ], + "score": 0.91, + "content": "\\mathbf { v a r } [ \\phi ( \\pmb { x } _ { i } + \\pmb { y } _ { i } ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". Since this lower bound holds", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 204, + 677 + ], + "score": 1.0, + "content": "for any fixed realization of", + "type": "text" + }, + { + "bbox": [ + 204, + 666, + 211, + 675 + ], + "score": 0.82, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 664, + 363, + 677 + ], + "score": 1.0, + "content": ", it still holds after taking expectation over", + "type": "text" + }, + { + "bbox": [ + 364, + 667, + 371, + 675 + ], + "score": 0.79, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 664, + 474, + 677 + ], + "score": 1.0, + "content": "; which concludes the proof.", + "type": "text" + }, + { + "bbox": [ + 495, + 665, + 505, + 675 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 103, + 687, + 484, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 475, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 475, + 700 + ], + "score": 1.0, + "content": "The next two lemmas are helper results for Lemma B.7 and are provided for the sake of completeness.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 104, + 701, + 503, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 267, + 713 + ], + "score": 1.0, + "content": "Lemma B.8 (Law of total covariance). Let", + "type": "text" + }, + { + "bbox": [ + 267, + 703, + 285, + 712 + ], + "score": 0.52, + "content": "\\mathbf { \\nabla } _ { \\mathbf { x } , \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 700, + 417, + 713 + ], + "score": 1.0, + "content": "be two random vectors and assume", + "type": "text" + }, + { + "bbox": [ + 417, + 703, + 425, + 712 + ], + "score": 0.76, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 700, + 504, + 713 + ], + "score": 1.0, + "content": "has finite covariance.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 710, + 128, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 710, + 128, + 723 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 719, + 372, + 734 + ], + "lines": [ + { + "bbox": [ + 239, + 719, + 372, + 734 + ], + "spans": [ + { + "bbox": [ + 239, + 719, + 372, + 734 + ], + "score": 0.91, + "content": "\\pmb { \\Sigma } [ \\pmb { y } ] = \\mathbb { E } [ \\pmb { \\Sigma } [ \\pmb { y } \\mid \\pmb { x } ] ] + \\pmb { \\Sigma } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] .", + "type": "interline_equation", + "image_path": "44d55fbe10b1d640569c7c17cc2cc9aef0b7ce68a2a43581bc9b22c8c3a059d0.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 239, + 719, + 372, + 734 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 275, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 496, + 276, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 496, + 276, + 505, + 287 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 472, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 496, + 472, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 496, + 472, + 505, + 483 + ], + "score": 0.993, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 455, + 96 + ], + "score": 1.0, + "content": "Theorem B.5 (State-vector lower bound). Consider the nonlinear state equation (1.1) with", + "type": "text" + }, + { + "bbox": [ + 456, + 82, + 488, + 95 + ], + "score": 0.57, + "content": "\\{ { \\pmb u } _ { t } \\} _ { t \\ge 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 79, + 505, + 96 + ], + "score": 1.0, + "content": "i.i.d. ∼", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 141, + 104 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 93, + 177, + 105 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 177, + 94, + 184, + 104 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 93, + 199, + 105 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 200, + 94, + 207, + 104 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 93, + 344, + 105 + ], + "score": 1.0, + "content": "-increasing function for some constant", + "type": "text" + }, + { + "bbox": [ + 345, + 93, + 368, + 104 + ], + "score": 0.9, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 93, + 401, + 105 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 402, + 94, + 423, + 104 + ], + "score": 0.89, + "content": "t \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ", the state vector obeys", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 79, + 505, + 105 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 109, + 358, + 123 + ], + "lines": [ + { + "bbox": [ + 253, + 109, + 358, + 123 + ], + "spans": [ + { + "bbox": [ + 253, + 109, + 358, + 123 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\Sigma [ h _ { t } ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B B ^ { T } ) I _ { n } . } \\end{array}", + "type": "interline_equation", + "image_path": "c0020b9459ed9c728ec193009573a65ddca6e99eb0404fba1e9b573aeb7d0647.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 253, + 109, + 358, + 123 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 506, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 391, + 146 + ], + "score": 1.0, + "content": "Proof. The proof is an application of Lemma B.7. The main idea is to write", + "type": "text" + }, + { + "bbox": [ + 391, + 135, + 402, + 144 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 134, + 505, + 146 + ], + "score": 1.0, + "content": "as sum of two independent", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 142, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 437, + 158 + ], + "score": 1.0, + "content": "vectors, one of which has independent entries. Consider a multivariate Gaussian vector", + "type": "text" + }, + { + "bbox": [ + 437, + 144, + 491, + 155 + ], + "score": 0.89, + "content": "\\mathbf { \\sigma } _ { \\mathbf { \\mathcal { g } } } \\sim \\mathcal { N } ( \\bar { 0 , } \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 142, + 496, + 158 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 146, + 504, + 155 + ], + "score": 0.46, + "content": "\\pmb { g }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 205, + 167 + ], + "score": 1.0, + "content": "is statistically identical to", + "type": "text" + }, + { + "bbox": [ + 206, + 155, + 238, + 165 + ], + "score": 0.9, + "content": "{ \\pmb g } _ { 1 } + { \\pmb g } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 153, + 265, + 167 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 266, + 154, + 358, + 165 + ], + "score": 0.91, + "content": "{ \\pmb g } _ { 1 } \\sim \\mathcal { N } ( 0 , s _ { \\mathrm { m i n } } ( { \\pmb \\Sigma } ) { \\pmb I } _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 153, + 376, + 167 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 377, + 154, + 489, + 165 + ], + "score": 0.89, + "content": "g _ { 2 } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } - s _ { \\mathrm { m i n } } ( \\pmb { \\Sigma } ) \\pmb { I } _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 239, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 239, + 176 + ], + "score": 1.0, + "content": "independent multivariate Gaussians.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 134, + 506, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 129, + 192 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 130, + 179, + 209, + 191 + ], + "score": 0.91, + "content": "B { \\boldsymbol { u } } _ { t } \\sim \\mathcal { N } ( 0 , B B ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 178, + 239, + 192 + ], + "score": 1.0, + "content": ", setting", + "type": "text" + }, + { + "bbox": [ + 240, + 179, + 284, + 189 + ], + "score": 0.91, + "content": "\\pmb { \\Sigma } = \\pmb { B } \\pmb { B } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 178, + 301, + 192 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 180, + 364, + 190 + ], + "score": 0.91, + "content": "s _ { \\mathrm { m i n } } = s _ { \\mathrm { m i n } } ( \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 178, + 417, + 192 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 417, + 181, + 479, + 191 + ], + "score": 0.94, + "content": "B u _ { t } \\sim g _ { 1 } + g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 178, + 506, + 192 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 498, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 130, + 201 + ], + "score": 0.89, + "content": "{ \\bf { \\mathit { g } } } _ { 1 } , { \\bf { \\mathit { g } } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 189, + 205, + 203 + ], + "score": 1.0, + "content": "are independent and", + "type": "text" + }, + { + "bbox": [ + 206, + 190, + 279, + 201 + ], + "score": 0.9, + "content": "{ \\pmb g } _ { 1 } \\sim \\mathcal { N } ( 0 , s _ { \\mathrm { m i n } } { \\pmb I } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 189, + 295, + 203 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 296, + 190, + 389, + 201 + ], + "score": 0.86, + "content": "\\pmb { g } _ { 2 } \\sim \\mathcal { N } ( 0 , \\pmb { \\Sigma } - s _ { \\mathrm { m i n } } \\pmb { I } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 189, + 498, + 203 + ], + "score": 1.0, + "content": ". Consequently, we may write", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 178, + 506, + 203 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 206, + 366, + 218 + ], + "lines": [ + { + "bbox": [ + 244, + 206, + 366, + 218 + ], + "spans": [ + { + "bbox": [ + 244, + 206, + 366, + 218 + ], + "score": 0.88, + "content": "B u _ { t } + A h _ { t } \\sim g _ { 1 } + g _ { 2 } + A h _ { t } .", + "type": "interline_equation", + "image_path": "3e308247203f6acd9824321cbae0181bc44ed86a7383abeaa717e1069498cb00.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 244, + 206, + 366, + 218 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 293, + 235 + ], + "score": 1.0, + "content": "For lower bound, the crucial component will be the", + "type": "text" + }, + { + "bbox": [ + 294, + 225, + 303, + 233 + ], + "score": 0.85, + "content": "\\pmb { g } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "term; which has i.i.d. entries. Applying Lemma B.7 by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 132, + 244 + ], + "score": 1.0, + "content": "setting", + "type": "text" + }, + { + "bbox": [ + 132, + 234, + 161, + 243 + ], + "score": 0.9, + "content": "{ \\pmb x } = { \\pmb g } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 232, + 177, + 244 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 233, + 234, + 243 + ], + "score": 0.91, + "content": "\\pmb { y } = \\pmb { g } _ { 2 } + \\pmb { A } \\hat { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 232, + 318, + 244 + ], + "score": 1.0, + "content": ", and using the fact that", + "type": "text" + }, + { + "bbox": [ + 318, + 234, + 355, + 243 + ], + "score": 0.92, + "content": "h _ { t } , g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "are all independent of each other, we find", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 248, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 209, + 254 + ], + "score": 1.0, + "content": "the advertised bound, for all", + "type": "text" + }, + { + "bbox": [ + 210, + 244, + 231, + 253 + ], + "score": 0.89, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 243, + 248, + 254 + ], + "score": 1.0, + "content": ", via", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 222, + 505, + 254 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 257, + 396, + 271 + ], + "lines": [ + { + "bbox": [ + 214, + 257, + 396, + 271 + ], + "spans": [ + { + "bbox": [ + 214, + 257, + 396, + 271 + ], + "score": 0.91, + "content": "\\Sigma [ h _ { t + 1 } ] = \\Sigma [ \\phi ( g _ { 1 } + g _ { 2 } + A h _ { t } ) ] \\succeq \\beta ^ { 2 } s _ { \\mathrm { m i n } } I _ { n } .", + "type": "interline_equation", + "image_path": "0005d34fc28006e4db2441493c0c34e267513d7bf3abd84d80c4f4177a93b2ae.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 214, + 257, + 396, + 271 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 401, + 309 + ], + "score": 1.0, + "content": "The next theorem applies to multiple-input-single-output (MISO) systems where", + "type": "text" + }, + { + "bbox": [ + 401, + 299, + 410, + 307 + ], + "score": 0.81, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 298, + 464, + 309 + ], + "score": 1.0, + "content": "is a scalar and", + "type": "text" + }, + { + "bbox": [ + 464, + 299, + 473, + 307 + ], + "score": 0.82, + "content": "_ B", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "is a row", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 308, + 327, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 327, + 319 + ], + "score": 1.0, + "content": "vector. The goal is refining the lower bound of Theorem B.5.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 106, + 298, + 505, + 319 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 321, + 504, + 342 + ], + "lines": [ + { + "bbox": [ + 104, + 319, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 319, + 447, + 335 + ], + "score": 1.0, + "content": "Theorem B.6 (MISO lower bound). Consider the setup of Theorem B.5 with single output i.e.", + "type": "text" + }, + { + "bbox": [ + 447, + 323, + 471, + 331 + ], + "score": 0.87, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 319, + 506, + 335 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 331, + 211, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 128, + 342 + ], + "score": 0.88, + "content": "t \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 331, + 211, + 344 + ], + "score": 1.0, + "content": ", the state vector obeys", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 319, + 506, + 344 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 340, + 371, + 366 + ], + "lines": [ + { + "bbox": [ + 239, + 340, + 371, + 366 + ], + "spans": [ + { + "bbox": [ + 239, + 340, + 371, + 366 + ], + "score": 0.94, + "content": "\\mathbf { v a r } [ \\pmb { h } _ { t } ] \\geq \\beta ^ { 2 } \\| \\pmb { B } \\| _ { \\ell _ { 2 } } ^ { 2 } \\frac { 1 - ( \\beta | \\pmb { A } | ) ^ { 2 t } } { 1 - \\beta ^ { 2 } | \\pmb { A } | ^ { 2 } } .", + "type": "interline_equation", + "image_path": "44abcb6d72d25dbd8fdd2dfddaf0193181e028550cd028b31632dddd3d835111.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 239, + 340, + 371, + 366 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 506, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 224, + 390 + ], + "score": 1.0, + "content": "Proof. For any random variable", + "type": "text" + }, + { + "bbox": [ + 225, + 379, + 234, + 387 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 376, + 362, + 390 + ], + "score": 1.0, + "content": ", applying Lemma B.7, we have va", + "type": "text" + }, + { + "bbox": [ + 362, + 377, + 439, + 389 + ], + "score": 0.91, + "content": "{ \\mathfrak { r } } [ \\phi ( X ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 376, + 505, + 390 + ], + "score": 1.0, + "content": ". Recursively, this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 103, + 387, + 132, + 400 + ], + "spans": [ + { + "bbox": [ + 103, + 387, + 132, + 400 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 103, + 376, + 505, + 400 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 403, + 469, + 417 + ], + "lines": [ + { + "bbox": [ + 140, + 403, + 469, + 417 + ], + "spans": [ + { + "bbox": [ + 140, + 403, + 469, + 417 + ], + "score": 0.88, + "content": "\\mathbf { v a r } [ h _ { t + 1 } ] = \\mathbf { v a r } [ \\phi ( A h _ { t } + B u _ { t } ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ A h _ { t } + B u _ { t } ] = \\beta ^ { 2 } ( | A | ^ { 2 } \\mathbf { v a r } [ h _ { t } ] + \\| B \\| _ { \\ell _ { 2 } } ^ { 2 } ) .", + "type": "interline_equation", + "image_path": "6633a7ff85f49fee6a9ec18e775d5263ccd2d49a8e76ae5a546b7efdd0a23caf.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 140, + 403, + 469, + 417 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 343, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 343, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 224, + 434 + ], + "score": 1.0, + "content": "Expanding these inequalities till", + "type": "text" + }, + { + "bbox": [ + 224, + 423, + 235, + 432 + ], + "score": 0.86, + "content": "\\scriptstyle h _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 421, + 343, + 434 + ], + "score": 1.0, + "content": ", we obtain the desired bound", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 421, + 343, + 434 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 437, + 370, + 468 + ], + "lines": [ + { + "bbox": [ + 240, + 437, + 370, + 468 + ], + "spans": [ + { + "bbox": [ + 240, + 437, + 370, + 468 + ], + "score": 0.94, + "content": "\\mathbf { v a r } [ \\pmb { h } _ { t } ] \\geq \\sum _ { i = 1 } ^ { t } ( \\beta ^ { i } | \\pmb { A } | ^ { i - 1 } \\| \\pmb { B } \\| _ { \\ell _ { 2 } } ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "8c144c4eb62532a3c0625658554e3fae1da1c6c7a3ab9e7c59eca69b3fc6a5c8.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 437, + 370, + 452.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 240, + 452.5, + 370, + 468.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 488, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 270, + 501 + ], + "score": 1.0, + "content": "Lemma B.7 (Vector lower bound). Suppose", + "type": "text" + }, + { + "bbox": [ + 271, + 489, + 277, + 499 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 488, + 293, + 501 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 293, + 489, + 300, + 499 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 488, + 391, + 501 + ], + "score": 1.0, + "content": "-increasing function. Let", + "type": "text" + }, + { + "bbox": [ + 392, + 488, + 461, + 500 + ], + "score": 0.89, + "content": "{ \\pmb x } = [ { \\pmb x } _ { 1 } ~ . ~ . ~ { \\pmb x } _ { n } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "be a vector", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 438, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 221, + 511 + ], + "score": 1.0, + "content": "with i.i.d. entries distributed as", + "type": "text" + }, + { + "bbox": [ + 221, + 499, + 251, + 509 + ], + "score": 0.91, + "content": "x _ { i } \\sim X", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 497, + 269, + 511 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 269, + 501, + 276, + 509 + ], + "score": 0.73, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 497, + 404, + 511 + ], + "score": 1.0, + "content": "be a random vector independent of", + "type": "text" + }, + { + "bbox": [ + 405, + 501, + 411, + 508 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 497, + 438, + 511 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 488, + 505, + 511 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 513, + 361, + 528 + ], + "lines": [ + { + "bbox": [ + 250, + 513, + 361, + 528 + ], + "spans": [ + { + "bbox": [ + 250, + 513, + 361, + 528 + ], + "score": 0.91, + "content": "\\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) ] \\succeq \\beta ^ { 2 } { \\mathbf { v a r } } [ X ] { \\cal I } _ { n } .", + "type": "interline_equation", + "image_path": "213d762ed80863dc48729ddbb319e2c4bcc6340c7ddb9f0e650fd1c34f13944a.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 250, + 513, + 361, + 528 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 552 + ], + "score": 1.0, + "content": "Proof. We first apply law of total covariance (e.g. Lemma B.8) to simplify the problem using the following", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 547, + 296, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 262, + 561 + ], + "score": 1.0, + "content": "lower bound based on the independence of", + "type": "text" + }, + { + "bbox": [ + 262, + 551, + 269, + 558 + ], + "score": 0.8, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 547, + 285, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 551, + 292, + 559 + ], + "score": 0.81, + "content": "\\pmb { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 547, + 296, + 561 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 537, + 505, + 561 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 563, + 376, + 592 + ], + "lines": [ + { + "bbox": [ + 234, + 563, + 376, + 592 + ], + "spans": [ + { + "bbox": [ + 234, + 563, + 376, + 592 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) ] \\succeq \\mathbb { E } _ { { \\pmb y } } [ \\Sigma [ \\phi ( { \\pmb x } + { \\pmb y } ) | { \\pmb y } ] ] } \\\\ { = \\mathbb { E } _ { { \\pmb y } } [ \\Sigma _ { \\pmb x } [ \\phi ( { \\pmb x } + { \\pmb y } ) ] ] . } \\end{array}", + "type": "interline_equation", + "image_path": "5f584816276ff3a576ef62768cc88df1104585866e70470c1b160e6a3e30c5bb.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 563, + 376, + 577.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 234, + 577.5, + 376, + 592.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 234, + 608 + ], + "score": 1.0, + "content": "Now, focusing on the covariance", + "type": "text" + }, + { + "bbox": [ + 235, + 595, + 290, + 606 + ], + "score": 0.92, + "content": "\\Sigma _ { x } [ \\phi ( { \\pmb x } + { \\pmb y } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 595, + 380, + 608 + ], + "score": 1.0, + "content": ", fixing a realization of", + "type": "text" + }, + { + "bbox": [ + 381, + 597, + 388, + 606 + ], + "score": 0.79, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 595, + 481, + 608 + ], + "score": 1.0, + "content": ", and using the fact that", + "type": "text" + }, + { + "bbox": [ + 481, + 597, + 488, + 605 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 154, + 617 + ], + "score": 1.0, + "content": "i.i.d. entries;", + "type": "text" + }, + { + "bbox": [ + 155, + 606, + 191, + 617 + ], + "score": 0.92, + "content": "\\phi ( { \\pmb x } + { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 605, + 290, + 617 + ], + "score": 1.0, + "content": "has independent entries as", + "type": "text" + }, + { + "bbox": [ + 291, + 606, + 297, + 616 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 605, + 434, + 617 + ], + "score": 1.0, + "content": "applies entry-wise. This implies that", + "type": "text" + }, + { + "bbox": [ + 434, + 605, + 489, + 617 + ], + "score": 0.92, + "content": "\\Sigma _ { \\pmb { x } } [ \\phi ( \\pmb { x } + \\pmb { y } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 616, + 447, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 447, + 627 + ], + "score": 1.0, + "content": "diagonal matrix. Consequently, its lowest eigenvalue is the minimum variance over all entries,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 595, + 506, + 627 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 630, + 387, + 649 + ], + "lines": [ + { + "bbox": [ + 223, + 630, + 387, + 649 + ], + "spans": [ + { + "bbox": [ + 223, + 630, + 387, + 649 + ], + "score": 0.92, + "content": "\\pmb { \\Sigma } _ { \\pmb { x } } [ \\phi ( \\pmb { x } + \\pmb { y } ) ] \\succeq \\operatorname* { m i n } _ { 1 \\leq i \\leq n } \\mathbf { v a r } [ \\phi ( \\pmb { x } _ { i } + \\pmb { y } _ { i } ) ] \\pmb { I } _ { n } .", + "type": "interline_equation", + "image_path": "26d2978b2dd2794600d62a5d4ab004125b839f14e7d5879deb1a0553ba151b7f.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 223, + 630, + 387, + 649 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 504, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 286, + 667 + ], + "score": 1.0, + "content": "Fortunately, Lemma B.9 provides the lower bound", + "type": "text" + }, + { + "bbox": [ + 287, + 654, + 396, + 666 + ], + "score": 0.91, + "content": "\\mathbf { v a r } [ \\phi ( \\pmb { x } _ { i } + \\pmb { y } _ { i } ) ] \\geq \\beta ^ { 2 } \\mathbf { v a r } [ X ]", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". Since this lower bound holds", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 204, + 677 + ], + "score": 1.0, + "content": "for any fixed realization of", + "type": "text" + }, + { + "bbox": [ + 204, + 666, + 211, + 675 + ], + "score": 0.82, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 664, + 363, + 677 + ], + "score": 1.0, + "content": ", it still holds after taking expectation over", + "type": "text" + }, + { + "bbox": [ + 364, + 667, + 371, + 675 + ], + "score": 0.79, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 664, + 474, + 677 + ], + "score": 1.0, + "content": "; which concludes the proof.", + "type": "text" + }, + { + "bbox": [ + 495, + 665, + 505, + 675 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 654, + 506, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 687, + 484, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 475, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 475, + 700 + ], + "score": 1.0, + "content": "The next two lemmas are helper results for Lemma B.7 and are provided for the sake of completeness.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 686, + 475, + 700 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 701, + 503, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 504, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 267, + 713 + ], + "score": 1.0, + "content": "Lemma B.8 (Law of total covariance). Let", + "type": "text" + }, + { + "bbox": [ + 267, + 703, + 285, + 712 + ], + "score": 0.52, + "content": "\\mathbf { \\nabla } _ { \\mathbf { x } , \\mathbf { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 700, + 417, + 713 + ], + "score": 1.0, + "content": "be two random vectors and assume", + "type": "text" + }, + { + "bbox": [ + 417, + 703, + 425, + 712 + ], + "score": 0.76, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 700, + 504, + 713 + ], + "score": 1.0, + "content": "has finite covariance.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 710, + 128, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 710, + 128, + 723 + ], + "score": 1.0, + "content": "Then", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 700, + 504, + 723 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 719, + 372, + 734 + ], + "lines": [ + { + "bbox": [ + 239, + 719, + 372, + 734 + ], + "spans": [ + { + "bbox": [ + 239, + 719, + 372, + 734 + ], + "score": 0.91, + "content": "\\pmb { \\Sigma } [ \\pmb { y } ] = \\mathbb { E } [ \\pmb { \\Sigma } [ \\pmb { y } \\mid \\pmb { x } ] ] + \\pmb { \\Sigma } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] .", + "type": "interline_equation", + "image_path": "44d55fbe10b1d640569c7c17cc2cc9aef0b7ce68a2a43581bc9b22c8c3a059d0.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 239, + 719, + 372, + 734 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 497, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 498, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 175, + 97 + ], + "score": 1.0, + "content": "Proof. First, write", + "type": "text" + }, + { + "bbox": [ + 176, + 82, + 288, + 94 + ], + "score": 0.91, + "content": "\\Sigma [ { \\pmb y } ] = \\mathbb { E } [ { \\pmb y } { \\pmb y } ^ { T } ] - \\mathbb { E } [ { \\pmb y } ] \\mathbb { E } [ { \\pmb y } ^ { T } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 81, + 498, + 97 + ], + "score": 1.0, + "content": ". Then, applying the law of total expectation to each term,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 96, + 397, + 110 + ], + "lines": [ + { + "bbox": [ + 210, + 96, + 397, + 110 + ], + "spans": [ + { + "bbox": [ + 210, + 96, + 397, + 110 + ], + "score": 0.66, + "content": "\\pmb { \\Sigma } [ \\pmb { y } ] = \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] - \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } ^ { T } \\mid \\pmb { x } ] ]", + "type": "interline_equation", + "image_path": "ae3d5e164ef16e7b936f71d53a104985a9d5229e1585e41ca785e85df5460e82.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 210, + 96, + 397, + 110 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 134 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 288, + 124 + ], + "score": 1.0, + "content": "Next, we can write the conditional expectation as", + "type": "text" + }, + { + "bbox": [ + 288, + 111, + 490, + 124 + ], + "score": 0.89, + "content": "\\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] = \\mathbb { E } [ \\pmb { \\Sigma } [ \\pmb { y } \\mid \\pmb { x } ] ] + \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] ^ { T } .", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 109, + 506, + 124 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 345, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 244, + 135 + ], + "score": 1.0, + "content": "conclude, we obtain the covariance of", + "type": "text" + }, + { + "bbox": [ + 245, + 122, + 276, + 135 + ], + "score": 0.92, + "content": "\\mathbb { E } [ { \\pmb y } \\mid { \\pmb x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 122, + 345, + 135 + ], + "score": 1.0, + "content": "via the difference,", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 136, + 426, + 150 + ], + "lines": [ + { + "bbox": [ + 184, + 136, + 426, + 150 + ], + "spans": [ + { + "bbox": [ + 184, + 136, + 426, + 150 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] ^ { T } - \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] = \\pmb { \\Sigma } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] , } \\end{array}", + "type": "interline_equation", + "image_path": "23772338c41be9a58f0e01f27b4886b0c960b4747eec6e9dcbfeca8a85e1ba99.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 184, + 136, + 426, + 150 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 221, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 222, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 222, + 162 + ], + "score": 1.0, + "content": "which yields the desired bound.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 105, + 166, + 504, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 270, + 178 + ], + "score": 1.0, + "content": "Lemma B.9 (Scalar lower bound). Suppose", + "type": "text" + }, + { + "bbox": [ + 270, + 168, + 277, + 177 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 165, + 293, + 178 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 293, + 167, + 300, + 177 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 165, + 392, + 178 + ], + "score": 1.0, + "content": "-increasing function with", + "type": "text" + }, + { + "bbox": [ + 393, + 167, + 416, + 177 + ], + "score": 0.91, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "as defined in Definition", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 176, + 313, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 215, + 188 + ], + "score": 1.0, + "content": "3.1. Given a random variable", + "type": "text" + }, + { + "bbox": [ + 215, + 177, + 224, + 186 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 176, + 272, + 188 + ], + "score": 1.0, + "content": "and a scalar", + "type": "text" + }, + { + "bbox": [ + 272, + 180, + 278, + 187 + ], + "score": 0.78, + "content": "_ y", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 176, + 313, + 188 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 187, + 359, + 201 + ], + "lines": [ + { + "bbox": [ + 251, + 187, + 359, + 201 + ], + "spans": [ + { + "bbox": [ + 251, + 187, + 359, + 201 + ], + "score": 0.87, + "content": "\\mathbf { v a r } [ { \\phi } ( X + y ) ] \\geq { \\beta } ^ { 2 } \\mathbf { v a r } [ X ] .", + "type": "interline_equation", + "image_path": "d91a8d679f115481ec73193dc053402121b809b0fe0f607b2d2702a944d7d2ca.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 251, + 187, + 359, + 201 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 504, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 155, + 224 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 155, + 213, + 162, + 222 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 210, + 171, + 224 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 171, + 212, + 178, + 222 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 210, + 286, + 224 + ], + "score": 1.0, + "content": "-increasing, it is invertible and", + "type": "text" + }, + { + "bbox": [ + 286, + 211, + 302, + 222 + ], + "score": 0.91, + "content": "\\phi ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 210, + 427, + 224 + ], + "score": 1.0, + "content": "is strictly increasing. 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This section shows that, for stable systems, the impact of the past states decay", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "exponentially fast and the system can be approximated by using the recent inputs only. We first define the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 402, + 213, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 213, + 412 + ], + "score": 1.0, + "content": "truncation of the state vector.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 290, + 425 + ], + "score": 1.0, + "content": "Definition C.1 (Truncated state vector). Suppose", + "type": "text" + }, + { + "bbox": [ + 290, + 413, + 325, + 424 + ], + "score": 0.92, + "content": "\\phi ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 412, + 390, + 425 + ], + "score": 1.0, + "content": ", initial condition", + "type": "text" + }, + { + "bbox": [ + 390, + 414, + 418, + 423 + ], + "score": 0.89, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 412, + 506, + 425 + ], + "score": 1.0, + "content": ", and consider the state", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 503, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 233, + 435 + ], + "score": 1.0, + "content": "equation (1.1). Given a timestamp", + "type": "text" + }, + { + "bbox": [ + 233, + 424, + 237, + 432 + ], + "score": 0.34, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 422, + 241, + 435 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 241, + 424, + 248, + 432 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 422, + 357, + 435 + ], + "score": 1.0, + "content": "-truncation of the state vector", + "type": "text" + }, + { + "bbox": [ + 357, + 424, + 368, + 433 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 422, + 419, + 435 + ], + "score": 1.0, + "content": "is denoted by", + "type": "text" + }, + { + "bbox": [ + 419, + 423, + 437, + 434 + ], + "score": 0.9, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 422, + 495, + 435 + ], + "score": 1.0, + "content": "and is equal to", + "type": "text" + }, + { + "bbox": [ + 495, + 426, + 503, + 434 + ], + "score": 0.66, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\eta } } \\mathbf { q } _ { t }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 132, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 132, + 444 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 444, + 377, + 457 + ], + "lines": [ + { + "bbox": [ + 234, + 444, + 377, + 457 + ], + "spans": [ + { + "bbox": [ + 234, + 444, + 377, + 457 + ], + "score": 0.91, + "content": "\\pmb { q } \\tau + 1 = \\phi ( \\pmb { A } \\pmb { q } _ { \\tau } + \\pmb { B } \\pmb { u } _ { \\tau } ^ { \\prime } ) \\quad , \\quad q _ { 0 } = 0", + "type": "interline_equation", + "image_path": "446a96b3504994399aa5ffecbf626bffb6669f479f0405ec924e2a4bf13b05b4.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 234, + 444, + 377, + 457 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 459, + 307, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 308, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 258, + 472 + ], + "score": 1.0, + "content": "is the state vector generated by the inputs", + "type": "text" + }, + { + "bbox": [ + 258, + 459, + 270, + 469 + ], + "score": 0.88, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 457, + 308, + 472 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 470, + 355, + 500 + ], + "lines": [ + { + "bbox": [ + 255, + 470, + 355, + 500 + ], + "spans": [ + { + "bbox": [ + 255, + 470, + 355, + 500 + ], + "score": 0.93, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime } = \\left\\{ \\begin{array} { l l } { 0 i f \\tau < t - L } \\\\ { { \\pmb u } _ { \\tau } e l s e } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "85b7b37191a5fc384df5b34ce1bf6e9203b7a99510a176013e8bcac592b30617.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 255, + 470, + 355, + 485.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 255, + 485.0, + 355, + 500.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 102, + 507, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 142, + 519 + ], + "score": 1.0, + "content": "In words,", + "type": "text" + }, + { + "bbox": [ + 142, + 508, + 150, + 517 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 506, + 228, + 519 + ], + "score": 1.0, + "content": "truncated state vector", + "type": "text" + }, + { + "bbox": [ + 228, + 507, + 246, + 518 + ], + "score": 0.91, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 506, + 334, + 519 + ], + "score": 1.0, + "content": "is obtained by unrolling", + "type": "text" + }, + { + "bbox": [ + 334, + 508, + 344, + 517 + ], + "score": 0.87, + "content": "\\mathbf { \\delta } _ { h _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 506, + 382, + 519 + ], + "score": 1.0, + "content": "until time", + "type": "text" + }, + { + "bbox": [ + 383, + 508, + 405, + 517 + ], + "score": 0.88, + "content": "t - L", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "and setting the contribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 514, + 429, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 172, + 532 + ], + "score": 1.0, + "content": "of the state vector", + "type": "text" + }, + { + "bbox": [ + 173, + 518, + 194, + 528 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { t - L }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 514, + 251, + 532 + ], + "score": 1.0, + "content": "to 0. This way,", + "type": "text" + }, + { + "bbox": [ + 251, + 517, + 269, + 528 + ], + "score": 0.9, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 514, + 379, + 532 + ], + "score": 1.0, + "content": "depends only on the variables", + "type": "text" + }, + { + "bbox": [ + 379, + 517, + 424, + 529 + ], + "score": 0.92, + "content": "\\{ u _ { \\tau } \\} _ { \\tau = t - L } ^ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 514, + 429, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 500, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 456, + 547 + ], + "score": 1.0, + "content": "The following lemma states that impact of truncation can be made fairly small for stable systems", + "type": "text" + }, + { + "bbox": [ + 456, + 534, + 496, + 546 + ], + "score": 0.88, + "content": "( \\left. A \\right. < 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 533, + 500, + 547 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 380, + 558 + ], + "score": 1.0, + "content": "Lemma C.2 (Truncation impact – deterministic). Consider the state vector", + "type": "text" + }, + { + "bbox": [ + 380, + 547, + 390, + 556 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 546, + 418, + 558 + ], + "score": 1.0, + "content": "and its", + "type": "text" + }, + { + "bbox": [ + 419, + 547, + 425, + 556 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 546, + 467, + 558 + ], + "score": 1.0, + "content": "-truncation", + "type": "text" + }, + { + "bbox": [ + 467, + 546, + 484, + 558 + ], + "score": 0.9, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 557, + 303, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 194, + 568 + ], + "score": 1.0, + "content": "Definition C.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 195, + 558, + 201, + 567 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 557, + 303, + 568 + ], + "score": 1.0, + "content": "is 1-Lipschitz. We have that", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 567, + 388, + 598 + ], + "lines": [ + { + "bbox": [ + 222, + 567, + 388, + 598 + ], + "spans": [ + { + "bbox": [ + 222, + 567, + 388, + 598 + ], + "score": 0.94, + "content": "\\| h _ { t } - \\bar { h } _ { t , L } \\| _ { \\ell _ { 2 } } \\leq \\left\\{ \\begin{array} { l l } { 0 i f t \\leq L } \\\\ { \\| A \\| ^ { L } \\| h _ { t - L } \\| _ { \\ell _ { 2 } } e l s e } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "9d8edc1566a5e0b82315ed1e75bc76aafb951a6daf4e2ed43431f0c8c9f0601e.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 222, + 567, + 388, + 582.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 222, + 582.5, + 388, + 598.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 158, + 621 + ], + "score": 1.0, + "content": "Proof. When", + "type": "text" + }, + { + "bbox": [ + 158, + 610, + 182, + 620 + ], + "score": 0.89, + "content": "t \\leq L", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 608, + 271, + 621 + ], + "score": 1.0, + "content": ", Definition C.1 implies", + "type": "text" + }, + { + "bbox": [ + 271, + 609, + 308, + 620 + ], + "score": 0.92, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime } = { \\pmb u } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 608, + 333, + 621 + ], + "score": 1.0, + "content": "hence", + "type": "text" + }, + { + "bbox": [ + 334, + 609, + 397, + 620 + ], + "score": 0.92, + "content": "{ \\pmb h } _ { t } = { \\pmb q } _ { t } = \\bar { { \\pmb h } } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 608, + 426, + 621 + ], + "score": 1.0, + "content": ". 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{ \\pmb q } _ { \\tau } \\| _ { \\ell _ { 2 } } = \\| \\phi ( { \\pmb A } h _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) - \\phi ( { \\pmb A } { \\pmb q } _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\le \\| ( { \\pmb A } h _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) - ( { \\pmb A } { \\pmb q } _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\le \\| { \\pmb A } ( h _ { \\tau - 1 } - { \\pmb q } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } \\le \\| { \\pmb A } \\| \\| h _ { \\tau - 1 } - { \\pmb q } _ { \\tau - 1 } \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "237f124d4e461a8981e182c3c3d4c9af98e84254ca898c20dce707d0ef9ffdf5.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 179, + 640, + 430, + 653.3333333333334 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 179, + 653.3333333333334, + 430, + 666.6666666666667 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 179, + 666.6666666666667, + 430, + 680.0000000000001 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 110, + 680, + 503, + 691 + ], + "lines": [ + { + "bbox": [ + 108, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 108, + 678, + 225, + 693 + ], + "score": 1.0, + "content": "Applying this recursion between", + "type": "text" + }, + { + "bbox": [ + 226, + 681, + 280, + 690 + ], + "score": 0.89, + "content": "t - L < \\tau \\leq t", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 678, + 362, + 693 + ], + "score": 1.0, + "content": "and using the fact that", + "type": "text" + }, + { + "bbox": [ + 363, + 681, + 400, + 691 + ], + "score": 0.89, + "content": "\\pmb q _ { t - L } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "implies the advertised result", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 691, + 381, + 722 + ], + "lines": [ + { + "bbox": [ + 230, + 691, + 381, + 722 + ], + "spans": [ + { + "bbox": [ + 230, + 691, + 381, + 722 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { \\| \\pmb { h } _ { t } - \\pmb { q } _ { t } \\| _ { \\ell _ { 2 } } \\leq \\| \\pmb { A } \\| ^ { L } \\| \\pmb { h } _ { t - L } - \\pmb { q } _ { t - L } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\pmb { A } \\| ^ { L } \\| \\pmb { h } _ { t - L } \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "94e0305accc9c22f5eaf86897b61391a47c90a0fa63e2ad21af6875fb2667f92.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 691, + 381, + 706.5 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 230, + 706.5, + 381, + 722.0 + ], + "spans": [], + "index": 46 + } + ] + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 322, + 505, + 332 + ], + "lines": [ + { + "bbox": [ + 496, + 323, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 496, + 323, + 505, + 334 + ], + "score": 0.993, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 150, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 496, + 151, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 496, + 151, + 505, + 162 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 721, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 496, + 721, + 505, + 733 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 497, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 498, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 175, + 97 + ], + "score": 1.0, + "content": "Proof. First, write", + "type": "text" + }, + { + "bbox": [ + 176, + 82, + 288, + 94 + ], + "score": 0.91, + "content": "\\Sigma [ { \\pmb y } ] = \\mathbb { E } [ { \\pmb y } { \\pmb y } ^ { T } ] - \\mathbb { E } [ { \\pmb y } ] \\mathbb { E } [ { \\pmb y } ^ { T } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 81, + 498, + 97 + ], + "score": 1.0, + "content": ". Then, applying the law of total expectation to each term,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 498, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 96, + 397, + 110 + ], + "lines": [ + { + "bbox": [ + 210, + 96, + 397, + 110 + ], + "spans": [ + { + "bbox": [ + 210, + 96, + 397, + 110 + ], + "score": 0.66, + "content": "\\pmb { \\Sigma } [ \\pmb { y } ] = \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] - \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } ^ { T } \\mid \\pmb { x } ] ]", + "type": "interline_equation", + "image_path": "ae3d5e164ef16e7b936f71d53a104985a9d5229e1585e41ca785e85df5460e82.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 210, + 96, + 397, + 110 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 134 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 288, + 124 + ], + "score": 1.0, + "content": "Next, we can write the conditional expectation as", + "type": "text" + }, + { + "bbox": [ + 288, + 111, + 490, + 124 + ], + "score": 0.89, + "content": "\\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] = \\mathbb { E } [ \\pmb { \\Sigma } [ \\pmb { y } \\mid \\pmb { x } ] ] + \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] ^ { T } .", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 109, + 506, + 124 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 345, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 244, + 135 + ], + "score": 1.0, + "content": "conclude, we obtain the covariance of", + "type": "text" + }, + { + "bbox": [ + 245, + 122, + 276, + 135 + ], + "score": 0.92, + "content": "\\mathbb { E } [ { \\pmb y } \\mid { \\pmb x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 122, + 345, + 135 + ], + "score": 1.0, + "content": "via the difference,", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 109, + 506, + 135 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 136, + 426, + 150 + ], + "lines": [ + { + "bbox": [ + 184, + 136, + 426, + 150 + ], + "spans": [ + { + "bbox": [ + 184, + 136, + 426, + 150 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] ^ { T } - \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] \\mathbb { E } [ \\mathbb { E } [ \\pmb { y } ^ { T } \\mid \\pmb { x } ] ] = \\pmb { \\Sigma } [ \\mathbb { E } [ \\pmb { y } \\mid \\pmb { x } ] ] , } \\end{array}", + "type": "interline_equation", + "image_path": "23772338c41be9a58f0e01f27b4886b0c960b4747eec6e9dcbfeca8a85e1ba99.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 184, + 136, + 426, + 150 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 221, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 222, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 222, + 162 + ], + "score": 1.0, + "content": "which yields the desired bound.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 150, + 222, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 166, + 504, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 270, + 178 + ], + "score": 1.0, + "content": "Lemma B.9 (Scalar lower bound). Suppose", + "type": "text" + }, + { + "bbox": [ + 270, + 168, + 277, + 177 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 165, + 293, + 178 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 293, + 167, + 300, + 177 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 165, + 392, + 178 + ], + "score": 1.0, + "content": "-increasing function with", + "type": "text" + }, + { + "bbox": [ + 393, + 167, + 416, + 177 + ], + "score": 0.91, + "content": "\\beta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "as defined in Definition", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 176, + 313, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 215, + 188 + ], + "score": 1.0, + "content": "3.1. 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\\mathbb { E } [ \\phi ( X + y ) ] ) ^ { 2 } } \\\\ & { \\qquad \\geq \\beta ^ { 2 } \\mathbb { E } ( ( X + y ) - \\phi ^ { - 1 } ( \\mathbb { E } [ \\phi ( X + y ) ] ) ) ^ { 2 } } \\\\ & { \\qquad \\geq \\beta ^ { 2 } \\mathbb { E } ( X + y - \\mathbb { E } [ X + y ] ) ^ { 2 } } \\\\ & { \\qquad = \\beta ^ { 2 } \\mathbb { E } ( X - \\mathbb { E } X ) ^ { 2 } = \\beta ^ { 2 } \\mathbf { v a r } [ X ] . } \\end{array}", + "type": "interline_equation", + "image_path": "59d943e38de5e0ffe9fb584eafd9376cd2fefc44425f91dc32a27433d99ec06c.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 262, + 412, + 277.25 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 198, + 277.25, + 412, + 292.5 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 198, + 292.5, + 412, + 307.75 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 198, + 307.75, + 412, + 323.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 322, + 283, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 284, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 284, + 334 + ], + "score": 1.0, + "content": "Note that, the final line is the desired conclusion.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 322, + 284, + 334 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 348, + 291, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 293, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 293, + 362 + ], + "score": 1.0, + "content": "C TRUNCATING STABLE SYSTEMS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "One of the challenges in analyzing dynamical systems is the fact that samples from the same trajectory", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "have temporal dependence. This section shows that, for stable systems, the impact of the past states decay", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "exponentially fast and the system can be approximated by using the recent inputs only. We first define the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 402, + 213, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 213, + 412 + ], + "score": 1.0, + "content": "truncation of the state vector.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 370, + 505, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 504, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 290, + 425 + ], + "score": 1.0, + "content": "Definition C.1 (Truncated state vector). Suppose", + "type": "text" + }, + { + "bbox": [ + 290, + 413, + 325, + 424 + ], + "score": 0.92, + "content": "\\phi ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 412, + 390, + 425 + ], + "score": 1.0, + "content": ", initial condition", + "type": "text" + }, + { + "bbox": [ + 390, + 414, + 418, + 423 + ], + "score": 0.89, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 412, + 506, + 425 + ], + "score": 1.0, + "content": ", and consider the state", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 503, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 233, + 435 + ], + "score": 1.0, + "content": "equation (1.1). Given a timestamp", + "type": "text" + }, + { + "bbox": [ + 233, + 424, + 237, + 432 + ], + "score": 0.34, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 422, + 241, + 435 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 241, + 424, + 248, + 432 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 422, + 357, + 435 + ], + "score": 1.0, + "content": "-truncation of the state vector", + "type": "text" + }, + { + "bbox": [ + 357, + 424, + 368, + 433 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 422, + 419, + 435 + ], + "score": 1.0, + "content": "is denoted by", + "type": "text" + }, + { + "bbox": [ + 419, + 423, + 437, + 434 + ], + "score": 0.9, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 422, + 495, + 435 + ], + "score": 1.0, + "content": "and is equal to", + "type": "text" + }, + { + "bbox": [ + 495, + 426, + 503, + 434 + ], + "score": 0.66, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\eta } } \\mathbf { q } _ { t }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 132, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 132, + 444 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 412, + 506, + 444 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 444, + 377, + 457 + ], + "lines": [ + { + "bbox": [ + 234, + 444, + 377, + 457 + ], + "spans": [ + { + "bbox": [ + 234, + 444, + 377, + 457 + ], + "score": 0.91, + "content": "\\pmb { q } \\tau + 1 = \\phi ( \\pmb { A } \\pmb { q } _ { \\tau } + \\pmb { B } \\pmb { u } _ { \\tau } ^ { \\prime } ) \\quad , \\quad q _ { 0 } = 0", + "type": "interline_equation", + "image_path": "446a96b3504994399aa5ffecbf626bffb6669f479f0405ec924e2a4bf13b05b4.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 234, + 444, + 377, + 457 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 459, + 307, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 308, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 258, + 472 + ], + "score": 1.0, + "content": "is the state vector generated by the inputs", + "type": "text" + }, + { + "bbox": [ + 258, + 459, + 270, + 469 + ], + "score": 0.88, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 457, + 308, + 472 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 457, + 308, + 472 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 470, + 355, + 500 + ], + "lines": [ + { + "bbox": [ + 255, + 470, + 355, + 500 + ], + "spans": [ + { + "bbox": [ + 255, + 470, + 355, + 500 + ], + "score": 0.93, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime } = \\left\\{ \\begin{array} { l l } { 0 i f \\tau < t - L } \\\\ { { \\pmb u } _ { \\tau } e l s e } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "85b7b37191a5fc384df5b34ce1bf6e9203b7a99510a176013e8bcac592b30617.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 255, + 470, + 355, + 485.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 255, + 485.0, + 355, + 500.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 102, + 507, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 142, + 519 + ], + "score": 1.0, + "content": "In words,", + "type": "text" + }, + { + "bbox": [ + 142, + 508, + 150, + 517 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 506, + 228, + 519 + ], + "score": 1.0, + "content": "truncated state vector", + "type": "text" + }, + { + "bbox": [ + 228, + 507, + 246, + 518 + ], + "score": 0.91, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 506, + 334, + 519 + ], + "score": 1.0, + "content": "is obtained by unrolling", + "type": "text" + }, + { + "bbox": [ + 334, + 508, + 344, + 517 + ], + "score": 0.87, + "content": "\\mathbf { \\delta } _ { h _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 506, + 382, + 519 + ], + "score": 1.0, + "content": "until time", + "type": "text" + }, + { + "bbox": [ + 383, + 508, + 405, + 517 + ], + "score": 0.88, + "content": "t - L", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "and setting the contribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 514, + 429, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 172, + 532 + ], + "score": 1.0, + "content": "of the state vector", + "type": "text" + }, + { + "bbox": [ + 173, + 518, + 194, + 528 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { t - L }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 514, + 251, + 532 + ], + "score": 1.0, + "content": "to 0. This way,", + "type": "text" + }, + { + "bbox": [ + 251, + 517, + 269, + 528 + ], + "score": 0.9, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 514, + 379, + 532 + ], + "score": 1.0, + "content": "depends only on the variables", + "type": "text" + }, + { + "bbox": [ + 379, + 517, + 424, + 529 + ], + "score": 0.92, + "content": "\\{ u _ { \\tau } \\} _ { \\tau = t - L } ^ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 514, + 429, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 506, + 505, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 500, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 456, + 547 + ], + "score": 1.0, + "content": "The following lemma states that impact of truncation can be made fairly small for stable systems", + "type": "text" + }, + { + "bbox": [ + 456, + 534, + 496, + 546 + ], + "score": 0.88, + "content": "( \\left. A \\right. < 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 533, + 500, + 547 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 380, + 558 + ], + "score": 1.0, + "content": "Lemma C.2 (Truncation impact – deterministic). Consider the state vector", + "type": "text" + }, + { + "bbox": [ + 380, + 547, + 390, + 556 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 546, + 418, + 558 + ], + "score": 1.0, + "content": "and its", + "type": "text" + }, + { + "bbox": [ + 419, + 547, + 425, + 556 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 546, + 467, + 558 + ], + "score": 1.0, + "content": "-truncation", + "type": "text" + }, + { + "bbox": [ + 467, + 546, + 484, + 558 + ], + "score": 0.9, + "content": "\\bar { h } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 557, + 303, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 194, + 568 + ], + "score": 1.0, + "content": "Definition C.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 195, + 558, + 201, + 567 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 557, + 303, + 568 + ], + "score": 1.0, + "content": "is 1-Lipschitz. We have that", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 533, + 505, + 568 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 567, + 388, + 598 + ], + "lines": [ + { + "bbox": [ + 222, + 567, + 388, + 598 + ], + "spans": [ + { + "bbox": [ + 222, + 567, + 388, + 598 + ], + "score": 0.94, + "content": "\\| h _ { t } - \\bar { h } _ { t , L } \\| _ { \\ell _ { 2 } } \\leq \\left\\{ \\begin{array} { l l } { 0 i f t \\leq L } \\\\ { \\| A \\| ^ { L } \\| h _ { t - L } \\| _ { \\ell _ { 2 } } e l s e } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "9d8edc1566a5e0b82315ed1e75bc76aafb951a6daf4e2ed43431f0c8c9f0601e.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 222, + 567, + 388, + 582.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 222, + 582.5, + 388, + 598.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 158, + 621 + ], + "score": 1.0, + "content": "Proof. When", + "type": "text" + }, + { + "bbox": [ + 158, + 610, + 182, + 620 + ], + "score": 0.89, + "content": "t \\leq L", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 608, + 271, + 621 + ], + "score": 1.0, + "content": ", Definition C.1 implies", + "type": "text" + }, + { + "bbox": [ + 271, + 609, + 308, + 620 + ], + "score": 0.92, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime } = { \\pmb u } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 608, + 333, + 621 + ], + "score": 1.0, + "content": "hence", + "type": "text" + }, + { + "bbox": [ + 334, + 609, + 397, + 620 + ], + "score": 0.92, + "content": "{ \\pmb h } _ { t } = { \\pmb q } _ { t } = \\bar { { \\pmb h } } _ { t , L }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 608, + 426, + 621 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 426, + 610, + 451, + 619 + ], + "score": 0.89, + "content": "t > L", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 608, + 505, + 621 + ], + "score": 1.0, + "content": ", we again use", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 214, + 631 + ], + "score": 1.0, + "content": "Definition C.1 and recall that", + "type": "text" + }, + { + "bbox": [ + 214, + 619, + 243, + 630 + ], + "score": 0.92, + "content": "{ \\pmb u } _ { \\tau } ^ { \\prime } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 618, + 281, + 631 + ], + "score": 1.0, + "content": "until time", + "type": "text" + }, + { + "bbox": [ + 281, + 620, + 336, + 630 + ], + "score": 0.89, + "content": "\\tau = t - L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 618, + 366, + 631 + ], + "score": 1.0, + "content": ". For all", + "type": "text" + }, + { + "bbox": [ + 366, + 621, + 421, + 630 + ], + "score": 0.91, + "content": "t - L < \\tau \\leq t", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 618, + 505, + 631 + ], + "score": 1.0, + "content": ", using 1-Lipschitzness", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 629, + 174, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 116, + 640 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 630, + 123, + 640 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 629, + 174, + 640 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 608, + 505, + 640 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 640, + 430, + 680 + ], + "lines": [ + { + "bbox": [ + 179, + 640, + 430, + 680 + ], + "spans": [ + { + "bbox": [ + 179, + 640, + 430, + 680 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| h _ { \\tau } - { \\pmb q } _ { \\tau } \\| _ { \\ell _ { 2 } } = \\| \\phi ( { \\pmb A } h _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) - \\phi ( { \\pmb A } { \\pmb q } _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\le \\| ( { \\pmb A } h _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) - ( { \\pmb A } { \\pmb q } _ { \\tau - 1 } + { \\pmb B } { \\pmb u } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\le \\| { \\pmb A } ( h _ { \\tau - 1 } - { \\pmb q } _ { \\tau - 1 } ) \\| _ { \\ell _ { 2 } } \\le \\| { \\pmb A } \\| \\| h _ { \\tau - 1 } - { \\pmb q } _ { \\tau - 1 } \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "237f124d4e461a8981e182c3c3d4c9af98e84254ca898c20dce707d0ef9ffdf5.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 179, + 640, + 430, + 653.3333333333334 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 179, + 653.3333333333334, + 430, + 666.6666666666667 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 179, + 666.6666666666667, + 430, + 680.0000000000001 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 110, + 680, + 503, + 691 + ], + "lines": [ + { + "bbox": [ + 108, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 108, + 678, + 225, + 693 + ], + "score": 1.0, + "content": "Applying this recursion between", + "type": "text" + }, + { + "bbox": [ + 226, + 681, + 280, + 690 + ], + "score": 0.89, + "content": "t - L < \\tau \\leq t", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 678, + 362, + 693 + ], + "score": 1.0, + "content": "and using the fact that", + "type": "text" + }, + { + "bbox": [ + 363, + 681, + 400, + 691 + ], + "score": 0.89, + "content": "\\pmb q _ { t - L } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "implies the advertised result", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 108, + 678, + 505, + 693 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 691, + 381, + 722 + ], + "lines": [ + { + "bbox": [ + 230, + 691, + 381, + 722 + ], + "spans": [ + { + "bbox": [ + 230, + 691, + 381, + 722 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { \\| \\pmb { h } _ { t } - \\pmb { q } _ { t } \\| _ { \\ell _ { 2 } } \\leq \\| \\pmb { A } \\| ^ { L } \\| \\pmb { h } _ { t - L } - \\pmb { q } _ { t - L } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\pmb { A } \\| ^ { L } \\| \\pmb { h } _ { t - L } \\| _ { \\ell _ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "94e0305accc9c22f5eaf86897b61391a47c90a0fa63e2ad21af6875fb2667f92.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 691, + 381, + 706.5 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 230, + 706.5, + 381, + 722.0 + ], + "spans": [], + "index": 46 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 101, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "We will now argue that, for stable systems, a single trajectory can be split into multiple nearly independent", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 112, + 364, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 364, + 124 + ], + "score": 1.0, + "content": "trajectories. First, we describe how the sub-trajectories are constructed.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 504, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 503, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 292, + 138 + ], + "score": 1.0, + "content": "Definition C.3 (Sub-trajectory). Let sampling rate", + "type": "text" + }, + { + "bbox": [ + 292, + 126, + 316, + 136 + ], + "score": 0.88, + "content": "L \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 125, + 354, + 138 + ], + "score": 1.0, + "content": "and offset", + "type": "text" + }, + { + "bbox": [ + 354, + 126, + 396, + 136 + ], + "score": 0.9, + "content": "1 \\le \\bar { \\tau } \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 125, + 470, + 138 + ], + "score": 1.0, + "content": "be two integers. Let", + "type": "text" + }, + { + "bbox": [ + 470, + 126, + 503, + 136 + ], + "score": 0.9, + "content": "\\bar { N } = \\bar { N } _ { \\bar { \\tau } }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 221, + 149 + ], + "score": 1.0, + "content": "be the largest integer obeying", + "type": "text" + }, + { + "bbox": [ + 222, + 136, + 303, + 147 + ], + "score": 0.91, + "content": "( { \\bar { N } } - 1 ) { \\bar { L } } + { \\bar { \\tau } } \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 136, + 407, + 149 + ], + "score": 1.0, + "content": ". We sample the trajectory", + "type": "text" + }, + { + "bbox": [ + 408, + 136, + 453, + 147 + ], + "score": 0.93, + "content": "\\{ h _ { t } , \\boldsymbol { u } _ { t } \\} _ { t = 0 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "at the points", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 145, + 362, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 230, + 157 + ], + "score": 0.92, + "content": "\\bar { \\tau } , \\bar { \\tau } + L , \\dots , \\bar { \\tau } + ( \\bar { N } - 1 ) L + \\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 145, + 362, + 158 + ], + "score": 1.0, + "content": "and define the τ¯th sub-trajectory as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 162, + 414, + 176 + ], + "lines": [ + { + "bbox": [ + 196, + 162, + 414, + 176 + ], + "spans": [ + { + "bbox": [ + 196, + 162, + 414, + 176 + ], + "score": 0.89, + "content": "( { \\pmb h } ^ { ( i ) } , { \\pmb u } ^ { ( i ) } ) : = ( { \\pmb h } ^ { ( i , \\bar { \\tau } ) } , { \\pmb u } ^ { ( i , \\bar { \\tau } ) } ) = ( { \\pmb h } _ { ( i - 1 ) L + \\bar { \\tau } } , { \\pmb u } _ { ( i - 1 ) L + \\bar { \\tau } } ) .", + "type": "interline_equation", + "image_path": "cfb605345930fd8d087ff1239d81c2fb3dd9aec4377a75146c63ceb6f04c964f.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 196, + 162, + 414, + 176 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 504, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "score": 1.0, + "content": "Definition C.4 (Truncated sub-trajectory). Consider the state equation (1.1) and recall Definition C.1. Given", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 102, + 189, + 508, + 207 + ], + "spans": [ + { + "bbox": [ + 102, + 189, + 127, + 207 + ], + "score": 1.0, + "content": "offset", + "type": "text" + }, + { + "bbox": [ + 128, + 193, + 134, + 201 + ], + "score": 0.73, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 189, + 201, + 207 + ], + "score": 1.0, + "content": "and sampling rate", + "type": "text" + }, + { + "bbox": [ + 201, + 193, + 208, + 201 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 189, + 224, + 207 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 224, + 191, + 265, + 202 + ], + "score": 0.9, + "content": "1 \\leq i \\leq \\bar { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 189, + 418, + 207 + ], + "score": 1.0, + "content": ", the ith truncated sub-trajectory states are", + "type": "text" + }, + { + "bbox": [ + 418, + 191, + 455, + 203 + ], + "score": 0.93, + "content": "\\{ \\bar { h } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 189, + 508, + 207 + ], + "score": 1.0, + "content": "where the ith", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 173, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 173, + 213 + ], + "score": 1.0, + "content": "state is defined as", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 210, + 349, + 226 + ], + "lines": [ + { + "bbox": [ + 261, + 210, + 349, + 226 + ], + "spans": [ + { + "bbox": [ + 261, + 210, + 349, + 226 + ], + "score": 0.91, + "content": "\\bar { \\pmb { h } } ^ { ( i ) } = \\bar { \\pmb { h } } _ { L ( i - 1 ) + \\bar { \\tau } , L - 1 } .", + "type": "interline_equation", + "image_path": "696e48086acd07d35240914316a784acf6191a950542cb24728e28c4b4156742.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 261, + 210, + 349, + 226 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 401, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 401, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 401, + 245 + ], + "score": 1.0, + "content": "The truncated samples are independent of each other as shown in the next lemma.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 506, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 507, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 507, + 258 + ], + "score": 1.0, + "content": "Lemma C.5. Consider the truncated states of Definition C.4. If (1.1) is generated by independent vec-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 254, + 507, + 273 + ], + "spans": [ + { + "bbox": [ + 102, + 254, + 379, + 273 + ], + "score": 1.0, + "content": "tors {ut}t≥0, for any offset τ¯ and sampling rate L, the vectors {h¯ (i)}N¯i=1", + "type": "text" + }, + { + "bbox": [ + 342, + 257, + 420, + 270 + ], + "score": 0.86, + "content": "\\{ \\bar { \\pmb { h } } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } } , \\{ \\pmb { u } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 258, + 507, + 271 + ], + "score": 1.0, + "content": "are all independent of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 149, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 149, + 279 + ], + "score": 1.0, + "content": "each other.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 101, + 282, + 510, + 310 + ], + "spans": [ + { + "bbox": [ + 101, + 282, + 192, + 310 + ], + "score": 1.0, + "content": "Proof. By construction", + "type": "text" + }, + { + "bbox": [ + 192, + 290, + 208, + 300 + ], + "score": 0.89, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 282, + 310, + 310 + ], + "score": 1.0, + "content": "only depends on the vectors", + "type": "text" + }, + { + "bbox": [ + 311, + 289, + 383, + 304 + ], + "score": 0.94, + "content": "\\left\\{ \\pmb { u } _ { \\tau } \\right\\} _ { \\tau = L ( i - 2 ) + \\bar { \\tau } + 1 } ^ { L ( i - 1 ) + \\bar { \\tau } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 282, + 510, + 310 + ], + "score": 1.0, + "content": ". Note that the dependence ranges", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 300, + 508, + 319 + ], + "spans": [ + { + "bbox": [ + 107, + 305, + 244, + 316 + ], + "score": 0.92, + "content": "[ L ( i - 2 ) + \\bar { \\tau } + 1 , L ( i - 1 ) + \\bar { \\tau } - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 300, + 363, + 319 + ], + "score": 1.0, + "content": "are disjoint intervals for different", + "type": "text" + }, + { + "bbox": [ + 364, + 306, + 368, + 314 + ], + "score": 0.61, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 300, + 401, + 319 + ], + "score": 1.0, + "content": "’s; hence", + "type": "text" + }, + { + "bbox": [ + 401, + 303, + 436, + 316 + ], + "score": 0.91, + "content": "( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 300, + 508, + 319 + ], + "score": 1.0, + "content": "are independent of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 313, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 255, + 330 + ], + "score": 1.0, + "content": "each other. To show the independence of", + "type": "text" + }, + { + "bbox": [ + 255, + 316, + 271, + 326 + ], + "score": 0.89, + "content": "\\mathbf { \\pmb { u } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 313, + 288, + 330 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 316, + 303, + 326 + ], + "score": 0.88, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 313, + 377, + 330 + ], + "score": 1.0, + "content": "; observe that inputs", + "type": "text" + }, + { + "bbox": [ + 378, + 316, + 445, + 329 + ], + "score": 0.92, + "content": "\\pmb { u } ^ { ( i ) } = \\pmb { u } _ { L ( i - 1 ) + \\hat { \\tau } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 313, + 506, + 330 + ], + "score": 1.0, + "content": "have timestamp", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 113, + 339 + ], + "score": 0.71, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 320, + 144, + 346 + ], + "score": 1.0, + "content": "modulo", + "type": "text" + }, + { + "bbox": [ + 144, + 330, + 151, + 339 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 320, + 332, + 346 + ], + "score": 1.0, + "content": "; which is not covered by the dependence range of", + "type": "text" + }, + { + "bbox": [ + 333, + 327, + 367, + 340 + ], + "score": 0.93, + "content": "( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 320, + 375, + 346 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 329, + 506, + 340 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 105, + 352, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 445, + 363 + ], + "score": 1.0, + "content": "If the input is randomly generated, Lemma C.2 can be combined with a probabilistic bound on", + "type": "text" + }, + { + "bbox": [ + 445, + 353, + 456, + 362 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 352, + 505, + 363 + ], + "score": 1.0, + "content": ", to show that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 325, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 164, + 375 + ], + "score": 1.0, + "content": "truncated states", + "type": "text" + }, + { + "bbox": [ + 164, + 362, + 180, + 373 + ], + "score": 0.88, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 361, + 305, + 375 + ], + "score": 1.0, + "content": "are fairly close to the actual states", + "type": "text" + }, + { + "bbox": [ + 306, + 362, + 321, + 373 + ], + "score": 0.89, + "content": "\\mathbf { \\delta } _ { h } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 361, + 325, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 506, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 319, + 389 + ], + "score": 1.0, + "content": "Lemma C.6 (Truncation impact – random). Given offset", + "type": "text" + }, + { + "bbox": [ + 320, + 379, + 326, + 387 + ], + "score": 0.7, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 376, + 397, + 389 + ], + "score": 1.0, + "content": "and sampling rate", + "type": "text" + }, + { + "bbox": [ + 397, + 378, + 404, + 387 + ], + "score": 0.71, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 376, + 506, + 389 + ], + "score": 1.0, + "content": ", consider the state vectors", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 103, + 386, + 503, + 404 + ], + "spans": [ + { + "bbox": [ + 103, + 386, + 183, + 404 + ], + "score": 1.0, + "content": "of the sub-trajectory", + "type": "text" + }, + { + "bbox": [ + 184, + 388, + 221, + 401 + ], + "score": 0.92, + "content": "\\{ h ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 386, + 239, + 404 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 390, + 262, + 400 + ], + "score": 0.83, + "content": "L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 386, + 308, + 404 + ], + "score": 1.0, + "content": "-truncations", + "type": "text" + }, + { + "bbox": [ + 309, + 388, + 343, + 401 + ], + "score": 0.92, + "content": "( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 386, + 381, + 404 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 381, + 388, + 503, + 401 + ], + "score": 0.67, + "content": "\\{ \\pmb { u } _ { t } \\} _ { t \\geq 0 } \\stackrel { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } ) , \\lVert \\pmb { A } \\rVert < 1", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 107, + 401, + 135, + 410 + ], + "score": 0.86, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 399, + 138, + 413 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 138, + 401, + 145, + 411 + ], + "score": 0.77, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 399, + 215, + 413 + ], + "score": 1.0, + "content": "is 1-Lipschitz, and", + "type": "text" + }, + { + "bbox": [ + 215, + 401, + 250, + 411 + ], + "score": 0.91, + "content": "\\phi ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 399, + 426, + 413 + ], + "score": 1.0, + "content": ". Also suppose upper bound (4.3) of Assumption", + "type": "text" + }, + { + "bbox": [ + 426, + 402, + 431, + 409 + ], + "score": 0.38, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 399, + 488, + 413 + ], + "score": 1.0, + "content": "holds for some", + "type": "text" + }, + { + "bbox": [ + 489, + 401, + 505, + 410 + ], + "score": 0.84, + "content": "\\theta \\leq", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 107, + 411, + 153, + 421 + ], + "score": 0.89, + "content": "{ \\sqrt { n } } , \\gamma _ { + } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 410, + 278, + 423 + ], + "score": 1.0, + "content": ". There exists an absolute constant", + "type": "text" + }, + { + "bbox": [ + 278, + 411, + 300, + 420 + ], + "score": 0.89, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 410, + 424, + 423 + ], + "score": 1.0, + "content": "such that with probability at least", + "type": "text" + }, + { + "bbox": [ + 424, + 410, + 503, + 421 + ], + "score": 0.9, + "content": "1 - 2 \\bar { N } \\mathrm { \\bar { e x p } } ( - 1 0 0 n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 410, + 506, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 273, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 131, + 432 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 131, + 420, + 172, + 430 + ], + "score": 0.91, + "content": "1 \\leq i \\leq \\bar { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 420, + 273, + 432 + ], + "score": 1.0, + "content": ", the following bound holds", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 435, + 376, + 450 + ], + "lines": [ + { + "bbox": [ + 234, + 435, + 376, + 450 + ], + "spans": [ + { + "bbox": [ + 234, + 435, + 376, + 450 + ], + "score": 0.9, + "content": "\\| h ^ { ( i ) } - \\bar { h } ^ { ( i ) } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { n } \\| A \\| ^ { L - 1 } \\sqrt { \\gamma _ { + } } .", + "type": "interline_equation", + "image_path": "8e187455e36ffdd751ac435d8796a1fc724da5d51e0b78956a59b374e08d727f.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 234, + 435, + 376, + 450 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 331, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 332, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 227, + 469 + ], + "score": 1.0, + "content": "In particular, we can always pick", + "type": "text" + }, + { + "bbox": [ + 228, + 455, + 266, + 466 + ], + "score": 0.92, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 454, + 312, + 469 + ], + "score": 1.0, + "content": "(via Lemma", + "type": "text" + }, + { + "bbox": [ + 312, + 456, + 326, + 465 + ], + "score": 0.26, + "content": "B . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 454, + 332, + 469 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 104, + 477, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 349, + 494 + ], + "score": 1.0, + "content": "Proof. Using Assumption 1, we can apply Lemma F.3 on vectors", + "type": "text" + }, + { + "bbox": [ + 349, + 478, + 420, + 492 + ], + "score": 0.93, + "content": "\\{ h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 477, + 507, + 494 + ], + "score": 1.0, + "content": ". Using a union bound,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 254, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 254, + 503 + ], + "score": 1.0, + "content": "with desired probability, all vectors obey", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 505, + 407, + 518 + ], + "lines": [ + { + "bbox": [ + 203, + 505, + 407, + 518 + ], + "spans": [ + { + "bbox": [ + 203, + 505, + 407, + 518 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } - \\mathbb { E } \\big [ h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\big ] \\| _ { \\ell _ { 2 } } \\leq ( c - 1 ) \\sqrt { n \\gamma _ { + } } , } \\end{array}", + "type": "interline_equation", + "image_path": "1102ab5eb959e5e9a6081c3a6a9df0efabf7791f01273a9dbb06cd61fe65c2c9.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 203, + 505, + 407, + 518 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 523, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 180, + 537 + ], + "score": 1.0, + "content": "for sufficiently large", + "type": "text" + }, + { + "bbox": [ + 181, + 525, + 186, + 532 + ], + "score": 0.66, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 522, + 211, + 537 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 242, + 534 + ], + "score": 0.91, + "content": "\\theta \\leq \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 522, + 340, + 537 + ], + "score": 1.0, + "content": ", triangle inequality implies", + "type": "text" + }, + { + "bbox": [ + 341, + 523, + 447, + 536 + ], + "score": 0.92, + "content": "\\| h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { n \\gamma _ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 522, + 505, + 537 + ], + "score": 1.0, + "content": ". Now, applying", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 252, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 177, + 546 + ], + "score": 1.0, + "content": "Lemma C.2, for all", + "type": "text" + }, + { + "bbox": [ + 178, + 534, + 219, + 545 + ], + "score": 0.92, + "content": "1 \\leq i \\leq \\bar { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 534, + 252, + 546 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 549, + 403, + 597 + ], + "lines": [ + { + "bbox": [ + 207, + 549, + 403, + 597 + ], + "spans": [ + { + "bbox": [ + 207, + 549, + 403, + 597 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| \\pmb { h } ^ { ( i ) } - \\bar { \\pmb { h } } ^ { ( i ) } \\| _ { \\ell _ { 2 } } = \\| \\pmb { h } _ { ( i - 1 ) L + \\bar { \\tau } } - \\bar { \\pmb { h } } _ { ( i - 1 ) L + \\bar { \\tau } , L - 1 } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\pmb { A } \\| ^ { L - 1 } \\| \\pmb { h } _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq c \\| \\pmb { A } \\| ^ { L - 1 } \\sqrt { n \\gamma + 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "5bf4df24229dac93eaa49840a340d125c6e63576b608103a3bd7da010fb011b6.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 207, + 549, + 403, + 565.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 207, + 565.0, + 403, + 581.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 207, + 581.0, + 403, + 597.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 625, + 305, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 305, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 305, + 639 + ], + "score": 1.0, + "content": "D PROPERTIES OF THE DATA MATRIX", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "This section utilizes the probabilistic estimates from Section B to provide bounds on the condition number of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 658, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 412, + 670 + ], + "score": 1.0, + "content": "data matrices obtained from the RNN trajectory (1.1). Following (2.2), these matrices", + "type": "text" + }, + { + "bbox": [ + 413, + 659, + 435, + 669 + ], + "score": 0.91, + "content": "_ { H , U }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 658, + 452, + 670 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 452, + 659, + 462, + 668 + ], + "score": 0.84, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 658, + 506, + 670 + ], + "score": 1.0, + "content": "are defined", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 670, + 118, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 118, + 680 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 682, + 453, + 697 + ], + "lines": [ + { + "bbox": [ + 159, + 682, + 453, + 697 + ], + "spans": [ + { + "bbox": [ + 159, + 682, + 453, + 697 + ], + "score": 0.88, + "content": "\\mathbf { { \\cal H } } = [ h _ { 1 } \\ \\dots \\ h _ { N } ] ^ { T } \\quad , \\quad \\mathbf { { \\cal U } } = \\mathbf { { \\cal H } } = [ u _ { 1 } \\ \\dots \\ u _ { N } ] ^ { T } \\quad , \\quad \\mathbf { { \\cal X } } = [ x _ { 1 } \\ \\dots \\ x _ { N } ] ^ { T } .", + "type": "interline_equation", + "image_path": "5d887163c25f83c4d904cc186b06cd4e8d76a784333b168f2b713d44414471d1.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 159, + 682, + 453, + 697 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 701, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 246, + 714 + ], + "score": 1.0, + "content": "The challenge is that, the state matrix", + "type": "text" + }, + { + "bbox": [ + 247, + 704, + 257, + 710 + ], + "score": 0.87, + "content": "_ H", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 699, + 505, + 714 + ], + "score": 1.0, + "content": "has dependent rows; which will be addressed by carefully splitting", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 709, + 507, + 725 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 156, + 725 + ], + "score": 1.0, + "content": "the trajectory", + "type": "text" + }, + { + "bbox": [ + 157, + 711, + 202, + 722 + ], + "score": 0.93, + "content": "\\{ u _ { t } , h _ { t } \\} _ { t = 0 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 709, + 507, + 725 + ], + "score": 1.0, + "content": "into multiple sub-trajectories which are internally weakly dependent as discussed in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 380, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 380, + 732 + ], + "score": 1.0, + "content": "Section C. We first define the matrices obtained from these sub-trajectories.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 599, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 496, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 496, + 599, + 505, + 611 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 109, + 82, + 326, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 82, + 328, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 82, + 328, + 95 + ], + "score": 1.0, + "content": "C.1 NEAR INDEPENDENCE OF SUB-TRAJECTORIES", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 101, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "We will now argue that, for stable systems, a single trajectory can be split into multiple nearly independent", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 112, + 364, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 364, + 124 + ], + "score": 1.0, + "content": "trajectories. First, we describe how the sub-trajectories are constructed.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 101, + 505, + 124 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 504, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 503, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 292, + 138 + ], + "score": 1.0, + "content": "Definition C.3 (Sub-trajectory). Let sampling rate", + "type": "text" + }, + { + "bbox": [ + 292, + 126, + 316, + 136 + ], + "score": 0.88, + "content": "L \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 125, + 354, + 138 + ], + "score": 1.0, + "content": "and offset", + "type": "text" + }, + { + "bbox": [ + 354, + 126, + 396, + 136 + ], + "score": 0.9, + "content": "1 \\le \\bar { \\tau } \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 125, + 470, + 138 + ], + "score": 1.0, + "content": "be two integers. Let", + "type": "text" + }, + { + "bbox": [ + 470, + 126, + 503, + 136 + ], + "score": 0.9, + "content": "\\bar { N } = \\bar { N } _ { \\bar { \\tau } }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 221, + 149 + ], + "score": 1.0, + "content": "be the largest integer obeying", + "type": "text" + }, + { + "bbox": [ + 222, + 136, + 303, + 147 + ], + "score": 0.91, + "content": "( { \\bar { N } } - 1 ) { \\bar { L } } + { \\bar { \\tau } } \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 136, + 407, + 149 + ], + "score": 1.0, + "content": ". We sample the trajectory", + "type": "text" + }, + { + "bbox": [ + 408, + 136, + 453, + 147 + ], + "score": 0.93, + "content": "\\{ h _ { t } , \\boldsymbol { u } _ { t } \\} _ { t = 0 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "at the points", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 145, + 362, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 230, + 157 + ], + "score": 0.92, + "content": "\\bar { \\tau } , \\bar { \\tau } + L , \\dots , \\bar { \\tau } + ( \\bar { N } - 1 ) L + \\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 145, + 362, + 158 + ], + "score": 1.0, + "content": "and define the τ¯th sub-trajectory as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 125, + 506, + 158 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 162, + 414, + 176 + ], + "lines": [ + { + "bbox": [ + 196, + 162, + 414, + 176 + ], + "spans": [ + { + "bbox": [ + 196, + 162, + 414, + 176 + ], + "score": 0.89, + "content": "( { \\pmb h } ^ { ( i ) } , { \\pmb u } ^ { ( i ) } ) : = ( { \\pmb h } ^ { ( i , \\bar { \\tau } ) } , { \\pmb u } ^ { ( i , \\bar { \\tau } ) } ) = ( { \\pmb h } _ { ( i - 1 ) L + \\bar { \\tau } } , { \\pmb u } _ { ( i - 1 ) L + \\bar { \\tau } } ) .", + "type": "interline_equation", + "image_path": "cfb605345930fd8d087ff1239d81c2fb3dd9aec4377a75146c63ceb6f04c964f.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 196, + 162, + 414, + 176 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 504, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "score": 1.0, + "content": "Definition C.4 (Truncated sub-trajectory). Consider the state equation (1.1) and recall Definition C.1. Given", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 102, + 189, + 508, + 207 + ], + "spans": [ + { + "bbox": [ + 102, + 189, + 127, + 207 + ], + "score": 1.0, + "content": "offset", + "type": "text" + }, + { + "bbox": [ + 128, + 193, + 134, + 201 + ], + "score": 0.73, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 189, + 201, + 207 + ], + "score": 1.0, + "content": "and sampling rate", + "type": "text" + }, + { + "bbox": [ + 201, + 193, + 208, + 201 + ], + "score": 0.72, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 189, + 224, + 207 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 224, + 191, + 265, + 202 + ], + "score": 0.9, + "content": "1 \\leq i \\leq \\bar { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 189, + 418, + 207 + ], + "score": 1.0, + "content": ", the ith truncated sub-trajectory states are", + "type": "text" + }, + { + "bbox": [ + 418, + 191, + 455, + 203 + ], + "score": 0.93, + "content": "\\{ \\bar { h } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 189, + 508, + 207 + ], + "score": 1.0, + "content": "where the ith", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 173, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 173, + 213 + ], + "score": 1.0, + "content": "state is defined as", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 102, + 179, + 508, + 213 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 210, + 349, + 226 + ], + "lines": [ + { + "bbox": [ + 261, + 210, + 349, + 226 + ], + "spans": [ + { + "bbox": [ + 261, + 210, + 349, + 226 + ], + "score": 0.91, + "content": "\\bar { \\pmb { h } } ^ { ( i ) } = \\bar { \\pmb { h } } _ { L ( i - 1 ) + \\bar { \\tau } , L - 1 } .", + "type": "interline_equation", + "image_path": "696e48086acd07d35240914316a784acf6191a950542cb24728e28c4b4156742.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 261, + 210, + 349, + 226 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 233, + 401, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 401, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 401, + 245 + ], + "score": 1.0, + "content": "The truncated samples are independent of each other as shown in the next lemma.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 232, + 401, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 506, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 507, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 507, + 258 + ], + "score": 1.0, + "content": "Lemma C.5. Consider the truncated states of Definition C.4. If (1.1) is generated by independent vec-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 254, + 507, + 273 + ], + "spans": [ + { + "bbox": [ + 102, + 254, + 379, + 273 + ], + "score": 1.0, + "content": "tors {ut}t≥0, for any offset τ¯ and sampling rate L, the vectors {h¯ (i)}N¯i=1", + "type": "text" + }, + { + "bbox": [ + 342, + 257, + 420, + 270 + ], + "score": 0.86, + "content": "\\{ \\bar { \\pmb { h } } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } } , \\{ \\pmb { u } ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 258, + 507, + 271 + ], + "score": 1.0, + "content": "are all independent of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 149, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 149, + 279 + ], + "score": 1.0, + "content": "each other.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 102, + 245, + 507, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 101, + 282, + 510, + 310 + ], + "spans": [ + { + "bbox": [ + 101, + 282, + 192, + 310 + ], + "score": 1.0, + "content": "Proof. By construction", + "type": "text" + }, + { + "bbox": [ + 192, + 290, + 208, + 300 + ], + "score": 0.89, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 282, + 310, + 310 + ], + "score": 1.0, + "content": "only depends on the vectors", + "type": "text" + }, + { + "bbox": [ + 311, + 289, + 383, + 304 + ], + "score": 0.94, + "content": "\\left\\{ \\pmb { u } _ { \\tau } \\right\\} _ { \\tau = L ( i - 2 ) + \\bar { \\tau } + 1 } ^ { L ( i - 1 ) + \\bar { \\tau } - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 282, + 510, + 310 + ], + "score": 1.0, + "content": ". Note that the dependence ranges", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 300, + 508, + 319 + ], + "spans": [ + { + "bbox": [ + 107, + 305, + 244, + 316 + ], + "score": 0.92, + "content": "[ L ( i - 2 ) + \\bar { \\tau } + 1 , L ( i - 1 ) + \\bar { \\tau } - 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 300, + 363, + 319 + ], + "score": 1.0, + "content": "are disjoint intervals for different", + "type": "text" + }, + { + "bbox": [ + 364, + 306, + 368, + 314 + ], + "score": 0.61, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 300, + 401, + 319 + ], + "score": 1.0, + "content": "’s; hence", + "type": "text" + }, + { + "bbox": [ + 401, + 303, + 436, + 316 + ], + "score": 0.91, + "content": "( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 300, + 508, + 319 + ], + "score": 1.0, + "content": "are independent of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 313, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 255, + 330 + ], + "score": 1.0, + "content": "each other. To show the independence of", + "type": "text" + }, + { + "bbox": [ + 255, + 316, + 271, + 326 + ], + "score": 0.89, + "content": "\\mathbf { \\pmb { u } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 313, + 288, + 330 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 288, + 316, + 303, + 326 + ], + "score": 0.88, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 313, + 377, + 330 + ], + "score": 1.0, + "content": "; observe that inputs", + "type": "text" + }, + { + "bbox": [ + 378, + 316, + 445, + 329 + ], + "score": 0.92, + "content": "\\pmb { u } ^ { ( i ) } = \\pmb { u } _ { L ( i - 1 ) + \\hat { \\tau } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 313, + 506, + 330 + ], + "score": 1.0, + "content": "have timestamp", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 320, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 113, + 339 + ], + "score": 0.71, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 320, + 144, + 346 + ], + "score": 1.0, + "content": "modulo", + "type": "text" + }, + { + "bbox": [ + 144, + 330, + 151, + 339 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 320, + 332, + 346 + ], + "score": 1.0, + "content": "; which is not covered by the dependence range of", + "type": "text" + }, + { + "bbox": [ + 333, + 327, + 367, + 340 + ], + "score": 0.93, + "content": "( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 320, + 375, + 346 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 329, + 506, + 340 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 101, + 282, + 510, + 346 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 352, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 445, + 363 + ], + "score": 1.0, + "content": "If the input is randomly generated, Lemma C.2 can be combined with a probabilistic bound on", + "type": "text" + }, + { + "bbox": [ + 445, + 353, + 456, + 362 + ], + "score": 0.87, + "content": "\\pmb { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 352, + 505, + 363 + ], + "score": 1.0, + "content": ", to show that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 325, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 164, + 375 + ], + "score": 1.0, + "content": "truncated states", + "type": "text" + }, + { + "bbox": [ + 164, + 362, + 180, + 373 + ], + "score": 0.88, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 361, + 305, + 375 + ], + "score": 1.0, + "content": "are fairly close to the actual states", + "type": "text" + }, + { + "bbox": [ + 306, + 362, + 321, + 373 + ], + "score": 0.89, + "content": "\\mathbf { \\delta } _ { h } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 361, + 325, + 375 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 352, + 505, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 506, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 319, + 389 + ], + "score": 1.0, + "content": "Lemma C.6 (Truncation impact – random). Given offset", + "type": "text" + }, + { + "bbox": [ + 320, + 379, + 326, + 387 + ], + "score": 0.7, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 376, + 397, + 389 + ], + "score": 1.0, + "content": "and sampling rate", + "type": "text" + }, + { + "bbox": [ + 397, + 378, + 404, + 387 + ], + "score": 0.71, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 376, + 506, + 389 + ], + "score": 1.0, + "content": ", consider the state vectors", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 103, + 386, + 503, + 404 + ], + "spans": [ + { + "bbox": [ + 103, + 386, + 183, + 404 + ], + "score": 1.0, + "content": "of the sub-trajectory", + "type": "text" + }, + { + "bbox": [ + 184, + 388, + 221, + 401 + ], + "score": 0.92, + "content": "\\{ h ^ { ( i ) } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 386, + 239, + 404 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 390, + 262, + 400 + ], + "score": 0.83, + "content": "L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 386, + 308, + 404 + ], + "score": 1.0, + "content": "-truncations", + "type": "text" + }, + { + "bbox": [ + 309, + 388, + 343, + 401 + ], + "score": 0.92, + "content": "( \\bar { \\pmb { h } } ^ { ( i ) } ) _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 386, + 381, + 404 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 381, + 388, + 503, + 401 + ], + "score": 0.67, + "content": "\\{ \\pmb { u } _ { t } \\} _ { t \\geq 0 } \\stackrel { i . i . d . } { \\sim } \\mathcal { N } ( 0 , \\pmb { I } _ { p } ) , \\lVert \\pmb { A } \\rVert < 1", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 107, + 401, + 135, + 410 + ], + "score": 0.86, + "content": "h _ { 0 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 399, + 138, + 413 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 138, + 401, + 145, + 411 + ], + "score": 0.77, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 399, + 215, + 413 + ], + "score": 1.0, + "content": "is 1-Lipschitz, and", + "type": "text" + }, + { + "bbox": [ + 215, + 401, + 250, + 411 + ], + "score": 0.91, + "content": "\\phi ( 0 ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 399, + 426, + 413 + ], + "score": 1.0, + "content": ". Also suppose upper bound (4.3) of Assumption", + "type": "text" + }, + { + "bbox": [ + 426, + 402, + 431, + 409 + ], + "score": 0.38, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 399, + 488, + 413 + ], + "score": 1.0, + "content": "holds for some", + "type": "text" + }, + { + "bbox": [ + 489, + 401, + 505, + 410 + ], + "score": 0.84, + "content": "\\theta \\leq", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 107, + 411, + 153, + 421 + ], + "score": 0.89, + "content": "{ \\sqrt { n } } , \\gamma _ { + } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 410, + 278, + 423 + ], + "score": 1.0, + "content": ". There exists an absolute constant", + "type": "text" + }, + { + "bbox": [ + 278, + 411, + 300, + 420 + ], + "score": 0.89, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 410, + 424, + 423 + ], + "score": 1.0, + "content": "such that with probability at least", + "type": "text" + }, + { + "bbox": [ + 424, + 410, + 503, + 421 + ], + "score": 0.9, + "content": "1 - 2 \\bar { N } \\mathrm { \\bar { e x p } } ( - 1 0 0 n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 410, + 506, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 273, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 131, + 432 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 131, + 420, + 172, + 430 + ], + "score": 0.91, + "content": "1 \\leq i \\leq \\bar { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 420, + 273, + 432 + ], + "score": 1.0, + "content": ", the following bound holds", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 103, + 376, + 506, + 432 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 435, + 376, + 450 + ], + "lines": [ + { + "bbox": [ + 234, + 435, + 376, + 450 + ], + "spans": [ + { + "bbox": [ + 234, + 435, + 376, + 450 + ], + "score": 0.9, + "content": "\\| h ^ { ( i ) } - \\bar { h } ^ { ( i ) } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { n } \\| A \\| ^ { L - 1 } \\sqrt { \\gamma _ { + } } .", + "type": "interline_equation", + "image_path": "8e187455e36ffdd751ac435d8796a1fc724da5d51e0b78956a59b374e08d727f.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 234, + 435, + 376, + 450 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 331, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 332, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 227, + 469 + ], + "score": 1.0, + "content": "In particular, we can always pick", + "type": "text" + }, + { + "bbox": [ + 228, + 455, + 266, + 466 + ], + "score": 0.92, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 454, + 312, + 469 + ], + "score": 1.0, + "content": "(via Lemma", + "type": "text" + }, + { + "bbox": [ + 312, + 456, + 326, + 465 + ], + "score": 0.26, + "content": "B . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 454, + 332, + 469 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 454, + 332, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 104, + 477, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 349, + 494 + ], + "score": 1.0, + "content": "Proof. Using Assumption 1, we can apply Lemma F.3 on vectors", + "type": "text" + }, + { + "bbox": [ + 349, + 478, + 420, + 492 + ], + "score": 0.93, + "content": "\\{ h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\} _ { i = 1 } ^ { \\bar { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 477, + 507, + 494 + ], + "score": 1.0, + "content": ". Using a union bound,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 254, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 254, + 503 + ], + "score": 1.0, + "content": "with desired probability, all vectors obey", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 477, + 507, + 503 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 505, + 407, + 518 + ], + "lines": [ + { + "bbox": [ + 203, + 505, + 407, + 518 + ], + "spans": [ + { + "bbox": [ + 203, + 505, + 407, + 518 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } - \\mathbb { E } \\big [ h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\big ] \\| _ { \\ell _ { 2 } } \\leq ( c - 1 ) \\sqrt { n \\gamma _ { + } } , } \\end{array}", + "type": "interline_equation", + "image_path": "1102ab5eb959e5e9a6081c3a6a9df0efabf7791f01273a9dbb06cd61fe65c2c9.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 203, + 505, + 407, + 518 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 523, + 504, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 180, + 537 + ], + "score": 1.0, + "content": "for sufficiently large", + "type": "text" + }, + { + "bbox": [ + 181, + 525, + 186, + 532 + ], + "score": 0.66, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 522, + 211, + 537 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 211, + 523, + 242, + 534 + ], + "score": 0.91, + "content": "\\theta \\leq \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 522, + 340, + 537 + ], + "score": 1.0, + "content": ", triangle inequality implies", + "type": "text" + }, + { + "bbox": [ + 341, + 523, + 447, + 536 + ], + "score": 0.92, + "content": "\\| h _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\| _ { \\ell _ { 2 } } \\leq c \\sqrt { n \\gamma _ { + } }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 522, + 505, + 537 + ], + "score": 1.0, + "content": ". Now, applying", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 252, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 177, + 546 + ], + "score": 1.0, + "content": "Lemma C.2, for all", + "type": "text" + }, + { + "bbox": [ + 178, + 534, + 219, + 545 + ], + "score": 0.92, + "content": "1 \\leq i \\leq \\bar { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 534, + 252, + 546 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 522, + 505, + 546 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 549, + 403, + 597 + ], + "lines": [ + { + "bbox": [ + 207, + 549, + 403, + 597 + ], + "spans": [ + { + "bbox": [ + 207, + 549, + 403, + 597 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| \\pmb { h } ^ { ( i ) } - \\bar { \\pmb { h } } ^ { ( i ) } \\| _ { \\ell _ { 2 } } = \\| \\pmb { h } _ { ( i - 1 ) L + \\bar { \\tau } } - \\bar { \\pmb { h } } _ { ( i - 1 ) L + \\bar { \\tau } , L - 1 } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq \\| \\pmb { A } \\| ^ { L - 1 } \\| \\pmb { h } _ { ( i - 2 ) L + \\bar { \\tau } + 1 } \\| _ { \\ell _ { 2 } } } \\\\ & { \\qquad \\leq c \\| \\pmb { A } \\| ^ { L - 1 } \\sqrt { n \\gamma + 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "5bf4df24229dac93eaa49840a340d125c6e63576b608103a3bd7da010fb011b6.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 207, + 549, + 403, + 565.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 207, + 565.0, + 403, + 581.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 207, + 581.0, + 403, + 597.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 625, + 305, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 305, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 305, + 639 + ], + "score": 1.0, + "content": "D PROPERTIES OF THE DATA MATRIX", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "This section utilizes the probabilistic estimates from Section B to provide bounds on the condition number of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 658, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 412, + 670 + ], + "score": 1.0, + "content": "data matrices obtained from the RNN trajectory (1.1). Following (2.2), these matrices", + "type": "text" + }, + { + "bbox": [ + 413, + 659, + 435, + 669 + ], + "score": 0.91, + "content": "_ { H , U }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 658, + 452, + 670 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 452, + 659, + 462, + 668 + ], + "score": 0.84, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 658, + 506, + 670 + ], + "score": 1.0, + "content": "are defined", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 670, + 118, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 118, + 680 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 649, + 506, + 680 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 682, + 453, + 697 + ], + "lines": [ + { + "bbox": [ + 159, + 682, + 453, + 697 + ], + "spans": [ + { + "bbox": [ + 159, + 682, + 453, + 697 + ], + "score": 0.88, + "content": "\\mathbf { { \\cal H } } = [ h _ { 1 } \\ \\dots \\ h _ { N } ] ^ { T } \\quad , \\quad \\mathbf { { \\cal U } } = \\mathbf { { \\cal H } } = [ u _ { 1 } \\ \\dots \\ u _ { N } ] ^ { T } \\quad , \\quad \\mathbf { { \\cal X } } = [ x _ { 1 } \\ \\dots \\ x _ { N } ] ^ { T } .", + "type": "interline_equation", + "image_path": "5d887163c25f83c4d904cc186b06cd4e8d76a784333b168f2b713d44414471d1.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 159, + 682, + 453, + 697 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 701, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 246, + 714 + ], + "score": 1.0, + "content": "The challenge is that, the state matrix", + "type": "text" + }, + { + "bbox": [ + 247, + 704, + 257, + 710 + ], + "score": 0.87, + "content": "_ H", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 699, + 505, + 714 + ], + "score": 1.0, + "content": "has dependent rows; which will be addressed by carefully splitting", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 709, + 507, + 725 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 156, + 725 + ], + "score": 1.0, + "content": "the trajectory", + "type": "text" + }, + { + "bbox": [ + 157, + 711, + 202, + 722 + ], + "score": 0.93, + "content": "\\{ u _ { t } , h _ { t } \\} _ { t = 0 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 709, + 507, + 725 + ], + "score": 1.0, + "content": "into multiple sub-trajectories which are internally weakly dependent as discussed in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 380, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 380, + 732 + ], + "score": 1.0, + "content": "Section C. 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I", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 151, + 376, + 169 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 376, + 153, + 502, + 167 + ], + "score": 0.92, + "content": "{ \\pmb Q } = [ \\gamma _ { + } ^ { - 1 / 2 } \\bar { \\pmb H } \\tilde { \\pmb U } ] \\in \\mathbb { R } ^ { \\bar { N } \\times ( n + p ) }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 135, + 165, + 513, + 207 + ], + "spans": [ + { + "bbox": [ + 135, + 165, + 179, + 207 + ], + "score": 1.0, + "content": "e Assumptiosuch that if √", + "type": "text" + }, + { + "bbox": [ + 186, + 170, + 191, + 177 + ], + "score": 0.53, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 169, + 226, + 179 + ], + "score": 0.81, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 169, + 244, + 179 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 165, + 307, + 207 + ], + "score": 1.0, + "content": "creasing, and , with probabil", + "type": "text" + }, + { + "bbox": [ + 307, + 166, + 370, + 180 + ], + "score": 0.87, + "content": "{ \\mathbf { \\mathscr { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\cal I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 165, + 460, + 207 + ], + "score": 1.0, + "content": "e exists an absol, for all matrices", + "type": "text" + }, + { + "bbox": [ + 473, + 165, + 513, + 207 + ], + "score": 1.0, + "content": "constantobeying", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 179, + 472, + 199 + ], + "spans": [ + { + "bbox": [ + 107, + 183, + 134, + 193 + ], + "score": 0.86, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 179, + 252, + 198 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\bar { N } \\ge C \\frac { \\gamma _ { + } ^ { 2 } } { \\gamma _ { - } ^ { 2 } } ( n + p ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 180, + 395, + 199 + ], + "score": 0.9, + "content": "1 - 8 \\exp ( - c \\frac { \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } ^ { 2 } } \\bar { N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 183, + 472, + 192 + ], + "score": 0.67, + "content": "M", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 198, + 319, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 198, + 187, + 216 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\| M - \\bar { H } \\| \\le \\frac { \\sqrt { \\gamma _ { - } \\bar { N } } } { 1 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 202, + 242, + 216 + ], + "score": 1.0, + "content": ", the perturbed", + "type": "text" + }, + { + "bbox": [ + 243, + 203, + 251, + 214 + ], + "score": 0.82, + "content": "\\ b { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 202, + 319, + 216 + ], + "score": 1.0, + "content": "matrices given by,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 220, + 343, + 235 + ], + "lines": [ + { + "bbox": [ + 268, + 220, + 343, + 235 + ], + "spans": [ + { + "bbox": [ + 268, + 220, + 343, + 235 + ], + "score": 0.93, + "content": "\\tilde { Q } = [ \\gamma _ { + } ^ { - 1 / 2 } M \\tilde { U } ] ,", + "type": "interline_equation", + "image_path": "3fdf1e5e331b950134e5e9a15c945d2c42688e7ebafaea0254cbd9a35b61e1f8.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 268, + 220, + 343, + 235 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 132, + 251 + ], + "lines": [ + { + "bbox": [ + 104, + 238, + 134, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 134, + 255 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 255, + 362, + 282 + ], + "lines": [ + { + "bbox": [ + 248, + 255, + 362, + 282 + ], + "spans": [ + { + "bbox": [ + 248, + 255, + 362, + 282 + ], + "score": 0.94, + "content": "( \\Theta + \\sqrt { 2 } ) ^ { 2 } \\succeq \\frac { \\tilde { Q } ^ { T } \\tilde { Q } } { \\bar { N } } \\succeq \\frac { \\gamma _ { - } } { 2 \\gamma _ { + } } .", + "type": "interline_equation", + "image_path": "31e590a2fd52e062ccef23b7723ae20f4145016590654297f00df4592938d594.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 248, + 255, + 362, + 282 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 506, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 303 + ], + "score": 1.0, + "content": "Proof. This result is a direct application of Theorem F.1 after determining minimum/maximum eigenvalues of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 295, + 316 + ], + "score": 1.0, + "content": "population covariance. The cross covariance obeys", + "type": "text" + }, + { + "bbox": [ + 296, + 302, + 349, + 315 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\bar { \\pmb { H } } ^ { T } \\tilde { \\pmb { U } } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 303, + 466, + 316 + ], + "score": 1.0, + "content": "due to independence. 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Using the assumed bound on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 469, + 168, + 483 + ], + "spans": [ + { + "bbox": [ + 107, + 472, + 124, + 482 + ], + "score": 0.92, + "content": "\\| \\pmb { E } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 469, + 168, + 483 + ], + "score": 1.0, + "content": ", this yields", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 480, + 397, + 505 + ], + "lines": [ + { + "bbox": [ + 214, + 480, + 397, + 505 + ], + "spans": [ + { + "bbox": [ + 214, + 480, + 397, + 505 + ], + "score": 0.93, + "content": "\\theta + \\sqrt { 2 } \\geq \\frac { 1 } { \\sqrt { \\bar { N } } } \\| \\tilde { \\pmb { Q } } \\| \\geq \\frac { 1 } { \\sqrt { \\bar { N } } } s _ { \\operatorname* { m i n } } ( \\tilde { \\pmb { Q } } ) \\geq \\sqrt { \\frac { \\gamma - \\underline { { \\mathbf { \\Lambda } } } } { 2 \\gamma + } } .", + "type": "interline_equation", + "image_path": "049cd6f4010c0350d142863ffbc3675008577aa10ed78380a97400da0e691af5.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 214, + 480, + 397, + 505 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 321, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 321, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 321, + 519 + ], + "score": 1.0, + "content": "This final inequality is identical to the desired bound (D.3).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 102, + 523, + 503, + 552 + ], + "spans": [ + { + "bbox": [ + 102, + 523, + 209, + 552 + ], + "score": 1.0, + "content": "Theorem D.3 (Data matrixdefine the condition number", + "type": "text" + }, + { + "bbox": [ + 239, + 523, + 348, + 552 + ], + "score": 1.0, + "content": "n). Consider the nonlinear s. For some absolute constants", + "type": "text" + }, + { + "bbox": [ + 381, + 523, + 447, + 552 + ], + "score": 1.0, + "content": "tion (1.1). Given , pick a trajectory", + "type": "text" + }, + { + "bbox": [ + 447, + 525, + 503, + 535 + ], + "score": 0.89, + "content": "\\gamma _ { + } \\geq \\gamma _ { - } > 0", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 209, + 534, + 479, + 548 + ], + "spans": [ + { + "bbox": [ + 209, + 534, + 239, + 548 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\rho = \\frac { \\gamma _ { + } } { \\gamma _ { - } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 534, + 381, + 545 + ], + "score": 0.9, + "content": "c , C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 536, + 479, + 543 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 553, + 408, + 577 + ], + "lines": [ + { + "bbox": [ + 202, + 553, + 408, + 577 + ], + "spans": [ + { + "bbox": [ + 202, + 553, + 408, + 577 + ], + "score": 0.91, + "content": "L = \\lceil 1 - \\frac { \\log { ( c n \\rho ) } } { \\log { \\| A \\| } } \\rceil \\quad , \\quad N _ { 0 } = \\lfloor \\frac { N } { L } \\rfloor \\ge C \\rho ^ { 2 } ( n + p ) ,", + "type": "interline_equation", + "image_path": "dbb4d709294ae8ab6863d574912596a40eed551b99fec5bc7606fda76508cbd2.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 202, + 553, + 408, + 577 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 172, + 601 + ], + "score": 1.0, + "content": "and pick scaling", + "type": "text" + }, + { + "bbox": [ + 172, + 585, + 213, + 600 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = \\frac { 1 } { \\sqrt { \\gamma _ { + } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 582, + 255, + 600 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 256, + 586, + 296, + 597 + ], + "score": 0.87, + "content": "\\| A \\| ~ < ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 582, + 300, + 600 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 300, + 586, + 308, + 597 + ], + "score": 0.62, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 582, + 319, + 600 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 319, + 586, + 326, + 597 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 582, + 372, + 600 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 373, + 584, + 437, + 597 + ], + "score": 0.87, + "content": "{ \\textbf { \\em u } } _ { t } \\stackrel { i . i . d . } { \\sim } { \\mathcal { N } } ( 0 , I _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 582, + 506, + 600 + ], + "score": 1.0, + "content": ", and Assumption", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 111, + 610 + ], + "score": 0.29, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 599, + 156, + 613 + ], + "score": 1.0, + "content": "holds with", + "type": "text" + }, + { + "bbox": [ + 156, + 601, + 203, + 611 + ], + "score": 0.89, + "content": "\\gamma _ { + } , \\gamma _ { - } , \\theta , L", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 599, + 239, + 613 + ], + "score": 1.0, + "content": ". Matrix", + "type": "text" + }, + { + "bbox": [ + 239, + 600, + 321, + 611 + ], + "score": 0.86, + "content": "\\mathbf { X } \\ = \\ [ \\pmb { x } _ { 1 } \\dots \\pmb { x } _ { N } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "of (D.1) satisfies the following with probability", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 609, + 281, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 276, + 622 + ], + "score": 0.83, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - \\mathcal { O } ( N _ { 0 } / \\rho ^ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 609, + 281, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 132, + 630, + 434, + 642 + ], + "lines": [ + { + "bbox": [ + 132, + 630, + 434, + 642 + ], + "spans": [ + { + "bbox": [ + 132, + 630, + 188, + 642 + ], + "score": 1.0, + "content": "• Each row of", + "type": "text" + }, + { + "bbox": [ + 189, + 631, + 198, + 640 + ], + "score": 0.75, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 630, + 214, + 642 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 214, + 631, + 223, + 640 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 630, + 274, + 642 + ], + "score": 1.0, + "content": "norm at most", + "type": "text" + }, + { + "bbox": [ + 274, + 631, + 313, + 642 + ], + "score": 0.92, + "content": "c _ { 0 } { \\sqrt { p + n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 630, + 337, + 642 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 337, + 632, + 347, + 641 + ], + "score": 0.85, + "content": "c _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 630, + 434, + 642 + ], + "score": 1.0, + "content": "is an absolute constant.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 132, + 648, + 230, + 660 + ], + "lines": [ + { + "bbox": [ + 132, + 647, + 231, + 662 + ], + "spans": [ + { + "bbox": [ + 132, + 647, + 142, + 662 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 649, + 168, + 659 + ], + "score": 0.88, + "content": "X ^ { T } X", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 647, + 231, + 662 + ], + "score": 1.0, + "content": "obeys the bound", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 665, + 404, + 689 + ], + "lines": [ + { + "bbox": [ + 243, + 665, + 404, + 689 + ], + "spans": [ + { + "bbox": [ + 243, + 665, + 404, + 689 + ], + "score": 0.94, + "content": "( \\Theta + \\sqrt { 2 } ) ^ { 2 } I _ { n + p } \\succeq \\frac { { \\bf { X } } ^ { T } { \\bf { X } } } { N } \\succeq \\rho ^ { - 1 } I _ { n + p } / 2 .", + "type": "interline_equation", + "image_path": "adb62a33f910556490a4a46f9a66ac4a1107349fa57e25d7c4e297a878fe775a.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 243, + 665, + 404, + 689 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 216, + 712 + ], + "score": 1.0, + "content": "Proof. The first statement on", + "type": "text" + }, + { + "bbox": [ + 216, + 700, + 225, + 709 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "-norm bound can be concluded from Lemma D.4 and holds with probability", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 204, + 720 + ], + "score": 0.89, + "content": "1 - 2 N \\exp ( - 1 0 0 ( n + p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 710, + 376, + 721 + ], + "score": 1.0, + "content": ". 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Observe that", + "type": "text" + }, + { + "bbox": [ + 290, + 723, + 299, + 730 + ], + "score": 0.84, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 718, + 506, + 734 + ], + "score": 1.0, + "content": "is obtained by merging multiple sub-trajectory matrices", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 506, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 496, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 496, + 507, + 505, + 518 + ], + "score": 0.99, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 117 + ], + "lines": [ + { + "bbox": [ + 102, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 102, + 80, + 243, + 96 + ], + "score": 1.0, + "content": "Definition D.1. Given sampling rate", + "type": "text" + }, + { + "bbox": [ + 243, + 84, + 250, + 92 + ], + "score": 0.68, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 80, + 288, + 96 + ], + "score": 1.0, + "content": "and offset", + "type": "text" + }, + { + "bbox": [ + 289, + 84, + 294, + 92 + ], + "score": 0.71, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 80, + 344, + 96 + ], + "score": 1.0, + "content": ", consider the", + "type": "text" + }, + { + "bbox": [ + 345, + 84, + 352, + 92 + ], + "score": 0.69, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 80, + 437, + 96 + ], + "score": 1.0, + "content": "-subsampled trajectory", + "type": "text" + }, + { + "bbox": [ + 437, + 83, + 493, + 94 + ], + "score": 0.92, + "content": "\\{ \\boldsymbol { h } ^ { ( i ) } , \\boldsymbol { u } ^ { ( i ) } \\} _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "i) }Ni=1 as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 318, + 107 + ], + "score": 1.0, + "content": "described in Definitions C.3 and C.4. Define the matrices", + "type": "text" + }, + { + "bbox": [ + 318, + 94, + 395, + 104 + ], + "score": 0.86, + "content": "\\bar { \\pmb { H } } = \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { \\pmb { N } } \\times { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 91, + 401, + 107 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 401, + 94, + 479, + 104 + ], + "score": 0.82, + "content": "\\tilde { \\pmb { H } } = \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { \\cal N } \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 91, + 484, + 107 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 485, + 95, + 506, + 106 + ], + "score": 0.81, + "content": "\\tilde { \\pmb { U } } =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 101, + 283, + 120 + ], + "spans": [ + { + "bbox": [ + 107, + 105, + 160, + 117 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { U } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { \\pmb { N } } \\times { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 101, + 180, + 120 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 180, + 105, + 283, + 117 + ], + "score": 0.89, + "content": "\\tilde { { \\cal X } } = \\tilde { { \\cal X } } ^ { ( \\bar { \\tau } ) } \\in \\mathbb { R } ^ { \\bar { N } \\times ( n + p ) } a s", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 102, + 80, + 506, + 120 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 122, + 471, + 137 + ], + "lines": [ + { + "bbox": [ + 138, + 122, + 471, + 137 + ], + "spans": [ + { + "bbox": [ + 138, + 122, + 471, + 137 + ], + "score": 0.88, + "content": "\\bar { \\pmb { H } } = [ \\bar { \\pmb { h } } ^ { ( 1 ) } \\dots \\bar { \\pmb { h } } ^ { ( \\bar { N } ) } ] ^ { T } , \\ \\tilde { \\pmb { H } } = [ \\pmb { h } ^ { ( 1 ) } \\dots \\pmb { h } ^ { ( \\bar { N } ) } ] ^ { T } , \\ \\tilde { \\pmb { U } } = [ \\pmb { u } ^ { ( 1 ) } \\dots \\pmb { u } ^ { ( \\bar { N } ) } ] ^ { T } , \\ \\tilde { \\pmb { X } } = [ \\mu \\tilde { \\pmb { H } } \\tilde { \\pmb { U } } ] .", + "type": "interline_equation", + "image_path": "79c3804269bf0100a77d8e21ea068947f21f4a2c6eefa3b110dc58dc617804aa.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 138, + 122, + 471, + 137 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 504, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 480, + 154 + ], + "score": 1.0, + "content": "Lemma D.2 (Handling perturbation). 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I", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 151, + 376, + 169 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 376, + 153, + 502, + 167 + ], + "score": 0.92, + "content": "{ \\pmb Q } = [ \\gamma _ { + } ^ { - 1 / 2 } \\bar { \\pmb H } \\tilde { \\pmb U } ] \\in \\mathbb { R } ^ { \\bar { N } \\times ( n + p ) }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 135, + 165, + 513, + 207 + ], + "spans": [ + { + "bbox": [ + 135, + 165, + 179, + 207 + ], + "score": 1.0, + "content": "e Assumptiosuch that if √", + "type": "text" + }, + { + "bbox": [ + 186, + 170, + 191, + 177 + ], + "score": 0.53, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 169, + 226, + 179 + ], + "score": 0.81, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 169, + 244, + 179 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 165, + 307, + 207 + ], + "score": 1.0, + "content": "creasing, and , with probabil", + "type": "text" + }, + { + "bbox": [ + 307, + 166, + 370, + 180 + ], + "score": 0.87, + "content": "{ \\mathbf { \\mathscr { u } } _ { t } } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\cal I } _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 165, + 460, + 207 + ], + "score": 1.0, + "content": "e exists an absol, for all matrices", + "type": "text" + }, + { + "bbox": [ + 473, + 165, + 513, + 207 + ], + "score": 1.0, + "content": "constantobeying", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 179, + 472, + 199 + ], + "spans": [ + { + "bbox": [ + 107, + 183, + 134, + 193 + ], + "score": 0.86, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 179, + 252, + 198 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\bar { N } \\ge C \\frac { \\gamma _ { + } ^ { 2 } } { \\gamma _ { - } ^ { 2 } } ( n + p ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 180, + 395, + 199 + ], + "score": 0.9, + "content": "1 - 8 \\exp ( - c \\frac { \\gamma _ { - } ^ { 2 } } { \\gamma _ { + } ^ { 2 } } \\bar { N } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 183, + 472, + 192 + ], + "score": 0.67, + "content": "M", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 198, + 319, + 216 + ], + "spans": [ + { + "bbox": [ + 107, + 198, + 187, + 216 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\| M - \\bar { H } \\| \\le \\frac { \\sqrt { \\gamma _ { - } \\bar { N } } } { 1 0 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 202, + 242, + 216 + ], + "score": 1.0, + "content": ", the perturbed", + "type": "text" + }, + { + "bbox": [ + 243, + 203, + 251, + 214 + ], + "score": 0.82, + "content": "\\ b { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 202, + 319, + 216 + ], + "score": 1.0, + "content": "matrices given by,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 102, + 142, + 513, + 216 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 220, + 343, + 235 + ], + "lines": [ + { + "bbox": [ + 268, + 220, + 343, + 235 + ], + "spans": [ + { + "bbox": [ + 268, + 220, + 343, + 235 + ], + "score": 0.93, + "content": "\\tilde { Q } = [ \\gamma _ { + } ^ { - 1 / 2 } M \\tilde { U } ] ,", + "type": "interline_equation", + "image_path": "3fdf1e5e331b950134e5e9a15c945d2c42688e7ebafaea0254cbd9a35b61e1f8.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 268, + 220, + 343, + 235 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 132, + 251 + ], + "lines": [ + { + "bbox": [ + 104, + 238, + 134, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 134, + 255 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 238, + 134, + 255 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 255, + 362, + 282 + ], + "lines": [ + { + "bbox": [ + 248, + 255, + 362, + 282 + ], + "spans": [ + { + "bbox": [ + 248, + 255, + 362, + 282 + ], + "score": 0.94, + "content": "( \\Theta + \\sqrt { 2 } ) ^ { 2 } \\succeq \\frac { \\tilde { Q } ^ { T } \\tilde { Q } } { \\bar { N } } \\succeq \\frac { \\gamma _ { - } } { 2 \\gamma _ { + } } .", + "type": "interline_equation", + "image_path": "31e590a2fd52e062ccef23b7723ae20f4145016590654297f00df4592938d594.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 248, + 255, + 362, + 282 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 506, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 303 + ], + "score": 1.0, + "content": "Proof. This result is a direct application of Theorem F.1 after determining minimum/maximum eigenvalues of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 295, + 316 + ], + "score": 1.0, + "content": "population covariance. The cross covariance obeys", + "type": "text" + }, + { + "bbox": [ + 296, + 302, + 349, + 315 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\bar { \\pmb { H } } ^ { T } \\tilde { \\pmb { U } } ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 303, + 466, + 316 + ], + "score": 1.0, + "content": "due to independence. Also, for", + "type": "text" + }, + { + "bbox": [ + 466, + 304, + 488, + 313 + ], + "score": 0.89, + "content": "i > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 303, + 506, + 316 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 312, + 508, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 312, + 182, + 329 + ], + "score": 1.0, + "content": "truncated state vector", + "type": "text" + }, + { + "bbox": [ + 183, + 314, + 198, + 325 + ], + "score": 0.9, + "content": "\\bar { \\pmb { h } } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 312, + 289, + 329 + ], + "score": 1.0, + "content": "is statistically identical to", + "type": "text" + }, + { + "bbox": [ + 290, + 316, + 312, + 326 + ], + "score": 0.9, + "content": "\\pmb { h } _ { L - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 312, + 335, + 329 + ], + "score": 1.0, + "content": "hence", + "type": "text" + }, + { + "bbox": [ + 335, + 314, + 397, + 326 + ], + "score": 0.92, + "content": "\\pmb { \\Sigma } [ \\bar { \\pmb { h } } ^ { ( i ) } ] \\succeq \\gamma _ { - } \\bar { \\pmb { I _ { n } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 312, + 452, + 329 + ], + "score": 1.0, + "content": ". 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Hence, setting", + "type": "text" + }, + { + "bbox": [ + 391, + 326, + 457, + 355 + ], + "score": 0.9, + "content": "\\pmb { q } _ { i } = \\left[ \\frac { 1 } { \\sqrt { \\gamma _ { + } } } \\bar { \\pmb { h } } ^ { ( i ) } \\right] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 333, + 483, + 345 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 484, + 335, + 505, + 344 + ], + "score": 0.85, + "content": "i > 1", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 292, + 508, + 355 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 359, + 347, + 381 + ], + "lines": [ + { + "bbox": [ + 263, + 359, + 347, + 381 + ], + "spans": [ + { + "bbox": [ + 263, + 359, + 347, + 381 + ], + "score": 0.92, + "content": "\\frac { \\gamma - } { \\gamma _ { + } } I _ { n } \\preceq \\Sigma [ { \\pmb q } _ { i } ] \\preceq I _ { n } .", + "type": "interline_equation", + "image_path": "7eaf3abc4e7b1cb90fa84636ed15ad747d18ad248e3888a3639a3b9aec9a7732.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 263, + 359, + 347, + 381 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 160, + 400 + ], + "score": 1.0, + "content": "Set the matrix", + "type": "text" + }, + { + "bbox": [ + 160, + 386, + 232, + 398 + ], + "score": 0.92, + "content": "\\bar { \\pmb { Q } } = [ \\pmb { q } _ { 2 } \\dotsm \\pmb { q } _ { \\bar { N } } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 385, + 283, + 400 + ], + "score": 1.0, + "content": "and note that", + "type": "text" + }, + { + "bbox": [ + 284, + 385, + 340, + 398 + ], + "score": 0.92, + "content": "Q = [ \\pmb { q } _ { 1 } \\bar { Q } ^ { T } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 385, + 442, + 400 + ], + "score": 1.0, + "content": ". Applying Theorem F.1 on", + "type": "text" + }, + { + "bbox": [ + 442, + 387, + 451, + 398 + ], + "score": 0.87, + "content": "\\bar { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "and Corollary", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 395, + 293, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 131, + 410 + ], + "score": 1.0, + "content": "F.2 on", + "type": "text" + }, + { + "bbox": [ + 131, + 398, + 140, + 408 + ], + "score": 0.85, + "content": "\\ b { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 395, + 293, + 410 + ], + "score": 1.0, + "content": ", we find that, with the desired probability,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 385, + 505, + 410 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 413, + 485, + 440 + ], + "lines": [ + { + "bbox": [ + 124, + 413, + 485, + 440 + ], + "spans": [ + { + "bbox": [ + 124, + 413, + 485, + 440 + ], + "score": 0.93, + "content": "\\theta + \\sqrt { 3 / 2 } \\geq \\frac { 1 } { \\sqrt { N } } \\| Q \\| \\geq \\frac { 1 } { \\sqrt { N } } s _ { \\operatorname* { m i n } } ( Q ) \\geq \\frac { 1 } { \\sqrt { N } } s _ { \\operatorname* { m i n } } ( \\bar { Q } ) \\geq \\sqrt { \\frac { N - 1 } { N } } \\sqrt { \\frac { 2 \\gamma _ { - } } { 3 \\gamma _ { + } } } \\geq 0 . 9 9 \\times \\sqrt { \\frac { 2 \\gamma _ { - } } { 3 \\gamma _ { + } } } .", + "type": "interline_equation", + "image_path": "afe580357b7156b889c09a393065f108ef780236beb64055ccd1dce53b577697.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 124, + 413, + 485, + 440 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 481 + ], + "lines": [ + { + "bbox": [ + 104, + 443, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 443, + 134, + 462 + ], + "score": 1.0, + "content": "Setting", + "type": "text" + }, + { + "bbox": [ + 135, + 446, + 188, + 457 + ], + "score": 0.91, + "content": "E = M - { \\bar { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 443, + 242, + 462 + ], + "score": 1.0, + "content": "and observing", + "type": "text" + }, + { + "bbox": [ + 242, + 445, + 324, + 459 + ], + "score": 0.91, + "content": "\\tilde { Q } = Q + [ \\gamma _ { + } ^ { - 1 / 2 } E \\mathrm { ~ 0 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 443, + 437, + 462 + ], + "score": 1.0, + "content": ", the impact of the perturbation", + "type": "text" + }, + { + "bbox": [ + 437, + 448, + 446, + 457 + ], + "score": 0.84, + "content": "\\pmb { \\cal E }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 443, + 507, + 462 + ], + "score": 1.0, + "content": "can be bounded", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 456, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 150, + 475 + ], + "score": 1.0, + "content": "naively via", + "type": "text" + }, + { + "bbox": [ + 150, + 459, + 392, + 472 + ], + "score": 0.91, + "content": "s _ { \\operatorname* { m i n } } ( \\pmb { Q } ) - \\gamma _ { + } ^ { - 1 / 2 } \\lVert \\pmb { E } \\rVert \\leq s _ { \\operatorname* { m i n } } ( \\tilde { \\pmb { Q } } ) \\leq \\lVert \\tilde { \\pmb { Q } } \\rVert \\leq \\lVert \\pmb { Q } \\rVert + \\gamma _ { + } ^ { - 1 / 2 } \\lVert \\pmb { E } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 456, + 507, + 475 + ], + "score": 1.0, + "content": ". Using the assumed bound on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 469, + 168, + 483 + ], + "spans": [ + { + "bbox": [ + 107, + 472, + 124, + 482 + ], + "score": 0.92, + "content": "\\| \\pmb { E } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 469, + 168, + 483 + ], + "score": 1.0, + "content": ", this yields", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 443, + 507, + 483 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 480, + 397, + 505 + ], + "lines": [ + { + "bbox": [ + 214, + 480, + 397, + 505 + ], + "spans": [ + { + "bbox": [ + 214, + 480, + 397, + 505 + ], + "score": 0.93, + "content": "\\theta + \\sqrt { 2 } \\geq \\frac { 1 } { \\sqrt { \\bar { N } } } \\| \\tilde { \\pmb { Q } } \\| \\geq \\frac { 1 } { \\sqrt { \\bar { N } } } s _ { \\operatorname* { m i n } } ( \\tilde { \\pmb { Q } } ) \\geq \\sqrt { \\frac { \\gamma - \\underline { { \\mathbf { \\Lambda } } } } { 2 \\gamma + } } .", + "type": "interline_equation", + "image_path": "049cd6f4010c0350d142863ffbc3675008577aa10ed78380a97400da0e691af5.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 214, + 480, + 397, + 505 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 321, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 321, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 321, + 519 + ], + "score": 1.0, + "content": "This final inequality is identical to the desired bound (D.3).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 506, + 321, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 102, + 523, + 503, + 552 + ], + "spans": [ + { + "bbox": [ + 102, + 523, + 209, + 552 + ], + "score": 1.0, + "content": "Theorem D.3 (Data matrixdefine the condition number", + "type": "text" + }, + { + "bbox": [ + 239, + 523, + 348, + 552 + ], + "score": 1.0, + "content": "n). Consider the nonlinear s. For some absolute constants", + "type": "text" + }, + { + "bbox": [ + 381, + 523, + 447, + 552 + ], + "score": 1.0, + "content": "tion (1.1). Given , pick a trajectory", + "type": "text" + }, + { + "bbox": [ + 447, + 525, + 503, + 535 + ], + "score": 0.89, + "content": "\\gamma _ { + } \\geq \\gamma _ { - } > 0", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 209, + 534, + 479, + 548 + ], + "spans": [ + { + "bbox": [ + 209, + 534, + 239, + 548 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\rho = \\frac { \\gamma _ { + } } { \\gamma _ { - } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 534, + 381, + 545 + ], + "score": 0.9, + "content": "c , C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 536, + 479, + 543 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 102, + 523, + 503, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 553, + 408, + 577 + ], + "lines": [ + { + "bbox": [ + 202, + 553, + 408, + 577 + ], + "spans": [ + { + "bbox": [ + 202, + 553, + 408, + 577 + ], + "score": 0.91, + "content": "L = \\lceil 1 - \\frac { \\log { ( c n \\rho ) } } { \\log { \\| A \\| } } \\rceil \\quad , \\quad N _ { 0 } = \\lfloor \\frac { N } { L } \\rfloor \\ge C \\rho ^ { 2 } ( n + p ) ,", + "type": "interline_equation", + "image_path": "dbb4d709294ae8ab6863d574912596a40eed551b99fec5bc7606fda76508cbd2.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 202, + 553, + 408, + 577 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 172, + 601 + ], + "score": 1.0, + "content": "and pick scaling", + "type": "text" + }, + { + "bbox": [ + 172, + 585, + 213, + 600 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = \\frac { 1 } { \\sqrt { \\gamma _ { + } } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 582, + 255, + 600 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 256, + 586, + 296, + 597 + ], + "score": 0.87, + "content": "\\| A \\| ~ < ~ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 582, + 300, + 600 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 300, + 586, + 308, + 597 + ], + "score": 0.62, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 582, + 319, + 600 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 319, + 586, + 326, + 597 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 582, + 372, + 600 + ], + "score": 1.0, + "content": "-increasing,", + "type": "text" + }, + { + "bbox": [ + 373, + 584, + 437, + 597 + ], + "score": 0.87, + "content": "{ \\textbf { \\em u } } _ { t } \\stackrel { i . i . d . } { \\sim } { \\mathcal { N } } ( 0 , I _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 582, + 506, + 600 + ], + "score": 1.0, + "content": ", and Assumption", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 111, + 610 + ], + "score": 0.29, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 599, + 156, + 613 + ], + "score": 1.0, + "content": "holds with", + "type": "text" + }, + { + "bbox": [ + 156, + 601, + 203, + 611 + ], + "score": 0.89, + "content": "\\gamma _ { + } , \\gamma _ { - } , \\theta , L", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 599, + 239, + 613 + ], + "score": 1.0, + "content": ". 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The first statement on", + "type": "text" + }, + { + "bbox": [ + 216, + 700, + 225, + 709 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "-norm bound can be concluded from Lemma D.4 and holds with probability", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 204, + 720 + ], + "score": 0.89, + "content": "1 - 2 N \\exp ( - 1 0 0 ( n + p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 710, + 376, + 721 + ], + "score": 1.0, + "content": ". To show the second statement, for a fixed offset", + "type": "text" + }, + { + "bbox": [ + 376, + 710, + 417, + 720 + ], + "score": 0.91, + "content": "1 \\le \\bar { \\tau } \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 710, + 505, + 721 + ], + "score": 1.0, + "content": ", consider Definition D.1", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 718, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 718, + 168, + 734 + ], + "score": 1.0, + "content": "and the matrices", + "type": "text" + }, + { + "bbox": [ + 169, + 720, + 236, + 732 + ], + "score": 0.92, + "content": "\\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } , \\tilde { \\pmb { U } } ^ { ( \\bar { \\tau } ) } , \\tilde { \\pmb { X } } ^ { ( \\bar { \\tau } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 718, + 289, + 734 + ], + "score": 1.0, + "content": ". Observe that", + "type": "text" + }, + { + "bbox": [ + 290, + 723, + 299, + 730 + ], + "score": 0.84, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 718, + 506, + 734 + ], + "score": 1.0, + "content": "is obtained by merging multiple sub-trajectory matrices", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 79, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 150, + 94 + ], + "score": 0.92, + "content": "\\{ \\tilde { X } ^ { ( \\bar { \\tau } ) } \\} _ { \\bar { \\tau } = 1 } ^ { L }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 151, + 79, + 360, + 97 + ], + "score": 1.0, + "content": ". 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We will first show the advertised bound for an individual", + "type": "text" + }, + { + "bbox": [ + 361, + 81, + 381, + 92 + ], + "score": 0.94, + "content": "\\tilde { X } ^ { ( \\bar { \\tau } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 79, + 507, + 97 + ], + "score": 1.0, + "content": "by applying Lemma D.2 and then", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 347, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 333, + 105 + ], + "score": 1.0, + "content": "apply Lemma A.1 to obtain the bound on the combined matrix", + "type": "text" + }, + { + "bbox": [ + 334, + 94, + 343, + 102 + ], + "score": 0.8, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 93, + 347, + 105 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 109, + 505, + 167 + ], + "lines": [ + { + "bbox": [ + 104, + 107, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 104, + 107, + 149, + 123 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + }, + { + "bbox": [ + 149, + 110, + 162, + 120 + ], + "score": 0.89, + "content": "\\bar { N } _ { \\bar { \\tau } }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 107, + 235, + 123 + ], + "score": 1.0, + "content": "is the length of the", + "type": "text" + }, + { + "bbox": [ + 235, + 112, + 241, + 119 + ], + "score": 0.48, + "content": "\\bar { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 107, + 389, + 123 + ], + "score": 1.0, + "content": "th sub-trajectory i.e. number of rows of", + "type": "text" + }, + { + "bbox": [ + 389, + 109, + 409, + 119 + ], + "score": 0.9, + "content": "\\tilde { X } ^ { ( \\bar { \\tau } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 107, + 476, + 123 + ], + "score": 1.0, + "content": ". By construction", + "type": "text" + }, + { + "bbox": [ + 476, + 110, + 505, + 121 + ], + "score": 0.86, + "content": "2 N _ { 0 } \\geq", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 119, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 107, + 121, + 144, + 132 + ], + "score": 0.92, + "content": "\\bar { N } _ { \\bar { \\tau } } \\geq N _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 119, + 169, + 134 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 169, + 122, + 211, + 132 + ], + "score": 0.9, + "content": "1 \\le \\bar { \\tau } \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 119, + 240, + 134 + ], + "score": 1.0, + "content": ". 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Applying Lemma C.6,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 299, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 213, + 235 + ], + "score": 1.0, + "content": "we find that, with probability", + "type": "text" + }, + { + "bbox": [ + 213, + 223, + 295, + 234 + ], + "score": 0.87, + "content": "1 - 2 \\bar { N } _ { \\bar { \\tau } } \\exp ( - 1 0 0 n )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 223, + 299, + 235 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 239, + 449, + 254 + ], + "lines": [ + { + "bbox": [ + 161, + 239, + 449, + 254 + ], + "spans": [ + { + "bbox": [ + 161, + 239, + 449, + 254 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\| \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } - \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\| \\leq \\sqrt { 2 N _ { 0 } } \\operatorname* { m a x } \\{ \\| \\pmb { h } ^ { ( i ) } - \\bar { \\pmb { h } } ^ { ( i ) } \\| _ { \\ell _ { 2 } } \\} \\leq c _ { 0 } \\sqrt { 2 N _ { 0 } } \\sqrt { n \\gamma _ { + } } \\| \\pmb { A } \\| ^ { L - 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "026f93854a61c2e7b5bc6bbc66d6fe9ed499bf3e12ea2a389e17b7994cc645f0.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 161, + 239, + 449, + 254 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 258, + 504, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 303, + 271 + ], + "score": 1.0, + "content": "Let us call this Event 2. We will show that our choice of", + "type": "text" + }, + { + "bbox": [ + 304, + 259, + 311, + 268 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "ensures right hand side is small enough and guarantees", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 267, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 107, + 268, + 228, + 281 + ], + "score": 0.91, + "content": "\\lVert \\bar { \\pmb { H } } ^ { ( \\bar { \\tau } ) } - \\tilde { \\pmb { H } } ^ { ( \\bar { \\tau } ) } \\rVert \\leq \\sqrt { \\gamma _ { - } N _ { 0 } } / 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 267, + 248, + 283 + ], + "score": 1.0, + "content": ". Set", + "type": "text" + }, + { + "bbox": [ + 248, + 269, + 325, + 281 + ], + "score": 0.91, + "content": "c = \\operatorname* { m a x } \\{ 2 0 0 c _ { 0 } ^ { 2 } , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 267, + 506, + 283 + ], + "score": 1.0, + "content": ". Desired claim follows by taking logarithms of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 331, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 248, + 292 + ], + "score": 1.0, + "content": "upper/lower bounds and cancelling out", + "type": "text" + }, + { + "bbox": [ + 248, + 281, + 268, + 291 + ], + "score": 0.91, + "content": "\\sqrt { N _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 280, + 331, + 292 + ], + "score": 1.0, + "content": "terms as follows", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 295, + 448, + 363 + ], + "lines": [ + { + "bbox": [ + 162, + 295, + 448, + 363 + ], + "spans": [ + { + "bbox": [ + 162, + 295, + 448, + 363 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { c _ { 0 } \\sqrt { n } \\| A \\| ^ { L - 1 } \\sqrt { \\gamma _ { + } } \\leq \\sqrt { \\gamma _ { - } } / 1 0 \\sqrt { 2 } \\iff ( L - 1 ) \\log \\| A \\| + \\log \\sqrt { c n \\rho } \\leq 0 } \\\\ & { \\iff - \\frac { \\log c n \\rho } { 2 \\log \\| A \\| } \\leq L - 1 } \\\\ & { \\iff L = \\lceil 1 - \\frac { \\log { ( c n \\rho ) } } { \\log \\| A \\| } \\rceil . } \\end{array}", + "type": "interline_equation", + "image_path": "2d649d4022ac6d43344413ab73349f4d80f20ea5385ff0cf9acb1543c2d92e86.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 162, + 295, + 448, + 317.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 162, + 317.6666666666667, + 448, + 340.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 162, + 340.33333333333337, + 448, + 363.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 198, + 378 + ], + "score": 1.0, + "content": "Here we use the fact that", + "type": "text" + }, + { + "bbox": [ + 199, + 366, + 248, + 377 + ], + "score": 0.91, + "content": "\\log \\| A \\| < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 365, + 269, + 378 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 270, + 366, + 304, + 377 + ], + "score": 0.91, + "content": "\\| A \\| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 365, + 321, + 378 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 366, + 353, + 376 + ], + "score": 0.89, + "content": "c n \\rho \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 365, + 506, + 378 + ], + "score": 1.0, + "content": ". 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Union", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 176, + 401 + ], + "score": 1.0, + "content": "bounding this over", + "type": "text" + }, + { + "bbox": [ + 176, + 389, + 217, + 399 + ], + "score": 0.91, + "content": "1 \\le \\bar { \\tau } \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 387, + 320, + 401 + ], + "score": 1.0, + "content": ", (D.5) uniformly holds with", + "type": "text" + }, + { + "bbox": [ + 320, + 388, + 361, + 399 + ], + "score": 0.92, + "content": "\\tilde { Q } = \\tilde { X } ^ { ( \\bar { \\tau } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 387, + 418, + 401 + ], + "score": 1.0, + "content": "and all rows of", + "type": "text" + }, + { + "bbox": [ + 418, + 389, + 428, + 398 + ], + "score": 0.82, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 387, + 442, + 401 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 443, + 389, + 451, + 399 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "-bounded with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 396, + 507, + 414 + ], + "spans": [ + { + "bbox": [ + 104, + 396, + 149, + 414 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 150, + 400, + 313, + 411 + ], + "score": 0.91, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - c _ { 1 } N _ { 0 } / \\rho ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 396, + 412, + 414 + ], + "score": 1.0, + "content": ". Applying Lemma A.1 on", + "type": "text" + }, + { + "bbox": [ + 412, + 399, + 453, + 411 + ], + "score": 0.92, + "content": "( \\tilde { X } ^ { ( \\bar { \\tau } ) } ) _ { \\bar { \\tau } = 1 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 396, + 507, + 414 + ], + "score": 1.0, + "content": ", we conclude", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 263, + 421 + ], + "score": 1.0, + "content": "with the bound (D.4) on the merged matrix", + "type": "text" + }, + { + "bbox": [ + 263, + 411, + 273, + 419 + ], + "score": 0.82, + "content": "\\pmb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 410, + 276, + 421 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 410, + 505, + 420 + ], + "score": 0.27, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 105, + 426, + 506, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 507, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 156, + 439 + ], + "score": 1.0, + "content": "Lemma D.4", + "type": "text" + }, + { + "bbox": [ + 157, + 428, + 165, + 437 + ], + "score": 0.83, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 426, + 417, + 439 + ], + "score": 1.0, + "content": "-bound on rows). Consider the setup of Theorem D.3. With probability √", + "type": "text" + }, + { + "bbox": [ + 417, + 426, + 507, + 438 + ], + "score": 0.85, + "content": "1 - 2 N \\exp ( - 1 0 0 ( n +", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 436, + 381, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 115, + 448 + ], + "score": 0.58, + "content": "p )", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 436, + 167, + 449 + ], + "score": 1.0, + "content": "), each row of", + "type": "text" + }, + { + "bbox": [ + 167, + 438, + 177, + 446 + ], + "score": 0.77, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 436, + 193, + 449 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 193, + 438, + 202, + 447 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 436, + 254, + 449 + ], + "score": 1.0, + "content": "-norm at most", + "type": "text" + }, + { + "bbox": [ + 254, + 437, + 288, + 448 + ], + "score": 0.91, + "content": "c { \\sqrt { p + n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 436, + 355, + 449 + ], + "score": 1.0, + "content": "for some constant", + "type": "text" + }, + { + "bbox": [ + 355, + 438, + 377, + 447 + ], + "score": 0.87, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 436, + 381, + 449 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 190, + 478 + ], + "score": 1.0, + "content": "Proof. The tth row of", + "type": "text" + }, + { + "bbox": [ + 191, + 465, + 200, + 473 + ], + "score": 0.83, + "content": "\\pmb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 460, + 243, + 478 + ], + "score": 1.0, + "content": "is equal to", + "type": "text" + }, + { + "bbox": [ + 244, + 460, + 311, + 478 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\pmb { x } _ { t } = [ \\frac { \\pmb { h } _ { t } ^ { T } } { \\sqrt { \\gamma _ { + } } } \\pmb { u } _ { t } ^ { T } ] ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 460, + 339, + 478 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 339, + 464, + 447, + 476 + ], + "score": 0.91, + "content": "\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( \\sqrt { \\gamma _ { + } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 460, + 464, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 464, + 464, + 505, + 475 + ], + "score": 0.89, + "content": "\\| u _ { t } \\| _ { \\psi _ { 2 } } ~ \\le", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 475, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 127, + 489 + ], + "score": 0.87, + "content": "\\mathcal { O } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 475, + 180, + 492 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 181, + 478, + 273, + 489 + ], + "score": 0.92, + "content": "\\| \\pmb { x } _ { t } - \\mathbb { E } [ \\pmb { x } _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 475, + 424, + 492 + ], + "score": 1.0, + "content": ". Now, applying Lemma F.3 on all rows", + "type": "text" + }, + { + "bbox": [ + 424, + 477, + 456, + 489 + ], + "score": 0.9, + "content": "\\{ \\pmb { x } _ { t } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 475, + 507, + 492 + ], + "score": 1.0, + "content": ", and using a√", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 245, + 501 + ], + "score": 1.0, + "content": "union bound, with probability at least", + "type": "text" + }, + { + "bbox": [ + 246, + 488, + 347, + 498 + ], + "score": 0.87, + "content": "1 - 2 \\dot { N } \\exp ( - 1 0 \\bar { 0 } ( n + p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 486, + 399, + 501 + ], + "score": 1.0, + "content": ", we have that √", + "type": "text" + }, + { + "bbox": [ + 399, + 489, + 504, + 500 + ], + "score": 0.89, + "content": "\\lVert \\mathbf { x } _ { t } - \\mathbb { E } [ \\mathbf { x } _ { t } ] \\rVert _ { \\ell _ { 2 } } \\leq c \\sqrt { n + p }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 131, + 511 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 131, + 500, + 135, + 507 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 497, + 221, + 511 + ], + "score": 1.0, + "content": ". To conclude, note that", + "type": "text" + }, + { + "bbox": [ + 221, + 498, + 379, + 510 + ], + "score": 0.9, + "content": "\\| \\mathbb { E } [ \\pmb { x } _ { t } ] \\| _ { \\ell _ { 2 } } = \\| \\mathbb { E } [ \\pmb { h } _ { t } ] \\| _ { \\ell _ { 2 } } / \\sqrt { \\gamma _ { + } } \\le \\theta \\le 3 \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 497, + 448, + 511 + ], + "score": 1.0, + "content": "via Assumption 1.", + "type": "text" + }, + { + "bbox": [ + 495, + 499, + 505, + 508 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 267, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 268, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 268, + 539 + ], + "score": 1.0, + "content": "E PROOFS OF MAIN RESULTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 548, + 228, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 229, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 229, + 561 + ], + "score": 1.0, + "content": "E.1 PROOF OF LEMMA 3.2", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 407, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 409, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 409, + 581 + ], + "score": 1.0, + "content": "The statement follows from upper bound Lemma B.3 and lower bound Lemma B.5.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 238, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 239, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 239, + 604 + ], + "score": 1.0, + "content": "E.2 PROOF OF THEOREM 4.2", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "Proof. To prove this theorem, we combine Theorem D.3 with deterministic SGD convergence result of Theorem", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 429, + 633 + ], + "score": 1.0, + "content": "4.1. Applying Theorem D.3, with the desired probability, inequality (D.4) holds and for all", + "type": "text" + }, + { + "bbox": [ + 430, + 623, + 434, + 631 + ], + "score": 0.65, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 621, + 505, + 633 + ], + "score": 1.0, + "content": ", input data satisfies", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 145, + 646 + ], + "score": 1.0, + "content": "the bound", + "type": "text" + }, + { + "bbox": [ + 145, + 632, + 246, + 645 + ], + "score": 0.91, + "content": "\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } \\leq \\sqrt { ( n + p ) / ( 2 c _ { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 632, + 362, + 646 + ], + "score": 1.0, + "content": "for a sufficiently small constant", + "type": "text" + }, + { + "bbox": [ + 362, + 633, + 388, + 643 + ], + "score": 0.9, + "content": "c _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 632, + 505, + 646 + ], + "score": 1.0, + "content": ". As the next step, we will argue", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 644, + 298, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 298, + 654 + ], + "score": 1.0, + "content": "that these two events imply the convergence of SGD.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 659, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 121, + 673 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 659, + 186, + 671 + ], + "score": 0.87, + "content": "\\pmb { \\theta } ^ { ( i ) } , \\pmb { c } ^ { ( i ) } \\in \\mathbb { R } ^ { n + p }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 657, + 267, + 673 + ], + "score": 1.0, + "content": "denote the ith rows of", + "type": "text" + }, + { + "bbox": [ + 268, + 661, + 289, + 671 + ], + "score": 0.72, + "content": "\\Theta , C", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 657, + 505, + 673 + ], + "score": 1.0, + "content": "respectively. Observe that the square-loss is separable along", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 668, + 507, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 150, + 685 + ], + "score": 1.0, + "content": "the rows of", + "type": "text" + }, + { + "bbox": [ + 150, + 672, + 159, + 681 + ], + "score": 0.83, + "content": "_ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 668, + 174, + 685 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 174, + 671, + 309, + 683 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\Theta - C \\| _ { F } ^ { 2 } = \\sum _ { i = 1 } ^ { n } \\| \\pmb { \\theta } ^ { ( i ) } - \\pmb { c } ^ { ( i ) } \\| _ { \\ell _ { 2 } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 668, + 428, + 685 + ], + "score": 1.0, + "content": ". 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Set", + "type": "text" + }, + { + "bbox": [ + 248, + 269, + 325, + 281 + ], + "score": 0.91, + "content": "c = \\operatorname* { m a x } \\{ 2 0 0 c _ { 0 } ^ { 2 } , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 267, + 506, + 283 + ], + "score": 1.0, + "content": ". Desired claim follows by taking logarithms of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 331, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 248, + 292 + ], + "score": 1.0, + "content": "upper/lower bounds and cancelling out", + "type": "text" + }, + { + "bbox": [ + 248, + 281, + 268, + 291 + ], + "score": 0.91, + "content": "\\sqrt { N _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 280, + 331, + 292 + ], + "score": 1.0, + "content": "terms as follows", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 257, + 506, + 292 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 295, + 448, + 363 + ], + "lines": [ + { + "bbox": [ + 162, + 295, + 448, + 363 + ], + "spans": [ + { + "bbox": [ + 162, + 295, + 448, + 363 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { c _ { 0 } \\sqrt { n } \\| A \\| ^ { L - 1 } \\sqrt { \\gamma _ { + } } \\leq \\sqrt { \\gamma _ { - } } / 1 0 \\sqrt { 2 } \\iff ( L - 1 ) \\log \\| A \\| + \\log \\sqrt { c n \\rho } \\leq 0 } \\\\ & { \\iff - \\frac { \\log c n \\rho } { 2 \\log \\| A \\| } \\leq L - 1 } \\\\ & { \\iff L = \\lceil 1 - \\frac { \\log { ( c n \\rho ) } } { \\log \\| A \\| } \\rceil . } \\end{array}", + "type": "interline_equation", + "image_path": "2d649d4022ac6d43344413ab73349f4d80f20ea5385ff0cf9acb1543c2d92e86.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 162, + 295, + 448, + 317.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 162, + 317.6666666666667, + 448, + 340.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 162, + 340.33333333333337, + 448, + 363.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 198, + 378 + ], + "score": 1.0, + "content": "Here we use the fact that", + "type": "text" + }, + { + "bbox": [ + 199, + 366, + 248, + 377 + ], + "score": 0.91, + "content": "\\log \\| A \\| < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 365, + 269, + 378 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 270, + 366, + 304, + 377 + ], + "score": 0.91, + "content": "\\| A \\| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 365, + 321, + 378 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 366, + 353, + 376 + ], + "score": 0.89, + "content": "c n \\rho \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 365, + 506, + 378 + ], + "score": 1.0, + "content": ". 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Union", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 176, + 401 + ], + "score": 1.0, + "content": "bounding this over", + "type": "text" + }, + { + "bbox": [ + 176, + 389, + 217, + 399 + ], + "score": 0.91, + "content": "1 \\le \\bar { \\tau } \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 387, + 320, + 401 + ], + "score": 1.0, + "content": ", (D.5) uniformly holds with", + "type": "text" + }, + { + "bbox": [ + 320, + 388, + 361, + 399 + ], + "score": 0.92, + "content": "\\tilde { Q } = \\tilde { X } ^ { ( \\bar { \\tau } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 387, + 418, + 401 + ], + "score": 1.0, + "content": "and all rows of", + "type": "text" + }, + { + "bbox": [ + 418, + 389, + 428, + 398 + ], + "score": 0.82, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 387, + 442, + 401 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 443, + 389, + 451, + 399 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "-bounded with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 396, + 507, + 414 + ], + "spans": [ + { + "bbox": [ + 104, + 396, + 149, + 414 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 150, + 400, + 313, + 411 + ], + "score": 0.91, + "content": "1 - 4 N \\exp ( - 1 0 0 n ) - 8 L \\exp ( - c _ { 1 } N _ { 0 } / \\rho ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 396, + 412, + 414 + ], + "score": 1.0, + "content": ". Applying Lemma A.1 on", + "type": "text" + }, + { + "bbox": [ + 412, + 399, + 453, + 411 + ], + "score": 0.92, + "content": "( \\tilde { X } ^ { ( \\bar { \\tau } ) } ) _ { \\bar { \\tau } = 1 } ^ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 396, + 507, + 414 + ], + "score": 1.0, + "content": ", we conclude", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 263, + 421 + ], + "score": 1.0, + "content": "with the bound (D.4) on the merged matrix", + "type": "text" + }, + { + "bbox": [ + 263, + 411, + 273, + 419 + ], + "score": 0.82, + "content": "\\pmb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 410, + 276, + 421 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 410, + 505, + 420 + ], + "score": 0.27, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 365, + 507, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 426, + 506, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 507, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 156, + 439 + ], + "score": 1.0, + "content": "Lemma D.4", + "type": "text" + }, + { + "bbox": [ + 157, + 428, + 165, + 437 + ], + "score": 0.83, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 426, + 417, + 439 + ], + "score": 1.0, + "content": "-bound on rows). Consider the setup of Theorem D.3. With probability √", + "type": "text" + }, + { + "bbox": [ + 417, + 426, + 507, + 438 + ], + "score": 0.85, + "content": "1 - 2 N \\exp ( - 1 0 0 ( n +", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 436, + 381, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 115, + 448 + ], + "score": 0.58, + "content": "p )", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 436, + 167, + 449 + ], + "score": 1.0, + "content": "), each row of", + "type": "text" + }, + { + "bbox": [ + 167, + 438, + 177, + 446 + ], + "score": 0.77, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 436, + 193, + 449 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 193, + 438, + 202, + 447 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 436, + 254, + 449 + ], + "score": 1.0, + "content": "-norm at most", + "type": "text" + }, + { + "bbox": [ + 254, + 437, + 288, + 448 + ], + "score": 0.91, + "content": "c { \\sqrt { p + n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 436, + 355, + 449 + ], + "score": 1.0, + "content": "for some constant", + "type": "text" + }, + { + "bbox": [ + 355, + 438, + 377, + 447 + ], + "score": 0.87, + "content": "c > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 436, + 381, + 449 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 426, + 507, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 190, + 478 + ], + "score": 1.0, + "content": "Proof. The tth row of", + "type": "text" + }, + { + "bbox": [ + 191, + 465, + 200, + 473 + ], + "score": 0.83, + "content": "\\pmb { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 460, + 243, + 478 + ], + "score": 1.0, + "content": "is equal to", + "type": "text" + }, + { + "bbox": [ + 244, + 460, + 311, + 478 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\pmb { x } _ { t } = [ \\frac { \\pmb { h } _ { t } ^ { T } } { \\sqrt { \\gamma _ { + } } } \\pmb { u } _ { t } ^ { T } ] ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 460, + 339, + 478 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 339, + 464, + 447, + 476 + ], + "score": 0.91, + "content": "\\| h _ { t } - \\mathbb { E } [ h _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( \\sqrt { \\gamma _ { + } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 460, + 464, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 464, + 464, + 505, + 475 + ], + "score": 0.89, + "content": "\\| u _ { t } \\| _ { \\psi _ { 2 } } ~ \\le", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 475, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 127, + 489 + ], + "score": 0.87, + "content": "\\mathcal { O } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 475, + 180, + 492 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 181, + 478, + 273, + 489 + ], + "score": 0.92, + "content": "\\| \\pmb { x } _ { t } - \\mathbb { E } [ \\pmb { x } _ { t } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 475, + 424, + 492 + ], + "score": 1.0, + "content": ". Now, applying Lemma F.3 on all rows", + "type": "text" + }, + { + "bbox": [ + 424, + 477, + 456, + 489 + ], + "score": 0.9, + "content": "\\{ \\pmb { x } _ { t } \\} _ { t = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 475, + 507, + 492 + ], + "score": 1.0, + "content": ", and using a√", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 245, + 501 + ], + "score": 1.0, + "content": "union bound, with probability at least", + "type": "text" + }, + { + "bbox": [ + 246, + 488, + 347, + 498 + ], + "score": 0.87, + "content": "1 - 2 \\dot { N } \\exp ( - 1 0 \\bar { 0 } ( n + p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 486, + 399, + 501 + ], + "score": 1.0, + "content": ", we have that √", + "type": "text" + }, + { + "bbox": [ + 399, + 489, + 504, + 500 + ], + "score": 0.89, + "content": "\\lVert \\mathbf { x } _ { t } - \\mathbb { E } [ \\mathbf { x } _ { t } ] \\rVert _ { \\ell _ { 2 } } \\leq c \\sqrt { n + p }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 131, + 511 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 131, + 500, + 135, + 507 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 497, + 221, + 511 + ], + "score": 1.0, + "content": ". To conclude, note that", + "type": "text" + }, + { + "bbox": [ + 221, + 498, + 379, + 510 + ], + "score": 0.9, + "content": "\\| \\mathbb { E } [ \\pmb { x } _ { t } ] \\| _ { \\ell _ { 2 } } = \\| \\mathbb { E } [ \\pmb { h } _ { t } ] \\| _ { \\ell _ { 2 } } / \\sqrt { \\gamma _ { + } } \\le \\theta \\le 3 \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 497, + 448, + 511 + ], + "score": 1.0, + "content": "via Assumption 1.", + "type": "text" + }, + { + "bbox": [ + 495, + 499, + 505, + 508 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 460, + 507, + 511 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 267, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 268, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 268, + 539 + ], + "score": 1.0, + "content": "E PROOFS OF MAIN RESULTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 548, + 228, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 229, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 229, + 561 + ], + "score": 1.0, + "content": "E.1 PROOF OF LEMMA 3.2", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 407, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 409, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 409, + 581 + ], + "score": 1.0, + "content": "The statement follows from upper bound Lemma B.3 and lower bound Lemma B.5.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 568, + 409, + 581 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 238, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 239, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 239, + 604 + ], + "score": 1.0, + "content": "E.2 PROOF OF THEOREM 4.2", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "Proof. To prove this theorem, we combine Theorem D.3 with deterministic SGD convergence result of Theorem", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 429, + 633 + ], + "score": 1.0, + "content": "4.1. Applying Theorem D.3, with the desired probability, inequality (D.4) holds and for all", + "type": "text" + }, + { + "bbox": [ + 430, + 623, + 434, + 631 + ], + "score": 0.65, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 621, + 505, + 633 + ], + "score": 1.0, + "content": ", input data satisfies", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 145, + 646 + ], + "score": 1.0, + "content": "the bound", + "type": "text" + }, + { + "bbox": [ + 145, + 632, + 246, + 645 + ], + "score": 0.91, + "content": "\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } \\leq \\sqrt { ( n + p ) / ( 2 c _ { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 632, + 362, + 646 + ], + "score": 1.0, + "content": "for a sufficiently small constant", + "type": "text" + }, + { + "bbox": [ + 362, + 633, + 388, + 643 + ], + "score": 0.9, + "content": "c _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 632, + 505, + 646 + ], + "score": 1.0, + "content": ". As the next step, we will argue", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 644, + 298, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 298, + 654 + ], + "score": 1.0, + "content": "that these two events imply the convergence of SGD.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 611, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 659, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 121, + 673 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 659, + 186, + 671 + ], + "score": 0.87, + "content": "\\pmb { \\theta } ^ { ( i ) } , \\pmb { c } ^ { ( i ) } \\in \\mathbb { R } ^ { n + p }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 657, + 267, + 673 + ], + "score": 1.0, + "content": "denote the ith rows of", + "type": "text" + }, + { + "bbox": [ + 268, + 661, + 289, + 671 + ], + "score": 0.72, + "content": "\\Theta , C", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 657, + 505, + 673 + ], + "score": 1.0, + "content": "respectively. Observe that the square-loss is separable along", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 668, + 507, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 150, + 685 + ], + "score": 1.0, + "content": "the rows of", + "type": "text" + }, + { + "bbox": [ + 150, + 672, + 159, + 681 + ], + "score": 0.83, + "content": "_ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 668, + 174, + 685 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 174, + 671, + 309, + 683 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\| \\Theta - C \\| _ { F } ^ { 2 } = \\sum _ { i = 1 } ^ { n } \\| \\pmb { \\theta } ^ { ( i ) } - \\pmb { c } ^ { ( i ) } \\| _ { \\ell _ { 2 } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 668, + 428, + 685 + ], + "score": 1.0, + "content": ". 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Also denote the label", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 212, + 104 + ], + "score": 1.0, + "content": "corresponding to ith row by", + "type": "text" + }, + { + "bbox": [ + 213, + 95, + 249, + 104 + ], + "score": 0.86, + "content": "y _ { t } = y _ { t , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 93, + 505, + 104 + ], + "score": 1.0, + "content": ". 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Applying Lemmas B.3 and 3.2, independent of", + "type": "text" + }, + { + "bbox": [ + 305, + 299, + 312, + 308 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 297, + 450, + 311 + ], + "score": 1.0, + "content": ", Assumption 1 holds with parameters", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 311, + 424, + 325 + ], + "lines": [ + { + "bbox": [ + 185, + 311, + 424, + 325 + ], + "spans": [ + { + "bbox": [ + 185, + 311, + 424, + 325 + ], + "score": 0.88, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 } \\quad , \\quad \\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } \\quad , \\quad \\theta = \\sqrt { 6 n } - \\sqrt { 2 } \\geq \\sqrt { n } .", + "type": "interline_equation", + "image_path": "bf81b85bc5403bd484b6f8d4f718375aec9603d21852fc0ff9a982dc793e4df6.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 185, + 311, + 424, + 325 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 104, + 325, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 325, + 482, + 345 + ], + "score": 1.0, + "content": "This yields (θ + 2)2 = 6n. Hence, we can apply Theorem 4.2 with the learning rate η = c0 β26ρn(n+p)", + "type": "text" + }, + { + "bbox": [ + 479, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 344, + 353, + 369 + ], + "lines": [ + { + "bbox": [ + 257, + 344, + 353, + 369 + ], + "spans": [ + { + "bbox": [ + 257, + 344, + 353, + 369 + ], + "score": 0.94, + "content": "\\rho = \\frac { B _ { \\infty } ^ { 2 } } { \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } } = \\frac { \\gamma _ { + } } { \\gamma _ { - } } ,", + "type": "interline_equation", + "image_path": "df580270ea63b2cf7c717a642d6652bf23e3cc5c5b6752bd2654c8fc9574fdd6.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 257, + 344, + 353, + 369 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 373, + 497, + 386 + ], + "lines": [ + { + "bbox": [ + 103, + 367, + 491, + 389 + ], + "spans": [ + { + "bbox": [ + 103, + 367, + 182, + 389 + ], + "score": 1.0, + "content": "and convergence rate", + "type": "text" + }, + { + "bbox": [ + 182, + 371, + 213, + 387 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\frac { \\beta ^ { 2 } \\eta } { 2 \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 367, + 444, + 389 + ], + "score": 1.0, + "content": ". To conclude with the stated result, we use the change of variable", + "type": "text" + }, + { + "bbox": [ + 444, + 374, + 485, + 385 + ], + "score": 0.91, + "content": "c _ { 0 } / 6 \\to c _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 367, + 491, + 389 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 396, + 246, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 247, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 247, + 408 + ], + "score": 1.0, + "content": "E.3.2 PROOF OF THEOREM 3.4", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 502, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 495, + 428 + ], + "score": 1.0, + "content": "Proof. The proof is similar to that of Theorem 3.3. Applying Lemmas B.3, B.4, and 3.2, independent of", + "type": "text" + }, + { + "bbox": [ + 496, + 416, + 502, + 424 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 426, + 240, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 240, + 437 + ], + "score": 1.0, + "content": "Assumption 1 holds with parameters", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 437, + 390, + 451 + ], + "lines": [ + { + "bbox": [ + 220, + 437, + 390, + 451 + ], + "spans": [ + { + "bbox": [ + 220, + 437, + 390, + 451 + ], + "score": 0.91, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 } \\quad , \\quad \\gamma _ { - } = s _ { \\mathrm { m i n } } ( B ) ^ { 2 } \\quad , \\quad \\theta = 0 .", + "type": "interline_equation", + "image_path": "73f78ed1f120e6b20278d967d2729c7d15ff90e1628a632a598d54e5c2af9a47.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 220, + 437, + 390, + 451 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 103, + 452, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 103, + 452, + 379, + 471 + ], + "score": 1.0, + "content": "Hence, we again apply Theorem 4.2 with the learning rate η = c0 β22ρ(n+p)", + "type": "text" + }, + { + "bbox": [ + 378, + 456, + 402, + 467 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 403, + 458, + 409, + 466 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "is given by (E.2). Use the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 107, + 468, + 174, + 479 + ], + "score": 1.0, + "content": "change of variable", + "type": "text" + }, + { + "bbox": [ + 175, + 468, + 216, + 479 + ], + "score": 0.92, + "content": "c _ { 0 } / 2 \\to c _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 468, + 339, + 479 + ], + "score": 1.0, + "content": "to conclude with the stated result.", + "type": "text" + }, + { + "bbox": [ + 495, + 469, + 505, + 479 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 491, + 268, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 269, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 269, + 503 + ], + "score": 1.0, + "content": "E.4 LEARNING UNSTABLE SYSTEMS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 504, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "In a similar fashion to Section 4, we provide a more general result on unstable systems that makes a parametric", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 521, + 316, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 316, + 532 + ], + "score": 1.0, + "content": "assumption on the statistical properties of the state vector.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 504, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 401, + 545 + ], + "score": 1.0, + "content": "Assumption 2 (Well-behaved state vector – single timestamp). Given timestamp √", + "type": "text" + }, + { + "bbox": [ + 401, + 534, + 429, + 543 + ], + "score": 0.89, + "content": "T _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 533, + 505, + 545 + ], + "score": 1.0, + "content": ", there exists positive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 543, + 450, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 134, + 555 + ], + "score": 1.0, + "content": "scalars", + "type": "text" + }, + { + "bbox": [ + 135, + 544, + 171, + 554 + ], + "score": 0.9, + "content": "\\gamma _ { + } , \\gamma _ { - } , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 543, + 264, + 555 + ], + "score": 1.0, + "content": "and an absolute constant", + "type": "text" + }, + { + "bbox": [ + 264, + 544, + 289, + 553 + ], + "score": 0.89, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 543, + 325, + 555 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 543, + 361, + 554 + ], + "score": 0.92, + "content": "{ \\dot { \\theta } } \\leq 3 { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 543, + 450, + 555 + ], + "score": 1.0, + "content": "and the following holds", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 556, + 462, + 568 + ], + "lines": [ + { + "bbox": [ + 131, + 556, + 462, + 568 + ], + "spans": [ + { + "bbox": [ + 131, + 556, + 462, + 568 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\gamma _ { + } I _ { n } \\succeq \\Sigma [ h _ { T _ { 0 } } ] \\succeq \\gamma _ { - } I _ { n } \\quad , \\quad \\| h _ { T _ { 0 } } - \\mathbb { E } [ h _ { T _ { 0 } } ] \\| _ { \\psi _ { 2 } } \\leq C \\sqrt { \\gamma _ { + } } \\quad a n d \\quad \\| \\mathbb { E } [ h _ { t } ] \\| _ { \\ell _ { 2 } } \\leq \\theta \\sqrt { \\gamma _ { + } } . } \\end{array}", + "type": "interline_equation", + "image_path": "9438b91dda38899e0a2d67a2c2af2ddd25b8dff5fc8f27ecc69daf059185b4cc.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 131, + 556, + 462, + 568 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 575, + 459, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 458, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 458, + 588 + ], + "score": 1.0, + "content": "The next theorem provides the parametrized result on unstable systems based on this assumption.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 105, + 590, + 504, + 613 + ], + "lines": [ + { + "bbox": [ + 102, + 583, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 102, + 583, + 347, + 607 + ], + "score": 1.0, + "content": "Theorem E.1 (Unstable system - general). Suppose we are given", + "type": "text" + }, + { + "bbox": [ + 347, + 591, + 356, + 600 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 583, + 449, + 607 + ], + "score": 1.0, + "content": "independent trajectories", + "type": "text" + }, + { + "bbox": [ + 449, + 588, + 504, + 602 + ], + "score": 0.92, + "content": "( { h _ { t } ^ { ( i ) } } , { u _ { t } ^ { ( i ) } } ) _ { t \\geq 0 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 601, + 496, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 119, + 614 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 119, + 601, + 161, + 612 + ], + "score": 0.9, + "content": "1 \\leq i \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 601, + 277, + 614 + ], + "score": 1.0, + "content": ". Sample each trajectory at time", + "type": "text" + }, + { + "bbox": [ + 277, + 602, + 288, + 611 + ], + "score": 0.84, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 601, + 323, + 614 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 323, + 602, + 333, + 611 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 601, + 365, + 614 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 366, + 601, + 421, + 613 + ], + "score": 0.93, + "content": "( { \\pmb y } _ { i } , { \\pmb h } _ { i } , { \\pmb u } _ { i } ) _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 601, + 496, + 614 + ], + "score": 1.0, + "content": "where ith sample is", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 614, + 368, + 630 + ], + "lines": [ + { + "bbox": [ + 240, + 614, + 368, + 630 + ], + "spans": [ + { + "bbox": [ + 240, + 614, + 368, + 630 + ], + "score": 0.88, + "content": "( { \\pmb y } _ { i } , { \\pmb h } _ { i } , { \\pmb u } _ { i } ) = ( { \\pmb h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \\pmb h } _ { T _ { 0 } } ^ { ( i ) } , { \\pmb u } _ { T _ { 0 } } ^ { ( i ) } ) .", + "type": "interline_equation", + "image_path": "a9a3926ae266e800ef88f4df5fedb737db10ee409ea1235fc9b294e04220741a.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 240, + 614, + 368, + 630 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 630, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 120, + 642 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 631, + 160, + 641 + ], + "score": 0.91, + "content": "C , c _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 629, + 328, + 642 + ], + "score": 1.0, + "content": "be absolute constants. Suppose Assumption", + "type": "text" + }, + { + "bbox": [ + 328, + 631, + 334, + 640 + ], + "score": 0.55, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 629, + 377, + 642 + ], + "score": 1.0, + "content": "holds with", + "type": "text" + }, + { + "bbox": [ + 377, + 631, + 387, + 640 + ], + "score": 0.84, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 629, + 483, + 642 + ], + "score": 1.0, + "content": "and sample size satisfies", + "type": "text" + }, + { + "bbox": [ + 484, + 631, + 505, + 641 + ], + "score": 0.81, + "content": "N \\geq", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 636, + 503, + 658 + ], + "spans": [ + { + "bbox": [ + 107, + 642, + 152, + 655 + ], + "score": 0.92, + "content": "C \\rho ^ { 2 } ( n + p )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 636, + 177, + 658 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 178, + 643, + 222, + 655 + ], + "score": 0.91, + "content": "\\rho = \\gamma _ { + } / \\gamma _ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 636, + 256, + 658 + ], + "score": 1.0, + "content": ". Assume", + "type": "text" + }, + { + "bbox": [ + 256, + 644, + 263, + 654 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 636, + 272, + 658 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 272, + 644, + 279, + 654 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 636, + 443, + 658 + ], + "score": 1.0, + "content": "-increasing, zero initial state conditions, and", + "type": "text" + }, + { + "bbox": [ + 443, + 641, + 503, + 654 + ], + "score": 0.91, + "content": "{ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 655, + 504, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 389, + 671 + ], + "score": 1.0, + "content": "Set scaling to be µ = 1/ γ+ and learning rate to be η = c0 β ρ(θ+√2)2(n+p) .", + "type": "text" + }, + { + "bbox": [ + 386, + 656, + 441, + 669 + ], + "score": 1.0, + "content": "Starting from", + "type": "text" + }, + { + "bbox": [ + 441, + 657, + 453, + 667 + ], + "score": 0.85, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 656, + 485, + 669 + ], + "score": 1.0, + "content": ", we run", + "type": "text" + }, + { + "bbox": [ + 486, + 657, + 504, + 666 + ], + "score": 0.58, + "content": "S G D", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 334, + 684 + ], + "score": 1.0, + "content": "over the equations described in (2.2) and (2.3). With probability", + "type": "text" + }, + { + "bbox": [ + 334, + 669, + 503, + 684 + ], + "score": 0.86, + "content": "1 - 2 \\dot { N } \\exp ( - 1 0 0 ( n + p ) ) - 4 \\exp ( - \\mathcal { O } ( \\textstyle { \\frac { N } { \\rho ^ { 2 } } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 669, + 506, + 684 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 681, + 172, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 172, + 693 + ], + "score": 1.0, + "content": "all iterates satisfy", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 695, + 424, + 720 + ], + "lines": [ + { + "bbox": [ + 186, + 695, + 424, + 720 + ], + "spans": [ + { + "bbox": [ + 186, + 695, + 424, + 720 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\Theta _ { i } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "40a187810ccb779821ff164c2bc79e99bdffd289d5c17f904282562bb74864e7.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 186, + 695, + 424, + 720 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 721, + 345, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 346, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 346, + 733 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 374, + 504, + 384 + ], + "lines": [ + { + "bbox": [ + 496, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 496, + 374, + 505, + 386 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 114 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 126, + 94 + ], + "score": 1.0, + "content": "truth", + "type": "text" + }, + { + "bbox": [ + 126, + 84, + 135, + 92 + ], + "score": 0.77, + "content": "_ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 81, + 203, + 94 + ], + "score": 1.0, + "content": ". Pick a row index", + "type": "text" + }, + { + "bbox": [ + 203, + 84, + 207, + 92 + ], + "score": 0.63, + "content": "_ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 83, + 252, + 93 + ], + "score": 0.84, + "content": "( 1 \\leq i \\leq n )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 81, + 269, + 94 + ], + "score": 1.0, + "content": ", set", + "type": "text" + }, + { + "bbox": [ + 270, + 82, + 301, + 93 + ], + "score": 0.92, + "content": "\\mathbf { c } = \\mathbf { c } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 81, + 383, + 94 + ], + "score": 1.0, + "content": "and denote ith row of", + "type": "text" + }, + { + "bbox": [ + 384, + 83, + 397, + 93 + ], + "score": 0.89, + "content": "\\Theta _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 81, + 410, + 94 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 410, + 83, + 421, + 93 + ], + "score": 0.86, + "content": "\\pmb { \\theta } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 81, + 506, + 94 + ], + "score": 1.0, + "content": ". Also denote the label", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 212, + 104 + ], + "score": 1.0, + "content": "corresponding to ith row by", + "type": "text" + }, + { + "bbox": [ + 213, + 95, + 249, + 104 + ], + "score": 0.86, + "content": "y _ { t } = y _ { t , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 93, + 505, + 104 + ], + "score": 1.0, + "content": ". With this notation, SGD over (2.3) runs SGD over the ith row with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 289, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 143, + 114 + ], + "score": 1.0, + "content": "equations", + "type": "text" + }, + { + "bbox": [ + 144, + 103, + 202, + 114 + ], + "score": 0.95, + "content": "y _ { t } \\overset { \\cdot } { = } \\phi ( \\langle c , \\pmb { x } _ { t } \\rangle )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 103, + 289, + 114 + ], + "score": 1.0, + "content": "and with loss functions", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 114 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 115, + 411, + 146 + ], + "lines": [ + { + "bbox": [ + 200, + 115, + 411, + 146 + ], + "spans": [ + { + "bbox": [ + 200, + 115, + 411, + 146 + ], + "score": 0.93, + "content": "\\mathcal { L } ( \\pmb { \\theta } ) = N ^ { - 1 } \\sum _ { t = 1 } ^ { N } \\mathcal { L } _ { t } ( \\pmb { \\theta } ) , \\mathcal { L } _ { t } ( \\pmb { \\theta } ) = \\frac { 1 } { 2 } \\big ( y _ { t } - \\phi ( \\langle \\pmb { \\theta } , \\pmb { x } _ { t } \\rangle ) \\big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "81edf3d5401aa679cfd7b1a43b894284bc107a49259a5f91279b7a26002bbec8.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 115, + 411, + 130.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 200, + 130.5, + 411, + 146.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 148, + 506, + 185 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 267, + 160 + ], + "score": 1.0, + "content": "Substituting our high-probability bounds on", + "type": "text" + }, + { + "bbox": [ + 267, + 150, + 278, + 159 + ], + "score": 0.85, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 147, + 429, + 160 + ], + "score": 1.0, + "content": "(e.g. (D.4)) into Theorem 4.1, we can set", + "type": "text" + }, + { + "bbox": [ + 429, + 148, + 503, + 159 + ], + "score": 0.87, + "content": "B = ( n + p ) / ( 2 c _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 147, + 507, + 160 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 156, + 507, + 177 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 170, + 173 + ], + "score": 0.92, + "content": "\\gamma _ { + } = ( \\theta + \\sqrt { 2 } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 156, + 189, + 177 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 189, + 160, + 237, + 172 + ], + "score": 0.93, + "content": "\\gamma _ { - } = \\rho ^ { - 1 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 156, + 375, + 177 + ], + "score": 1.0, + "content": ". Consequently, using the learning rate", + "type": "text" + }, + { + "bbox": [ + 375, + 159, + 452, + 176 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\eta = c _ { 0 } \\frac { \\beta ^ { 2 } \\rho ^ { - 1 } } { ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 156, + 479, + 177 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 480, + 162, + 502, + 172 + ], + "score": 0.83, + "content": "\\tau \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 156, + 507, + 177 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 222, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 119, + 185 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 120, + 176, + 125, + 183 + ], + "score": 0.65, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 172, + 186, + 185 + ], + "score": 1.0, + "content": "th SGD iteration", + "type": "text" + }, + { + "bbox": [ + 187, + 174, + 197, + 184 + ], + "score": 0.86, + "content": "\\pmb { \\theta } _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 172, + 222, + 185 + ], + "score": 1.0, + "content": "obeys", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 147, + 507, + 185 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 186, + 416, + 212 + ], + "lines": [ + { + "bbox": [ + 193, + 186, + 416, + 212 + ], + "spans": [ + { + "bbox": [ + 193, + 186, + 416, + 212 + ], + "score": 0.92, + "content": "\\mathbb { E } [ \\| \\pmb { \\theta } _ { \\tau } - \\pmb { c } \\| _ { \\ell _ { 2 } } ^ { 2 } ] \\le \\| \\pmb { \\theta } _ { 0 } - \\pmb { c } \\| _ { \\ell _ { 2 } } ^ { 2 } ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } \\rho ^ { - 2 } } { 2 ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } ,", + "type": "interline_equation", + "image_path": "a2a34de338e81f6f5bfb551642d33031ebb1a1cbff91529d11c41478fa0cfbbd.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 193, + 186, + 416, + 212 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 506, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "where the expectation is over the random selection of SGD updates. This establishes the convergence for a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 221, + 507, + 239 + ], + "spans": [ + { + "bbox": [ + 103, + 221, + 172, + 239 + ], + "score": 1.0, + "content": "particular row of", + "type": "text" + }, + { + "bbox": [ + 172, + 226, + 181, + 235 + ], + "score": 0.81, + "content": "_ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 221, + 376, + 239 + ], + "score": 1.0, + "content": ". Summing up these inequalities (E.1) over all rows", + "type": "text" + }, + { + "bbox": [ + 377, + 223, + 429, + 236 + ], + "score": 0.94, + "content": "\\pmb { \\theta } _ { \\tau } ^ { ( 1 ) } , \\dots , \\pmb { \\theta } _ { \\tau } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 221, + 507, + 239 + ], + "score": 1.0, + "content": "(which converge to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 234, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 107, + 235, + 158, + 247 + ], + "score": 0.91, + "content": "\\bar { \\mathbf { c } } ^ { ( 1 ) } , \\ldots , \\mathbf { c } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 234, + 322, + 250 + ], + "score": 1.0, + "content": "respectively) yields the targeted bound (4.4).", + "type": "text" + }, + { + "bbox": [ + 495, + 237, + 505, + 247 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 103, + 213, + 507, + 250 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 260, + 338, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 339, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 339, + 272 + ], + "score": 1.0, + "content": "E.3 PROOFS OF MAIN RESULTS ON STABLE SYSTEMS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 279, + 246, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 246, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 246, + 291 + ], + "score": 1.0, + "content": "E.3.1 PROOF OF THEOREM 3.3", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 449, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 450, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 304, + 311 + ], + "score": 1.0, + "content": "Proof. Applying Lemmas B.3 and 3.2, independent of", + "type": "text" + }, + { + "bbox": [ + 305, + 299, + 312, + 308 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 297, + 450, + 311 + ], + "score": 1.0, + "content": ", Assumption 1 holds with parameters", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 297, + 450, + 311 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 311, + 424, + 325 + ], + "lines": [ + { + "bbox": [ + 185, + 311, + 424, + 325 + ], + "spans": [ + { + "bbox": [ + 185, + 311, + 424, + 325 + ], + "score": 0.88, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 } \\quad , \\quad \\gamma _ { - } = \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } \\quad , \\quad \\theta = \\sqrt { 6 n } - \\sqrt { 2 } \\geq \\sqrt { n } .", + "type": "interline_equation", + "image_path": "bf81b85bc5403bd484b6f8d4f718375aec9603d21852fc0ff9a982dc793e4df6.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 185, + 311, + 424, + 325 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 104, + 325, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 325, + 482, + 345 + ], + "score": 1.0, + "content": "This yields (θ + 2)2 = 6n. Hence, we can apply Theorem 4.2 with the learning rate η = c0 β26ρn(n+p)", + "type": "text" + }, + { + "bbox": [ + 479, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 325, + 505, + 345 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 344, + 353, + 369 + ], + "lines": [ + { + "bbox": [ + 257, + 344, + 353, + 369 + ], + "spans": [ + { + "bbox": [ + 257, + 344, + 353, + 369 + ], + "score": 0.94, + "content": "\\rho = \\frac { B _ { \\infty } ^ { 2 } } { \\beta ^ { 2 } s _ { \\mathrm { m i n } } ( B ) ^ { 2 } } = \\frac { \\gamma _ { + } } { \\gamma _ { - } } ,", + "type": "interline_equation", + "image_path": "df580270ea63b2cf7c717a642d6652bf23e3cc5c5b6752bd2654c8fc9574fdd6.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 257, + 344, + 353, + 369 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 373, + 497, + 386 + ], + "lines": [ + { + "bbox": [ + 103, + 367, + 491, + 389 + ], + "spans": [ + { + "bbox": [ + 103, + 367, + 182, + 389 + ], + "score": 1.0, + "content": "and convergence rate", + "type": "text" + }, + { + "bbox": [ + 182, + 371, + 213, + 387 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 1 - \\frac { \\beta ^ { 2 } \\eta } { 2 \\rho } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 367, + 444, + 389 + ], + "score": 1.0, + "content": ". To conclude with the stated result, we use the change of variable", + "type": "text" + }, + { + "bbox": [ + 444, + 374, + 485, + 385 + ], + "score": 0.91, + "content": "c _ { 0 } / 6 \\to c _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 367, + 491, + 389 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 103, + 367, + 491, + 389 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 396, + 246, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 247, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 247, + 408 + ], + "score": 1.0, + "content": "E.3.2 PROOF OF THEOREM 3.4", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 502, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 495, + 428 + ], + "score": 1.0, + "content": "Proof. The proof is similar to that of Theorem 3.3. Applying Lemmas B.3, B.4, and 3.2, independent of", + "type": "text" + }, + { + "bbox": [ + 496, + 416, + 502, + 424 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 426, + 240, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 240, + 437 + ], + "score": 1.0, + "content": "Assumption 1 holds with parameters", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 414, + 502, + 437 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 437, + 390, + 451 + ], + "lines": [ + { + "bbox": [ + 220, + 437, + 390, + 451 + ], + "spans": [ + { + "bbox": [ + 220, + 437, + 390, + 451 + ], + "score": 0.91, + "content": "\\gamma _ { + } = B _ { \\infty } ^ { 2 } \\quad , \\quad \\gamma _ { - } = s _ { \\mathrm { m i n } } ( B ) ^ { 2 } \\quad , \\quad \\theta = 0 .", + "type": "interline_equation", + "image_path": "73f78ed1f120e6b20278d967d2729c7d15ff90e1628a632a598d54e5c2af9a47.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 220, + 437, + 390, + 451 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 103, + 452, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 103, + 452, + 379, + 471 + ], + "score": 1.0, + "content": "Hence, we again apply Theorem 4.2 with the learning rate η = c0 β22ρ(n+p)", + "type": "text" + }, + { + "bbox": [ + 378, + 456, + 402, + 467 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 403, + 458, + 409, + 466 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "is given by (E.2). Use the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 107, + 468, + 174, + 479 + ], + "score": 1.0, + "content": "change of variable", + "type": "text" + }, + { + "bbox": [ + 175, + 468, + 216, + 479 + ], + "score": 0.92, + "content": "c _ { 0 } / 2 \\to c _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 468, + 339, + 479 + ], + "score": 1.0, + "content": "to conclude with the stated result.", + "type": "text" + }, + { + "bbox": [ + 495, + 469, + 505, + 479 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 103, + 452, + 505, + 479 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 491, + 268, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 269, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 269, + 503 + ], + "score": 1.0, + "content": "E.4 LEARNING UNSTABLE SYSTEMS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 504, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "In a similar fashion to Section 4, we provide a more general result on unstable systems that makes a parametric", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 521, + 316, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 316, + 532 + ], + "score": 1.0, + "content": "assumption on the statistical properties of the state vector.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 510, + 505, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 504, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 401, + 545 + ], + "score": 1.0, + "content": "Assumption 2 (Well-behaved state vector – single timestamp). Given timestamp √", + "type": "text" + }, + { + "bbox": [ + 401, + 534, + 429, + 543 + ], + "score": 0.89, + "content": "T _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 533, + 505, + 545 + ], + "score": 1.0, + "content": ", there exists positive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 543, + 450, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 134, + 555 + ], + "score": 1.0, + "content": "scalars", + "type": "text" + }, + { + "bbox": [ + 135, + 544, + 171, + 554 + ], + "score": 0.9, + "content": "\\gamma _ { + } , \\gamma _ { - } , \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 543, + 264, + 555 + ], + "score": 1.0, + "content": "and an absolute constant", + "type": "text" + }, + { + "bbox": [ + 264, + 544, + 289, + 553 + ], + "score": 0.89, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 543, + 325, + 555 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 325, + 543, + 361, + 554 + ], + "score": 0.92, + "content": "{ \\dot { \\theta } } \\leq 3 { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 543, + 450, + 555 + ], + "score": 1.0, + "content": "and the following holds", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 533, + 505, + 555 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 556, + 462, + 568 + ], + "lines": [ + { + "bbox": [ + 131, + 556, + 462, + 568 + ], + "spans": [ + { + "bbox": [ + 131, + 556, + 462, + 568 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\gamma _ { + } I _ { n } \\succeq \\Sigma [ h _ { T _ { 0 } } ] \\succeq \\gamma _ { - } I _ { n } \\quad , \\quad \\| h _ { T _ { 0 } } - \\mathbb { E } [ h _ { T _ { 0 } } ] \\| _ { \\psi _ { 2 } } \\leq C \\sqrt { \\gamma _ { + } } \\quad a n d \\quad \\| \\mathbb { E } [ h _ { t } ] \\| _ { \\ell _ { 2 } } \\leq \\theta \\sqrt { \\gamma _ { + } } . } \\end{array}", + "type": "interline_equation", + "image_path": "9438b91dda38899e0a2d67a2c2af2ddd25b8dff5fc8f27ecc69daf059185b4cc.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 131, + 556, + 462, + 568 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 575, + 459, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 458, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 458, + 588 + ], + "score": 1.0, + "content": "The next theorem provides the parametrized result on unstable systems based on this assumption.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 574, + 458, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 590, + 504, + 613 + ], + "lines": [ + { + "bbox": [ + 102, + 583, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 102, + 583, + 347, + 607 + ], + "score": 1.0, + "content": "Theorem E.1 (Unstable system - general). Suppose we are given", + "type": "text" + }, + { + "bbox": [ + 347, + 591, + 356, + 600 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 583, + 449, + 607 + ], + "score": 1.0, + "content": "independent trajectories", + "type": "text" + }, + { + "bbox": [ + 449, + 588, + 504, + 602 + ], + "score": 0.92, + "content": "( { h _ { t } ^ { ( i ) } } , { u _ { t } ^ { ( i ) } } ) _ { t \\geq 0 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 601, + 496, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 119, + 614 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 119, + 601, + 161, + 612 + ], + "score": 0.9, + "content": "1 \\leq i \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 601, + 277, + 614 + ], + "score": 1.0, + "content": ". Sample each trajectory at time", + "type": "text" + }, + { + "bbox": [ + 277, + 602, + 288, + 611 + ], + "score": 0.84, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 601, + 323, + 614 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 323, + 602, + 333, + 611 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 601, + 365, + 614 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 366, + 601, + 421, + 613 + ], + "score": 0.93, + "content": "( { \\pmb y } _ { i } , { \\pmb h } _ { i } , { \\pmb u } _ { i } ) _ { i = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 601, + 496, + 614 + ], + "score": 1.0, + "content": "where ith sample is", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 102, + 583, + 504, + 614 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 614, + 368, + 630 + ], + "lines": [ + { + "bbox": [ + 240, + 614, + 368, + 630 + ], + "spans": [ + { + "bbox": [ + 240, + 614, + 368, + 630 + ], + "score": 0.88, + "content": "( { \\pmb y } _ { i } , { \\pmb h } _ { i } , { \\pmb u } _ { i } ) = ( { \\pmb h } _ { T _ { 0 } + 1 } ^ { ( i ) } , { \\pmb h } _ { T _ { 0 } } ^ { ( i ) } , { \\pmb u } _ { T _ { 0 } } ^ { ( i ) } ) .", + "type": "interline_equation", + "image_path": "a9a3926ae266e800ef88f4df5fedb737db10ee409ea1235fc9b294e04220741a.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 240, + 614, + 368, + 630 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 630, + 506, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 120, + 642 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 631, + 160, + 641 + ], + "score": 0.91, + "content": "C , c _ { 0 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 629, + 328, + 642 + ], + "score": 1.0, + "content": "be absolute constants. Suppose Assumption", + "type": "text" + }, + { + "bbox": [ + 328, + 631, + 334, + 640 + ], + "score": 0.55, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 629, + 377, + 642 + ], + "score": 1.0, + "content": "holds with", + "type": "text" + }, + { + "bbox": [ + 377, + 631, + 387, + 640 + ], + "score": 0.84, + "content": "T _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 629, + 483, + 642 + ], + "score": 1.0, + "content": "and sample size satisfies", + "type": "text" + }, + { + "bbox": [ + 484, + 631, + 505, + 641 + ], + "score": 0.81, + "content": "N \\geq", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 636, + 503, + 658 + ], + "spans": [ + { + "bbox": [ + 107, + 642, + 152, + 655 + ], + "score": 0.92, + "content": "C \\rho ^ { 2 } ( n + p )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 636, + 177, + 658 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 178, + 643, + 222, + 655 + ], + "score": 0.91, + "content": "\\rho = \\gamma _ { + } / \\gamma _ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 636, + 256, + 658 + ], + "score": 1.0, + "content": ". Assume", + "type": "text" + }, + { + "bbox": [ + 256, + 644, + 263, + 654 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 636, + 272, + 658 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 272, + 644, + 279, + 654 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 636, + 443, + 658 + ], + "score": 1.0, + "content": "-increasing, zero initial state conditions, and", + "type": "text" + }, + { + "bbox": [ + 443, + 641, + 503, + 654 + ], + "score": 0.91, + "content": "{ \\pmb u } _ { t } \\overset { i . i . d . } { \\sim } \\mathcal { N } ( 0 , { \\pmb I } _ { p } )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 655, + 504, + 671 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 389, + 671 + ], + "score": 1.0, + "content": "Set scaling to be µ = 1/ γ+ and learning rate to be η = c0 β ρ(θ+√2)2(n+p) .", + "type": "text" + }, + { + "bbox": [ + 386, + 656, + 441, + 669 + ], + "score": 1.0, + "content": "Starting from", + "type": "text" + }, + { + "bbox": [ + 441, + 657, + 453, + 667 + ], + "score": 0.85, + "content": "\\Theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 656, + 485, + 669 + ], + "score": 1.0, + "content": ", we run", + "type": "text" + }, + { + "bbox": [ + 486, + 657, + 504, + 666 + ], + "score": 0.58, + "content": "S G D", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 334, + 684 + ], + "score": 1.0, + "content": "over the equations described in (2.2) and (2.3). With probability", + "type": "text" + }, + { + "bbox": [ + 334, + 669, + 503, + 684 + ], + "score": 0.86, + "content": "1 - 2 \\dot { N } \\exp ( - 1 0 0 ( n + p ) ) - 4 \\exp ( - \\mathcal { O } ( \\textstyle { \\frac { N } { \\rho ^ { 2 } } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 669, + 506, + 684 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 681, + 172, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 172, + 693 + ], + "score": 1.0, + "content": "all iterates satisfy", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 629, + 506, + 693 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 695, + 424, + 720 + ], + "lines": [ + { + "bbox": [ + 186, + 695, + 424, + 720 + ], + "spans": [ + { + "bbox": [ + 186, + 695, + 424, + 720 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\Theta _ { i } - C \\| _ { F } ^ { 2 } ] \\le ( 1 - c _ { 0 } \\frac { \\beta ^ { 4 } } { 2 \\rho ^ { 2 } ( \\theta + \\sqrt { 2 } ) ^ { 2 } ( n + p ) } ) ^ { \\tau } \\| \\Theta _ { 0 } - C \\| _ { F } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "40a187810ccb779821ff164c2bc79e99bdffd289d5c17f904282562bb74864e7.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 186, + 695, + 424, + 720 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 721, + 345, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 346, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 346, + 733 + ], + "score": 1.0, + "content": "where the expectation is over the randomness of the SGD updates.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 720, + 346, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 103, + 78, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 103, + 78, + 147, + 97 + ], + "score": 1.0, + "content": "Proof. Set", + "type": "text" + }, + { + "bbox": [ + 148, + 81, + 231, + 95 + ], + "score": 0.92, + "content": "\\pmb { x } _ { i } = [ \\gamma _ { + } ^ { - 1 / 2 } \\pmb { h } _ { i } ^ { T } \\ \\pmb { u } _ { i } ^ { T } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 78, + 248, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 82, + 324, + 94 + ], + "score": 0.92, + "content": "\\pmb { X } = [ \\pmb { x } _ { 1 } ~ . ~ . ~ \\pmb { x } _ { N } ] ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 78, + 351, + 97 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 351, + 83, + 362, + 92 + ], + "score": 0.78, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 78, + 507, + 97 + ], + "score": 1.0, + "content": "has i.i.d. rows, we can apply Theorem", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 321, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 321, + 104 + ], + "score": 1.0, + "content": "F.1 and Lemma F.3 to find with the desired probability that", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 133, + 117, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 132, + 115, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 132, + 115, + 176, + 131 + ], + "score": 1.0, + "content": "• Rows of", + "type": "text" + }, + { + "bbox": [ + 177, + 120, + 187, + 128 + ], + "score": 0.83, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 115, + 216, + 131 + ], + "score": 1.0, + "content": "satisfy", + "type": "text" + }, + { + "bbox": [ + 216, + 118, + 309, + 129 + ], + "score": 0.89, + "content": "\\| \\pmb { x } _ { i } - \\mathbb { E } [ \\pmb { x } _ { i } ] \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 115, + 327, + 131 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 327, + 118, + 398, + 129 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } ] \\le 3 \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 115, + 469, + 131 + ], + "score": 1.0, + "content": ", hence all rows of", + "type": "text" + }, + { + "bbox": [ + 470, + 119, + 479, + 127 + ], + "score": 0.82, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 115, + 505, + 131 + ], + "score": 1.0, + "content": "obeys", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 143, + 129, + 248, + 142 + ], + "spans": [ + { + "bbox": [ + 143, + 129, + 244, + 141 + ], + "score": 0.86, + "content": "\\| \\pmb { x } _ { i } \\| _ { \\ell _ { 2 } } \\leq \\sqrt { ( n + p ) / ( 2 c _ { 0 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 129, + 248, + 142 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 133, + 148, + 185, + 159 + ], + "lines": [ + { + "bbox": [ + 132, + 147, + 185, + 160 + ], + "spans": [ + { + "bbox": [ + 132, + 147, + 142, + 160 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 149, + 153, + 158 + ], + "score": 0.78, + "content": "\\boldsymbol { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 147, + 185, + 160 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 157, + 383, + 180 + ], + "lines": [ + { + "bbox": [ + 263, + 157, + 383, + 180 + ], + "spans": [ + { + "bbox": [ + 263, + 157, + 383, + 180 + ], + "score": 0.91, + "content": "( \\theta + \\sqrt { 2 } ) ^ { 2 } \\succeq \\frac { X ^ { T } X } { N } \\succeq \\rho ^ { - 1 } / 2 .", + "type": "interline_equation", + "image_path": "3d224d945f8ed522394feb5bf26a9b05f69d8d27c01ea6e360fb5f732a91d988.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 263, + 157, + 383, + 180 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 194, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 171, + 208 + ], + "score": 1.0, + "content": "To proceed, using", + "type": "text" + }, + { + "bbox": [ + 172, + 194, + 220, + 206 + ], + "score": 0.79, + "content": "\\gamma _ { - } = \\rho ^ { - 1 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 193, + 223, + 208 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 223, + 195, + 297, + 206 + ], + "score": 0.87, + "content": "B = ( n + p ) / ( 2 c _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 193, + 315, + 208 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 316, + 194, + 379, + 206 + ], + "score": 0.93, + "content": "\\gamma _ { + } = ( \\theta + \\sqrt { 2 } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 193, + 506, + 208 + ], + "score": 1.0, + "content": ", we apply Theorem 4.1 on the loss", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 204, + 503, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 275, + 217 + ], + "score": 1.0, + "content": "function (2.3); which yields the desired result.", + "type": "text" + }, + { + "bbox": [ + 497, + 208, + 503, + 213 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 107, + 229, + 237, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 238, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 238, + 242 + ], + "score": 1.0, + "content": "E.5 PROOF OF THEOREM 5.1", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "Proof. The proof is a corollary of Theorem E.1. We need to substitute the proper values in Assumption 2.√ √", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 260, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 252, + 273 + ], + "score": 1.0, + "content": "Applying Lemma B.3, we can substitute", + "type": "text" + }, + { + "bbox": [ + 253, + 260, + 292, + 272 + ], + "score": 0.93, + "content": "\\gamma _ { + } = B _ { T _ { 0 } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 260, + 308, + 273 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 260, + 394, + 272 + ], + "score": 0.93, + "content": "\\theta = \\sqrt { 6 n } - \\sqrt { 2 } \\geq \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 260, + 506, + 273 + ], + "score": 1.0, + "content": ". Next, we need to find a lower", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 269, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 228, + 284 + ], + "score": 1.0, + "content": "bound. Applying Lemma 3.2 for", + "type": "text" + }, + { + "bbox": [ + 228, + 272, + 252, + 281 + ], + "score": 0.89, + "content": "n > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 269, + 327, + 284 + ], + "score": 1.0, + "content": "and Lemma B.6 for", + "type": "text" + }, + { + "bbox": [ + 327, + 272, + 351, + 281 + ], + "score": 0.87, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 269, + 420, + 284 + ], + "score": 1.0, + "content": ", we can substitute", + "type": "text" + }, + { + "bbox": [ + 420, + 271, + 465, + 282 + ], + "score": 0.92, + "content": "\\gamma _ { - } = \\gamma _ { + } / \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 269, + 497, + 284 + ], + "score": 1.0, + "content": "with the", + "type": "text" + }, + { + "bbox": [ + 498, + 273, + 504, + 282 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 281, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 439, + 292 + ], + "score": 1.0, + "content": "definition of (5.2). With these, the result follows as an immediate corollary of Theorem E.1.", + "type": "text" + }, + { + "bbox": [ + 496, + 281, + 505, + 292 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 308, + 336, + 321 + ], + "lines": [ + { + "bbox": [ + 104, + 306, + 336, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 336, + 322 + ], + "score": 1.0, + "content": "F SUPPLEMENTARY STATISTICAL RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 105, + 332, + 504, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "score": 1.0, + "content": "The following theorem bounds the empirical covariance of matrices with independent subgaussian rows. Given", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 342, + 376, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 167, + 355 + ], + "score": 1.0, + "content": "a random vector", + "type": "text" + }, + { + "bbox": [ + 167, + 345, + 174, + 352 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 342, + 300, + 355 + ], + "score": 1.0, + "content": ", define the de-biasing operation as", + "type": "text" + }, + { + "bbox": [ + 300, + 343, + 373, + 354 + ], + "score": 0.91, + "content": "\\mathbf { z m } ( \\pmb { x } ) = \\pmb { x } - \\mathbb { E } [ \\pmb { x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 342, + 376, + 355 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 354, + 507, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 354, + 183, + 372 + ], + "score": 1.0, + "content": "Theorem F.1. Let", + "type": "text" + }, + { + "bbox": [ + 183, + 357, + 235, + 367 + ], + "score": 0.89, + "content": "\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 354, + 432, + 372 + ], + "score": 1.0, + "content": "be a matrix with independent subgaussian rows", + "type": "text" + }, + { + "bbox": [ + 432, + 358, + 464, + 369 + ], + "score": 0.91, + "content": "\\{ { \\pmb a } _ { i } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 354, + 507, + 372 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 368, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 107, + 368, + 189, + 379 + ], + "score": 0.89, + "content": "\\| \\mathbf { z } \\mathbf { m } ( \\mathbf { { a } } _ { i } ) \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 368, + 207, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 368, + 263, + 379 + ], + "score": 0.91, + "content": "\\Sigma [ { \\pmb a } _ { i } ] \\preceq K ^ { 2 } { \\pmb I } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 368, + 298, + 380 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 299, + 369, + 325, + 378 + ], + "score": 0.89, + "content": "K > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 368, + 342, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 343, + 369, + 397, + 379 + ], + "score": 0.92, + "content": "\\| \\mathbb { E } [ \\pmb { a } _ { i } ] \\| _ { \\ell _ { 2 } } \\le \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 368, + 434, + 380 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 434, + 369, + 483, + 379 + ], + "score": 0.87, + "content": "\\pmb { \\Sigma } [ \\pmb { a } _ { i } ] \\succeq \\lambda \\pmb { I } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 368, + 506, + 380 + ], + "score": 1.0, + "content": ". Sup-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 125, + 390 + ], + "score": 1.0, + "content": "pose", + "type": "text" + }, + { + "bbox": [ + 126, + 378, + 190, + 389 + ], + "score": 0.91, + "content": "n \\ge \\bar { \\mathcal { O } } ( K ^ { 4 } d / \\lambda ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 377, + 414, + 390 + ], + "score": 1.0, + "content": ". Then, each of the following happens with probability at least", + "type": "text" + }, + { + "bbox": [ + 415, + 379, + 502, + 389 + ], + "score": 0.86, + "content": "1 - 2 \\exp ( - c K ^ { - 4 } \\lambda ^ { 2 } n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 377, + 506, + 390 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 398, + 236, + 414 + ], + "lines": [ + { + "bbox": [ + 139, + 398, + 236, + 414 + ], + "spans": [ + { + "bbox": [ + 139, + 398, + 236, + 414 + ], + "score": 0.76, + "content": "\\begin{array} { r } { \\theta + \\sqrt { 3 / 2 } K \\geq \\frac { 1 } { \\sqrt { n } } \\| A \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "1d6c6892247eec94579b048e95bc254ffa5b0901d3e00f09933ceef579b9d2da.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 139, + 398, + 236, + 414 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 133, + 421, + 425, + 436 + ], + "lines": [ + { + "bbox": [ + 131, + 420, + 423, + 438 + ], + "spans": [ + { + "bbox": [ + 131, + 420, + 215, + 438 + ], + "score": 1.0, + "content": "• Suppose all rows of", + "type": "text" + }, + { + "bbox": [ + 215, + 424, + 224, + 432 + ], + "score": 0.7, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 420, + 336, + 438 + ], + "score": 1.0, + "content": "have equal expectations. Then", + "type": "text" + }, + { + "bbox": [ + 337, + 421, + 423, + 436 + ], + "score": 0.91, + "content": "\\textstyle { \\frac { 1 } { \\sqrt { n } } } s _ { \\operatorname* { m i n } } ( A ) \\geq { \\sqrt { 2 \\lambda / 3 } }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 506, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 147, + 464 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 148, + 451, + 309, + 463 + ], + "score": 0.56, + "content": "\\pmb { E } = \\mathbb { E } [ \\pmb { A } ] , \\ \\bar { \\pmb { A } } = \\pmb { A } - \\mathbb { E } [ \\pmb { A } ] , \\ \\bar { \\pmb { a } } _ { i } = \\mathbf { z m } ( \\pmb { a } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 451, + 386, + 464 + ], + "score": 1.0, + "content": ". We will decompose", + "type": "text" + }, + { + "bbox": [ + 387, + 452, + 435, + 462 + ], + "score": 0.92, + "content": "\\pmb { A } = \\bar { \\pmb { A } } + \\pmb { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "hence we will first", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 503, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 314, + 476 + ], + "score": 1.0, + "content": "focus on bounding the upper and lower singular values of", + "type": "text" + }, + { + "bbox": [ + 315, + 462, + 323, + 471 + ], + "score": 0.84, + "content": "\\bar { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 461, + 447, + 476 + ], + "score": 1.0, + "content": "by studying the random processes", + "type": "text" + }, + { + "bbox": [ + 448, + 461, + 503, + 474 + ], + "score": 0.93, + "content": "X _ { v } = \\| \\bar { A } v \\| _ { \\ell _ { 2 } } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 472, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 123, + 487 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 474, + 199, + 485 + ], + "score": 0.91, + "content": "Y _ { v } = X _ { v } - \\mathbb { E } [ X _ { v } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 472, + 279, + 487 + ], + "score": 1.0, + "content": "over the unit sphere", + "type": "text" + }, + { + "bbox": [ + 279, + 473, + 300, + 484 + ], + "score": 0.9, + "content": "S ^ { d - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 472, + 506, + 487 + ], + "score": 1.0, + "content": ". First, we provide a deviation bound for the quantity", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 483, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 107, + 485, + 167, + 496 + ], + "score": 0.88, + "content": "{ \\operatorname* { s u p } } _ { v \\in { \\mathcal { S } } ^ { d - 1 } } | Y _ { v } |", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 483, + 492, + 497 + ], + "score": 1.0, + "content": ". To achieve this, we will utilize Talagrand’s mixed tail bound and show that increments of", + "type": "text" + }, + { + "bbox": [ + 493, + 485, + 504, + 494 + ], + "score": 0.87, + "content": "Y _ { v }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 495, + 460, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 252, + 508 + ], + "score": 1.0, + "content": "are subexpoential. 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We have that", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 513, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 111, + 513, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 111, + 513, + 505, + 542 + ], + "score": 0.89, + "content": "{ \\displaystyle { \\cal X } _ { u } - { \\cal X } _ { v } = \\| \\bar { \\cal A } u \\| _ { \\ell _ { 2 } } ^ { 2 } - \\| \\bar { \\cal A } v \\| _ { \\ell _ { 2 } } ^ { 2 } = \\| \\bar { \\cal A } ( x + y ) / 2 \\| _ { \\ell _ { 2 } } ^ { 2 } - \\| \\bar { \\cal A } ( x - y ) / 2 \\| _ { \\ell _ { 2 } } ^ { 2 } = x ^ { T } \\bar { \\cal A } ^ { T } \\bar { \\cal A } y = \\sum _ { i = 1 } ^ { n } ( \\bar { a } _ { i } ^ { T } x ) ( \\bar { a } _ { i } ^ { T } y ) . }", + "type": "interline_equation", + "image_path": "7ea56592793109b32bd96fd4d8a6dbb46b2cd41baf9db46d71e52ec2cc025a6d.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 111, + 513, + 505, + 522.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 111, + 522.6666666666666, + 505, + 532.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 111, + 532.3333333333333, + 505, + 541.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 549, + 472, + 561 + ], + "lines": [ + { + "bbox": [ + 104, + 547, + 474, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 135, + 563 + ], + "score": 1.0, + "content": "Letting", + "type": "text" + }, + { + "bbox": [ + 135, + 550, + 243, + 561 + ], + "score": 0.91, + "content": "\\hat { \\pmb x } = { \\pmb x } / \\| { \\pmb x } \\| _ { \\ell _ { 2 } } , \\hat { \\pmb y } = { \\pmb y } / \\| { \\pmb y } \\| _ { \\ell _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 547, + 405, + 563 + ], + "score": 1.0, + "content": ", observe that, multiplication of subgaussians", + "type": "text" + }, + { + "bbox": [ + 406, + 549, + 452, + 560 + ], + "score": 0.92, + "content": "\\pmb { x } ^ { T } \\bar { \\pmb { a } } _ { i } , \\pmb { y } ^ { T } \\bar { \\pmb { a } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 547, + 474, + 563 + ], + "score": 1.0, + "content": "obey", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 567, + 419, + 581 + ], + "lines": [ + { + "bbox": [ + 192, + 567, + 419, + 581 + ], + "spans": [ + { + "bbox": [ + 192, + 567, + 419, + 581 + ], + "score": 0.89, + "content": "\\| ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ( \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) \\| _ { \\psi _ { 1 } } \\leq \\mathcal { O } ( \\| \\boldsymbol { x } \\| _ { \\ell _ { 2 } } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } K ^ { 2 } ) \\leq \\mathcal { O } ( K ^ { 2 } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } ) .", + "type": "interline_equation", + "image_path": "901e3c5ba18531c25f17d53ac587a6b5dcf583757c69fee376a1a60fe94034ca.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 192, + 567, + 419, + 581 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "Centering this subexponential variable around zero introduces a factor of 2 when bounding subexponential norm", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 147, + 610 + ], + "score": 1.0, + "content": "and yields", + "type": "text" + }, + { + "bbox": [ + 148, + 596, + 365, + 609 + ], + "score": 0.71, + "content": "\\| ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) \\dot { \\langle } \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } \\rangle - \\mathbb { E } [ ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ( \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ] \\| _ { \\psi _ { 1 } } \\le \\mathcal { O } ( K ^ { 2 } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 597, + 461, + 610 + ], + "score": 1.0, + "content": ". 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which yields the desired result.", + "type": "text" + }, + { + "bbox": [ + 497, + 208, + 503, + 213 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 193, + 506, + 217 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 229, + 237, + 241 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 238, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 238, + 242 + ], + "score": 1.0, + "content": "E.5 PROOF OF THEOREM 5.1", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "Proof. The proof is a corollary of Theorem E.1. We need to substitute the proper values in Assumption 2.√ √", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 260, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 252, + 273 + ], + "score": 1.0, + "content": "Applying Lemma B.3, we can substitute", + "type": "text" + }, + { + "bbox": [ + 253, + 260, + 292, + 272 + ], + "score": 0.93, + "content": "\\gamma _ { + } = B _ { T _ { 0 } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 260, + 308, + 273 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 260, + 394, + 272 + ], + "score": 0.93, + "content": "\\theta = \\sqrt { 6 n } - \\sqrt { 2 } \\geq \\sqrt { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 260, + 506, + 273 + ], + "score": 1.0, + "content": ". Next, we need to find a lower", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 269, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 228, + 284 + ], + "score": 1.0, + "content": "bound. Applying Lemma 3.2 for", + "type": "text" + }, + { + "bbox": [ + 228, + 272, + 252, + 281 + ], + "score": 0.89, + "content": "n > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 269, + 327, + 284 + ], + "score": 1.0, + "content": "and Lemma B.6 for", + "type": "text" + }, + { + "bbox": [ + 327, + 272, + 351, + 281 + ], + "score": 0.87, + "content": "n = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 269, + 420, + 284 + ], + "score": 1.0, + "content": ", we can substitute", + "type": "text" + }, + { + "bbox": [ + 420, + 271, + 465, + 282 + ], + "score": 0.92, + "content": "\\gamma _ { - } = \\gamma _ { + } / \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 269, + 497, + 284 + ], + "score": 1.0, + "content": "with the", + "type": "text" + }, + { + "bbox": [ + 498, + 273, + 504, + 282 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 281, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 439, + 292 + ], + "score": 1.0, + "content": "definition of (5.2). With these, the result follows as an immediate corollary of Theorem E.1.", + "type": "text" + }, + { + "bbox": [ + 496, + 281, + 505, + 292 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 249, + 506, + 292 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 308, + 336, + 321 + ], + "lines": [ + { + "bbox": [ + 104, + 306, + 336, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 336, + 322 + ], + "score": 1.0, + "content": "F SUPPLEMENTARY STATISTICAL RESULTS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 105, + 332, + 504, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 345 + ], + "score": 1.0, + "content": "The following theorem bounds the empirical covariance of matrices with independent subgaussian rows. Given", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 342, + 376, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 167, + 355 + ], + "score": 1.0, + "content": "a random vector", + "type": "text" + }, + { + "bbox": [ + 167, + 345, + 174, + 352 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 342, + 300, + 355 + ], + "score": 1.0, + "content": ", define the de-biasing operation as", + "type": "text" + }, + { + "bbox": [ + 300, + 343, + 373, + 354 + ], + "score": 0.91, + "content": "\\mathbf { z m } ( \\pmb { x } ) = \\pmb { x } - \\mathbb { E } [ \\pmb { x } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 342, + 376, + 355 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 331, + 505, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 354, + 507, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 354, + 183, + 372 + ], + "score": 1.0, + "content": "Theorem F.1. Let", + "type": "text" + }, + { + "bbox": [ + 183, + 357, + 235, + 367 + ], + "score": 0.89, + "content": "\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 354, + 432, + 372 + ], + "score": 1.0, + "content": "be a matrix with independent subgaussian rows", + "type": "text" + }, + { + "bbox": [ + 432, + 358, + 464, + 369 + ], + "score": 0.91, + "content": "\\{ { \\pmb a } _ { i } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 354, + 507, + 372 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 368, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 107, + 368, + 189, + 379 + ], + "score": 0.89, + "content": "\\| \\mathbf { z } \\mathbf { m } ( \\mathbf { { a } } _ { i } ) \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 368, + 207, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 368, + 263, + 379 + ], + "score": 0.91, + "content": "\\Sigma [ { \\pmb a } _ { i } ] \\preceq K ^ { 2 } { \\pmb I } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 368, + 298, + 380 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 299, + 369, + 325, + 378 + ], + "score": 0.89, + "content": "K > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 368, + 342, + 380 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 343, + 369, + 397, + 379 + ], + "score": 0.92, + "content": "\\| \\mathbb { E } [ \\pmb { a } _ { i } ] \\| _ { \\ell _ { 2 } } \\le \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 368, + 434, + 380 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 434, + 369, + 483, + 379 + ], + "score": 0.87, + "content": "\\pmb { \\Sigma } [ \\pmb { a } _ { i } ] \\succeq \\lambda \\pmb { I } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 368, + 506, + 380 + ], + "score": 1.0, + "content": ". Sup-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 125, + 390 + ], + "score": 1.0, + "content": "pose", + "type": "text" + }, + { + "bbox": [ + 126, + 378, + 190, + 389 + ], + "score": 0.91, + "content": "n \\ge \\bar { \\mathcal { O } } ( K ^ { 4 } d / \\lambda ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 377, + 414, + 390 + ], + "score": 1.0, + "content": ". Then, each of the following happens with probability at least", + "type": "text" + }, + { + "bbox": [ + 415, + 379, + 502, + 389 + ], + "score": 0.86, + "content": "1 - 2 \\exp ( - c K ^ { - 4 } \\lambda ^ { 2 } n )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 377, + 506, + 390 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 354, + 507, + 390 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 398, + 236, + 414 + ], + "lines": [ + { + "bbox": [ + 139, + 398, + 236, + 414 + ], + "spans": [ + { + "bbox": [ + 139, + 398, + 236, + 414 + ], + "score": 0.76, + "content": "\\begin{array} { r } { \\theta + \\sqrt { 3 / 2 } K \\geq \\frac { 1 } { \\sqrt { n } } \\| A \\| . } \\end{array}", + "type": "interline_equation", + "image_path": "1d6c6892247eec94579b048e95bc254ffa5b0901d3e00f09933ceef579b9d2da.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 139, + 398, + 236, + 414 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 133, + 421, + 425, + 436 + ], + "lines": [ + { + "bbox": [ + 131, + 420, + 423, + 438 + ], + "spans": [ + { + "bbox": [ + 131, + 420, + 215, + 438 + ], + "score": 1.0, + "content": "• Suppose all rows of", + "type": "text" + }, + { + "bbox": [ + 215, + 424, + 224, + 432 + ], + "score": 0.7, + "content": "\\pmb { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 420, + 336, + 438 + ], + "score": 1.0, + "content": "have equal expectations. Then", + "type": "text" + }, + { + "bbox": [ + 337, + 421, + 423, + 436 + ], + "score": 0.91, + "content": "\\textstyle { \\frac { 1 } { \\sqrt { n } } } s _ { \\operatorname* { m i n } } ( A ) \\geq { \\sqrt { 2 \\lambda / 3 } }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 131, + 420, + 423, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 506, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 147, + 464 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 148, + 451, + 309, + 463 + ], + "score": 0.56, + "content": "\\pmb { E } = \\mathbb { E } [ \\pmb { A } ] , \\ \\bar { \\pmb { A } } = \\pmb { A } - \\mathbb { E } [ \\pmb { A } ] , \\ \\bar { \\pmb { a } } _ { i } = \\mathbf { z m } ( \\pmb { a } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 451, + 386, + 464 + ], + "score": 1.0, + "content": ". We will decompose", + "type": "text" + }, + { + "bbox": [ + 387, + 452, + 435, + 462 + ], + "score": 0.92, + "content": "\\pmb { A } = \\bar { \\pmb { A } } + \\pmb { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "hence we will first", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 503, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 314, + 476 + ], + "score": 1.0, + "content": "focus on bounding the upper and lower singular values of", + "type": "text" + }, + { + "bbox": [ + 315, + 462, + 323, + 471 + ], + "score": 0.84, + "content": "\\bar { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 461, + 447, + 476 + ], + "score": 1.0, + "content": "by studying the random processes", + "type": "text" + }, + { + "bbox": [ + 448, + 461, + 503, + 474 + ], + "score": 0.93, + "content": "X _ { v } = \\| \\bar { A } v \\| _ { \\ell _ { 2 } } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 472, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 123, + 487 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 474, + 199, + 485 + ], + "score": 0.91, + "content": "Y _ { v } = X _ { v } - \\mathbb { E } [ X _ { v } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 472, + 279, + 487 + ], + "score": 1.0, + "content": "over the unit sphere", + "type": "text" + }, + { + "bbox": [ + 279, + 473, + 300, + 484 + ], + "score": 0.9, + "content": "S ^ { d - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 472, + 506, + 487 + ], + "score": 1.0, + "content": ". First, we provide a deviation bound for the quantity", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 483, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 107, + 485, + 167, + 496 + ], + "score": 0.88, + "content": "{ \\operatorname* { s u p } } _ { v \\in { \\mathcal { S } } ^ { d - 1 } } | Y _ { v } |", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 483, + 492, + 497 + ], + "score": 1.0, + "content": ". To achieve this, we will utilize Talagrand’s mixed tail bound and show that increments of", + "type": "text" + }, + { + "bbox": [ + 493, + 485, + 504, + 494 + ], + "score": 0.87, + "content": "Y _ { v }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 495, + 460, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 252, + 508 + ], + "score": 1.0, + "content": "are subexpoential. Pick two unit vectors", + "type": "text" + }, + { + "bbox": [ + 253, + 495, + 291, + 506 + ], + "score": 0.92, + "content": "\\pmb { v } , \\pmb { u } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 495, + 318, + 508 + ], + "score": 1.0, + "content": ". Write", + "type": "text" + }, + { + "bbox": [ + 318, + 496, + 407, + 506 + ], + "score": 0.91, + "content": "\\pmb { x } = \\pmb { u } + \\pmb { v } , \\pmb { y } = \\pmb { u } - \\pmb { v } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 495, + 460, + 508 + ], + "score": 1.0, + "content": ". We have that", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 451, + 506, + 508 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 513, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 111, + 513, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 111, + 513, + 505, + 542 + ], + "score": 0.89, + "content": "{ \\displaystyle { \\cal X } _ { u } - { \\cal X } _ { v } = \\| \\bar { \\cal A } u \\| _ { \\ell _ { 2 } } ^ { 2 } - \\| \\bar { \\cal A } v \\| _ { \\ell _ { 2 } } ^ { 2 } = \\| \\bar { \\cal A } ( x + y ) / 2 \\| _ { \\ell _ { 2 } } ^ { 2 } - \\| \\bar { \\cal A } ( x - y ) / 2 \\| _ { \\ell _ { 2 } } ^ { 2 } = x ^ { T } \\bar { \\cal A } ^ { T } \\bar { \\cal A } y = \\sum _ { i = 1 } ^ { n } ( \\bar { a } _ { i } ^ { T } x ) ( \\bar { a } _ { i } ^ { T } y ) . }", + "type": "interline_equation", + "image_path": "7ea56592793109b32bd96fd4d8a6dbb46b2cd41baf9db46d71e52ec2cc025a6d.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 111, + 513, + 505, + 522.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 111, + 522.6666666666666, + 505, + 532.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 111, + 532.3333333333333, + 505, + 541.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 549, + 472, + 561 + ], + "lines": [ + { + "bbox": [ + 104, + 547, + 474, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 135, + 563 + ], + "score": 1.0, + "content": "Letting", + "type": "text" + }, + { + "bbox": [ + 135, + 550, + 243, + 561 + ], + "score": 0.91, + "content": "\\hat { \\pmb x } = { \\pmb x } / \\| { \\pmb x } \\| _ { \\ell _ { 2 } } , \\hat { \\pmb y } = { \\pmb y } / \\| { \\pmb y } \\| _ { \\ell _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 547, + 405, + 563 + ], + "score": 1.0, + "content": ", observe that, multiplication of subgaussians", + "type": "text" + }, + { + "bbox": [ + 406, + 549, + 452, + 560 + ], + "score": 0.92, + "content": "\\pmb { x } ^ { T } \\bar { \\pmb { a } } _ { i } , \\pmb { y } ^ { T } \\bar { \\pmb { a } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 547, + 474, + 563 + ], + "score": 1.0, + "content": "obey", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 547, + 474, + 563 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 567, + 419, + 581 + ], + "lines": [ + { + "bbox": [ + 192, + 567, + 419, + 581 + ], + "spans": [ + { + "bbox": [ + 192, + 567, + 419, + 581 + ], + "score": 0.89, + "content": "\\| ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ( \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) \\| _ { \\psi _ { 1 } } \\leq \\mathcal { O } ( \\| \\boldsymbol { x } \\| _ { \\ell _ { 2 } } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } K ^ { 2 } ) \\leq \\mathcal { O } ( K ^ { 2 } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } ) .", + "type": "interline_equation", + "image_path": "901e3c5ba18531c25f17d53ac587a6b5dcf583757c69fee376a1a60fe94034ca.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 192, + 567, + 419, + 581 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 587, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 599 + ], + "score": 1.0, + "content": "Centering this subexponential variable around zero introduces a factor of 2 when bounding subexponential norm", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 147, + 610 + ], + "score": 1.0, + "content": "and yields", + "type": "text" + }, + { + "bbox": [ + 148, + 596, + 365, + 609 + ], + "score": 0.71, + "content": "\\| ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) \\dot { \\langle } \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } \\rangle - \\mathbb { E } [ ( \\boldsymbol { x } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ( \\boldsymbol { y } ^ { T } \\bar { \\boldsymbol { a } } _ { i } ) ] \\| _ { \\psi _ { 1 } } \\le \\mathcal { O } ( K ^ { 2 } \\| \\boldsymbol { y } \\| _ { \\ell _ { 2 } } ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 597, + 461, + 610 + ], + "score": 1.0, + "content": ". 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Hence, again using", + "type": "text" + }, + { + "bbox": [ + 365, + 246, + 426, + 258 + ], + "score": 0.92, + "content": "n \\geq C K ^ { 4 } \\lambda ^ { - 2 } d", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 245, + 507, + 261 + ], + "score": 1.0, + "content": "for sufficiently large", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 256, + 507, + 271 + ], + "spans": [ + { + "bbox": [ + 107, + 258, + 133, + 267 + ], + "score": 0.88, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 256, + 234, + 271 + ], + "score": 1.0, + "content": ", applying Lemma F.3 with", + "type": "text" + }, + { + "bbox": [ + 234, + 258, + 316, + 267 + ], + "score": 0.86, + "content": "m = c _ { 0 } K ^ { - 4 } \\lambda ^ { 2 } n > d", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 256, + 464, + 271 + ], + "score": 1.0, + "content": "by picking a sufficiently small constant", + "type": "text" + }, + { + "bbox": [ + 464, + 258, + 502, + 269 + ], + "score": 0.93, + "content": "c _ { 0 } > 1 / \\bar { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 256, + 507, + 271 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 266, + 300, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 194, + 280 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 194, + 267, + 300, + 279 + ], + "score": 0.83, + "content": "1 - 2 \\exp ( - 1 0 0 c _ { 0 } K ^ { - 4 } \\lambda ^ { 2 } n )", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 284, + 413, + 312 + ], + "lines": [ + { + "bbox": [ + 199, + 284, + 413, + 312 + ], + "spans": [ + { + "bbox": [ + 199, + 284, + 413, + 312 + ], + "score": 0.9, + "content": "\\frac { 1 } { \\sqrt { n } } \\operatorname* { s u p } _ { \\| v \\| _ { 2 } = 1 } | Z _ { v } | = \\frac { 1 } { n } \\| \\bar { \\cal A } ^ { T } { \\bf 1 } _ { n } \\| _ { \\ell _ { 2 } } \\leq \\frac { 1 } { 1 2 } { \\cal K } { \\cal K } ^ { - 2 } \\lambda \\leq \\frac { \\sqrt { \\lambda } } { 1 2 } .", + "type": "interline_equation", + "image_path": "ef8ed4bc3249b9fef4bb30908aff4d969b4de6464af0b98c10a45ec75772d77c.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 199, + 284, + 413, + 312 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 318, + 504, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 121, + 332 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 318, + 195, + 331 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\cal P } = I _ { n } - \\frac { 1 } { n } { \\bf 1 } _ { n } { \\bf 1 } _ { n } ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "be the projection onto the orthogonal complement of the all ones vector. Note that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 328, + 498, + 342 + ], + "spans": [ + { + "bbox": [ + 107, + 330, + 146, + 339 + ], + "score": 0.88, + "content": "P E v = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 328, + 199, + 342 + ], + "score": 1.0, + "content": "as the rows of", + "type": "text" + }, + { + "bbox": [ + 200, + 330, + 208, + 339 + ], + "score": 0.85, + "content": "\\pmb { { \\cal E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 328, + 486, + 342 + ], + "score": 1.0, + "content": "are equal. With this observation, with desired probability, for any unit length", + "type": "text" + }, + { + "bbox": [ + 487, + 331, + 493, + 339 + ], + "score": 0.76, + "content": "_ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 328, + 498, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 343, + 419, + 381 + ], + "lines": [ + { + "bbox": [ + 191, + 343, + 419, + 381 + ], + "spans": [ + { + "bbox": [ + 191, + 343, + 419, + 381 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\| \\pmb { A } \\pmb { v } \\| _ { \\ell _ { 2 } } \\ge \\| \\pmb { P } \\pmb { A } \\pmb { v } \\| _ { \\ell _ { 2 } } = \\| \\pmb { P } \\pmb { \\bar { A } } \\pmb { v } \\| _ { \\ell _ { 2 } } \\ge \\| \\pmb { \\bar { A } } \\pmb { v } \\| _ { \\ell _ { 2 } } - | Z _ { \\pmb { v } } | } \\\\ & { \\qquad \\ge s _ { \\operatorname* { m i n } } ( \\pmb { \\bar { A } } ) - \\ \\underset { \\pmb { v } \\in { S } ^ { d - 1 } } { \\operatorname* { s u p } } | Z _ { \\pmb { v } } | \\ge ( \\sqrt { 7 / 8 } - 1 / 1 2 ) \\sqrt { \\lambda n } , } \\end{array}", + "type": "interline_equation", + "image_path": "362cedfdc475dad89402176709671c2e32d457104a23fc049106e0daf5f007d5.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 343, + 419, + 362.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 191, + 362.0, + 419, + 381.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 254, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 254, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 159, + 401 + ], + "score": 1.0, + "content": "which implies", + "type": "text" + }, + { + "bbox": [ + 159, + 387, + 251, + 400 + ], + "score": 0.91, + "content": "s _ { \\mathrm { m i n } } ( A ) / \\sqrt { n } \\geq \\sqrt { 2 \\lambda / 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 387, + 254, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 503, + 435 + ], + "lines": [ + { + "bbox": [ + 103, + 408, + 504, + 431 + ], + "spans": [ + { + "bbox": [ + 103, + 408, + 396, + 431 + ], + "score": 1.0, + "content": "The corollary below is obtained by slightly modifying the proof above by using", + "type": "text" + }, + { + "bbox": [ + 396, + 412, + 504, + 426 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\| \\bar { \\mathbfcal A } ^ { T } \\bar { \\mathbfcal A } - \\mathbb { E } [ \\bar { \\mathbfcal A } ^ { T } \\bar { \\mathbfcal A } ] \\| \\leq \\frac { K ^ { 2 } } { 8 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 425, + 320, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 320, + 436 + ], + "score": 1.0, + "content": "in line (F.2) and only focusing on the spectral norm bound.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 185, + 452 + ], + "score": 1.0, + "content": "Corollary F.2. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 437, + 236, + 448 + ], + "score": 0.89, + "content": "\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 435, + 360, + 452 + ], + "score": 1.0, + "content": "be a matrix with independent", + "type": "text" + }, + { + "bbox": [ + 361, + 439, + 392, + 449 + ], + "score": 0.91, + "content": "\\{ { \\pmb a } _ { i } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 435, + 506, + 452 + ], + "score": 1.0, + "content": "subgaussian rows satisfying", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 447, + 502, + 461 + ], + "spans": [ + { + "bbox": [ + 107, + 448, + 192, + 460 + ], + "score": 0.89, + "content": "\\| \\mathbf { z } \\mathbf { m } ( \\mathbf { { a } } _ { i } ) \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 447, + 210, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 211, + 448, + 270, + 460 + ], + "score": 0.92, + "content": "\\Sigma [ { \\pmb a } _ { i } ] \\preceq K ^ { 2 } { \\pmb I } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 447, + 307, + 461 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 307, + 449, + 336, + 458 + ], + "score": 0.91, + "content": "K > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 447, + 354, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 449, + 411, + 460 + ], + "score": 0.89, + "content": "\\| \\mathbb { E } [ \\mathbf { \\underline { { a } } } _ { i } ] \\rVert _ { \\ell _ { 2 } } \\leq \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 447, + 451, + 461 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 451, + 448, + 502, + 460 + ], + "score": 0.91, + "content": "\\pmb { \\Sigma } [ \\pmb { a } _ { i } ] \\succeq \\lambda \\pmb { I } _ { d }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 457, + 388, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 138, + 470 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 138, + 459, + 189, + 469 + ], + "score": 0.89, + "content": "n \\geq \\mathcal { O } ( K ^ { 2 } d )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 457, + 304, + 470 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 304, + 459, + 382, + 469 + ], + "score": 0.86, + "content": "1 - 4 \\exp ( - c K ^ { - 2 } n )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 457, + 388, + 470 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 474, + 353, + 498 + ], + "lines": [ + { + "bbox": [ + 257, + 474, + 353, + 498 + ], + "spans": [ + { + "bbox": [ + 257, + 474, + 353, + 498 + ], + "score": 0.93, + "content": "\\theta + { \\sqrt { 3 / 2 } } K \\geq { \\frac { 1 } { \\sqrt { n } } } \\| A \\| .", + "type": "interline_equation", + "image_path": "f1f28689a063fcd9425a36acbd0539e8f550978cfda9084a87afeb7030e5a357.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 257, + 474, + 353, + 498 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 406, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 406, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 406, + 522 + ], + "score": 1.0, + "content": "The following lemma is fairly standard and is proved for the sake of completeness.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 502, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 503, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 276, + 536 + ], + "score": 1.0, + "content": "Lemma F.3 (Subgaussian vector length). Let", + "type": "text" + }, + { + "bbox": [ + 276, + 523, + 306, + 532 + ], + "score": 0.91, + "content": "\\pmb { a } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 520, + 458, + 536 + ], + "score": 1.0, + "content": "be a zero-mean subgaussian vector with", + "type": "text" + }, + { + "bbox": [ + 459, + 523, + 503, + 534 + ], + "score": 0.92, + "content": "\\| \\pmb { a } \\| _ { \\psi _ { 2 } } \\le L", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 532, + 291, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 156, + 544 + ], + "score": 1.0, + "content": "Then, for any", + "type": "text" + }, + { + "bbox": [ + 156, + 533, + 184, + 543 + ], + "score": 0.91, + "content": "m \\geq n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 532, + 229, + 544 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 229, + 533, + 254, + 542 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 532, + 291, + 544 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 548, + 388, + 561 + ], + "lines": [ + { + "bbox": [ + 222, + 548, + 388, + 561 + ], + "spans": [ + { + "bbox": [ + 222, + 548, + 388, + 561 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } ( \\| \\pmb { a } \\| _ { \\ell _ { 2 } } \\le C L \\sqrt { m } ) \\ge 1 - 2 \\exp ( - 1 0 0 m ) . } \\end{array}", + "type": "interline_equation", + "image_path": "87a698569083f817b8ddfe1cebd33738e9ba8f8d4f8cd25e08ad33cc93fccb48.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 222, + 548, + 388, + 561 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 506, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 186, + 585 + ], + "score": 1.0, + "content": "Proof. We can pick a", + "type": "text" + }, + { + "bbox": [ + 186, + 573, + 201, + 583 + ], + "score": 0.8, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 571, + 224, + 585 + ], + "score": 1.0, + "content": "cover", + "type": "text" + }, + { + "bbox": [ + 225, + 573, + 231, + 582 + ], + "score": 0.82, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 571, + 271, + 585 + ], + "score": 1.0, + "content": "of the unit", + "type": "text" + }, + { + "bbox": [ + 271, + 573, + 280, + 582 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 571, + 342, + 585 + ], + "score": 1.0, + "content": "-sphere with size", + "type": "text" + }, + { + "bbox": [ + 342, + 572, + 390, + 583 + ], + "score": 0.92, + "content": "\\log | { \\mathcal { C } } | \\leq 2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 571, + 423, + 585 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 424, + 573, + 447, + 582 + ], + "score": 0.89, + "content": "{ \\pmb v } \\in \\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 571, + 505, + 585 + ], + "score": 1.0, + "content": ", subgaussianity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 509, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 138, + 603 + ], + "score": 1.0, + "content": "implies,", + "type": "text" + }, + { + "bbox": [ + 138, + 583, + 251, + 597 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { P } ( | v ^ { T } \\pmb { a } | \\geq t ) \\leq 2 \\exp ( - \\frac { c t ^ { 2 } } { 2 L ^ { 2 } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 579, + 283, + 603 + ], + "score": 1.0, + "content": ". Setting", + "type": "text" + }, + { + "bbox": [ + 283, + 585, + 329, + 596 + ], + "score": 0.93, + "content": "t = C L { \\sqrt { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 579, + 438, + 603 + ], + "score": 1.0, + "content": "for sufficiently large constant", + "type": "text" + }, + { + "bbox": [ + 438, + 585, + 463, + 595 + ], + "score": 0.89, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 579, + 509, + 603 + ], + "score": 1.0, + "content": ", and union", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 595, + 228, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 172, + 606 + ], + "score": 1.0, + "content": "bounding over all", + "type": "text" + }, + { + "bbox": [ + 172, + 596, + 195, + 605 + ], + "score": 0.86, + "content": "\\pmb { v } \\in \\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 595, + 228, + 606 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 610, + 453, + 639 + ], + "lines": [ + { + "bbox": [ + 157, + 610, + 453, + 639 + ], + "spans": [ + { + "bbox": [ + 157, + 610, + 453, + 639 + ], + "score": 0.93, + "content": "\\mathbb { P } ( \\bigcap _ { v \\in \\mathcal { C } } \\| v \\| _ { \\ell _ { 2 } } \\leq C L \\sqrt { m } ) \\geq 1 - 2 \\exp ( 2 n - \\frac { c C ^ { 2 } L ^ { 2 } m } { 2 L ^ { 2 } } ) \\leq 1 - 2 \\exp ( - 1 0 0 m ) .", + "type": "interline_equation", + "image_path": "905ac3d7de230ea2bfd0b2e10b7518950215c6b624da0400760313bcceb7d4cf.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 157, + 610, + 453, + 639 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 644, + 446, + 659 + ], + "lines": [ + { + "bbox": [ + 101, + 636, + 450, + 665 + ], + "spans": [ + { + "bbox": [ + 101, + 636, + 450, + 665 + ], + "score": 1.0, + "content": "To conclude, let v(a) ∈ C be a’s neighbor satisfying kv − akak`2 k`2 ≤ 1/2. 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Hence, again using", + "type": "text" + }, + { + "bbox": [ + 365, + 246, + 426, + 258 + ], + "score": 0.92, + "content": "n \\geq C K ^ { 4 } \\lambda ^ { - 2 } d", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 245, + 507, + 261 + ], + "score": 1.0, + "content": "for sufficiently large", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 256, + 507, + 271 + ], + "spans": [ + { + "bbox": [ + 107, + 258, + 133, + 267 + ], + "score": 0.88, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 256, + 234, + 271 + ], + "score": 1.0, + "content": ", applying Lemma F.3 with", + "type": "text" + }, + { + "bbox": [ + 234, + 258, + 316, + 267 + ], + "score": 0.86, + "content": "m = c _ { 0 } K ^ { - 4 } \\lambda ^ { 2 } n > d", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 256, + 464, + 271 + ], + "score": 1.0, + "content": "by picking a sufficiently small constant", + "type": "text" + }, + { + "bbox": [ + 464, + 258, + 502, + 269 + ], + "score": 0.93, + "content": "c _ { 0 } > 1 / \\bar { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 256, + 507, + 271 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 266, + 300, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 194, + 280 + ], + "score": 1.0, + "content": "with probability at least", + "type": "text" + }, + { + "bbox": [ + 194, + 267, + 300, + 279 + ], + "score": 0.83, + "content": "1 - 2 \\exp ( - 1 0 0 c _ { 0 } K ^ { - 4 } \\lambda ^ { 2 } n )", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 102, + 225, + 507, + 280 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 284, + 413, + 312 + ], + "lines": [ + { + "bbox": [ + 199, + 284, + 413, + 312 + ], + "spans": [ + { + "bbox": [ + 199, + 284, + 413, + 312 + ], + "score": 0.9, + "content": "\\frac { 1 } { \\sqrt { n } } \\operatorname* { s u p } _ { \\| v \\| _ { 2 } = 1 } | Z _ { v } | = \\frac { 1 } { n } \\| \\bar { \\cal A } ^ { T } { \\bf 1 } _ { n } \\| _ { \\ell _ { 2 } } \\leq \\frac { 1 } { 1 2 } { \\cal K } { \\cal K } ^ { - 2 } \\lambda \\leq \\frac { \\sqrt { \\lambda } } { 1 2 } .", + "type": "interline_equation", + "image_path": "ef8ed4bc3249b9fef4bb30908aff4d969b4de6464af0b98c10a45ec75772d77c.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 199, + 284, + 413, + 312 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 318, + 504, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 121, + 332 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 318, + 195, + 331 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\cal P } = I _ { n } - \\frac { 1 } { n } { \\bf 1 } _ { n } { \\bf 1 } _ { n } ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "be the projection onto the orthogonal complement of the all ones vector. Note that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 328, + 498, + 342 + ], + "spans": [ + { + "bbox": [ + 107, + 330, + 146, + 339 + ], + "score": 0.88, + "content": "P E v = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 328, + 199, + 342 + ], + "score": 1.0, + "content": "as the rows of", + "type": "text" + }, + { + "bbox": [ + 200, + 330, + 208, + 339 + ], + "score": 0.85, + "content": "\\pmb { { \\cal E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 328, + 486, + 342 + ], + "score": 1.0, + "content": "are equal. With this observation, with desired probability, for any unit length", + "type": "text" + }, + { + "bbox": [ + 487, + 331, + 493, + 339 + ], + "score": 0.76, + "content": "_ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 328, + 498, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 318, + 506, + 342 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 343, + 419, + 381 + ], + "lines": [ + { + "bbox": [ + 191, + 343, + 419, + 381 + ], + "spans": [ + { + "bbox": [ + 191, + 343, + 419, + 381 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\| \\pmb { A } \\pmb { v } \\| _ { \\ell _ { 2 } } \\ge \\| \\pmb { P } \\pmb { A } \\pmb { v } \\| _ { \\ell _ { 2 } } = \\| \\pmb { P } \\pmb { \\bar { A } } \\pmb { v } \\| _ { \\ell _ { 2 } } \\ge \\| \\pmb { \\bar { A } } \\pmb { v } \\| _ { \\ell _ { 2 } } - | Z _ { \\pmb { v } } | } \\\\ & { \\qquad \\ge s _ { \\operatorname* { m i n } } ( \\pmb { \\bar { A } } ) - \\ \\underset { \\pmb { v } \\in { S } ^ { d - 1 } } { \\operatorname* { s u p } } | Z _ { \\pmb { v } } | \\ge ( \\sqrt { 7 / 8 } - 1 / 1 2 ) \\sqrt { \\lambda n } , } \\end{array}", + "type": "interline_equation", + "image_path": "362cedfdc475dad89402176709671c2e32d457104a23fc049106e0daf5f007d5.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 343, + 419, + 362.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 191, + 362.0, + 419, + 381.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 254, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 254, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 159, + 401 + ], + "score": 1.0, + "content": "which implies", + "type": "text" + }, + { + "bbox": [ + 159, + 387, + 251, + 400 + ], + "score": 0.91, + "content": "s _ { \\mathrm { m i n } } ( A ) / \\sqrt { n } \\geq \\sqrt { 2 \\lambda / 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 387, + 254, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 387, + 254, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 503, + 435 + ], + "lines": [ + { + "bbox": [ + 103, + 408, + 504, + 431 + ], + "spans": [ + { + "bbox": [ + 103, + 408, + 396, + 431 + ], + "score": 1.0, + "content": "The corollary below is obtained by slightly modifying the proof above by using", + "type": "text" + }, + { + "bbox": [ + 396, + 412, + 504, + 426 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { n } \\| \\bar { \\mathbfcal A } ^ { T } \\bar { \\mathbfcal A } - \\mathbb { E } [ \\bar { \\mathbfcal A } ^ { T } \\bar { \\mathbfcal A } ] \\| \\leq \\frac { K ^ { 2 } } { 8 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 425, + 320, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 320, + 436 + ], + "score": 1.0, + "content": "in line (F.2) and only focusing on the spectral norm bound.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 103, + 408, + 504, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 185, + 452 + ], + "score": 1.0, + "content": "Corollary F.2. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 437, + 236, + 448 + ], + "score": 0.89, + "content": "\\textbf { \\textit { A } } \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 435, + 360, + 452 + ], + "score": 1.0, + "content": "be a matrix with independent", + "type": "text" + }, + { + "bbox": [ + 361, + 439, + 392, + 449 + ], + "score": 0.91, + "content": "\\{ { \\pmb a } _ { i } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 435, + 506, + 452 + ], + "score": 1.0, + "content": "subgaussian rows satisfying", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 447, + 502, + 461 + ], + "spans": [ + { + "bbox": [ + 107, + 448, + 192, + 460 + ], + "score": 0.89, + "content": "\\| \\mathbf { z } \\mathbf { m } ( \\mathbf { { a } } _ { i } ) \\| _ { \\psi _ { 2 } } \\leq \\mathcal { O } ( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 447, + 210, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 211, + 448, + 270, + 460 + ], + "score": 0.92, + "content": "\\Sigma [ { \\pmb a } _ { i } ] \\preceq K ^ { 2 } { \\pmb I } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 447, + 307, + 461 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 307, + 449, + 336, + 458 + ], + "score": 0.91, + "content": "K > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 447, + 354, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 355, + 449, + 411, + 460 + ], + "score": 0.89, + "content": "\\| \\mathbb { E } [ \\mathbf { \\underline { { a } } } _ { i } ] \\rVert _ { \\ell _ { 2 } } \\leq \\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 447, + 451, + 461 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 451, + 448, + 502, + 460 + ], + "score": 0.91, + "content": "\\pmb { \\Sigma } [ \\pmb { a } _ { i } ] \\succeq \\lambda \\pmb { I } _ { d }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 457, + 388, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 138, + 470 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 138, + 459, + 189, + 469 + ], + "score": 0.89, + "content": "n \\geq \\mathcal { O } ( K ^ { 2 } d )", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 457, + 304, + 470 + ], + "score": 1.0, + "content": ". Then, with probability at least", + "type": "text" + }, + { + "bbox": [ + 304, + 459, + 382, + 469 + ], + "score": 0.86, + "content": "1 - 4 \\exp ( - c K ^ { - 2 } n )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 457, + 388, + 470 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 435, + 506, + 470 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 474, + 353, + 498 + ], + "lines": [ + { + "bbox": [ + 257, + 474, + 353, + 498 + ], + "spans": [ + { + "bbox": [ + 257, + 474, + 353, + 498 + ], + "score": 0.93, + "content": "\\theta + { \\sqrt { 3 / 2 } } K \\geq { \\frac { 1 } { \\sqrt { n } } } \\| A \\| .", + "type": "interline_equation", + "image_path": "f1f28689a063fcd9425a36acbd0539e8f550978cfda9084a87afeb7030e5a357.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 257, + 474, + 353, + 498 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 406, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 406, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 406, + 522 + ], + "score": 1.0, + "content": "The following lemma is fairly standard and is proved for the sake of completeness.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 507, + 406, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 502, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 503, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 276, + 536 + ], + "score": 1.0, + "content": "Lemma F.3 (Subgaussian vector length). Let", + "type": "text" + }, + { + "bbox": [ + 276, + 523, + 306, + 532 + ], + "score": 0.91, + "content": "\\pmb { a } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 520, + 458, + 536 + ], + "score": 1.0, + "content": "be a zero-mean subgaussian vector with", + "type": "text" + }, + { + "bbox": [ + 459, + 523, + 503, + 534 + ], + "score": 0.92, + "content": "\\| \\pmb { a } \\| _ { \\psi _ { 2 } } \\le L", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 532, + 291, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 156, + 544 + ], + "score": 1.0, + "content": "Then, for any", + "type": "text" + }, + { + "bbox": [ + 156, + 533, + 184, + 543 + ], + "score": 0.91, + "content": "m \\geq n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 532, + 229, + 544 + ], + "score": 1.0, + "content": "there exists", + "type": "text" + }, + { + "bbox": [ + 229, + 533, + 254, + 542 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 532, + 291, + 544 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 520, + 503, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 548, + 388, + 561 + ], + "lines": [ + { + "bbox": [ + 222, + 548, + 388, + 561 + ], + "spans": [ + { + "bbox": [ + 222, + 548, + 388, + 561 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathbb { P } ( \\| \\pmb { a } \\| _ { \\ell _ { 2 } } \\le C L \\sqrt { m } ) \\ge 1 - 2 \\exp ( - 1 0 0 m ) . } \\end{array}", + "type": "interline_equation", + "image_path": "87a698569083f817b8ddfe1cebd33738e9ba8f8d4f8cd25e08ad33cc93fccb48.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 222, + 548, + 388, + 561 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 506, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 186, + 585 + ], + "score": 1.0, + "content": "Proof. We can pick a", + "type": "text" + }, + { + "bbox": [ + 186, + 573, + 201, + 583 + ], + "score": 0.8, + "content": "1 / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 571, + 224, + 585 + ], + "score": 1.0, + "content": "cover", + "type": "text" + }, + { + "bbox": [ + 225, + 573, + 231, + 582 + ], + "score": 0.82, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 571, + 271, + 585 + ], + "score": 1.0, + "content": "of the unit", + "type": "text" + }, + { + "bbox": [ + 271, + 573, + 280, + 582 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 571, + 342, + 585 + ], + "score": 1.0, + "content": "-sphere with size", + "type": "text" + }, + { + "bbox": [ + 342, + 572, + 390, + 583 + ], + "score": 0.92, + "content": "\\log | { \\mathcal { C } } | \\leq 2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 571, + 423, + 585 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 424, + 573, + 447, + 582 + ], + "score": 0.89, + "content": "{ \\pmb v } \\in \\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 571, + 505, + 585 + ], + "score": 1.0, + "content": ", subgaussianity", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 509, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 138, + 603 + ], + "score": 1.0, + "content": "implies,", + "type": "text" + }, + { + "bbox": [ + 138, + 583, + 251, + 597 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { P } ( | v ^ { T } \\pmb { a } | \\geq t ) \\leq 2 \\exp ( - \\frac { c t ^ { 2 } } { 2 L ^ { 2 } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 579, + 283, + 603 + ], + "score": 1.0, + "content": ". Setting", + "type": "text" + }, + { + "bbox": [ + 283, + 585, + 329, + 596 + ], + "score": 0.93, + "content": "t = C L { \\sqrt { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 579, + 438, + 603 + ], + "score": 1.0, + "content": "for sufficiently large constant", + "type": "text" + }, + { + "bbox": [ + 438, + 585, + 463, + 595 + ], + "score": 0.89, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 579, + 509, + 603 + ], + "score": 1.0, + "content": ", and union", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 595, + 228, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 172, + 606 + ], + "score": 1.0, + "content": "bounding over all", + "type": "text" + }, + { + "bbox": [ + 172, + 596, + 195, + 605 + ], + "score": 0.86, + "content": "\\pmb { v } \\in \\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 595, + 228, + 606 + ], + "score": 1.0, + "content": ", we find", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 571, + 509, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 610, + 453, + 639 + ], + "lines": [ + { + "bbox": [ + 157, + 610, + 453, + 639 + ], + "spans": [ + { + "bbox": [ + 157, + 610, + 453, + 639 + ], + "score": 0.93, + "content": "\\mathbb { P } ( \\bigcap _ { v \\in \\mathcal { C } } \\| v \\| _ { \\ell _ { 2 } } \\leq C L \\sqrt { m } ) \\geq 1 - 2 \\exp ( 2 n - \\frac { c C ^ { 2 } L ^ { 2 } m } { 2 L ^ { 2 } } ) \\leq 1 - 2 \\exp ( - 1 0 0 m ) .", + "type": "interline_equation", + "image_path": "905ac3d7de230ea2bfd0b2e10b7518950215c6b624da0400760313bcceb7d4cf.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 157, + 610, + 453, + 639 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 644, + 446, + 659 + ], + "lines": [ + { + "bbox": [ + 101, + 636, + 450, + 665 + ], + "spans": [ + { + "bbox": [ + 101, + 636, + 450, + 665 + ], + "score": 1.0, + "content": "To conclude, let v(a) ∈ C be a’s neighbor satisfying kv − akak`2 k`2 ≤ 1/2. Hence, we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 101, + 636, + 450, + 665 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 665, + 473, + 679 + ], + "lines": [ + { + "bbox": [ + 138, + 665, + 473, + 679 + ], + "spans": [ + { + "bbox": [ + 138, + 665, + 473, + 679 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\| a \\| _ { \\ell _ { 2 } } \\leq \\| ( a - v ( a ) ) ^ { T } a \\| _ { \\ell _ { 2 } } + \\| v ^ { T } a \\| _ { \\ell _ { 2 } } \\leq \\| a \\| _ { \\ell _ { 2 } } / 2 + C L \\sqrt { m } \\implies \\| a \\| _ { \\ell _ { 2 } } \\leq 2 C L \\sqrt { m } . } \\end{array}", + "type": "interline_equation", + "image_path": "4b3f1cd8afcc305b2e98880ecf87a213468ff4e7b8aa7c70ea3ed937bf4985d7.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 138, + 665, + 473, + 679 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 684, + 292, + 695 + ], + 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0000000000000000000000000000000000000000..7904bdc6575e50c569c5691c018885c8716446f9 --- /dev/null +++ b/parse/train/tHgJoMfy6nI/tHgJoMfy6nI.md @@ -0,0 +1,290 @@ +# REMEMBERING FOR THE RIGHT REASONS: EXPLANATIONS REDUCE CATASTROPHIC FORGETTING + +Sayna Ebrahimi1, Suzanne Petryk1, Akash Gokul1, William Gan1, Joseph E. Gonzalez1, Marcus Rohrbach2, Trevor Darrell1 + +1UC Berkeley, 2 Facebook AI Research +{sayna,spetryk,akashgokul,wjgan,jegonzal,trevordarrell}@berkeley.edu +mrf@fb.com + +# ABSTRACT + +The goal of continual learning (CL) is to learn a sequence of tasks without suffering from the phenomenon of catastrophic forgetting. Previous work has shown that leveraging memory in the form of a replay buffer can reduce performance degradation on prior tasks. We hypothesize that forgetting can be further reduced when the model is encouraged to remember the evidence for previously made decisions. As a first step towards exploring this hypothesis, we propose a simple novel training paradigm, called Remembering for the Right Reasons (RRR), that additionally stores visual model explanations for each example in the buffer and ensures the model has “the right reasons” for its predictions by encouraging its explanations to remain consistent with those used to make decisions at training time. Without this constraint, there is a drift in explanations and increase in forgetting as conventional continual learning algorithms learn new tasks. We demonstrate how RRR can be easily added to any memory or regularizationbased approach and results in reduced forgetting, and more importantly, improved model explanations. We have evaluated our approach in the standard and few-shot settings and observed a consistent improvement across various CL approaches using different architectures and techniques to generate model explanations and demonstrated our approach showing a promising connection between explainability and continual learning. Our code is available at https://github.com/ SaynaEbrahimi/Remembering-for-the-Right-Reasons. + +# 1 INTRODUCTION + +Humans are capable of continuously learning novel tasks by leveraging their lifetime knowledge and expanding them when they encounter a new experience. They can remember the majority of their prior knowledge despite the never-ending nature of their learning process by simply keeping a running tally of the observations distributed over time or presented in summary form. The field of continual learning or lifelong learning (Thrun & Mitchell, 1995; Silver et al., 2013) aims at maintaining previous performance and avoiding so-called catastrophic forgetting of previous experience (McCloskey & Cohen, 1989; McClelland et al., 1995) when learning new skills. The goal is to develop algorithms that continually update or add parameters to accommodate an online stream of data over time. + +An active line of research in continual learning explores the effectiveness of using small memory budgets to store data points from the training set (Castro et al., 2018; Rajasegaran et al., 2020; Rebuffi et al., 2017; Wu et al., 2019), gradients (Lopez-Paz et al., 2017), or storing an online generative model that can fake them later (Shin et al., 2017). Memory has been also exploited in the form of accommodating space for architecture growth and storage to fully recover the old performance when needed (Ebrahimi et al., 2020b; Rusu et al., 2016). Some methods store an old snapshot of the model to distill the features (Li & Hoiem, 2016) or attention maps (Dhar et al., 2019) between the teacher and student models. + +![](images/d0281df291d1cac11c700eef36b889185db8c8c6f64b2f631bab1a588c539ccf.jpg) +Figure 1: An illustration of applying RRR paradigm. (Left) In a typical experience replay scenario, samples from prior tasks are kept in a memory buffer $\mathcal { M } ^ { \mathrm { r e p } }$ and revisited during training. (Right) In our proposed idea (RRR), in addition to ${ \mathcal { M } } ^ { \mathrm { r e p } }$ , we also store model explanations (saliency maps) as $\mathcal { M } ^ { \mathrm { R R R } }$ for those samples and encourage the model to remember the original reasoning for the prediction. Note that the saliency maps are small masks resulting in a negligible memory overhead (see Section 4.1). + +The internal reasoning process of deep models is often treated as a black box and remains hidden from the user. However, recent work in explainable artificial intelligence (XAI) has developed methods to create human-interpretable explanations for model decisions (Simonyan et al., 2013; Zhang et al., 2018; Petsiuk et al., 2018; Zhou et al., 2016; Selvaraju et al., 2017). We posit that the catastrophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was used for a previously seen observation. Therefore, we hypothesize that forgetting can be mitigated when the model is encouraged to remember the evidence for previously made decisions. In other words, a model which can remember its final decision and can reconstruct the same prior reasoning. Based on this approach, we develop a novel strategy to exploit explainable models for improving performance. + +Among the various explainability techniques proposed in XAI, saliency methods have emerged as a popular tool to identify the support of a model prediction in terms of relevant features in the input. These methods produce saliency maps, defined as regions of visual evidence upon which a network makes a decision. Our goal is to investigate whether augmenting experience replay with explanation replay reduces forgetting and how enforcing to remember the explanations will affect the explanations themselves. Figure 1 illustrates our proposed method. + +In this work, we propose RRR, a training strategy guided by model explanations generated by any white-box differentiable explanation method; RRR adds an explanation loss to continual learning. White-box methods generate an explanation by using some internal state of the model, such as gradients, enabling their use in end-to-end training. We evaluate our approach using various popular explanation methods including vanilla backpropagation (Zeiler & Fergus, 2014), backpropagation with smoothing gradients (Smoothgrad) (Smilkov et al., 2017), Guided Backpropagation (Springenberg et al., 2014), and Gradient Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017) and compare their performance versus their computational feasibility. We integrate RRR into several state of the art class incremental learning (CIL) methods, including iTAML (Rajasegaran et al., 2020), EEIl (Castro et al., 2018), BiC (Wu et al., 2019), TOPIC (Tao et al., 2020), iCaRL (Rebuffi et al., 2017), EWC (Kirkpatrick et al., 2017), and LwF (Li & Hoiem, 2016). Note that RRR does not require task IDs at test time. We qualitatively and quantitatively analyze model explanations in the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence of tasks due to the requirement to focus on the the right evidence. We empirically show the effect of RRR in standard and few-shot class incremental learning (CIL) scenarios on popular benchmark datasets including CIFAR100, ImageNet100, and Caltech-UCSD Birds 200 using different network architectures where RRR improves overall accuracy and forgetting over experience replay and other memory-based method. + +Our contribution is threefold: we first propose our novel, simple, yet effective memory constraint, which we call Remembering for the Right Reasons (RRR), and show that it reduces catastrophic forgetting by encouraging the model to look at the same explanations it initially found for its decisions. Second, we show how RRR can be readily combined with memory-based and regularization-based + +CL methods to improve performance. Third, we demonstrate how guiding a continual learner to remember its explanations can improve the quality of the explanations themselves; i.e., the model looks at the right region in an image when making correct decisions while it focuses its maximum attention on the background when it misclassifies an object. + +# 2 BACKGROUND: WHITE-BOX EXPLANABILITY TECHNIQUES + +Here we briefly review the explainability methods we have evaluated our approach with. The core idea behind RRR is to guide explanations or saliency maps during training to preserve their values. Hence, only gradient-based saliency techniques can be used which are differentiable and hence trainable during training for the mainstream task as opposed to black-box saliency methods which can be used only after training to determine important parts of an image. + +Vanilla Backpropagation (Zeiler & Fergus, 2014): The simplest way to understand and visualize which pixels are most salient in an image is to look at the gradients. This is typically done by making a forward pass through the model and taking the gradient of the given output class with respect to the input. Those pixel-wise derivative values can be rendered as a normalized heatmap representing the amount of change in the output probability of a class caused by perturbing that pixel. To store a saliency map for each RGB image of size $3 \times W \times H$ , we need an equivalent memory size of storing $W \times H$ pixel values. + +Backpropagation with SmoothGrad: Smilkov et al. (2017) showed that the saliency maps obtained using raw gradients are visually noisy and using them as a proxy for feature importance is sub-optimal. They proposed a simple technique for denoising the gradients that adds pixel-wise Gaussian noise to $n$ copies of the image, and simply averages the resulting gradients. SmoothGrad requires the same amount of memory to store the saliency maps while it takes $n$ times longer to repeat generating each saliency map. We found $n = 4 0$ to be large enough to make a noticeable change in the saliencies in our experiments. + +Gradient-weighted Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017): is a whitebox explainability technique which uses gradients to determine the influence of specific feature map activations on a given prediction. Because later layers in a convolutional neural network are known to encode higher-level semantics, taking the gradient of a model output with respect to the activations of these feature maps discovers which high-level semantics are important for the prediction. We refer to this layer as the target layer in our analysis. For example, when using Grad-CAM to visualize explanations for image classification, taking the gradient of the correct class prediction with respect to the last convolutional layer highlights class-discriminative regions in the image (such as the wings of a bird when identifying bird species). + +Consider the pre-softmax score $y _ { c }$ for class $c$ in an image classification output. In general, any differentiable activation can be used. Consider also a single convolutional layer with $K$ feature maps, with a single feature map noted as $A ^ { k } \in \mathbb { R } ^ { u \times v }$ . Grad-CAM takes the derivative of $y _ { c }$ with respect to each feature map $A ^ { k }$ . It then performs global average pooling over the height and width dimensions for each of these feature map gradients, getting a vector of length $K$ . Each element in this vector is used as a weight $\alpha _ { k } ^ { c }$ , indicating the importance of feature map $k$ for the prediction $y _ { c }$ . Next, these importance weights are used in computing a linear combination of the feature maps. Followed by a ReLU (Jarrett et al., 2009) to zero-out any activations with a negative influence on the prediction of class $c$ , the final Grad-CAM output (s) is as below with $A _ { i j } ^ { k }$ defined at location $( i , j )$ in feature map $A ^ { k }$ . + +$$ +\begin{array} { r l } { \alpha _ { k } ^ { c } = } & { { } \frac { 1 } { u v } \displaystyle \sum _ { i = 1 } ^ { u } \displaystyle \sum _ { j = 1 } ^ { v } \frac { \partial y _ { c } } { \partial A _ { i j } ^ { k } } \qquad s _ { G r a d - C A M } ^ { c } = R e L U \left( \displaystyle \sum _ { k = 1 } ^ { K } \alpha _ { k } ^ { c } A ^ { k } \right) } \end{array} +$$ + +Unlike the common saliency map techniques of Guided BackProp (Springenberg et al., 2014), Guided GradCAM (Selvaraju et al., 2016), Integrated Gradients (Sundararajan et al., 2017b), Gradient $\odot$ Input (Shrikumar et al., 2016), Backpropagation with SmoothGrad (Smilkov et al., 2017) etc., vanilla Backpropagation and Grad-CAM pass important “sanity checks” regarding their sensitivity to data and model parameters (Adebayo et al., 2018). We will compare using vanilla Backpropagation, Backpropagation with SmoothGrad, and Grad-CAM in RRR in Section 4. We will refer to the function that computes the output $s$ of these saliency method as $ { \mathcal { X } } { \mathcal { A } } { \mathcal { T } }$ . + +Algorithm 1 Remembering for the Right Reasons (RRR) for Continual Learning +1: function TRAIN $( f _ { \theta } , \mathcal { D } ^ { t r } , \mathcal { D } ^ { t s } )$ function UPDATE MEM(f kθ , Dtrk , Mrep, MRRR) +2: $T$ : # of tasks, $n$ : # of samples in task (xi, k, yi) ∼ Dtrk +3: R ← 0 ∈ R T ×T Mrep ← Mrep ∪ {(xi, k, yi)} +4: Mrep ← {} sˆ ← X AI(f kθ (xi, k)) +5: $\mathcal { M } ^ { \mathrm { R R R } } \{ \}$ MRRR ← MRRR ∪ {sˆ} +6: for $k = 1$ to T do return Mrep, MRRR +7: for $i = 1$ to n do end function +8: Compute cross entropy on task $( \mathcal { L } _ { \mathrm { t a s k } } )$ +9: Compute $\mathcal { L } _ { \mathrm { R R R } }$ using Eq. 2 function EVAL(f kθ , Dts{1···k}) +10: $\theta ^ { \prime } \theta - \alpha \nabla _ { \theta } ( \mathcal { L } _ { \mathrm { t a s k } } + \mathcal { L } _ { \mathrm { R R R } } )$ for $i = 1$ to $k$ do +11: end for Rk,i = Accuracy $( f _ { \theta } ^ { k } ( x , i ) , y \vert \forall ( x , y ) \in \mathcal { D } _ { i } ^ { t s } )$ +12: $\begin{array} { r l } & { \mathcal { M } ^ { \mathrm { r e p } } , \mathcal { M } ^ { \mathrm { R R R } } \gets \mathrm { U P D A T E ~ M E M } ( f _ { \theta } ^ { k } , \mathcal { D } _ { k } ^ { t r } , \mathcal { M } ^ { \mathrm { r e p } } , } \\ & { } \\ & { R _ { k , \{ 1 \cdots k \} } \gets \mathrm { E V A L } \left( f _ { \theta } ^ { k } , \mathcal { D } _ { \{ 1 \cdots k \} } ^ { t s } \right) } \end{array}$ , end for +return $R$ +13: end function +14: end for +15: return $f _ { \boldsymbol { \theta } } , R$ +16: end function + +# 3 REMEMBERING FOR THE RIGHT REASONS (RRR) + +Memory-based methods in continual learning have achieved high performance on vision benchmarks using a small amount of memory, i.e. storing a few samples from the training data into the replay buffer to directly train with them when learning new tasks. This simple method, known as experience replay, has been explored and shown to be effective (Rebuffi et al., 2017; Wu et al., 2019; Castro et al., 2018; Rajasegaran et al., 2020; Ebrahimi et al., 2020b; Hayes et al., 2019; Riemer et al., 2018). In this work we aim to go one step further and investigate the role of explanations in continual learning, particularly on mitigating forgetting and change of model explanations. + +We consider the problem of learning a sequence of $T$ data distributions $\mathcal { D } ^ { t r } = \{ \mathcal { D } _ { 1 } ^ { t r } , \cdot \cdot \cdot , \mathcal { D } _ { T } ^ { t r } \}$ , where $\mathcal { D } _ { k } ^ { t r } = \{ ( x _ { i } ^ { k } , y _ { i } ^ { k } ) _ { i = 1 } ^ { n _ { k } } \}$ is the data distribution for task $k$ with $n$ sample tuples of input $( \mathbf { x } ^ { k } \subset \mathcal { X } )$ and set of output labels $( \mathbf { y } ^ { k } \subset \mathcal { V } )$ . The goal is to sequentially learn the model $f _ { \theta } : \mathcal { X } \times \mathcal { T } \mathcal { Y }$ for each task that can map each input $x$ to its target output, $y$ , while maintaining its performance on all prior tasks. We aim to achieve this by using memory to enhance better knowledge transfer as well as better avoidance of catastrophic forgetting. We assume two limited memory pools $\mathcal { M } ^ { \mathrm { r e p } }$ for raw samples and $\mathcal { M } ^ { \mathrm { R R R } }$ for model explanations. In particular, $\mathcal { M } ^ { \mathrm { r e p } } = \{ ( x _ { i } ^ { j } , y _ { i } ^ { j } ) _ { i = 1 } ^ { m } \stackrel { \cdot } { \sim } \mathcal { D } _ { j = 1 \cdots k - 1 } ^ { t r } \}$ stores $m$ samples in total from all prior tasks to $k$ . Similarly $\mathcal { M } ^ { \mathrm { R R R } }$ stores the saliency maps generated based on $f _ { \theta } ^ { k }$ by one of the explanation methods $( { \mathcal { X } } { \mathcal { A } } { \mathcal { T } } )$ discussed in Section 2 for images in $\mathcal { M } ^ { \mathrm { r e p } }$ where $f _ { \theta } ^ { k }$ is $f _ { \theta }$ being trained for task $k$ . We use a single-head architecture where the task ID integer $t$ is not required at test time. + +Upon finishing the $k ^ { t h }$ task, we randomly select $m / ( k { - } 1 )$ samples per task from its training data and update our replay buffer memory ${ \mathcal { M } } ^ { \mathrm { r e p } }$ . RRR uses model explanations on memory samples to perform continual learning such that the model preserves its reasoning for previously seen observations. We explore several explanation techniques to compute saliency maps using $f _ { \theta } ^ { \mathbf { \bar { k } } }$ for the stored samples in the replay buffer to populate the xai buffer memory $\mathcal { M } ^ { \mathrm { x a i } }$ . The stored saliency maps will serve as reference explanations during the learning of future tasks to prevent model parameters from being altered resulting in different reasoning for the same samples. We implement RRR using an L1 loss on the error in saliency maps generated after training a new task with respect to their stored reference evidence. + +$$ +\mathcal { L } _ { \mathrm { R R R } } \big ( f _ { \theta } , \mathcal { M } ^ { \mathrm { r e p } } , \mathcal { M } ^ { \mathrm { R R R } } \big ) = \mathbb { E } _ { ( ( x , y ) , \hat { s } ) \sim \left( \mathcal { M } ^ { \mathrm { r e p } } , \mathcal { M } ^ { \mathrm { R R R } } \right) } \vert \vert \mathcal { X } \mathcal { A } \mathcal { Z } \big ( f _ { \theta } ^ { k } ( x ) \big ) - \hat { s } \vert \vert _ { 1 } +$$ + +where $\mathcal { X } \mathcal { A } \mathcal { T } ( \cdot )$ denotes the explanation method used to compute saliency maps using the model trained on the last seen example from task $k$ , and $\hat { s }$ are the reference saliency maps generated by $\chi \mathcal { A } \mathcal { T } ( f _ { \theta } ^ { k } )$ upon learning each task prior to $T$ and stored in to the memory. We show below that combining RRR into the objective function of state-of-the-art memory and regularization-based methods results in significant performance improvements. The full algorithm for RRR is given in Alg. 1. + +![](images/cb3d7154cc20f9b0d8c9e69038bd7edd29572c0f83a7677e8527ca6aa7b2c07f.jpg) +Figure 2: Few-shot CIL learning of CUB200 in 11 tasks where each point shows the classification accuracy on all seen classes so far. (Left) Shows ER with and without $\mathcal { L } _ { \mathrm { R R R } }$ using different backbone architectures and saliency map techniques. (Right) Performance of the state-of-the-art existing approaches with and without $\mathcal { L } _ { \mathrm { R R R } }$ on CUB200 including TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), iCaRL (Rebuffi et al., 2017). Joint training serves as the upper bound. Results for baselines are obtained using their original implementation. All results are averaged over 3 runs and mean and standard deviation values are given in the appendix. Best viewed in color. + +# 4 EXPERIMENTS + +In this section, we apply RRR in two distinct learning regimes: standard and few-shot class incremental learning. These are the most challenging CL scenarios, in which task descriptions are not available at test time. We first explore the effect of backbone architecture and the saliency map technique on RRR performance. We then report our obtained results integrating $\mathcal { L } _ { \mathrm { R R R } }$ into existing memory-based and regularization-based methods. + +# 4.1 FEW-SHOT CIL PERFORMANCE + +We first explore CIL of low-data regimes where preventing overfitting to few-shot new classes is another challenge to overcome in addition to avoiding catastrophic forgetting of old classes. We use $C$ classes and $K$ training samples per class as the $C$ -way $K$ -shot few-shot class incrementally learning setting where we have a set of $b$ base classes to learn as the first task while the remaining classes are learned with only a few randomly selected samples. In order to provide a direct comparison to the state-of-the-art work of Tao et al. (2020) we precisely followed their setup and and used the same Caltech-UCSD Birds dataset (Wah et al., 2011), divided into 11 disjoint tasks and a 10-way 5-shot setting, where the first task contains $b = 1 0 0$ base classes resulting in 3000 samples for training and 2834 images for testing. The remaining 100 classes are divided into 10 tasks where 5 samples per class are randomly selected as the training set, while the test set is kept intact containing near 300 images per task. The images in CUB200 are resized to $2 5 6 \times 2 5 6$ and then randomly cropped to $2 2 4 \times 2 2 4$ for training. We store 4 images per class from base classes in the first task and 1 sample per each few-shot class in the remaining 10 tasks (Tao et al., 2020). We used the RAdam (Liu et al., 2019) optimizer with a learning rate of 0.001 which was reduced by 0.2 at epochs 20, 40, and 60 and trained for a total of 70 epochs with a batch size of 128 for the first and 10 for the remaining tasks. + +Figure 2 (left) shows results for ER with and without $\mathcal { L } _ { \mathrm { R R R } }$ using different backbone architectures and saliency map techniques. Among the tested saliency map methods, Grad-CAM on ResNet18 outperforms Vanilla Backpropagation and SmoothGrad by $2 { - } 3 \%$ while SmoothGrad and vanilla Backpropagation achieve similar CL performance. To compute the memory overhead of storing the output for a saliency method, if we assume the memory required to store an image is $M$ , vanilla Backpropagation and SmoothGrad generate a pixel-wise saliency map that occupies $M / 3$ of memory. However, in Grad-CAM the saliency map size is equal to the feature map of the target layer in the architecture. In our study with Grad-CAM we chose our target layer to be the last convolution layer before the fully-connected layers. For instance using ResNet18 for colored $2 2 4 \times 2 2 4$ images results in the Grad-CAM output of $7 \times 7$ occupying 196B. Table 2 shows the target layer name and saliency map size for other network architectures used in this work (AlexNet and SqueezeNet1 1) as well. + +![](images/c256de790417db433077726ead7e5e774741a08099f8b9eb519352448f894179.jpg) +Figure 3: Effect of RRR on existing methods for CIL on CIFAR100 in (a) 10 and (b) 20 tasks and (c) ImageNet100 in 10 tasks. Each point shows the classification accuracy on all seen classes so far. Results for iTAML, BiC, and EEIL are produced with their original implementation while EWC and LwF are re-implemented by us. All results are averaged over 3 runs and mean and standard deviation values are given in the appendix. Best viewed in color. + +Figure 2 (right) shows the effect of adding $\mathcal { L } _ { \mathrm { R R R } }$ on existing recent state-of-the-art methods such as TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), and iCaRL (Rebuffi et al., 2017). Tao et al. (2020) used a neural gas network (Martinetz et al., 1991; Fritzke et al., 1995) which can learn and preserve the topology of the feature manifold formed by different classes and we have followed their experimental protocol for our CUB200 experiment by using identical samples drawn in each task which are used across all the baselines for fair comparison. Adding $\mathcal { L } _ { \mathrm { R R R } }$ improves the performance of all the baselines; TOPIC becomes nearly on-par with joint training which serves as the upper bound and does not adhere to continual learning. The gap between ER and iCaRL is also reduced when ER uses $\mathcal { L } _ { \mathrm { R R R } }$ . + +# 4.2 STANDARD CIL PERFORMANCE + +In order to provide a direct comparison to the recent work of Rajasegaran et al. (2020) we perform our standard CIL experiment on CIFAR100 (Krizhevsky & Hinton, 2009) and ImageNet100 where we assume a memory budget of 2000 samples which are identical across all the baselines. Following Rajasegaran et al. (2020) we divide CIFAR100 to 10 and 20 disjoint tasks with 10 and 5 classes at a time. Figures 3a and 3b show the classification accuracy upon learning each task on all seen classes. We see a consistent average improvement of $2 - 4 \%$ when $\mathcal { L } _ { \mathrm { R R R } }$ is added as an additional constraint to preserve the model explanations across all methods, from the most naive memory-based method, experience replay (ER), to more sophisticated approaches which store a set of old class exemplars along with meta-learning (iTAML), correct bias for new classes (BiC), and fine tune on the exemplar set (EEIL). We also applied the RRR constraint on regularization-based methods such as EWC and LwF with no memory used as a replay buffer. The accuracy for both improves despite not benefiting from revisiting the raw data. However, they fall behind all memory-based methods with or without $\mathcal { L } _ { \mathrm { R R R } }$ . The final accuracy on the entire sequence for joint training (multi-task learning) with RAdam optimizer (Liu et al., 2019) is $8 0 . 0 3 \%$ which serves as an upper bound as it has access to data from all tasks at all time. + +Figure 3c shows our results on learning ImageNet100 in 10 tasks. The effect of adding $\mathcal { L } _ { \mathrm { R R R } }$ to baselines on the ImageNet100 experiment is more significant $( 3 - 6 \% )$ compared to CIFAR100. This is mainly due to the larger size and better quality of images in ImageNet100, resulting in generating larger Grad-CAM saliency maps. These experiments clearly reveal the effectiveness of $\mathcal { L } _ { \mathrm { R R R } }$ on model explanations in a continual learning problem at nearly zero cost of memory overhead when a memory buffer is already created and applied as a catastrophic forgetting avoidance strategy. This makes Grad-CAM the ideal approach to generate saliency maps when applying the RRR training strategy, as it achieves the highest accuracy while utilizing the least storage space to store saliencies. Note that we adopt Grad-CAM to generate saliency maps in the remaining experiments in this paper. We also keep using only ResNet18 for a fair comparison with the literature. + +(a) PG localization accuracy and backward transfer + +
MethodsPG-ACC (%)PG-BWT (%)
ER54.0-17.4
ER+RRR58.5-15.6
TOPIC72.7-0.9
TOPIC+RRR74.2-2.1
+ +Table 1: PG experiment results on few-shot CIL CUB200 measuring (a) PG-ACC $( \% )$ and PG-BWT $( \% )$ and (b) precision and recall averaged over all tasks. $P r _ { i , i }$ and $R e _ { i , i }$ evaluate the pointing game on each task $\mathbf { t } ^ { \mathrm { i } }$ directly after the model has been trained on $\mathbf { t } ^ { \mathbf { i } }$ . $P r _ { T , i }$ and $R e _ { T , i }$ are obtained by the evaluation for task $\mathbf { t ^ { i } }$ using the model trained for all $T$ tasks. +(b) Precision and recall using PG experiment + +
PrecisionRecall
MethodsPri,iPrT,iRei,iReT,i
ER80.068.964.165.1
ER+RRR82.170.364.266.8
TOPIC91.088.498.197.4
TOPIC+RRR92.889.199.699.2
+ +# 5 ANALYSIS OF MODEL EXPLANATIONS + +In this section we want to answer the question “How often does the model remember its decision for the right reason upon learning a sequence of tasks?”. In particular, we want to evaluate how often the model is “pointing” at the right evidence for its predictions, instead of focusing its maximum attention on the background or other objects in the image. We use the Pointing Game experiment (PG) (Zhang et al., 2018) for this evaluation, which was introduced to measure the discriminativeness of a visualization method for target object localization. Here, we use ground truth segmentation annotation labels provided with the CUB-200 dataset to define the true object region. + +First, we look into hits and misses defined by the PG experiment. When the location of the maximum in a predicted saliency map falls inside the object, it is considered as a hit and otherwise it is a miss. Figure 5 shows an example from CUB200 where the segmentation annotation is used to determine whether the peak of the predicted saliency map (marked with red cross) falls inside the object region. This example is regarded as hit as the red cross is inside the segmentation mask for the bird. PG localization accuracy is defined as the number of hits over the total number of predictions. We would like to measure both the overall PG performance of a continual learner as well as how much learning new tasks causes it to forget its ability to hit the target object. For these metrics, inspired by (Lopez-Paz et al., 2017), we define $\begin{array} { r } { \mathrm { P G } { \mathrm { - } } \mathrm { A C C } = \frac { 1 } { T } \sum _ { i = 1 } ^ { \bar { T } } \dot { R _ { T , i } } } \end{array}$ as the average PG localization accuracy computed over all prior tasks after training for each new task and $\begin{array} { r } { \mathrm { P G - B W T } = \frac { 1 } { T - 1 } \sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } } \end{array}$ (backward transfer) which indicates how much learning new tasks has influenced the PG localization accuracy on previous tasks where $R _ { n , i }$ is the on task $i$ after learning the $n ^ { \mathrm { t h } }$ task. Results for ER and TOPIC with and without $\mathcal { L } _ { \mathrm { R R R } }$ on CUB200 are shown in Table 1a. It shows how constraining different models to remember their initial evidence can lead to better localization of the bird across learning new tasks. + +However, PG performance does not capture all of our desired properties for a continual learner. Ideally, we not only want a model to predict the object correctly if it is looking at the right evidence, but also we want it to not predict an object if it is not able to find the right evidence for it. To measure how close our baselines can get to this ideal model when they are combined with $\mathcal { L } _ { \mathrm { R R R } }$ , we measure the precision as $\mathrm { t p / ( t p / \Delta \mathfrak { p } ) }$ , and recall as $\mathrm { t p / ( t p / \Delta \mathfrak { t } f n ) }$ . We evaluate these metrics once immediately after learning each task, denoted as $P r _ { i , i }$ and $R e _ { i , i }$ , respectively, and again at the end of the learning process of final task $T$ denoted as $P r _ { T , i }$ and $R e _ { T , i }$ , where the first subscript refers to the model ID and the second subscript is the test dataset ID on which the model is evaluated. The higher the precision for a model is, the less often it has made the right decision without looking at the right evidence. On the other hand, the higher the recall, the less often it makes a wrong decision despite looking at the correct evidence. We show our evaluation on these metrics in Table 1b for ER and TOPIC with and without $\mathcal { L } _ { \mathrm { R R R } }$ on CUB200 where $\mathcal { L } _ { \mathrm { R R R } }$ increases both precision and recall across all methods, demonstrating that our approach continually makes better predictions because it finds the right evidence for its decisions. + +In our final analysis, we would like to visualize the evolution of saliency maps across learning a sequence of tasks. Figure 4 illustrates the evolution of saliency maps for an image from the test-set of the second task, which both ER without $\mathcal { L } _ { \mathrm { R R R } }$ (top row) and with $\mathcal { L } _ { \mathrm { R R R } }$ (bottom row) have seen during training for the second task. We only visualize the generated saliencies after finishing tasks #2, #5, #7, #9, and #11 for simplicity. We indicate the correctness of the prediction made by each model with ‘correct’ or ‘incorrect’ written on top of their corresponding saliency map. Our goal is to visualize if adding the loss term $\mathcal { L } _ { \mathrm { R R R } }$ prevents the drifting of explanations. Given the same input image, the ER without $\mathcal { L } _ { \mathrm { R R R } }$ model makes an incorrect prediction after being continually trained for 11 tasks while never recovering from its mistake. On the other hand, when it is combined with $\mathcal { L } _ { \mathrm { R R R } }$ . it is able to recover from an early mistake after task 5. Considering the saliency map obtained after finishing task one as a reference evidence, we can see that ER’s evidence drifts further from the reference. On the bottom row, the region of focus of ER+RRR remains nearly identical to its initial evidence, apart from one incorrect prediction. As applying $\mathcal { L } _ { \mathrm { R R R } }$ corrects its saliency back to the original, this prediction is corrected as well. This supports the conclusion that retaining the original saliency is important for retaining the original accuracy. + +![](images/09c0ee24c47d173aa2e6b5de78cbf95caefed73a635754290317195ebd291fa3.jpg) +Figure 4: An illustration of the progression of saliencies on an image from the test set of the second task, evaluated after the model is trained on tasks #2, #5, #7, #9, and #11 on CUB200. Failure case for ER w.o. $\mathcal { L } _ { \mathrm { R R R } }$ (top row), where saliency drifts from the original and the prediction becomes incorrect. $_ \mathrm { E R + R R R }$ (bottom row) retains close to the original saliency as the model trains on more tasks, with the exception of Task #5 which it is able to correct later on. Its performance is retained as well, for saliencies that are close to the original. + +# 6 RELATED WORK + +Continual learning: Past work in CL has generally made use of either memory, model structure, or regularization to prevent catastrophic forgetting. Memory-based methods store some form of past experience into a replay buffer. However, the definition of “experience” varies between methods. Rehearsal-based methods use episodic memories as raw samples (Robins, 1995; Rebuffi et al., 2017; Riemer et al., 2018) or their gradients (Lopez-Paz et al., 2017; Chaudhry et al., 2019) for the model to revisit. Incremental Classifier and Representation Learning (iCaRL) (Rebuffi et al., 2017), is a class-incremental learner that uses a nearest-exemplar algorithm for classification and prevents catastrophic forgetting by using an episodic memory. iTAML (Rajasegaran et al., 2020) is a task-agnostic meta-learning algorithm that uses a momentum based strategy for meta-update and in addition to the object classification task, it predicts task labels during inference. An end-to-end incremental learning framework (EEIL) (Castro et al., 2018) also uses an exemplar set along with data augmentation and balanced fine-tuning to alleviate the imbalance between the old and new classes. Bias Correction Method (BiC) (Wu et al., 2019) is another class-incremental learning algorithm for large datasets in which a linear model is used to correct bias towards new classes using a fully connected layer. In contrast, pseudo-rehearsal methods generate the replay samples using a generative model such as an autoencoder (Kemker & Kanan, 2017) or a GAN (Kamra et al., 2017; Shin et al., 2017). Regularization-based methods define different metrics to measure importance and limit the changes on parameters accordingly (Kirkpatrick et al., 2017; Zenke et al., 2017; Ebrahimi et al., 2020a; Serra et al., 2018; Li & Hoiem, 2016; Dhar et al., 2019) but in general these methods have limited capacity. Structure-based methods control which portions of a model are responsible for specific tasks such that the model increases its capacity in a controlled fashion as more tasks are added. Inference for different tasks can be restricted to various neurons (Fernando et al., 2017; Yoon et al., 2018), columns (Rusu et al., 2016), task-specific modules (Ebrahimi et al., 2020b), or parameters selected by a mask (Mallya & Lazebnik, 2018; Serra et al., 2018). In RRR we explored the addition of explanations to replay buffer and showed that saliency-based explanations offer performance upgrade as well as improvement in explanations across all memory-based and regularization-based baselines we tried. + +Visual explanation approaches or saliency methods attempt to produce a posterior explanation or a pseudo-probability map for the detected signals from the target object in the input image. These approaches can be broadly divided into three categories including activation, gradient, and perturbation based methods. Activation-based methods (Zhou et al., 2016; Selvaraju et al., 2017; Chattopadhay et al., 2018) use a weighted linear combination of feature maps whereas gradient-based methods (Baehrens et al., 2010; Sundararajan et al., 2017a; Springenberg et al., 2014; Shrikumar et al., 2017; Zhang et al., 2018) use the derivative of outputs w.r.t the input image to compute pixel-wise importance scores to generate attention maps. Methods in these categories are only applicable to differentiable models, including convolutional neural networks (CNNs). In contrast, perturbation-based methods are model-agnostic and produce saliency maps by observing the change in prediction when the input is perturbed (Petsiuk et al., 2018; Ribeiro et al., 2016; Ross et al., 2017; Zhou et al., 2014; Seo et al., 2018). While these methods attempt to identify if models are right for the wrong reason, Ross et al. (2017) took a step further and applied penalties to correct the explanations provided in supervised/unsupervised fashion during training. Selvaraju et al. (2019) used human explanations in the form of question and answering to bring model explanations closer to human answers. + +# 7 CONCLUSION + +In this paper, we proposed the use of model explanations with continual learning algorithms to enhance better knowledge transfer as well as better recall of the previous tasks. The intuition behind our method is that encouraging a model to remember its evidence will increase the generalisability and rationality of recalled predictions and help retrieving the relevant aspects of each task. We advocate for the use of explainable AI as a tool to improve model performance, rather than as an artifact or interpretation of the model itself. We demonstrate that models which incorporate a “remember for the right reasons” constraint as part of a continual learning process can both be interpretable and more accurate. We empirically demonstrated the effectiveness of our approach in a variety of settings and provided an analysis of improved performance and explainability. + +# REFERENCES + +Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. 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In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2921–2929, 2016. + +# A APPENDIX + +# A.1 GRAD-CAM TARGET LAYERS + +Table 2 shows the target layer names used in Grad-CAM for different network architectures according to their standard PyTorch (Paszke et al., 2017) implementations. Saliency map size is equal to the activation map of the target layers. + +Table 2: Target layer names and activation maps size for saliencies generated by different network architectures in Grad-CAM. + +
Target layer name in PyTorch torchvision packageSaliency map size
SqueezeNet1_1features.0.12.expand3x313 ×13
AlexNetfeatures.0.1013 ×13
ResNet18features.7.1.conv27×7
+ +# B POINTING GAME VISUALIZATION + +Figure 5 shows an example from CUB200 in the Pointing Game. We used the segmentation annotation to verify whether the peak of the predicted saliency map (marked with red cross) falls inside the object region. It is regarded as hit as the red cross is inside the segmentation mask for the bird. + +![](images/3e59b0dc386b86facc06d523b925d24c5ac95c5b1d5165a09da8cda4bb4f9375.jpg) +Figure 5: An example of PG evaluation as hit for an image in CUB200. Left: image saliency map overlaid on the image. Right: the segmentation label where the red cross marks the peak saliency. + +# C TABULAR RESULTS + +In this section, we have tabulated results shown in Figure 2 and Figure 3 with means and standard deviations averaged over 3 runs. + +Table 3: Classification accuracy of few-shot CIL learning of CUB200 at the end of 11 tasks for ER with and without $\mathcal { L } _ { \mathrm { R R R } }$ using different backbone architectures and saliency map techniques. Results are averaged over 3 runs. Figure 2 (left) in the main paper is generated using numbers in this Table. + +
234567891011
RN18-RRR-GCam67.8±0.853.5±0.745.6±0.6 39.6±0.735.3±0.932.3± 1.129.4±0.925.9±0.825.7±0.626.3±0.723.6±0.7
RN18-ER67.8±0.849.7 ±0.941.7 ±0.835.8 ±0.731.4 ± 0.928.5±0.825.5±0.822.1±0.821.8±0.822.5 ± 1.119.8± 0.9
RN18-RRR-Smooth67.8± 0.850.9±0.643.5 ± 0.937.0±0.833.0±0.729.5± 0.626.8±0.823.9 ±0.823.9±0.823.4± 0.821.5± 0.5
RN18-RRR-BP67.8±0.850.8±0.843.9 ±0.636.6 ±0.432.7±0.628.9±0.627.2± 0.523.8 ±0.623.8±0.624.0± 0.421.5± 0.6
RN18-Finetune67.8± 0.844.8 ± 0.632.2± 0.525.8±0.725.6 ± 0.725.2± 0.720.8±0.616.8 ± 0.718.8± 0.518.3 ± 0.417.1 ± 0.6
Alex-RRR-GCam56.7±0.746.6±0.543.9±0.741.3± 0.733.7 ± 0.527.4± 0.725.3±0.722.0±0.521.5±0.621.4± 0.621.2± 0.6
Alex-ER56.7± 0.744.6 ± 0.741.3 ± 0.738.7±0.731.1 ± 0.724.5± 0.722.6± 0.719.6 ± 0.619.1± 0.818.7 ± 0.819.1± 0.8
Alex-Finetune56.7±0.742.8 ± 0.839.6±0.836.9±0.829.5 ± 0.723.3±0.621.4± 0.817.9 ± 0.718.0 ±0.717.0 ± 0.516.9 ± 0.4
SQ-RRR-GCam46.8± 0.536.2 ±0.430.1±0.628.3±0.425.1 ± 0.523.4± 0.519.3± 0.619.0± 0.618.5± 0.518.4± 0.518.2 ±0.6
SQ-ER46.8 ± 0.533.2±0.5 27.1±0.625.3±0.622.1±0.5220.5±0.516.3± 0.416.0±0.615.5± 0.615.4 ± 0.615.2 ± 0.7
SQ-Finetune46.8 ± 0.532.0±0.7 25.2±0.723.9±0.720.2±0.8119.4 ± 0.414.9 ± 0.414.4 ± 0.513.8± 0.414.2 ± 0.513.7±0.6
+ +Table 4: Performance of the state-of-the-art existing approaches with and without $\mathcal { L } _ { \mathrm { R R R } }$ on CUB200 including TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), iCaRL (Rebuffi et al., 2017). Results for baselines are obtained using their original implementation. Results are averaged over 3 runs. Figure 2 (right) in the main paper is generated using numbers in this Table. + +
1234567891011
EEIL68.6±0.453.6± 0.447.9 ± 0.344.2±0.836.3±0.927.4 ± 1.225.9±0.724.7±0.523.9±0.724.1± 0.722.1 ± 0.5
EEIL+RRR68.6±0.456.6±0.550.9±0.648.3± 0.539.7 ± 1.231.4± 0.728.3± 1.228.0±0.626.5±0.627.4± 0.625.2±0.9
iCaRL68.6±0.452.6±0.748.6± 1.244.1 ± 0.536.6±0.329.5±0.927.8± 0.426.2±0.524.0±0.623.8±0.621.1± 0.7
iCaRL+RRR68.6±0.455.6± 1.253.6±0.747.1 ± 0.839.6±0.532.5±0.831.8 ± 0.429.2±0.627.0±0.827.8±0.624.1 ± 0.3
TOPIC68.6 ± 0.462.4 ± 0.854.8 ± 0.449.9 ± 1.245.2 ±0.641.4± 0.338.3±0.835.3±0.632.2± 0.328.3±0.626.2 ± 1.2
TOPIC+RRR68.6±0.462.5 ± 0.956.8 ± 0.451.5 ± 0.548.2 ± 0.444.4 ± 0.442.3±0.738.3±0.635.2±0.932.3±0.929.2 ± 0.5
FT68.6±0.444.8 ± 0.532.2±0.825.8± 0.425.6 ± 1.125.2± 0.720.8± 1.116.7± 0.418.8 ± 1.118.2± 0.317.1 ± 0.8
ER67.8±0.849.7± 0.941.7 ± 0.835.8 ±0.731.4± 0.928.5±0.825.5±0.822.1± 0.821.8± 0.622.5 ± 1.119.8±0.9
RRR67.8±0.853.5± 0.745.6± 0.639.6± 0.735.3±0.932.3 ± 1.129.4± 0.925.9± 0.825.7±0.626.3±0.723.6±0.7
JT68.6±0.462.4± 0.457.2 ± 0.452.8±0.549.5 ± 0.946.1 ± 0.542.8 ± 1.140.1±0.838.7±0.737.1± 0.535.6±0.9
+ +Table 5: Performance of the state-of-the-art existing approaches with and without $\mathcal { L } _ { \mathrm { R R R } }$ on CIFAR100 in 10 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3 runs. Figure 3a in the main paper is generated using numbers in this Table. + +
12345678910
iTAML+RRR89.2± 0.592.3±0.789.5 ± 1.287.5 ± 1.284.1±0.883.5±0.983.9±0.781.2± 0.379.6 ± 0.979.7±0.5
iTAML89.2± 0.588.9±0.587.0 ± 1.185.7 ± 1.184.1 ± 1.181.8 ± 0.380.0±0.679.0± 0.378.6 ±0.877.8 ±0.6
BiC90.3±0.782.1 ± 0.775.1± 0.469.8 ± 1.265.3±0.861.3± 0.957.4± 0.754.9± 0.553.2 ±0.950.3±0.7
BiC+RRR90.3±0.784.9 ± 1.176.4± 0.669.3±0.365.1±0.963.3± 0.459.7 ± 1.155.4± 0.855.8 ± 0.752.1 ± 0.5
EEIL80.0±0.780.5± 1.275.5± 0.971.5± 0.468.0± 1.262.0±0.959.0± 0.755.1 ± 1.251.4 ± 0.848.7 ± 0.4
EEIL+RRR80.0±0.783.5± 0.378.7 ± 1.274.0 ± 1.271.7± 0.365.1 ± 0.461.2± 0.557.6± 0.554.1 ± 0.451.7± 0.3
LwF86.1 ± 1.269.0±0.755.0±0.345.8± 0.340.4± 0.536.7±0.930.8 ±0.728.6±0.526.1 ± 0.724.2 ±0.7
LwF+RRR86.1 ± 1.272.4± 0.857.0 ± 1.148.3 ± 0.343.2 ±0.839.3 ±0.534.1 ± 0.632.1 ± 1.129.8 ±0.727.1± 0.6
EWC86.1 ± 1.252.6 ± 0.448.6± 0.438.5±0.531.1 ± 0.926.5±0.321.7±0.620.0±0.718.9 ± 0.516.6 ±0.9
EWC+RRR86.1 ± 1.256.0± 0.453.9 ± 1.244.4 ± 0.935.1±0.528.6±0.625.1 ± 1.123.1± 0.518.8 ±0.919.0 ± 1.2
ER86.1 ± 1.274.5 ± 0.965.2±0.862.5± 0.856.7±0.750.5± 0.347.6 ± 0.443.4± 0.341.6 ± 0.938.1 ± 1.1
RRR86.1 ± 1.278.5± 0.969.2 ± 1.163.5 ± 1.258.7±0.853.5 ± 1.149.6± 0.744.4± 0.342.6 ± 1.239.1 ± 1.1
+ +Table 7: Performance of the state-of-the-art existing approaches with and without $\mathcal { L } _ { \mathrm { R R R } }$ on ImageNet100 in 10 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3 runs. Figure 3c in the main paper is generated using numbers in this Table. + +
12345678910
iTAML99.4± 0.896.4±0.994.4 ± 0.993.0±0.392.4± 1.290.6±0.389.9 ±0.490.3±0.890.3 ±1.189.8±0.4
iTAML+RRR99.4± 0.897.3± 0.596.6±0.796.3 ± 1.195.3 ± 0.593.1 ± 0.592.1 ± 0.692.1±0.692.9±0.991.9 ± 0.4
EEIL99.5 ± 0.498.8 ± 1.195.9±0.993.0 ±0.488.3 ±1.186.7 ± 1.283.0 ±1.281.1± 0.578.2 ±0.775.4± 0.4
EEIL+RRR99.5± 0.498.1 ± 0.797.4 ± 1.196.7 ± 0.493.3 ± 0.589.4 ± 1.186.5±0.386.1 ± 1.181.8 ± 0.477.0± 0.3
BiC98.3±0.794.9 ±0.893.5±0.790.9 ± 1.289.0 ± 1.287.3± 0.685.2±0.783.2 ± 0.482.5±0.981.1 ± 1.1
BiC+RRR98.3±0.798.9 ±0.396.5±0.693.9 ± 0.492.0±0.789.3 ± 1.187.2 ±0.887.2 ± 1.185.5±0.984.1± 0.6
iCaRL99.3±0.497.2 ± 0.993.5±0.991.0 ± 0.387.5 ± 1.282.1 ± 1.277.1 ± 0.472.8± 0.667.1 ±0.863.5 ± 1.1
iCaRL+RRR99.3± 0.497.9 ± 1.294.1 ± 0.792.8 ±0.791.7 ± 0.985.7 ± 1.182.1 ± 0.674.4 ± 0.972.2 ± 0.868.1±0.9
LwF99.3±0.595.2 ± 0.985.9± 0.973.9 ± 1.163.7±0.854.8 ± 0.850.1 ± 0.644.5 ± 0.940.7 ± 0.536.7±0.3
LwF+RRR99.3±0.597.1 ± 1.289.3 ±0.679.1 ± 0.569.1 ± 1.159.4 ± 1.157.2 ± 0.748.2 ± 1.145.1 ± 0.641.5 ± 0.5
FT99.3± 0.549.4 ± 0.332.6±0.324.7 ± 0.620.0 ± 1.216.7 ± 0.313.9 ± 0.312.3 ± 0.711.1 ± 0.69.9 ±0.7
ER99.3± 0.595.2 ± 0.888.1±0.878.1± 0.972.5 ± 0.669.1± 0.867.1 ± 0.661.8 ±0.655.1± 0.350.1 ± 1.1
RRR99.3± 0.596.5 ± 0.393.4±0.884.8±0.778.7 ± 0.474.7 ± 0.473.1 ± 0.568.4±0.860.2±0.355.1 ±0.7
+ +Table 6: Performance of the state-of-the-art existing approaches with and without $\mathcal { L } _ { \mathrm { R R R } }$ on CIFAR100 in 20 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3 runs. Figure 3b in the main paper is generated using numbers in this Table. +(a) Tasks 1-10 + +
12345678910
iTAML84.7±0.685.7 ±0.486.5±0.386.5±0.886.3±1.285.7±0.884.9 ± 1.182.6±0.380.8±0.782.4± 0.3
iTAML+RRR84.7±0.689.9± 0.589.2±0.989.2±0.689.0 ± 1.187.2 ±0.688.0±0.485.6 ± 1.186.6±0.385.4±0.3
BiC95.7±0.690.3±0.980.9±0.875.8 ± 0.873.5±0.671.5 ± 1.267.8±0.465.4±0.862.7 ± 1.261.9 ± 1.2
BiC+RRR95.7± 0.693.3±0.684.7 ± 1.177.5 ± 0.973.4±0.674.8 ± 0.669.6± 0.767.4± 0.365.7± 0.564.9±0.6
EEIL81.9 ± 0.586.3±0.384.9 ± 0.480.7±0.377.7 ±0.674.9± 0.370.9±0.767.4± 0.764.9 ± 0.562.4±0.3
EEIL+RRR81.9±0.588.4±0.887.6±0.782.6 ± 1.278.5 ± 0.676.9 ± 0.471.2 ±0.767.3± 0.467.0 ± 1.264.5±0.3
LwF85.1 ± 0.768.8 ±0.958.6 ±1.150.5 ± 1.243.5±0.937.5 ± 0.633.7±0.930.4±0.926.8 ± 1.124.9 ± 0.7
LwF+RRR85.1±0.771.6 ± 0.661.8±0.754.2 ± 0.546.2±0.940.7 ±0.736.7 ± 1.234.4± 0.429.8±0.727.2 ± 1.2
EWC85.1± 0.761.3± 0.547.4± 0.836.2±0.331.3±0.627.9± 0.523.7 ± 1.122.5± 0.420.8±0.818.9±0.7
EWC+RRR85.1±0.768.9 ±0.552.2 ±0.939.9 ±0.935.2±0.330.0±0.324.3 ± 0.824.0±0.623.7 ± 0.421.0 ± 1.1
ER85.1± 0.783.1± 0.981.8±0.774.9 ± 0.370.4± 0.361.5 ± 1.260.8 ± 1.157.0±0.754.3 ± 0.448.2±0.6
RRR85.1± 0.785.1 ± 0.983.8±0.477.9 ± 0.472.4 ± 1.264.5 ± 0.762.8±0.759.0±0.357.3±0.851.2 ± 1.1
+ +(b) Tasks 11-20 + +
11121314151617181920
iTAML80.0 ± 1.180.6±0.574.3 ± 0.870.7±0.671.3 ± 1.168.3±0.570.3 ±0.868.3±0.669.5 ± 0.366.0±0.6
iTAML+RRR85.5±0.585.2±0.879.7± 0.674.3 ± 0.474.0± 0.973.4 ± 1.174.8± 0.974.4 ± 0.473.9 ±0.571.8±0.9
BiC59.2 ±0.457.0± 0.656.1 ± 1.255.7± 0.653.8±0.552.4 ± 1.249.7 ± 0.649.2 ± 1.247.7 ± 1.146.7 ± 1.2
BiC+RRR62.2 ± 0.559.1 ±0.758.2±0.557.8±0.554.4 ± 1.256.6±0.953.9 ±0.752.4 ± 1.149.5 ± 0.849.4± 0.9
EEIL60.9±0.659.5 ± 0.657.8±0.655.1 ± 0.353.9±0.551.7± 0.350.1±0.849.4± 0.547.4± 0.646.9 ± 0.9
EEIL+RRR63.7±0.662.9 ± 0.459.7 ± 0.457.0±0.355.6±0.853.5± 0.453.5±0.352.7 ± 0.449.1 ± 0.347.8 ± 0.4
LwF23.9 ±0.721.4± 0.720.0±0.719.1 ± 0.918.7±0.817.1 ± 0.815.6 ±0.814.7 ± 0.814.0 ± 0.813.7 ± 1.1
LwF+RRR27.7± 0.726.9 ±0.925.7±0.724.5 ± 1.223.6±0.622.6±0.719.5 ± 0.318.6 ± 0.519.7 ± 0.818.4± 1.2
EWC17.2 ± 1.116.0 ± 0.515.0± 0.814.5 ± 0.813.4 ± 1.112.4 ± 0.412.3 ± 0.411.5 ± 0.811.2 ± 0.89.44± 0.5
EWC+RRR20.7±0.319.5 ± 0.418.4± 0.717.3 ± 0.516.2 ± 0.415.8 ± 0.515.0 ± 0.716.6± 0.914.3 ± 0.413.2± 0.3
ER45.8 ± 0.642.7± 0.741.6 ± 0.641.2 ± 0.636.5±0.436.5±0.633.8± 0.432.4± 1.231.4± 0.730.2±0.5
RRR48.8± 0.346.7 ± 0.943.6 ± 1.144.2 ± 0.739.5±0.338.5±0.935.8±0.333.4± 0.332.4±0.331.2±0.3
\ No newline at end of file diff --git a/parse/train/tHgJoMfy6nI/tHgJoMfy6nI_content_list.json b/parse/train/tHgJoMfy6nI/tHgJoMfy6nI_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cdeba3122e626ed0b3d32de0d447deb6f08e8574 --- /dev/null +++ b/parse/train/tHgJoMfy6nI/tHgJoMfy6nI_content_list.json @@ -0,0 +1,1409 @@ +[ + { + "type": "text", + "text": "REMEMBERING FOR THE RIGHT REASONS: EXPLANATIONS REDUCE CATASTROPHIC FORGETTING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 689, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sayna Ebrahimi1, Suzanne Petryk1, Akash Gokul1, William Gan1, Joseph E. Gonzalez1, Marcus Rohrbach2, Trevor Darrell1 ", + "bbox": [ + 183, + 194, + 799, + 224 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1UC Berkeley, 2 Facebook AI Research \n{sayna,spetryk,akashgokul,wjgan,jegonzal,trevordarrell}@berkeley.edu \nmrf@fb.com ", + "bbox": [ + 186, + 228, + 784, + 272 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 310, + 544, + 325 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The goal of continual learning (CL) is to learn a sequence of tasks without suffering from the phenomenon of catastrophic forgetting. Previous work has shown that leveraging memory in the form of a replay buffer can reduce performance degradation on prior tasks. We hypothesize that forgetting can be further reduced when the model is encouraged to remember the evidence for previously made decisions. As a first step towards exploring this hypothesis, we propose a simple novel training paradigm, called Remembering for the Right Reasons (RRR), that additionally stores visual model explanations for each example in the buffer and ensures the model has “the right reasons” for its predictions by encouraging its explanations to remain consistent with those used to make decisions at training time. Without this constraint, there is a drift in explanations and increase in forgetting as conventional continual learning algorithms learn new tasks. We demonstrate how RRR can be easily added to any memory or regularizationbased approach and results in reduced forgetting, and more importantly, improved model explanations. We have evaluated our approach in the standard and few-shot settings and observed a consistent improvement across various CL approaches using different architectures and techniques to generate model explanations and demonstrated our approach showing a promising connection between explainability and continual learning. Our code is available at https://github.com/ SaynaEbrahimi/Remembering-for-the-Right-Reasons. ", + "bbox": [ + 233, + 342, + 764, + 619 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 648, + 336, + 664 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Humans are capable of continuously learning novel tasks by leveraging their lifetime knowledge and expanding them when they encounter a new experience. They can remember the majority of their prior knowledge despite the never-ending nature of their learning process by simply keeping a running tally of the observations distributed over time or presented in summary form. The field of continual learning or lifelong learning (Thrun & Mitchell, 1995; Silver et al., 2013) aims at maintaining previous performance and avoiding so-called catastrophic forgetting of previous experience (McCloskey & Cohen, 1989; McClelland et al., 1995) when learning new skills. The goal is to develop algorithms that continually update or add parameters to accommodate an online stream of data over time. ", + "bbox": [ + 174, + 680, + 825, + 805 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "An active line of research in continual learning explores the effectiveness of using small memory budgets to store data points from the training set (Castro et al., 2018; Rajasegaran et al., 2020; Rebuffi et al., 2017; Wu et al., 2019), gradients (Lopez-Paz et al., 2017), or storing an online generative model that can fake them later (Shin et al., 2017). Memory has been also exploited in the form of accommodating space for architecture growth and storage to fully recover the old performance when needed (Ebrahimi et al., 2020b; Rusu et al., 2016). Some methods store an old snapshot of the model to distill the features (Li & Hoiem, 2016) or attention maps (Dhar et al., 2019) between the teacher and student models. ", + "bbox": [ + 174, + 811, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d0281df291d1cac11c700eef36b889185db8c8c6f64b2f631bab1a588c539ccf.jpg", + "image_caption": [ + "Figure 1: An illustration of applying RRR paradigm. (Left) In a typical experience replay scenario, samples from prior tasks are kept in a memory buffer $\\mathcal { M } ^ { \\mathrm { r e p } }$ and revisited during training. (Right) In our proposed idea (RRR), in addition to ${ \\mathcal { M } } ^ { \\mathrm { r e p } }$ , we also store model explanations (saliency maps) as $\\mathcal { M } ^ { \\mathrm { R R R } }$ for those samples and encourage the model to remember the original reasoning for the prediction. Note that the saliency maps are small masks resulting in a negligible memory overhead (see Section 4.1). " + ], + "image_footnote": [], + "bbox": [ + 323, + 101, + 673, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The internal reasoning process of deep models is often treated as a black box and remains hidden from the user. However, recent work in explainable artificial intelligence (XAI) has developed methods to create human-interpretable explanations for model decisions (Simonyan et al., 2013; Zhang et al., 2018; Petsiuk et al., 2018; Zhou et al., 2016; Selvaraju et al., 2017). We posit that the catastrophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was used for a previously seen observation. Therefore, we hypothesize that forgetting can be mitigated when the model is encouraged to remember the evidence for previously made decisions. In other words, a model which can remember its final decision and can reconstruct the same prior reasoning. Based on this approach, we develop a novel strategy to exploit explainable models for improving performance. ", + "bbox": [ + 174, + 375, + 825, + 513 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Among the various explainability techniques proposed in XAI, saliency methods have emerged as a popular tool to identify the support of a model prediction in terms of relevant features in the input. These methods produce saliency maps, defined as regions of visual evidence upon which a network makes a decision. Our goal is to investigate whether augmenting experience replay with explanation replay reduces forgetting and how enforcing to remember the explanations will affect the explanations themselves. Figure 1 illustrates our proposed method. ", + "bbox": [ + 174, + 520, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we propose RRR, a training strategy guided by model explanations generated by any white-box differentiable explanation method; RRR adds an explanation loss to continual learning. White-box methods generate an explanation by using some internal state of the model, such as gradients, enabling their use in end-to-end training. We evaluate our approach using various popular explanation methods including vanilla backpropagation (Zeiler & Fergus, 2014), backpropagation with smoothing gradients (Smoothgrad) (Smilkov et al., 2017), Guided Backpropagation (Springenberg et al., 2014), and Gradient Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017) and compare their performance versus their computational feasibility. We integrate RRR into several state of the art class incremental learning (CIL) methods, including iTAML (Rajasegaran et al., 2020), EEIl (Castro et al., 2018), BiC (Wu et al., 2019), TOPIC (Tao et al., 2020), iCaRL (Rebuffi et al., 2017), EWC (Kirkpatrick et al., 2017), and LwF (Li & Hoiem, 2016). Note that RRR does not require task IDs at test time. We qualitatively and quantitatively analyze model explanations in the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence of tasks due to the requirement to focus on the the right evidence. We empirically show the effect of RRR in standard and few-shot class incremental learning (CIL) scenarios on popular benchmark datasets including CIFAR100, ImageNet100, and Caltech-UCSD Birds 200 using different network architectures where RRR improves overall accuracy and forgetting over experience replay and other memory-based method. ", + "bbox": [ + 173, + 611, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contribution is threefold: we first propose our novel, simple, yet effective memory constraint, which we call Remembering for the Right Reasons (RRR), and show that it reduces catastrophic forgetting by encouraging the model to look at the same explanations it initially found for its decisions. Second, we show how RRR can be readily combined with memory-based and regularization-based ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "CL methods to improve performance. Third, we demonstrate how guiding a continual learner to remember its explanations can improve the quality of the explanations themselves; i.e., the model looks at the right region in an image when making correct decisions while it focuses its maximum attention on the background when it misclassifies an object. ", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 BACKGROUND: WHITE-BOX EXPLANABILITY TECHNIQUES ", + "text_level": 1, + "bbox": [ + 174, + 179, + 700, + 196 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here we briefly review the explainability methods we have evaluated our approach with. The core idea behind RRR is to guide explanations or saliency maps during training to preserve their values. Hence, only gradient-based saliency techniques can be used which are differentiable and hence trainable during training for the mainstream task as opposed to black-box saliency methods which can be used only after training to determine important parts of an image. ", + "bbox": [ + 174, + 212, + 825, + 281 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Vanilla Backpropagation (Zeiler & Fergus, 2014): The simplest way to understand and visualize which pixels are most salient in an image is to look at the gradients. This is typically done by making a forward pass through the model and taking the gradient of the given output class with respect to the input. Those pixel-wise derivative values can be rendered as a normalized heatmap representing the amount of change in the output probability of a class caused by perturbing that pixel. To store a saliency map for each RGB image of size $3 \\times W \\times H$ , we need an equivalent memory size of storing $W \\times H$ pixel values. ", + "bbox": [ + 174, + 287, + 825, + 386 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Backpropagation with SmoothGrad: Smilkov et al. (2017) showed that the saliency maps obtained using raw gradients are visually noisy and using them as a proxy for feature importance is sub-optimal. They proposed a simple technique for denoising the gradients that adds pixel-wise Gaussian noise to $n$ copies of the image, and simply averages the resulting gradients. SmoothGrad requires the same amount of memory to store the saliency maps while it takes $n$ times longer to repeat generating each saliency map. We found $n = 4 0$ to be large enough to make a noticeable change in the saliencies in our experiments. ", + "bbox": [ + 174, + 392, + 825, + 491 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Gradient-weighted Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017): is a whitebox explainability technique which uses gradients to determine the influence of specific feature map activations on a given prediction. Because later layers in a convolutional neural network are known to encode higher-level semantics, taking the gradient of a model output with respect to the activations of these feature maps discovers which high-level semantics are important for the prediction. We refer to this layer as the target layer in our analysis. For example, when using Grad-CAM to visualize explanations for image classification, taking the gradient of the correct class prediction with respect to the last convolutional layer highlights class-discriminative regions in the image (such as the wings of a bird when identifying bird species). ", + "bbox": [ + 173, + 497, + 825, + 622 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Consider the pre-softmax score $y _ { c }$ for class $c$ in an image classification output. In general, any differentiable activation can be used. Consider also a single convolutional layer with $K$ feature maps, with a single feature map noted as $A ^ { k } \\in \\mathbb { R } ^ { u \\times v }$ . Grad-CAM takes the derivative of $y _ { c }$ with respect to each feature map $A ^ { k }$ . It then performs global average pooling over the height and width dimensions for each of these feature map gradients, getting a vector of length $K$ . Each element in this vector is used as a weight $\\alpha _ { k } ^ { c }$ , indicating the importance of feature map $k$ for the prediction $y _ { c }$ . Next, these importance weights are used in computing a linear combination of the feature maps. Followed by a ReLU (Jarrett et al., 2009) to zero-out any activations with a negative influence on the prediction of class $c$ , the final Grad-CAM output (s) is as below with $A _ { i j } ^ { k }$ defined at location $( i , j )$ in feature map $A ^ { k }$ . ", + "bbox": [ + 173, + 628, + 825, + 770 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/5911ba57e7dc3c03984f39cc0c9163a4456070f768cfefe513df46d7f04827a6.jpg", + "text": "$$\n\\begin{array} { r l } { \\alpha _ { k } ^ { c } = } & { { } \\frac { 1 } { u v } \\displaystyle \\sum _ { i = 1 } ^ { u } \\displaystyle \\sum _ { j = 1 } ^ { v } \\frac { \\partial y _ { c } } { \\partial A _ { i j } ^ { k } } \\qquad s _ { G r a d - C A M } ^ { c } = R e L U \\left( \\displaystyle \\sum _ { k = 1 } ^ { K } \\alpha _ { k } ^ { c } A ^ { k } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 289, + 772, + 709, + 813 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Unlike the common saliency map techniques of Guided BackProp (Springenberg et al., 2014), Guided GradCAM (Selvaraju et al., 2016), Integrated Gradients (Sundararajan et al., 2017b), Gradient $\\odot$ Input (Shrikumar et al., 2016), Backpropagation with SmoothGrad (Smilkov et al., 2017) etc., vanilla Backpropagation and Grad-CAM pass important “sanity checks” regarding their sensitivity to data and model parameters (Adebayo et al., 2018). We will compare using vanilla Backpropagation, Backpropagation with SmoothGrad, and Grad-CAM in RRR in Section 4. We will refer to the function that computes the output $s$ of these saliency method as $ { \\mathcal { X } } { \\mathcal { A } } { \\mathcal { T } }$ . ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 1 Remembering for the Right Reasons (RRR) for Continual Learning \n1: function TRAIN $( f _ { \\theta } , \\mathcal { D } ^ { t r } , \\mathcal { D } ^ { t s } )$ function UPDATE MEM(f kθ , Dtrk , Mrep, MRRR) \n2: $T$ : # of tasks, $n$ : # of samples in task (xi, k, yi) ∼ Dtrk \n3: R ← 0 ∈ R T ×T Mrep ← Mrep ∪ {(xi, k, yi)} \n4: Mrep ← {} sˆ ← X AI(f kθ (xi, k)) \n5: $\\mathcal { M } ^ { \\mathrm { R R R } } \\{ \\}$ MRRR ← MRRR ∪ {sˆ} \n6: for $k = 1$ to T do return Mrep, MRRR \n7: for $i = 1$ to n do end function \n8: Compute cross entropy on task $( \\mathcal { L } _ { \\mathrm { t a s k } } )$ \n9: Compute $\\mathcal { L } _ { \\mathrm { R R R } }$ using Eq. 2 function EVAL(f kθ , Dts{1···k}) \n10: $\\theta ^ { \\prime } \\theta - \\alpha \\nabla _ { \\theta } ( \\mathcal { L } _ { \\mathrm { t a s k } } + \\mathcal { L } _ { \\mathrm { R R R } } )$ for $i = 1$ to $k$ do \n11: end for Rk,i = Accuracy $( f _ { \\theta } ^ { k } ( x , i ) , y \\vert \\forall ( x , y ) \\in \\mathcal { D } _ { i } ^ { t s } )$ \n12: $\\begin{array} { r l } & { \\mathcal { M } ^ { \\mathrm { r e p } } , \\mathcal { M } ^ { \\mathrm { R R R } } \\gets \\mathrm { U P D A T E ~ M E M } ( f _ { \\theta } ^ { k } , \\mathcal { D } _ { k } ^ { t r } , \\mathcal { M } ^ { \\mathrm { r e p } } , } \\\\ & { } \\\\ & { R _ { k , \\{ 1 \\cdots k \\} } \\gets \\mathrm { E V A L } \\left( f _ { \\theta } ^ { k } , \\mathcal { D } _ { \\{ 1 \\cdots k \\} } ^ { t s } \\right) } \\end{array}$ , end for \nreturn $R$ \n13: end function \n14: end for \n15: return $f _ { \\boldsymbol { \\theta } } , R$ \n16: end function ", + "bbox": [ + 173, + 102, + 821, + 353 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 REMEMBERING FOR THE RIGHT REASONS (RRR) ", + "text_level": 1, + "bbox": [ + 174, + 375, + 619, + 391 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Memory-based methods in continual learning have achieved high performance on vision benchmarks using a small amount of memory, i.e. storing a few samples from the training data into the replay buffer to directly train with them when learning new tasks. This simple method, known as experience replay, has been explored and shown to be effective (Rebuffi et al., 2017; Wu et al., 2019; Castro et al., 2018; Rajasegaran et al., 2020; Ebrahimi et al., 2020b; Hayes et al., 2019; Riemer et al., 2018). In this work we aim to go one step further and investigate the role of explanations in continual learning, particularly on mitigating forgetting and change of model explanations. ", + "bbox": [ + 173, + 406, + 825, + 503 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We consider the problem of learning a sequence of $T$ data distributions $\\mathcal { D } ^ { t r } = \\{ \\mathcal { D } _ { 1 } ^ { t r } , \\cdot \\cdot \\cdot , \\mathcal { D } _ { T } ^ { t r } \\}$ , where $\\mathcal { D } _ { k } ^ { t r } = \\{ ( x _ { i } ^ { k } , y _ { i } ^ { k } ) _ { i = 1 } ^ { n _ { k } } \\}$ is the data distribution for task $k$ with $n$ sample tuples of input $( \\mathbf { x } ^ { k } \\subset \\mathcal { X } )$ and set of output labels $( \\mathbf { y } ^ { k } \\subset \\mathcal { V } )$ . The goal is to sequentially learn the model $f _ { \\theta } : \\mathcal { X } \\times \\mathcal { T } \\mathcal { Y }$ for each task that can map each input $x$ to its target output, $y$ , while maintaining its performance on all prior tasks. We aim to achieve this by using memory to enhance better knowledge transfer as well as better avoidance of catastrophic forgetting. We assume two limited memory pools $\\mathcal { M } ^ { \\mathrm { r e p } }$ for raw samples and $\\mathcal { M } ^ { \\mathrm { R R R } }$ for model explanations. In particular, $\\mathcal { M } ^ { \\mathrm { r e p } } = \\{ ( x _ { i } ^ { j } , y _ { i } ^ { j } ) _ { i = 1 } ^ { m } \\stackrel { \\cdot } { \\sim } \\mathcal { D } _ { j = 1 \\cdots k - 1 } ^ { t r } \\}$ stores $m$ samples in total from all prior tasks to $k$ . Similarly $\\mathcal { M } ^ { \\mathrm { R R R } }$ stores the saliency maps generated based on $f _ { \\theta } ^ { k }$ by one of the explanation methods $( { \\mathcal { X } } { \\mathcal { A } } { \\mathcal { T } } )$ discussed in Section 2 for images in $\\mathcal { M } ^ { \\mathrm { r e p } }$ where $f _ { \\theta } ^ { k }$ is $f _ { \\theta }$ being trained for task $k$ . We use a single-head architecture where the task ID integer $t$ is not required at test time. ", + "bbox": [ + 173, + 510, + 825, + 674 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Upon finishing the $k ^ { t h }$ task, we randomly select $m / ( k { - } 1 )$ samples per task from its training data and update our replay buffer memory ${ \\mathcal { M } } ^ { \\mathrm { r e p } }$ . RRR uses model explanations on memory samples to perform continual learning such that the model preserves its reasoning for previously seen observations. We explore several explanation techniques to compute saliency maps using $f _ { \\theta } ^ { \\mathbf { \\bar { k } } }$ for the stored samples in the replay buffer to populate the xai buffer memory $\\mathcal { M } ^ { \\mathrm { x a i } }$ . The stored saliency maps will serve as reference explanations during the learning of future tasks to prevent model parameters from being altered resulting in different reasoning for the same samples. We implement RRR using an L1 loss on the error in saliency maps generated after training a new task with respect to their stored reference evidence. ", + "bbox": [ + 173, + 679, + 825, + 808 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ca05e2d767f2280955f394944331ecddccfb977bf9d8c8fb3b870434e5a70c13.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { R R R } } \\big ( f _ { \\theta } , \\mathcal { M } ^ { \\mathrm { r e p } } , \\mathcal { M } ^ { \\mathrm { R R R } } \\big ) = \\mathbb { E } _ { ( ( x , y ) , \\hat { s } ) \\sim \\left( \\mathcal { M } ^ { \\mathrm { r e p } } , \\mathcal { M } ^ { \\mathrm { R R R } } \\right) } \\vert \\vert \\mathcal { X } \\mathcal { A } \\mathcal { Z } \\big ( f _ { \\theta } ^ { k } ( x ) \\big ) - \\hat { s } \\vert \\vert _ { 1 }\n$$", + "text_format": "latex", + "bbox": [ + 264, + 825, + 732, + 845 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { X } \\mathcal { A } \\mathcal { T } ( \\cdot )$ denotes the explanation method used to compute saliency maps using the model trained on the last seen example from task $k$ , and $\\hat { s }$ are the reference saliency maps generated by $\\chi \\mathcal { A } \\mathcal { T } ( f _ { \\theta } ^ { k } )$ upon learning each task prior to $T$ and stored in to the memory. We show below that combining RRR into the objective function of state-of-the-art memory and regularization-based methods results in significant performance improvements. The full algorithm for RRR is given in Alg. 1. ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/cb3d7154cc20f9b0d8c9e69038bd7edd29572c0f83a7677e8527ca6aa7b2c07f.jpg", + "image_caption": [ + "Figure 2: Few-shot CIL learning of CUB200 in 11 tasks where each point shows the classification accuracy on all seen classes so far. (Left) Shows ER with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ using different backbone architectures and saliency map techniques. (Right) Performance of the state-of-the-art existing approaches with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on CUB200 including TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), iCaRL (Rebuffi et al., 2017). Joint training serves as the upper bound. Results for baselines are obtained using their original implementation. All results are averaged over 3 runs and mean and standard deviation values are given in the appendix. Best viewed in color. " + ], + "image_footnote": [], + "bbox": [ + 207, + 104, + 789, + 260 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 401, + 326, + 417 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we apply RRR in two distinct learning regimes: standard and few-shot class incremental learning. These are the most challenging CL scenarios, in which task descriptions are not available at test time. We first explore the effect of backbone architecture and the saliency map technique on RRR performance. We then report our obtained results integrating $\\mathcal { L } _ { \\mathrm { R R R } }$ into existing memory-based and regularization-based methods. ", + "bbox": [ + 174, + 435, + 825, + 506 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 FEW-SHOT CIL PERFORMANCE ", + "text_level": 1, + "bbox": [ + 176, + 527, + 431, + 542 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We first explore CIL of low-data regimes where preventing overfitting to few-shot new classes is another challenge to overcome in addition to avoiding catastrophic forgetting of old classes. We use $C$ classes and $K$ training samples per class as the $C$ -way $K$ -shot few-shot class incrementally learning setting where we have a set of $b$ base classes to learn as the first task while the remaining classes are learned with only a few randomly selected samples. In order to provide a direct comparison to the state-of-the-art work of Tao et al. (2020) we precisely followed their setup and and used the same Caltech-UCSD Birds dataset (Wah et al., 2011), divided into 11 disjoint tasks and a 10-way 5-shot setting, where the first task contains $b = 1 0 0$ base classes resulting in 3000 samples for training and 2834 images for testing. The remaining 100 classes are divided into 10 tasks where 5 samples per class are randomly selected as the training set, while the test set is kept intact containing near 300 images per task. The images in CUB200 are resized to $2 5 6 \\times 2 5 6$ and then randomly cropped to $2 2 4 \\times 2 2 4$ for training. We store 4 images per class from base classes in the first task and 1 sample per each few-shot class in the remaining 10 tasks (Tao et al., 2020). We used the RAdam (Liu et al., 2019) optimizer with a learning rate of 0.001 which was reduced by 0.2 at epochs 20, 40, and 60 and trained for a total of 70 epochs with a batch size of 128 for the first and 10 for the remaining tasks. ", + "bbox": [ + 174, + 556, + 825, + 776 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 2 (left) shows results for ER with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ using different backbone architectures and saliency map techniques. Among the tested saliency map methods, Grad-CAM on ResNet18 outperforms Vanilla Backpropagation and SmoothGrad by $2 { - } 3 \\%$ while SmoothGrad and vanilla Backpropagation achieve similar CL performance. To compute the memory overhead of storing the output for a saliency method, if we assume the memory required to store an image is $M$ , vanilla Backpropagation and SmoothGrad generate a pixel-wise saliency map that occupies $M / 3$ of memory. However, in Grad-CAM the saliency map size is equal to the feature map of the target layer in the architecture. In our study with Grad-CAM we chose our target layer to be the last convolution layer before the fully-connected layers. For instance using ResNet18 for colored $2 2 4 \\times 2 2 4$ images results in the Grad-CAM output of $7 \\times 7$ occupying 196B. Table 2 shows the target layer name and saliency map size for other network architectures used in this work (AlexNet and SqueezeNet1 1) as well. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/c256de790417db433077726ead7e5e774741a08099f8b9eb519352448f894179.jpg", + "image_caption": [ + "Figure 3: Effect of RRR on existing methods for CIL on CIFAR100 in (a) 10 and (b) 20 tasks and (c) ImageNet100 in 10 tasks. Each point shows the classification accuracy on all seen classes so far. Results for iTAML, BiC, and EEIL are produced with their original implementation while EWC and LwF are re-implemented by us. All results are averaged over 3 runs and mean and standard deviation values are given in the appendix. Best viewed in color. " + ], + "image_footnote": [], + "bbox": [ + 184, + 101, + 812, + 227 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 347, + 821, + 376 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 2 (right) shows the effect of adding $\\mathcal { L } _ { \\mathrm { R R R } }$ on existing recent state-of-the-art methods such as TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), and iCaRL (Rebuffi et al., 2017). Tao et al. (2020) used a neural gas network (Martinetz et al., 1991; Fritzke et al., 1995) which can learn and preserve the topology of the feature manifold formed by different classes and we have followed their experimental protocol for our CUB200 experiment by using identical samples drawn in each task which are used across all the baselines for fair comparison. Adding $\\mathcal { L } _ { \\mathrm { R R R } }$ improves the performance of all the baselines; TOPIC becomes nearly on-par with joint training which serves as the upper bound and does not adhere to continual learning. The gap between ER and iCaRL is also reduced when ER uses $\\mathcal { L } _ { \\mathrm { R R R } }$ . ", + "bbox": [ + 174, + 382, + 825, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 STANDARD CIL PERFORMANCE ", + "text_level": 1, + "bbox": [ + 176, + 539, + 434, + 553 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In order to provide a direct comparison to the recent work of Rajasegaran et al. (2020) we perform our standard CIL experiment on CIFAR100 (Krizhevsky & Hinton, 2009) and ImageNet100 where we assume a memory budget of 2000 samples which are identical across all the baselines. Following Rajasegaran et al. (2020) we divide CIFAR100 to 10 and 20 disjoint tasks with 10 and 5 classes at a time. Figures 3a and 3b show the classification accuracy upon learning each task on all seen classes. We see a consistent average improvement of $2 - 4 \\%$ when $\\mathcal { L } _ { \\mathrm { R R R } }$ is added as an additional constraint to preserve the model explanations across all methods, from the most naive memory-based method, experience replay (ER), to more sophisticated approaches which store a set of old class exemplars along with meta-learning (iTAML), correct bias for new classes (BiC), and fine tune on the exemplar set (EEIL). We also applied the RRR constraint on regularization-based methods such as EWC and LwF with no memory used as a replay buffer. The accuracy for both improves despite not benefiting from revisiting the raw data. However, they fall behind all memory-based methods with or without $\\mathcal { L } _ { \\mathrm { R R R } }$ . The final accuracy on the entire sequence for joint training (multi-task learning) with RAdam optimizer (Liu et al., 2019) is $8 0 . 0 3 \\%$ which serves as an upper bound as it has access to data from all tasks at all time. ", + "bbox": [ + 174, + 569, + 825, + 776 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 3c shows our results on learning ImageNet100 in 10 tasks. The effect of adding $\\mathcal { L } _ { \\mathrm { R R R } }$ to baselines on the ImageNet100 experiment is more significant $( 3 - 6 \\% )$ compared to CIFAR100. This is mainly due to the larger size and better quality of images in ImageNet100, resulting in generating larger Grad-CAM saliency maps. These experiments clearly reveal the effectiveness of $\\mathcal { L } _ { \\mathrm { R R R } }$ on model explanations in a continual learning problem at nearly zero cost of memory overhead when a memory buffer is already created and applied as a catastrophic forgetting avoidance strategy. This makes Grad-CAM the ideal approach to generate saliency maps when applying the RRR training strategy, as it achieves the highest accuracy while utilizing the least storage space to store saliencies. Note that we adopt Grad-CAM to generate saliency maps in the remaining experiments in this paper. We also keep using only ResNet18 for a fair comparison with the literature. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/e2b1e19980c3f9b7ecb6e64f1143253a1cb79accb300528e13091a42fd07d04b.jpg", + "table_caption": [ + "(a) PG localization accuracy and backward transfer " + ], + "table_footnote": [], + "table_body": "
MethodsPG-ACC (%)PG-BWT (%)
ER54.0-17.4
ER+RRR58.5-15.6
TOPIC72.7-0.9
TOPIC+RRR74.2-2.1
", + "bbox": [ + 220, + 188, + 455, + 248 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/6f54cde8ae181ebad783837edda719aa3df99fab32acb6d83f9cc03ee08894b1.jpg", + "table_caption": [ + "Table 1: PG experiment results on few-shot CIL CUB200 measuring (a) PG-ACC $( \\% )$ and PG-BWT $( \\% )$ and (b) precision and recall averaged over all tasks. $P r _ { i , i }$ and $R e _ { i , i }$ evaluate the pointing game on each task $\\mathbf { t } ^ { \\mathrm { i } }$ directly after the model has been trained on $\\mathbf { t } ^ { \\mathbf { i } }$ . $P r _ { T , i }$ and $R e _ { T , i }$ are obtained by the evaluation for task $\\mathbf { t ^ { i } }$ using the model trained for all $T$ tasks. ", + "(b) Precision and recall using PG experiment " + ], + "table_footnote": [], + "table_body": "
PrecisionRecall
MethodsPri,iPrT,iRei,iReT,i
ER80.068.964.165.1
ER+RRR82.170.364.266.8
TOPIC91.088.498.197.4
TOPIC+RRR92.889.199.699.2
", + "bbox": [ + 537, + 179, + 794, + 258 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 ANALYSIS OF MODEL EXPLANATIONS ", + "text_level": 1, + "bbox": [ + 174, + 276, + 522, + 292 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section we want to answer the question “How often does the model remember its decision for the right reason upon learning a sequence of tasks?”. In particular, we want to evaluate how often the model is “pointing” at the right evidence for its predictions, instead of focusing its maximum attention on the background or other objects in the image. We use the Pointing Game experiment (PG) (Zhang et al., 2018) for this evaluation, which was introduced to measure the discriminativeness of a visualization method for target object localization. Here, we use ground truth segmentation annotation labels provided with the CUB-200 dataset to define the true object region. ", + "bbox": [ + 173, + 310, + 825, + 409 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "First, we look into hits and misses defined by the PG experiment. When the location of the maximum in a predicted saliency map falls inside the object, it is considered as a hit and otherwise it is a miss. Figure 5 shows an example from CUB200 where the segmentation annotation is used to determine whether the peak of the predicted saliency map (marked with red cross) falls inside the object region. This example is regarded as hit as the red cross is inside the segmentation mask for the bird. PG localization accuracy is defined as the number of hits over the total number of predictions. We would like to measure both the overall PG performance of a continual learner as well as how much learning new tasks causes it to forget its ability to hit the target object. For these metrics, inspired by (Lopez-Paz et al., 2017), we define $\\begin{array} { r } { \\mathrm { P G } { \\mathrm { - } } \\mathrm { A C C } = \\frac { 1 } { T } \\sum _ { i = 1 } ^ { \\bar { T } } \\dot { R _ { T , i } } } \\end{array}$ as the average PG localization accuracy computed over all prior tasks after training for each new task and $\\begin{array} { r } { \\mathrm { P G - B W T } = \\frac { 1 } { T - 1 } \\sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } } \\end{array}$ (backward transfer) which indicates how much learning new tasks has influenced the PG localization accuracy on previous tasks where $R _ { n , i }$ is the on task $i$ after learning the $n ^ { \\mathrm { t h } }$ task. Results for ER and TOPIC with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on CUB200 are shown in Table 1a. It shows how constraining different models to remember their initial evidence can lead to better localization of the bird across learning new tasks. ", + "bbox": [ + 174, + 415, + 825, + 631 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "However, PG performance does not capture all of our desired properties for a continual learner. Ideally, we not only want a model to predict the object correctly if it is looking at the right evidence, but also we want it to not predict an object if it is not able to find the right evidence for it. To measure how close our baselines can get to this ideal model when they are combined with $\\mathcal { L } _ { \\mathrm { R R R } }$ , we measure the precision as $\\mathrm { t p / ( t p / \\Delta \\mathfrak { p } ) }$ , and recall as $\\mathrm { t p / ( t p / \\Delta \\mathfrak { t } f n ) }$ . We evaluate these metrics once immediately after learning each task, denoted as $P r _ { i , i }$ and $R e _ { i , i }$ , respectively, and again at the end of the learning process of final task $T$ denoted as $P r _ { T , i }$ and $R e _ { T , i }$ , where the first subscript refers to the model ID and the second subscript is the test dataset ID on which the model is evaluated. The higher the precision for a model is, the less often it has made the right decision without looking at the right evidence. On the other hand, the higher the recall, the less often it makes a wrong decision despite looking at the correct evidence. We show our evaluation on these metrics in Table 1b for ER and TOPIC with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on CUB200 where $\\mathcal { L } _ { \\mathrm { R R R } }$ increases both precision and recall across all methods, demonstrating that our approach continually makes better predictions because it finds the right evidence for its decisions. ", + "bbox": [ + 173, + 638, + 825, + 832 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In our final analysis, we would like to visualize the evolution of saliency maps across learning a sequence of tasks. Figure 4 illustrates the evolution of saliency maps for an image from the test-set of the second task, which both ER without $\\mathcal { L } _ { \\mathrm { R R R } }$ (top row) and with $\\mathcal { L } _ { \\mathrm { R R R } }$ (bottom row) have seen during training for the second task. We only visualize the generated saliencies after finishing tasks #2, #5, #7, #9, and #11 for simplicity. We indicate the correctness of the prediction made by each model with ‘correct’ or ‘incorrect’ written on top of their corresponding saliency map. Our goal is to visualize if adding the loss term $\\mathcal { L } _ { \\mathrm { R R R } }$ prevents the drifting of explanations. Given the same input image, the ER without $\\mathcal { L } _ { \\mathrm { R R R } }$ model makes an incorrect prediction after being continually trained for 11 tasks while never recovering from its mistake. On the other hand, when it is combined with $\\mathcal { L } _ { \\mathrm { R R R } }$ . it is able to recover from an early mistake after task 5. Considering the saliency map obtained after finishing task one as a reference evidence, we can see that ER’s evidence drifts further from the reference. On the bottom row, the region of focus of ER+RRR remains nearly identical to its initial evidence, apart from one incorrect prediction. As applying $\\mathcal { L } _ { \\mathrm { R R R } }$ corrects its saliency back to the original, this prediction is corrected as well. This supports the conclusion that retaining the original saliency is important for retaining the original accuracy. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/09c0ee24c47d173aa2e6b5de78cbf95caefed73a635754290317195ebd291fa3.jpg", + "image_caption": [ + "Figure 4: An illustration of the progression of saliencies on an image from the test set of the second task, evaluated after the model is trained on tasks #2, #5, #7, #9, and #11 on CUB200. Failure case for ER w.o. $\\mathcal { L } _ { \\mathrm { R R R } }$ (top row), where saliency drifts from the original and the prediction becomes incorrect. $_ \\mathrm { E R + R R R }$ (bottom row) retains close to the original saliency as the model trains on more tasks, with the exception of Task #5 which it is able to correct later on. Its performance is retained as well, for saliencies that are close to the original. " + ], + "image_footnote": [], + "bbox": [ + 272, + 102, + 727, + 243 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 367, + 825, + 492 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 516, + 343, + 532 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Continual learning: Past work in CL has generally made use of either memory, model structure, or regularization to prevent catastrophic forgetting. Memory-based methods store some form of past experience into a replay buffer. However, the definition of “experience” varies between methods. Rehearsal-based methods use episodic memories as raw samples (Robins, 1995; Rebuffi et al., 2017; Riemer et al., 2018) or their gradients (Lopez-Paz et al., 2017; Chaudhry et al., 2019) for the model to revisit. Incremental Classifier and Representation Learning (iCaRL) (Rebuffi et al., 2017), is a class-incremental learner that uses a nearest-exemplar algorithm for classification and prevents catastrophic forgetting by using an episodic memory. iTAML (Rajasegaran et al., 2020) is a task-agnostic meta-learning algorithm that uses a momentum based strategy for meta-update and in addition to the object classification task, it predicts task labels during inference. An end-to-end incremental learning framework (EEIL) (Castro et al., 2018) also uses an exemplar set along with data augmentation and balanced fine-tuning to alleviate the imbalance between the old and new classes. Bias Correction Method (BiC) (Wu et al., 2019) is another class-incremental learning algorithm for large datasets in which a linear model is used to correct bias towards new classes using a fully connected layer. In contrast, pseudo-rehearsal methods generate the replay samples using a generative model such as an autoencoder (Kemker & Kanan, 2017) or a GAN (Kamra et al., 2017; Shin et al., 2017). Regularization-based methods define different metrics to measure importance and limit the changes on parameters accordingly (Kirkpatrick et al., 2017; Zenke et al., 2017; Ebrahimi et al., 2020a; Serra et al., 2018; Li & Hoiem, 2016; Dhar et al., 2019) but in general these methods have limited capacity. Structure-based methods control which portions of a model are responsible for specific tasks such that the model increases its capacity in a controlled fashion as more tasks are added. Inference for different tasks can be restricted to various neurons (Fernando et al., 2017; Yoon et al., 2018), columns (Rusu et al., 2016), task-specific modules (Ebrahimi et al., 2020b), or parameters selected by a mask (Mallya & Lazebnik, 2018; Serra et al., 2018). In RRR we explored the addition of explanations to replay buffer and showed that saliency-based explanations offer performance upgrade as well as improvement in explanations across all memory-based and regularization-based baselines we tried. ", + "bbox": [ + 173, + 550, + 825, + 922 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Visual explanation approaches or saliency methods attempt to produce a posterior explanation or a pseudo-probability map for the detected signals from the target object in the input image. These approaches can be broadly divided into three categories including activation, gradient, and perturbation based methods. Activation-based methods (Zhou et al., 2016; Selvaraju et al., 2017; Chattopadhay et al., 2018) use a weighted linear combination of feature maps whereas gradient-based methods (Baehrens et al., 2010; Sundararajan et al., 2017a; Springenberg et al., 2014; Shrikumar et al., 2017; Zhang et al., 2018) use the derivative of outputs w.r.t the input image to compute pixel-wise importance scores to generate attention maps. Methods in these categories are only applicable to differentiable models, including convolutional neural networks (CNNs). In contrast, perturbation-based methods are model-agnostic and produce saliency maps by observing the change in prediction when the input is perturbed (Petsiuk et al., 2018; Ribeiro et al., 2016; Ross et al., 2017; Zhou et al., 2014; Seo et al., 2018). While these methods attempt to identify if models are right for the wrong reason, Ross et al. (2017) took a step further and applied penalties to correct the explanations provided in supervised/unsupervised fashion during training. Selvaraju et al. (2019) used human explanations in the form of question and answering to bring model explanations closer to human answers. ", + "bbox": [ + 174, + 104, + 825, + 311 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 333, + 318, + 349 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we proposed the use of model explanations with continual learning algorithms to enhance better knowledge transfer as well as better recall of the previous tasks. The intuition behind our method is that encouraging a model to remember its evidence will increase the generalisability and rationality of recalled predictions and help retrieving the relevant aspects of each task. We advocate for the use of explainable AI as a tool to improve model performance, rather than as an artifact or interpretation of the model itself. We demonstrate that models which incorporate a “remember for the right reasons” constraint as part of a continual learning process can both be interpretable and more accurate. We empirically demonstrated the effectiveness of our approach in a variety of settings and provided an analysis of improved performance and explainability. ", + "bbox": [ + 174, + 366, + 825, + 491 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 512, + 285, + 527 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. 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Fewshot class-incremental learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12183–12192, 2020. ", + "bbox": [ + 174, + 193, + 823, + 236 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Sebastian Thrun and Tom M Mitchell. Lifelong robot learning. Robotics and autonomous systems, 15(1-2):25–46, 1995. ", + "bbox": [ + 173, + 244, + 823, + 272 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C. Wah, S. Branson, P. Welinder, P. Perona, and S. Belongie. The Caltech-UCSD Birds-200-2011 Dataset. Technical Report CNS-TR-2011-001, California Institute of Technology, 2011. ", + "bbox": [ + 174, + 281, + 823, + 310 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Yue Wu, Yinpeng Chen, Lijuan Wang, Yuancheng Ye, Zicheng Liu, Yandong Guo, and Yun Fu. Large scale incremental learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 374–382, 2019. 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", + "bbox": [ + 174, + 602, + 821, + 645 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 GRAD-CAM TARGET LAYERS ", + "text_level": 1, + "bbox": [ + 176, + 133, + 426, + 150 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Table 2 shows the target layer names used in Grad-CAM for different network architectures according to their standard PyTorch (Paszke et al., 2017) implementations. Saliency map size is equal to the activation map of the target layers. ", + "bbox": [ + 174, + 161, + 825, + 204 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/62b7e29ccbad9e6d2bd9434932e28de7bef6864f5424e9df20bf640dcc97999e.jpg", + "table_caption": [ + "Table 2: Target layer names and activation maps size for saliencies generated by different network architectures in Grad-CAM. " + ], + "table_footnote": [], + "table_body": "
Target layer name in PyTorch torchvision packageSaliency map size
SqueezeNet1_1features.0.12.expand3x313 ×13
AlexNetfeatures.0.1013 ×13
ResNet18features.7.1.conv27×7
", + "bbox": [ + 173, + 258, + 825, + 335 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B POINTING GAME VISUALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 382, + 490, + 398 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Figure 5 shows an example from CUB200 in the Pointing Game. We used the segmentation annotation to verify whether the peak of the predicted saliency map (marked with red cross) falls inside the object region. It is regarded as hit as the red cross is inside the segmentation mask for the bird. ", + "bbox": [ + 173, + 415, + 821, + 458 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/3e59b0dc386b86facc06d523b925d24c5ac95c5b1d5165a09da8cda4bb4f9375.jpg", + "image_caption": [ + "Figure 5: An example of PG evaluation as hit for an image in CUB200. Left: image saliency map overlaid on the image. Right: the segmentation label where the red cross marks the peak saliency. " + ], + "image_footnote": [], + "bbox": [ + 354, + 473, + 643, + 584 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C TABULAR RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 656, + 370, + 672 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section, we have tabulated results shown in Figure 2 and Figure 3 with means and standard deviations averaged over 3 runs. ", + "bbox": [ + 174, + 689, + 823, + 718 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/d91a0e49a90a15c0b2d1c86bd040faa5f4c70e3caf7ca5895edb811ccb835801.jpg", + "table_caption": [ + "Table 3: Classification accuracy of few-shot CIL learning of CUB200 at the end of 11 tasks for ER with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ using different backbone architectures and saliency map techniques. Results are averaged over 3 runs. Figure 2 (left) in the main paper is generated using numbers in this Table. " + ], + "table_footnote": [], + "table_body": "
234567891011
RN18-RRR-GCam67.8±0.853.5±0.745.6±0.6 39.6±0.735.3±0.932.3± 1.129.4±0.925.9±0.825.7±0.626.3±0.723.6±0.7
RN18-ER67.8±0.849.7 ±0.941.7 ±0.835.8 ±0.731.4 ± 0.928.5±0.825.5±0.822.1±0.821.8±0.822.5 ± 1.119.8± 0.9
RN18-RRR-Smooth67.8± 0.850.9±0.643.5 ± 0.937.0±0.833.0±0.729.5± 0.626.8±0.823.9 ±0.823.9±0.823.4± 0.821.5± 0.5
RN18-RRR-BP67.8±0.850.8±0.843.9 ±0.636.6 ±0.432.7±0.628.9±0.627.2± 0.523.8 ±0.623.8±0.624.0± 0.421.5± 0.6
RN18-Finetune67.8± 0.844.8 ± 0.632.2± 0.525.8±0.725.6 ± 0.725.2± 0.720.8±0.616.8 ± 0.718.8± 0.518.3 ± 0.417.1 ± 0.6
Alex-RRR-GCam56.7±0.746.6±0.543.9±0.741.3± 0.733.7 ± 0.527.4± 0.725.3±0.722.0±0.521.5±0.621.4± 0.621.2± 0.6
Alex-ER56.7± 0.744.6 ± 0.741.3 ± 0.738.7±0.731.1 ± 0.724.5± 0.722.6± 0.719.6 ± 0.619.1± 0.818.7 ± 0.819.1± 0.8
Alex-Finetune56.7±0.742.8 ± 0.839.6±0.836.9±0.829.5 ± 0.723.3±0.621.4± 0.817.9 ± 0.718.0 ±0.717.0 ± 0.516.9 ± 0.4
SQ-RRR-GCam46.8± 0.536.2 ±0.430.1±0.628.3±0.425.1 ± 0.523.4± 0.519.3± 0.619.0± 0.618.5± 0.518.4± 0.518.2 ±0.6
SQ-ER46.8 ± 0.533.2±0.5 27.1±0.625.3±0.622.1±0.5220.5±0.516.3± 0.416.0±0.615.5± 0.615.4 ± 0.615.2 ± 0.7
SQ-Finetune46.8 ± 0.532.0±0.7 25.2±0.723.9±0.720.2±0.8119.4 ± 0.414.9 ± 0.414.4 ± 0.513.8± 0.414.2 ± 0.513.7±0.6
", + "bbox": [ + 173, + 786, + 825, + 905 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/b7ce24f962799d22920c982182f82f9d1f679676f53370079e058adf9746ce99.jpg", + "table_caption": [ + "Table 4: Performance of the state-of-the-art existing approaches with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on CUB200 including TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), iCaRL (Rebuffi et al., 2017). Results for baselines are obtained using their original implementation. Results are averaged over 3 runs. Figure 2 (right) in the main paper is generated using numbers in this Table. " + ], + "table_footnote": [], + "table_body": "
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EEIL68.6±0.453.6± 0.447.9 ± 0.344.2±0.836.3±0.927.4 ± 1.225.9±0.724.7±0.523.9±0.724.1± 0.722.1 ± 0.5
EEIL+RRR68.6±0.456.6±0.550.9±0.648.3± 0.539.7 ± 1.231.4± 0.728.3± 1.228.0±0.626.5±0.627.4± 0.625.2±0.9
iCaRL68.6±0.452.6±0.748.6± 1.244.1 ± 0.536.6±0.329.5±0.927.8± 0.426.2±0.524.0±0.623.8±0.621.1± 0.7
iCaRL+RRR68.6±0.455.6± 1.253.6±0.747.1 ± 0.839.6±0.532.5±0.831.8 ± 0.429.2±0.627.0±0.827.8±0.624.1 ± 0.3
TOPIC68.6 ± 0.462.4 ± 0.854.8 ± 0.449.9 ± 1.245.2 ±0.641.4± 0.338.3±0.835.3±0.632.2± 0.328.3±0.626.2 ± 1.2
TOPIC+RRR68.6±0.462.5 ± 0.956.8 ± 0.451.5 ± 0.548.2 ± 0.444.4 ± 0.442.3±0.738.3±0.635.2±0.932.3±0.929.2 ± 0.5
FT68.6±0.444.8 ± 0.532.2±0.825.8± 0.425.6 ± 1.125.2± 0.720.8± 1.116.7± 0.418.8 ± 1.118.2± 0.317.1 ± 0.8
ER67.8±0.849.7± 0.941.7 ± 0.835.8 ±0.731.4± 0.928.5±0.825.5±0.822.1± 0.821.8± 0.622.5 ± 1.119.8±0.9
RRR67.8±0.853.5± 0.745.6± 0.639.6± 0.735.3±0.932.3 ± 1.129.4± 0.925.9± 0.825.7±0.626.3±0.723.6±0.7
JT68.6±0.462.4± 0.457.2 ± 0.452.8±0.549.5 ± 0.946.1 ± 0.542.8 ± 1.140.1±0.838.7±0.737.1± 0.535.6±0.9
", + "bbox": [ + 173, + 167, + 825, + 273 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/a36c84c93cbd9f6f56147d35481bdbe9a921307573394a0ff7cc54dd55d7ce76.jpg", + "table_caption": [ + "Table 5: Performance of the state-of-the-art existing approaches with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on CIFAR100 in 10 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3 runs. Figure 3a in the main paper is generated using numbers in this Table. " + ], + "table_footnote": [], + "table_body": "
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iTAML+RRR89.2± 0.592.3±0.789.5 ± 1.287.5 ± 1.284.1±0.883.5±0.983.9±0.781.2± 0.379.6 ± 0.979.7±0.5
iTAML89.2± 0.588.9±0.587.0 ± 1.185.7 ± 1.184.1 ± 1.181.8 ± 0.380.0±0.679.0± 0.378.6 ±0.877.8 ±0.6
BiC90.3±0.782.1 ± 0.775.1± 0.469.8 ± 1.265.3±0.861.3± 0.957.4± 0.754.9± 0.553.2 ±0.950.3±0.7
BiC+RRR90.3±0.784.9 ± 1.176.4± 0.669.3±0.365.1±0.963.3± 0.459.7 ± 1.155.4± 0.855.8 ± 0.752.1 ± 0.5
EEIL80.0±0.780.5± 1.275.5± 0.971.5± 0.468.0± 1.262.0±0.959.0± 0.755.1 ± 1.251.4 ± 0.848.7 ± 0.4
EEIL+RRR80.0±0.783.5± 0.378.7 ± 1.274.0 ± 1.271.7± 0.365.1 ± 0.461.2± 0.557.6± 0.554.1 ± 0.451.7± 0.3
LwF86.1 ± 1.269.0±0.755.0±0.345.8± 0.340.4± 0.536.7±0.930.8 ±0.728.6±0.526.1 ± 0.724.2 ±0.7
LwF+RRR86.1 ± 1.272.4± 0.857.0 ± 1.148.3 ± 0.343.2 ±0.839.3 ±0.534.1 ± 0.632.1 ± 1.129.8 ±0.727.1± 0.6
EWC86.1 ± 1.252.6 ± 0.448.6± 0.438.5±0.531.1 ± 0.926.5±0.321.7±0.620.0±0.718.9 ± 0.516.6 ±0.9
EWC+RRR86.1 ± 1.256.0± 0.453.9 ± 1.244.4 ± 0.935.1±0.528.6±0.625.1 ± 1.123.1± 0.518.8 ±0.919.0 ± 1.2
ER86.1 ± 1.274.5 ± 0.965.2±0.862.5± 0.856.7±0.750.5± 0.347.6 ± 0.443.4± 0.341.6 ± 0.938.1 ± 1.1
RRR86.1 ± 1.278.5± 0.969.2 ± 1.163.5 ± 1.258.7±0.853.5 ± 1.149.6± 0.744.4± 0.342.6 ± 1.239.1 ± 1.1
", + "bbox": [ + 173, + 368, + 825, + 494 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/e9095d279ac45c96e65fcb06e92f7aa485b9946a5073f03f6f7fc7070deee7c0.jpg", + "table_caption": [ + "Table 7: Performance of the state-of-the-art existing approaches with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on ImageNet100 in 10 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3 runs. Figure 3c in the main paper is generated using numbers in this Table. " + ], + "table_footnote": [], + "table_body": "
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iTAML99.4± 0.896.4±0.994.4 ± 0.993.0±0.392.4± 1.290.6±0.389.9 ±0.490.3±0.890.3 ±1.189.8±0.4
iTAML+RRR99.4± 0.897.3± 0.596.6±0.796.3 ± 1.195.3 ± 0.593.1 ± 0.592.1 ± 0.692.1±0.692.9±0.991.9 ± 0.4
EEIL99.5 ± 0.498.8 ± 1.195.9±0.993.0 ±0.488.3 ±1.186.7 ± 1.283.0 ±1.281.1± 0.578.2 ±0.775.4± 0.4
EEIL+RRR99.5± 0.498.1 ± 0.797.4 ± 1.196.7 ± 0.493.3 ± 0.589.4 ± 1.186.5±0.386.1 ± 1.181.8 ± 0.477.0± 0.3
BiC98.3±0.794.9 ±0.893.5±0.790.9 ± 1.289.0 ± 1.287.3± 0.685.2±0.783.2 ± 0.482.5±0.981.1 ± 1.1
BiC+RRR98.3±0.798.9 ±0.396.5±0.693.9 ± 0.492.0±0.789.3 ± 1.187.2 ±0.887.2 ± 1.185.5±0.984.1± 0.6
iCaRL99.3±0.497.2 ± 0.993.5±0.991.0 ± 0.387.5 ± 1.282.1 ± 1.277.1 ± 0.472.8± 0.667.1 ±0.863.5 ± 1.1
iCaRL+RRR99.3± 0.497.9 ± 1.294.1 ± 0.792.8 ±0.791.7 ± 0.985.7 ± 1.182.1 ± 0.674.4 ± 0.972.2 ± 0.868.1±0.9
LwF99.3±0.595.2 ± 0.985.9± 0.973.9 ± 1.163.7±0.854.8 ± 0.850.1 ± 0.644.5 ± 0.940.7 ± 0.536.7±0.3
LwF+RRR99.3±0.597.1 ± 1.289.3 ±0.679.1 ± 0.569.1 ± 1.159.4 ± 1.157.2 ± 0.748.2 ± 1.145.1 ± 0.641.5 ± 0.5
FT99.3± 0.549.4 ± 0.332.6±0.324.7 ± 0.620.0 ± 1.216.7 ± 0.313.9 ± 0.312.3 ± 0.711.1 ± 0.69.9 ±0.7
ER99.3± 0.595.2 ± 0.888.1±0.878.1± 0.972.5 ± 0.669.1± 0.867.1 ± 0.661.8 ±0.655.1± 0.350.1 ± 1.1
RRR99.3± 0.596.5 ± 0.393.4±0.884.8±0.778.7 ± 0.474.7 ± 0.473.1 ± 0.568.4±0.860.2±0.355.1 ±0.7
", + "bbox": [ + 173, + 601, + 825, + 736 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/676956df21d0f00bae734752c33d3bc22a5457a67dca5308216eef3e529b5321.jpg", + "table_caption": [ + "Table 6: Performance of the state-of-the-art existing approaches with and without $\\mathcal { L } _ { \\mathrm { R R R } }$ on CIFAR100 in 20 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3 runs. Figure 3b in the main paper is generated using numbers in this Table. ", + "(a) Tasks 1-10 " + ], + "table_footnote": [], + "table_body": "
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iTAML84.7±0.685.7 ±0.486.5±0.386.5±0.886.3±1.285.7±0.884.9 ± 1.182.6±0.380.8±0.782.4± 0.3
iTAML+RRR84.7±0.689.9± 0.589.2±0.989.2±0.689.0 ± 1.187.2 ±0.688.0±0.485.6 ± 1.186.6±0.385.4±0.3
BiC95.7±0.690.3±0.980.9±0.875.8 ± 0.873.5±0.671.5 ± 1.267.8±0.465.4±0.862.7 ± 1.261.9 ± 1.2
BiC+RRR95.7± 0.693.3±0.684.7 ± 1.177.5 ± 0.973.4±0.674.8 ± 0.669.6± 0.767.4± 0.365.7± 0.564.9±0.6
EEIL81.9 ± 0.586.3±0.384.9 ± 0.480.7±0.377.7 ±0.674.9± 0.370.9±0.767.4± 0.764.9 ± 0.562.4±0.3
EEIL+RRR81.9±0.588.4±0.887.6±0.782.6 ± 1.278.5 ± 0.676.9 ± 0.471.2 ±0.767.3± 0.467.0 ± 1.264.5±0.3
LwF85.1 ± 0.768.8 ±0.958.6 ±1.150.5 ± 1.243.5±0.937.5 ± 0.633.7±0.930.4±0.926.8 ± 1.124.9 ± 0.7
LwF+RRR85.1±0.771.6 ± 0.661.8±0.754.2 ± 0.546.2±0.940.7 ±0.736.7 ± 1.234.4± 0.429.8±0.727.2 ± 1.2
EWC85.1± 0.761.3± 0.547.4± 0.836.2±0.331.3±0.627.9± 0.523.7 ± 1.122.5± 0.420.8±0.818.9±0.7
EWC+RRR85.1±0.768.9 ±0.552.2 ±0.939.9 ±0.935.2±0.330.0±0.324.3 ± 0.824.0±0.623.7 ± 0.421.0 ± 1.1
ER85.1± 0.783.1± 0.981.8±0.774.9 ± 0.370.4± 0.361.5 ± 1.260.8 ± 1.157.0±0.754.3 ± 0.448.2±0.6
RRR85.1± 0.785.1 ± 0.983.8±0.477.9 ± 0.472.4 ± 1.264.5 ± 0.762.8±0.759.0±0.357.3±0.851.2 ± 1.1
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iTAML80.0 ± 1.180.6±0.574.3 ± 0.870.7±0.671.3 ± 1.168.3±0.570.3 ±0.868.3±0.669.5 ± 0.366.0±0.6
iTAML+RRR85.5±0.585.2±0.879.7± 0.674.3 ± 0.474.0± 0.973.4 ± 1.174.8± 0.974.4 ± 0.473.9 ±0.571.8±0.9
BiC59.2 ±0.457.0± 0.656.1 ± 1.255.7± 0.653.8±0.552.4 ± 1.249.7 ± 0.649.2 ± 1.247.7 ± 1.146.7 ± 1.2
BiC+RRR62.2 ± 0.559.1 ±0.758.2±0.557.8±0.554.4 ± 1.256.6±0.953.9 ±0.752.4 ± 1.149.5 ± 0.849.4± 0.9
EEIL60.9±0.659.5 ± 0.657.8±0.655.1 ± 0.353.9±0.551.7± 0.350.1±0.849.4± 0.547.4± 0.646.9 ± 0.9
EEIL+RRR63.7±0.662.9 ± 0.459.7 ± 0.457.0±0.355.6±0.853.5± 0.453.5±0.352.7 ± 0.449.1 ± 0.347.8 ± 0.4
LwF23.9 ±0.721.4± 0.720.0±0.719.1 ± 0.918.7±0.817.1 ± 0.815.6 ±0.814.7 ± 0.814.0 ± 0.813.7 ± 1.1
LwF+RRR27.7± 0.726.9 ±0.925.7±0.724.5 ± 1.223.6±0.622.6±0.719.5 ± 0.318.6 ± 0.519.7 ± 0.818.4± 1.2
EWC17.2 ± 1.116.0 ± 0.515.0± 0.814.5 ± 0.813.4 ± 1.112.4 ± 0.412.3 ± 0.411.5 ± 0.811.2 ± 0.89.44± 0.5
EWC+RRR20.7±0.319.5 ± 0.418.4± 0.717.3 ± 0.516.2 ± 0.415.8 ± 0.515.0 ± 0.716.6± 0.914.3 ± 0.413.2± 0.3
ER45.8 ± 0.642.7± 0.741.6 ± 0.641.2 ± 0.636.5±0.436.5±0.633.8± 0.432.4± 1.231.4± 0.730.2±0.5
RRR48.8± 0.346.7 ± 0.943.6 ± 1.144.2 ± 0.739.5±0.338.5±0.935.8±0.333.4± 0.332.4±0.331.2±0.3
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Previous work has shown", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 293, + 470, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 293, + 470, + 306 + ], + "score": 1.0, + "content": "that leveraging memory in the form of a replay buffer can reduce performance", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 304, + 470, + 317 + ], + "spans": [ + { + "bbox": [ + 141, + 304, + 470, + 317 + ], + "score": 1.0, + "content": "degradation on prior tasks. 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As a first step towards exploring this hypothesis, we propose a sim-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 338, + 469, + 350 + ], + "spans": [ + { + "bbox": [ + 142, + 338, + 469, + 350 + ], + "score": 1.0, + "content": "ple novel training paradigm, called Remembering for the Right Reasons (RRR),", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 348, + 469, + 360 + ], + "spans": [ + { + "bbox": [ + 142, + 348, + 469, + 360 + ], + "score": 1.0, + "content": "that additionally stores visual model explanations for each example in the buffer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 358, + 470, + 373 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 470, + 373 + ], + "score": 1.0, + "content": "and ensures the model has “the right reasons” for its predictions by encourag-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 370, + 470, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 470, + 382 + ], + "score": 1.0, + "content": "ing its explanations to remain consistent with those used to make decisions at", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 381, + 469, + 394 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 469, + 394 + ], + "score": 1.0, + "content": "training time. Without this constraint, there is a drift in explanations and in-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 393, + 469, + 405 + ], + "spans": [ + { + "bbox": [ + 141, + 393, + 469, + 405 + ], + "score": 1.0, + "content": "crease in forgetting as conventional continual learning algorithms learn new tasks.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 403, + 469, + 416 + ], + "spans": [ + { + "bbox": [ + 141, + 403, + 469, + 416 + ], + "score": 1.0, + "content": "We demonstrate how RRR can be easily added to any memory or regularization-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 414, + 469, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 469, + 427 + ], + "score": 1.0, + "content": "based approach and results in reduced forgetting, and more importantly, improved", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 425, + 469, + 437 + ], + "spans": [ + { + "bbox": [ + 141, + 425, + 469, + 437 + ], + "score": 1.0, + "content": "model explanations. We have evaluated our approach in the standard and few-shot", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 437, + 469, + 448 + ], + "spans": [ + { + "bbox": [ + 141, + 437, + 469, + 448 + ], + "score": 1.0, + "content": "settings and observed a consistent improvement across various CL approaches", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 447, + 470, + 459 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 470, + 459 + ], + "score": 1.0, + "content": "using different architectures and techniques to generate model explanations and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 457, + 470, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 457, + 470, + 471 + ], + "score": 1.0, + "content": "demonstrated our approach showing a promising connection between explainabil-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 469, + 469, + 481 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 469, + 481 + ], + "score": 1.0, + "content": "ity and continual learning. 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The field of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "continual learning or lifelong learning (Thrun & Mitchell, 1995; Silver et al., 2013) aims at main-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "taining previous performance and avoiding so-called catastrophic forgetting of previous experience", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "(McCloskey & Cohen, 1989; McClelland et al., 1995) when learning new skills. The goal is to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 630 + ], + "score": 1.0, + "content": "develop algorithms that continually update or add parameters to accommodate an online stream of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 168, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 168, + 639 + ], + "score": 1.0, + "content": "data over time.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 540, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "An active line of research in continual learning explores the effectiveness of using small memory", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "budgets to store data points from the training set (Castro et al., 2018; Rajasegaran et al., 2020; Re-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "buffi et al., 2017; Wu et al., 2019), gradients (Lopez-Paz et al., 2017), or storing an online generative", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "model that can fake them later (Shin et al., 2017). 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(Left) In a typical experience replay scenario,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 326, + 232 + ], + "score": 1.0, + "content": "samples from prior tasks are kept in a memory buffer", + "type": "text" + }, + { + "bbox": [ + 326, + 219, + 348, + 229 + ], + "score": 0.89, + "content": "\\mathcal { M } ^ { \\mathrm { r e p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "and revisited during training. (Right)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 279, + 243 + ], + "score": 1.0, + "content": "In our proposed idea (RRR), in addition to", + "type": "text" + }, + { + "bbox": [ + 279, + 230, + 301, + 240 + ], + "score": 0.9, + "content": "{ \\mathcal { M } } ^ { \\mathrm { r e p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 228, + 506, + 243 + ], + "score": 1.0, + "content": ", we also store model explanations (saliency maps)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 238, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 118, + 255 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 240, + 146, + 251 + ], + "score": 0.89, + "content": "\\mathcal { M } ^ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 238, + 506, + 255 + ], + "score": 1.0, + "content": "for those samples and encourage the model to remember the original reasoning for the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "prediction. 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However, recent work in explainable artificial intelligence (XAI) has developed meth-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "ods to create human-interpretable explanations for model decisions (Simonyan et al., 2013; Zhang", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "et al., 2018; Petsiuk et al., 2018; Zhou et al., 2016; Selvaraju et al., 2017). We posit that the catas-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "trophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 364 + ], + "score": 1.0, + "content": "used for a previously seen observation. Therefore, we hypothesize that forgetting can be mitigated", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "when the model is encouraged to remember the evidence for previously made decisions. In other", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "words, a model which can remember its final decision and can reconstruct the same prior reasoning.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "Based on this approach, we develop a novel strategy to exploit explainable models for improving", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 162, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 162, + 408 + ], + "score": 1.0, + "content": "performance.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "Among the various explainability techniques proposed in XAI, saliency methods have emerged as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "a popular tool to identify the support of a model prediction in terms of relevant features in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "input. These methods produce saliency maps, defined as regions of visual evidence upon which a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "network makes a decision. Our goal is to investigate whether augmenting experience replay with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "explanation replay reduces forgetting and how enforcing to remember the explanations will affect", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 390, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 390, + 479 + ], + "score": 1.0, + "content": "the explanations themselves. Figure 1 illustrates our proposed method.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 504, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 504, + 498 + ], + "score": 1.0, + "content": "In this work, we propose RRR, a training strategy guided by model explanations generated by any", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 495, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 504, + 509 + ], + "score": 1.0, + "content": "white-box differentiable explanation method; RRR adds an explanation loss to continual learning.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "White-box methods generate an explanation by using some internal state of the model, such as gra-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "dients, enabling their use in end-to-end training. We evaluate our approach using various popular", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "explanation methods including vanilla backpropagation (Zeiler & Fergus, 2014), backpropagation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "with smoothing gradients (Smoothgrad) (Smilkov et al., 2017), Guided Backpropagation (Sprin-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "genberg et al., 2014), and Gradient Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "and compare their performance versus their computational feasibility. We integrate RRR into sev-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "eral state of the art class incremental learning (CIL) methods, including iTAML (Rajasegaran et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "2020), EEIl (Castro et al., 2018), BiC (Wu et al., 2019), TOPIC (Tao et al., 2020), iCaRL (Rebuffi", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "et al., 2017), EWC (Kirkpatrick et al., 2017), and LwF (Li & Hoiem, 2016). Note that RRR does", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "not require task IDs at test time. We qualitatively and quantitatively analyze model explanations in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "of tasks due to the requirement to focus on the the right evidence. We empirically show the effect", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "of RRR in standard and few-shot class incremental learning (CIL) scenarios on popular benchmark", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "datasets including CIFAR100, ImageNet100, and Caltech-UCSD Birds 200 using different network", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "score": 1.0, + "content": "architectures where RRR improves overall accuracy and forgetting over experience replay and other", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 671, + 203, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 203, + 682 + ], + "score": 1.0, + "content": "memory-based method.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Our contribution is threefold: we first propose our novel, simple, yet effective memory constraint,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "which we call Remembering for the Right Reasons (RRR), and show that it reduces catastrophic for-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "getting by encouraging the model to look at the same explanations it initially found for its decisions.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "Second, we show how RRR can be readily combined with memory-based and regularization-based", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 198, + 80, + 412, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 198, + 80, + 412, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 198, + 80, + 412, + 199 + ], + "spans": [ + { + "bbox": [ + 198, + 80, + 412, + 199 + ], + "score": 0.973, + "type": "image", + "image_path": "d0281df291d1cac11c700eef36b889185db8c8c6f64b2f631bab1a588c539ccf.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 198, + 80, + 412, + 93.22222222222223 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 198, + 93.22222222222223, + 412, + 106.44444444444446 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 198, + 106.44444444444446, + 412, + 119.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 198, + 119.66666666666669, + 412, + 132.8888888888889 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 198, + 132.8888888888889, + 412, + 146.11111111111114 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 198, + 146.11111111111114, + 412, + 159.33333333333337 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 198, + 159.33333333333337, + 412, + 172.5555555555556 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 198, + 172.5555555555556, + 412, + 185.77777777777783 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 198, + 185.77777777777783, + 412, + 199.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 274 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 222 + ], + "score": 1.0, + "content": "Figure 1: An illustration of applying RRR paradigm. (Left) In a typical experience replay scenario,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 326, + 232 + ], + "score": 1.0, + "content": "samples from prior tasks are kept in a memory buffer", + "type": "text" + }, + { + "bbox": [ + 326, + 219, + 348, + 229 + ], + "score": 0.89, + "content": "\\mathcal { M } ^ { \\mathrm { r e p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "and revisited during training. (Right)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 279, + 243 + ], + "score": 1.0, + "content": "In our proposed idea (RRR), in addition to", + "type": "text" + }, + { + "bbox": [ + 279, + 230, + 301, + 240 + ], + "score": 0.9, + "content": "{ \\mathcal { M } } ^ { \\mathrm { r e p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 228, + 506, + 243 + ], + "score": 1.0, + "content": ", we also store model explanations (saliency maps)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 238, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 118, + 255 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 240, + 146, + 251 + ], + "score": 0.89, + "content": "\\mathcal { M } ^ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 238, + 506, + 255 + ], + "score": 1.0, + "content": "for those samples and encourage the model to remember the original reasoning for the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "prediction. Note that the saliency maps are small masks resulting in a negligible memory overhead", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 177, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 177, + 274 + ], + "score": 1.0, + "content": "(see Section 4.1).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + } + ], + "index": 7.75 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "The internal reasoning process of deep models is often treated as a black box and remains hidden", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "from the user. However, recent work in explainable artificial intelligence (XAI) has developed meth-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "ods to create human-interpretable explanations for model decisions (Simonyan et al., 2013; Zhang", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "et al., 2018; Petsiuk et al., 2018; Zhou et al., 2016; Selvaraju et al., 2017). We posit that the catas-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "trophic forgetting phenomenon is due in part to not being able to rely on the same reasoning as was", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 364 + ], + "score": 1.0, + "content": "used for a previously seen observation. Therefore, we hypothesize that forgetting can be mitigated", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "when the model is encouraged to remember the evidence for previously made decisions. In other", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "words, a model which can remember its final decision and can reconstruct the same prior reasoning.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 399 + ], + "score": 1.0, + "content": "Based on this approach, we develop a novel strategy to exploit explainable models for improving", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 396, + 162, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 162, + 408 + ], + "score": 1.0, + "content": "performance.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 297, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "Among the various explainability techniques proposed in XAI, saliency methods have emerged as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "a popular tool to identify the support of a model prediction in terms of relevant features in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "input. These methods produce saliency maps, defined as regions of visual evidence upon which a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "network makes a decision. Our goal is to investigate whether augmenting experience replay with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "explanation replay reduces forgetting and how enforcing to remember the explanations will affect", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 390, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 390, + 479 + ], + "score": 1.0, + "content": "the explanations themselves. Figure 1 illustrates our proposed method.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 413, + 506, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 504, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 504, + 498 + ], + "score": 1.0, + "content": "In this work, we propose RRR, a training strategy guided by model explanations generated by any", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 495, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 504, + 509 + ], + "score": 1.0, + "content": "white-box differentiable explanation method; RRR adds an explanation loss to continual learning.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "White-box methods generate an explanation by using some internal state of the model, such as gra-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "dients, enabling their use in end-to-end training. We evaluate our approach using various popular", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "explanation methods including vanilla backpropagation (Zeiler & Fergus, 2014), backpropagation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "with smoothing gradients (Smoothgrad) (Smilkov et al., 2017), Guided Backpropagation (Sprin-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "genberg et al., 2014), and Gradient Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "and compare their performance versus their computational feasibility. We integrate RRR into sev-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "eral state of the art class incremental learning (CIL) methods, including iTAML (Rajasegaran et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "2020), EEIl (Castro et al., 2018), BiC (Wu et al., 2019), TOPIC (Tao et al., 2020), iCaRL (Rebuffi", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "et al., 2017), EWC (Kirkpatrick et al., 2017), and LwF (Li & Hoiem, 2016). Note that RRR does", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "not require task IDs at test time. We qualitatively and quantitatively analyze model explanations in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "the form of saliency maps and demonstrate that RRR remembers its earlier decisions in a sequence", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "of tasks due to the requirement to focus on the the right evidence. We empirically show the effect", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "of RRR in standard and few-shot class incremental learning (CIL) scenarios on popular benchmark", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "datasets including CIFAR100, ImageNet100, and Caltech-UCSD Birds 200 using different network", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 674 + ], + "score": 1.0, + "content": "architectures where RRR improves overall accuracy and forgetting over experience replay and other", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 671, + 203, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 203, + 682 + ], + "score": 1.0, + "content": "memory-based method.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 484, + 506, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Our contribution is threefold: we first propose our novel, simple, yet effective memory constraint,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "which we call Remembering for the Right Reasons (RRR), and show that it reduces catastrophic for-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "getting by encouraging the model to look at the same explanations it initially found for its decisions.", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "Second, we show how RRR can be readily combined with memory-based and regularization-based", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 687, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "CL methods to improve performance. Third, we demonstrate how guiding a continual learner to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "remember its explanations can improve the quality of the explanations themselves; i.e., the model", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "looks at the right region in an image when making correct decisions while it focuses its maximum", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 346, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 346, + 127 + ], + "score": 1.0, + "content": "attention on the background when it misclassifies an object.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 142, + 429, + 156 + ], + "lines": [ + { + "bbox": [ + 104, + 141, + 432, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 432, + 159 + ], + "score": 1.0, + "content": "2 BACKGROUND: WHITE-BOX EXPLANABILITY TECHNIQUES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "Here we briefly review the explainability methods we have evaluated our approach with. The core", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "idea behind RRR is to guide explanations or saliency maps during training to preserve their values.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "score": 1.0, + "content": "Hence, only gradient-based saliency techniques can be used which are differentiable and hence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "trainable during training for the mainstream task as opposed to black-box saliency methods which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 398, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 398, + 226 + ], + "score": 1.0, + "content": "can be used only after training to determine important parts of an image.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "Vanilla Backpropagation (Zeiler & Fergus, 2014): The simplest way to understand and visualize", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "which pixels are most salient in an image is to look at the gradients. This is typically done by making", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "a forward pass through the model and taking the gradient of the given output class with respect to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "the input. Those pixel-wise derivative values can be rendered as a normalized heatmap representing", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "the amount of change in the output probability of a class caused by perturbing that pixel. To store", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 288, + 297 + ], + "score": 1.0, + "content": "a saliency map for each RGB image of size", + "type": "text" + }, + { + "bbox": [ + 288, + 284, + 341, + 294 + ], + "score": 0.91, + "content": "3 \\times W \\times H", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 284, + 506, + 297 + ], + "score": 1.0, + "content": ", we need an equivalent memory size of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 223, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 137, + 307 + ], + "score": 1.0, + "content": "storing", + "type": "text" + }, + { + "bbox": [ + 137, + 295, + 170, + 305 + ], + "score": 0.9, + "content": "W \\times H", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 295, + 223, + 307 + ], + "score": 1.0, + "content": "pixel values.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 311, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 504, + 324 + ], + "score": 1.0, + "content": "Backpropagation with SmoothGrad: Smilkov et al. (2017) showed that the saliency maps ob-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "tained using raw gradients are visually noisy and using them as a proxy for feature importance is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "sub-optimal. They proposed a simple technique for denoising the gradients that adds pixel-wise", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 180, + 357 + ], + "score": 1.0, + "content": "Gaussian noise to", + "type": "text" + }, + { + "bbox": [ + 180, + 347, + 187, + 354 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "copies of the image, and simply averages the resulting gradients. SmoothGrad", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 431, + 368 + ], + "score": 1.0, + "content": "requires the same amount of memory to store the saliency maps while it takes", + "type": "text" + }, + { + "bbox": [ + 431, + 357, + 439, + 365 + ], + "score": 0.66, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "times longer to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 304, + 380 + ], + "score": 1.0, + "content": "repeat generating each saliency map. We found", + "type": "text" + }, + { + "bbox": [ + 304, + 367, + 338, + 376 + ], + "score": 0.89, + "content": "n = 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "to be large enough to make a noticeable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 283, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 283, + 390 + ], + "score": 1.0, + "content": "change in the saliencies in our experiments.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 504, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 504, + 406 + ], + "score": 1.0, + "content": "Gradient-weighted Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017): is a white-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 419 + ], + "score": 1.0, + "content": "box explainability technique which uses gradients to determine the influence of specific feature map", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 416, + 504, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 504, + 428 + ], + "score": 1.0, + "content": "activations on a given prediction. Because later layers in a convolutional neural network are known", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "to encode higher-level semantics, taking the gradient of a model output with respect to the activations", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "of these feature maps discovers which high-level semantics are important for the prediction. We refer", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "to this layer as the target layer in our analysis. For example, when using Grad-CAM to visualize", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "explanations for image classification, taking the gradient of the correct class prediction with respect", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "to the last convolutional layer highlights class-discriminative regions in the image (such as the wings", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 268, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 268, + 495 + ], + "score": 1.0, + "content": "of a bird when identifying bird species).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 238, + 512 + ], + "score": 1.0, + "content": "Consider the pre-softmax score", + "type": "text" + }, + { + "bbox": [ + 238, + 501, + 249, + 510 + ], + "score": 0.85, + "content": "y _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 498, + 289, + 512 + ], + "score": 1.0, + "content": "for class", + "type": "text" + }, + { + "bbox": [ + 290, + 501, + 296, + 509 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "in an image classification output. In general, any", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 462, + 523 + ], + "score": 1.0, + "content": "differentiable activation can be used. Consider also a single convolutional layer with", + "type": "text" + }, + { + "bbox": [ + 462, + 510, + 473, + 520 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "feature", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 518, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 275, + 534 + ], + "score": 1.0, + "content": "maps, with a single feature map noted as", + "type": "text" + }, + { + "bbox": [ + 276, + 520, + 326, + 531 + ], + "score": 0.91, + "content": "A ^ { k } \\in \\mathbb { R } ^ { u \\times v }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 518, + 472, + 534 + ], + "score": 1.0, + "content": ". Grad-CAM takes the derivative of", + "type": "text" + }, + { + "bbox": [ + 473, + 522, + 483, + 532 + ], + "score": 0.85, + "content": "y _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 518, + 506, + 534 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 219, + 545 + ], + "score": 1.0, + "content": "respect to each feature map", + "type": "text" + }, + { + "bbox": [ + 219, + 531, + 233, + 542 + ], + "score": 0.87, + "content": "A ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 531, + 505, + 545 + ], + "score": 1.0, + "content": ". 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Each element in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 234, + 567 + ], + "score": 1.0, + "content": "this vector is used as a weight", + "type": "text" + }, + { + "bbox": [ + 235, + 554, + 247, + 566 + ], + "score": 0.89, + "content": "\\alpha _ { k } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 553, + 422, + 567 + ], + "score": 1.0, + "content": ", indicating the importance of feature map", + "type": "text" + }, + { + "bbox": [ + 423, + 554, + 429, + 564 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "for the prediction", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 107, + 566, + 117, + 576 + ], + "score": 0.81, + "content": "y _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 564, + 505, + 578 + ], + "score": 1.0, + "content": ". Next, these importance weights are used in computing a linear combination of the feature maps.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "Followed by a ReLU (Jarrett et al., 2009) to zero-out any activations with a negative influence on the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 584, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 584, + 182, + 601 + ], + "score": 1.0, + "content": "prediction of class", + "type": "text" + }, + { + "bbox": [ + 183, + 589, + 189, + 596 + ], + "score": 0.68, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 584, + 388, + 601 + ], + "score": 1.0, + "content": ", the final Grad-CAM output (s) is as below with", + "type": "text" + }, + { + "bbox": [ + 388, + 586, + 403, + 600 + ], + "score": 0.92, + "content": "A _ { i j } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 584, + 483, + 601 + ], + "score": 1.0, + "content": "defined at location", + "type": "text" + }, + { + "bbox": [ + 483, + 587, + 504, + 599 + ], + "score": 0.91, + "content": "( i , j )", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 183, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 167, + 612 + ], + "score": 1.0, + "content": "in feature map", + "type": "text" + }, + { + "bbox": [ + 167, + 599, + 180, + 610 + ], + "score": 0.87, + "content": "A ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 599, + 183, + 612 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 612, + 434, + 644 + ], + "lines": [ + { + "bbox": [ + 177, + 612, + 434, + 644 + ], + "spans": [ + { + "bbox": [ + 177, + 612, + 434, + 644 + ], + "score": 0.93, + "content": "\\begin{array} { r l } { \\alpha _ { k } ^ { c } = } & { { } \\frac { 1 } { u v } \\displaystyle \\sum _ { i = 1 } ^ { u } \\displaystyle \\sum _ { j = 1 } ^ { v } \\frac { \\partial y _ { c } } { \\partial A _ { i j } ^ { k } } \\qquad s _ { G r a d - C A M } ^ { c } = R e L U \\left( \\displaystyle \\sum _ { k = 1 } ^ { K } \\alpha _ { k } ^ { c } A ^ { k } \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "5911ba57e7dc3c03984f39cc0c9163a4456070f768cfefe513df46d7f04827a6.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 177, + 612, + 434, + 622.6666666666666 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 177, + 622.6666666666666, + 434, + 633.3333333333333 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 177, + 633.3333333333333, + 434, + 643.9999999999999 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "score": 1.0, + "content": "Unlike the common saliency map techniques of Guided BackProp (Springenberg et al., 2014),", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "Guided GradCAM (Selvaraju et al., 2016), Integrated Gradients (Sundararajan et al., 2017b), Gradi-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 121, + 690 + ], + "score": 1.0, + "content": "ent", + "type": "text" + }, + { + "bbox": [ + 121, + 678, + 130, + 687 + ], + "score": 0.81, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "Input (Shrikumar et al., 2016), Backpropagation with SmoothGrad (Smilkov et al., 2017) etc.,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "vanilla Backpropagation and Grad-CAM pass important “sanity checks” regarding their sensitivity", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "to data and model parameters (Adebayo et al., 2018). We will compare using vanilla Backpropaga-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "tion, Backpropagation with SmoothGrad, and Grad-CAM in RRR in Section 4. We will refer to the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 720, + 390, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 243, + 733 + ], + "score": 1.0, + "content": "function that computes the output", + "type": "text" + }, + { + "bbox": [ + 243, + 722, + 249, + 730 + ], + "score": 0.52, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 720, + 362, + 733 + ], + "score": 1.0, + "content": "of these saliency method as", + "type": "text" + }, + { + "bbox": [ + 362, + 721, + 386, + 730 + ], + "score": 0.72, + "content": " { \\mathcal { X } } { \\mathcal { A } } { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 720, + 390, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "CL methods to improve performance. Third, we demonstrate how guiding a continual learner to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "remember its explanations can improve the quality of the explanations themselves; i.e., the model", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "looks at the right region in an image when making correct decisions while it focuses its maximum", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 346, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 346, + 127 + ], + "score": 1.0, + "content": "attention on the background when it misclassifies an object.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 127 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 142, + 429, + 156 + ], + "lines": [ + { + "bbox": [ + 104, + 141, + 432, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 432, + 159 + ], + "score": 1.0, + "content": "2 BACKGROUND: WHITE-BOX EXPLANABILITY TECHNIQUES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "Here we briefly review the explainability methods we have evaluated our approach with. The core", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "idea behind RRR is to guide explanations or saliency maps during training to preserve their values.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "score": 1.0, + "content": "Hence, only gradient-based saliency techniques can be used which are differentiable and hence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "trainable during training for the mainstream task as opposed to black-box saliency methods which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 398, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 398, + 226 + ], + "score": 1.0, + "content": "can be used only after training to determine important parts of an image.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 168, + 506, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "Vanilla Backpropagation (Zeiler & Fergus, 2014): The simplest way to understand and visualize", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 240, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 253 + ], + "score": 1.0, + "content": "which pixels are most salient in an image is to look at the gradients. This is typically done by making", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "a forward pass through the model and taking the gradient of the given output class with respect to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "the input. Those pixel-wise derivative values can be rendered as a normalized heatmap representing", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "the amount of change in the output probability of a class caused by perturbing that pixel. To store", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 288, + 297 + ], + "score": 1.0, + "content": "a saliency map for each RGB image of size", + "type": "text" + }, + { + "bbox": [ + 288, + 284, + 341, + 294 + ], + "score": 0.91, + "content": "3 \\times W \\times H", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 284, + 506, + 297 + ], + "score": 1.0, + "content": ", we need an equivalent memory size of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 223, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 137, + 307 + ], + "score": 1.0, + "content": "storing", + "type": "text" + }, + { + "bbox": [ + 137, + 295, + 170, + 305 + ], + "score": 0.9, + "content": "W \\times H", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 295, + 223, + 307 + ], + "score": 1.0, + "content": "pixel values.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 229, + 506, + 307 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 311, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 504, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 504, + 324 + ], + "score": 1.0, + "content": "Backpropagation with SmoothGrad: Smilkov et al. (2017) showed that the saliency maps ob-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "tained using raw gradients are visually noisy and using them as a proxy for feature importance is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "sub-optimal. They proposed a simple technique for denoising the gradients that adds pixel-wise", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 180, + 357 + ], + "score": 1.0, + "content": "Gaussian noise to", + "type": "text" + }, + { + "bbox": [ + 180, + 347, + 187, + 354 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "copies of the image, and simply averages the resulting gradients. SmoothGrad", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 431, + 368 + ], + "score": 1.0, + "content": "requires the same amount of memory to store the saliency maps while it takes", + "type": "text" + }, + { + "bbox": [ + 431, + 357, + 439, + 365 + ], + "score": 0.66, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "times longer to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 304, + 380 + ], + "score": 1.0, + "content": "repeat generating each saliency map. We found", + "type": "text" + }, + { + "bbox": [ + 304, + 367, + 338, + 376 + ], + "score": 0.89, + "content": "n = 4 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "to be large enough to make a noticeable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 283, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 283, + 390 + ], + "score": 1.0, + "content": "change in the saliencies in our experiments.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 311, + 505, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 394, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 504, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 504, + 406 + ], + "score": 1.0, + "content": "Gradient-weighted Class Activation Mapping (Grad-CAM) (Selvaraju et al., 2017): is a white-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 419 + ], + "score": 1.0, + "content": "box explainability technique which uses gradients to determine the influence of specific feature map", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 416, + 504, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 504, + 428 + ], + "score": 1.0, + "content": "activations on a given prediction. Because later layers in a convolutional neural network are known", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "to encode higher-level semantics, taking the gradient of a model output with respect to the activations", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "of these feature maps discovers which high-level semantics are important for the prediction. We refer", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "to this layer as the target layer in our analysis. For example, when using Grad-CAM to visualize", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "explanations for image classification, taking the gradient of the correct class prediction with respect", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "to the last convolutional layer highlights class-discriminative regions in the image (such as the wings", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 268, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 268, + 495 + ], + "score": 1.0, + "content": "of a bird when identifying bird species).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 394, + 506, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 498, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 238, + 512 + ], + "score": 1.0, + "content": "Consider the pre-softmax score", + "type": "text" + }, + { + "bbox": [ + 238, + 501, + 249, + 510 + ], + "score": 0.85, + "content": "y _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 498, + 289, + 512 + ], + "score": 1.0, + "content": "for class", + "type": "text" + }, + { + "bbox": [ + 290, + 501, + 296, + 509 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "in an image classification output. In general, any", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 462, + 523 + ], + "score": 1.0, + "content": "differentiable activation can be used. 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This simple method, known as ex-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "perience replay, has been explored and shown to be effective (Rebuffi et al., 2017; Wu et al., 2019;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "Castro et al., 2018; Rajasegaran et al., 2020; Ebrahimi et al., 2020b; Hayes et al., 2019; Riemer", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "et al., 2018). In this work we aim to go one step further and investigate the role of explanations in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 388, + 470, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 470, + 401 + ], + "score": 1.0, + "content": "continual learning, particularly on mitigating forgetting and change of model explanations.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 319, + 418 + ], + "score": 1.0, + "content": "We consider the problem of learning a sequence of", + "type": "text" + }, + { + "bbox": [ + 320, + 405, + 329, + 415 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 402, + 403, + 418 + ], + "score": 1.0, + "content": "data distributions", + "type": "text" + }, + { + "bbox": [ + 403, + 404, + 501, + 417 + ], + "score": 0.91, + "content": "\\mathcal { D } ^ { t r } = \\{ \\mathcal { D } _ { 1 } ^ { t r } , \\cdot \\cdot \\cdot , \\mathcal { D } _ { T } ^ { t r } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 402, + 506, + 418 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 413, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 413, + 133, + 434 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 416, + 219, + 430 + ], + "score": 0.93, + "content": "\\mathcal { D } _ { k } ^ { t r } = \\{ ( x _ { i } ^ { k } , y _ { i } ^ { k } ) _ { i = 1 } ^ { n _ { k } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 413, + 339, + 434 + ], + "score": 1.0, + "content": "is the data distribution for task", + "type": "text" + }, + { + "bbox": [ + 340, + 417, + 347, + 427 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 413, + 366, + 434 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 367, + 419, + 374, + 427 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 413, + 464, + 434 + ], + "score": 1.0, + "content": "sample tuples of input", + "type": "text" + }, + { + "bbox": [ + 464, + 417, + 504, + 428 + ], + "score": 0.86, + "content": "( \\mathbf { x } ^ { k } \\subset \\mathcal { X } )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 427, + 504, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 205, + 442 + ], + "score": 1.0, + "content": "and set of output labels", + "type": "text" + }, + { + "bbox": [ + 205, + 429, + 245, + 441 + ], + "score": 0.89, + "content": "( \\mathbf { y } ^ { k } \\subset \\mathcal { V } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 427, + 428, + 442 + ], + "score": 1.0, + "content": ". 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We aim to achieve this by using memory to enhance better knowledge transfer as", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 467, + 475 + ], + "score": 1.0, + "content": "well as better avoidance of catastrophic forgetting. 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In particular,", + "type": "text" + }, + { + "bbox": [ + 356, + 473, + 504, + 488 + ], + "score": 0.92, + "content": "\\mathcal { M } ^ { \\mathrm { r e p } } = \\{ ( x _ { i } ^ { j } , y _ { i } ^ { j } ) _ { i = 1 } ^ { m } \\stackrel { \\cdot } { \\sim } \\mathcal { D } _ { j = 1 \\cdots k - 1 } ^ { t r } \\}", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 484, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 132, + 503 + ], + "score": 1.0, + "content": "stores", + "type": "text" + }, + { + "bbox": [ + 133, + 490, + 143, + 498 + ], + "score": 0.7, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 484, + 302, + 503 + ], + "score": 1.0, + "content": "samples in total from all prior tasks to", + "type": "text" + }, + { + "bbox": [ + 302, + 488, + 309, + 498 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 484, + 354, + 503 + ], + "score": 1.0, + "content": ". 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The stored saliency maps will", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 607 + ], + "score": 1.0, + "content": "serve as reference explanations during the learning of future tasks to prevent model parameters from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 619 + ], + "score": 1.0, + "content": "being altered resulting in different reasoning for the same samples. 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We show below that com-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "bining RRR into the objective function of state-of-the-art memory and regularization-based methods", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 491, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 491, + 733 + ], + "score": 1.0, + "content": "results in significant performance improvements. 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This simple method, known as ex-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "perience replay, has been explored and shown to be effective (Rebuffi et al., 2017; Wu et al., 2019;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "Castro et al., 2018; Rajasegaran et al., 2020; Ebrahimi et al., 2020b; Hayes et al., 2019; Riemer", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "et al., 2018). 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We show below that com-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "bining RRR into the objective function of state-of-the-art memory and regularization-based methods", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 491, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 491, + 733 + ], + "score": 1.0, + "content": "results in significant performance improvements. 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(Left) Shows ER with and without", + "type": "text" + }, + { + "bbox": [ + 395, + 229, + 417, + 240 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "using different back-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 239, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 505, + 253 + ], + "score": 1.0, + "content": "bone architectures and saliency map techniques. (Right) Performance of the state-of-the-art existing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 226, + 263 + ], + "score": 1.0, + "content": "approaches with and without", + "type": "text" + }, + { + "bbox": [ + 226, + 251, + 249, + 262 + ], + "score": 0.54, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "on CUB200 including TOPIC (Tao et al., 2020), EEIL (Castro", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "et al., 2018), iCaRL (Rebuffi et al., 2017). Joint training serves as the upper bound. Results for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 273, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 284 + ], + "score": 1.0, + "content": "baselines are obtained using their original implementation. All results are averaged over 3 runs and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 284, + 442, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 442, + 295 + ], + "score": 1.0, + "content": "mean and standard deviation values are given in the appendix. Best viewed in color.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 200, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 201, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 201, + 333 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "In this section, we apply RRR in two distinct learning regimes: standard and few-shot class incre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "mental learning. These are the most challenging CL scenarios, in which task descriptions are not", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 382 + ], + "score": 1.0, + "content": "available at test time. We first explore the effect of backbone architecture and the saliency map", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 377, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 429, + 393 + ], + "score": 1.0, + "content": "technique on RRR performance. 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(2020) we precisely followed their setup and and used the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "Caltech-UCSD Birds dataset (Wah et al., 2011), divided into 11 disjoint tasks and a 10-way 5-shot", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 251, + 531 + ], + "score": 1.0, + "content": "setting, where the first task contains", + "type": "text" + }, + { + "bbox": [ + 251, + 518, + 285, + 528 + ], + "score": 0.9, + "content": "b = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "base classes resulting in 3000 samples for training and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "2834 images for testing. The remaining 100 classes are divided into 10 tasks where 5 samples per", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "class are randomly selected as the training set, while the test set is kept intact containing near 300", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 334, + 563 + ], + "score": 1.0, + "content": "images per task. The images in CUB200 are resized to", + "type": "text" + }, + { + "bbox": [ + 334, + 550, + 379, + 561 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "and then randomly cropped to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 150, + 572 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "for training. We store 4 images per class from base classes in the first task and 1 sample", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "per each few-shot class in the remaining 10 tasks (Tao et al., 2020). We used the RAdam (Liu et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "2019) optimizer with a learning rate of 0.001 which was reduced by 0.2 at epochs 20, 40, and 60", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "score": 1.0, + "content": "and trained for a total of 70 epochs with a batch size of 128 for the first and 10 for the remaining", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 603, + 132, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 132, + 617 + ], + "score": 1.0, + "content": "tasks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 325, + 634 + ], + "score": 1.0, + "content": "Figure 2 (left) shows results for ER with and without", + "type": "text" + }, + { + "bbox": [ + 325, + 622, + 347, + 633 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "using different backbone architectures", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "and saliency map techniques. Among the tested saliency map methods, Grad-CAM on ResNet18", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 350, + 657 + ], + "score": 1.0, + "content": "outperforms Vanilla Backpropagation and SmoothGrad by", + "type": "text" + }, + { + "bbox": [ + 351, + 644, + 374, + 655 + ], + "score": 0.87, + "content": "2 { - } 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "while SmoothGrad and vanilla", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Backpropagation achieve similar CL performance. To compute the memory overhead of storing", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 460, + 679 + ], + "score": 1.0, + "content": "the output for a saliency method, if we assume the memory required to store an image is", + "type": "text" + }, + { + "bbox": [ + 460, + 666, + 472, + 676 + ], + "score": 0.79, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 665, + 505, + 679 + ], + "score": 1.0, + "content": ", vanilla", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 449, + 690 + ], + "score": 1.0, + "content": "Backpropagation and SmoothGrad generate a pixel-wise saliency map that occupies", + "type": "text" + }, + { + "bbox": [ + 450, + 677, + 466, + 689 + ], + "score": 0.89, + "content": "M / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "of mem-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "ory. However, in Grad-CAM the saliency map size is equal to the feature map of the target layer in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "the architecture. In our study with Grad-CAM we chose our target layer to be the last convolution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 429, + 722 + ], + "score": 1.0, + "content": "layer before the fully-connected layers. 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(Left) Shows ER with and without", + "type": "text" + }, + { + "bbox": [ + 395, + 229, + 417, + 240 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "using different back-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 239, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 505, + 253 + ], + "score": 1.0, + "content": "bone architectures and saliency map techniques. (Right) Performance of the state-of-the-art existing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 226, + 263 + ], + "score": 1.0, + "content": "approaches with and without", + "type": "text" + }, + { + "bbox": [ + 226, + 251, + 249, + 262 + ], + "score": 0.54, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "on CUB200 including TOPIC (Tao et al., 2020), EEIL (Castro", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "et al., 2018), iCaRL (Rebuffi et al., 2017). Joint training serves as the upper bound. Results for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 273, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 284 + ], + "score": 1.0, + "content": "baselines are obtained using their original implementation. All results are averaged over 3 runs and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 284, + 442, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 442, + 295 + ], + "score": 1.0, + "content": "mean and standard deviation values are given in the appendix. Best viewed in color.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 200, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 201, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 201, + 333 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 358 + ], + "score": 1.0, + "content": "In this section, we apply RRR in two distinct learning regimes: standard and few-shot class incre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "mental learning. These are the most challenging CL scenarios, in which task descriptions are not", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 382 + ], + "score": 1.0, + "content": "available at test time. We first explore the effect of backbone architecture and the saliency map", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 377, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 429, + 393 + ], + "score": 1.0, + "content": "technique on RRR performance. We then report our obtained results integrating", + "type": "text" + }, + { + "bbox": [ + 429, + 379, + 451, + 390 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 377, + 505, + 393 + ], + "score": 1.0, + "content": "into existing", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 390, + 307, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 307, + 403 + ], + "score": 1.0, + "content": "memory-based and regularization-based methods.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 346, + 506, + 403 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 418, + 264, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 266, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 266, + 431 + ], + "score": 1.0, + "content": "4.1 FEW-SHOT CIL PERFORMANCE", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "We first explore CIL of low-data regimes where preventing overfitting to few-shot new classes is an-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 451, + 504, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 494, + 464 + ], + "score": 1.0, + "content": "other challenge to overcome in addition to avoiding catastrophic forgetting of old classes. We use", + "type": "text" + }, + { + "bbox": [ + 495, + 452, + 504, + 462 + ], + "score": 0.77, + "content": "C", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 461, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 153, + 477 + ], + "score": 1.0, + "content": "classes and", + "type": "text" + }, + { + "bbox": [ + 153, + 463, + 164, + 473 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 461, + 293, + 477 + ], + "score": 1.0, + "content": "training samples per class as the", + "type": "text" + }, + { + "bbox": [ + 294, + 463, + 302, + 473 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 461, + 323, + 477 + ], + "score": 1.0, + "content": "-way", + "type": "text" + }, + { + "bbox": [ + 324, + 463, + 334, + 473 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 461, + 505, + 477 + ], + "score": 1.0, + "content": "-shot few-shot class incrementally learning", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 228, + 487 + ], + "score": 1.0, + "content": "setting where we have a set of", + "type": "text" + }, + { + "bbox": [ + 228, + 474, + 234, + 483 + ], + "score": 0.73, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "base classes to learn as the first task while the remaining classes are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "learned with only a few randomly selected samples. In order to provide a direct comparison to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "state-of-the-art work of Tao et al. (2020) we precisely followed their setup and and used the same", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "Caltech-UCSD Birds dataset (Wah et al., 2011), divided into 11 disjoint tasks and a 10-way 5-shot", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 251, + 531 + ], + "score": 1.0, + "content": "setting, where the first task contains", + "type": "text" + }, + { + "bbox": [ + 251, + 518, + 285, + 528 + ], + "score": 0.9, + "content": "b = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "base classes resulting in 3000 samples for training and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "2834 images for testing. The remaining 100 classes are divided into 10 tasks where 5 samples per", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "class are randomly selected as the training set, while the test set is kept intact containing near 300", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 334, + 563 + ], + "score": 1.0, + "content": "images per task. The images in CUB200 are resized to", + "type": "text" + }, + { + "bbox": [ + 334, + 550, + 379, + 561 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "and then randomly cropped to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 150, + 572 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "for training. We store 4 images per class from base classes in the first task and 1 sample", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "per each few-shot class in the remaining 10 tasks (Tao et al., 2020). We used the RAdam (Liu et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "2019) optimizer with a learning rate of 0.001 which was reduced by 0.2 at epochs 20, 40, and 60", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "score": 1.0, + "content": "and trained for a total of 70 epochs with a batch size of 128 for the first and 10 for the remaining", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 603, + 132, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 132, + 617 + ], + "score": 1.0, + "content": "tasks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 440, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 325, + 634 + ], + "score": 1.0, + "content": "Figure 2 (left) shows results for ER with and without", + "type": "text" + }, + { + "bbox": [ + 325, + 622, + 347, + 633 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "using different backbone architectures", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "and saliency map techniques. Among the tested saliency map methods, Grad-CAM on ResNet18", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 350, + 657 + ], + "score": 1.0, + "content": "outperforms Vanilla Backpropagation and SmoothGrad by", + "type": "text" + }, + { + "bbox": [ + 351, + 644, + 374, + 655 + ], + "score": 0.87, + "content": "2 { - } 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "while SmoothGrad and vanilla", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Backpropagation achieve similar CL performance. To compute the memory overhead of storing", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 460, + 679 + ], + "score": 1.0, + "content": "the output for a saliency method, if we assume the memory required to store an image is", + "type": "text" + }, + { + "bbox": [ + 460, + 666, + 472, + 676 + ], + "score": 0.79, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 665, + 505, + 679 + ], + "score": 1.0, + "content": ", vanilla", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 449, + 690 + ], + "score": 1.0, + "content": "Backpropagation and SmoothGrad generate a pixel-wise saliency map that occupies", + "type": "text" + }, + { + "bbox": [ + 450, + 677, + 466, + 689 + ], + "score": 0.89, + "content": "M / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "of mem-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "ory. However, in Grad-CAM the saliency map size is equal to the feature map of the target layer in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "the architecture. In our study with Grad-CAM we chose our target layer to be the last convolution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 429, + 722 + ], + "score": 1.0, + "content": "layer before the fully-connected layers. 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Table 2 shows the target layer name and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "saliency map size for other network architectures used in this work (AlexNet and SqueezeNet1 1)", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 287, + 140, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 140, + 299 + ], + "score": 1.0, + "content": "as well.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 622, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 80, + 497, + 180 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 80, + 497, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 80, + 497, + 180 + ], + "spans": [ + { + "bbox": [ + 113, + 80, + 497, + 180 + ], + "score": 0.968, + "type": "image", + "image_path": "c256de790417db433077726ead7e5e774741a08099f8b9eb519352448f894179.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 80, + 497, + 113.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 113.33333333333334, + 497, + 146.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 146.66666666666669, + 497, + 180.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 189, + 505, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "Figure 3: Effect of RRR on existing methods for CIL on CIFAR100 in (a) 10 and (b) 20 tasks and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 201, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 506, + 213 + ], + "score": 1.0, + "content": "(c) ImageNet100 in 10 tasks. Each point shows the classification accuracy on all seen classes so", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "far. Results for iTAML, BiC, and EEIL are produced with their original implementation while EWC", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "and LwF are re-implemented by us. All results are averaged over 3 runs and mean and standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 233, + 365, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 365, + 245 + ], + "score": 1.0, + "content": "deviation values are given in the appendix. Best viewed in color.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 108, + 275, + 503, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 289 + ], + "score": 1.0, + "content": "saliency map size for other network architectures used in this work (AlexNet and SqueezeNet1 1)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 287, + 140, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 140, + 299 + ], + "score": 1.0, + "content": "as well.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 278, + 316 + ], + "score": 1.0, + "content": "Figure 2 (right) shows the effect of adding", + "type": "text" + }, + { + "bbox": [ + 278, + 304, + 301, + 315 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "on existing recent state-of-the-art methods such as", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "score": 1.0, + "content": "TOPIC (Tao et al., 2020), EEIL (Castro et al., 2018), and iCaRL (Rebuffi et al., 2017). Tao et al.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "(2020) used a neural gas network (Martinetz et al., 1991; Fritzke et al., 1995) which can learn and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "preserve the topology of the feature manifold formed by different classes and we have followed their", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "experimental protocol for our CUB200 experiment by using identical samples drawn in each task", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 375, + 372 + ], + "score": 1.0, + "content": "which are used across all the baselines for fair comparison. Adding", + "type": "text" + }, + { + "bbox": [ + 375, + 359, + 397, + 370 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 357, + 506, + 372 + ], + "score": 1.0, + "content": "improves the performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 383 + ], + "score": 1.0, + "content": "of all the baselines; TOPIC becomes nearly on-par with joint training which serves as the upper", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "bound and does not adhere to continual learning. The gap between ER and iCaRL is also reduced", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 192, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 165, + 405 + ], + "score": 1.0, + "content": "when ER uses", + "type": "text" + }, + { + "bbox": [ + 165, + 392, + 187, + 403 + ], + "score": 0.79, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 389, + 192, + 405 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 266, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 268, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 268, + 440 + ], + "score": 1.0, + "content": "4.2 STANDARD CIL PERFORMANCE", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "In order to provide a direct comparison to the recent work of Rajasegaran et al. (2020) we perform", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "our standard CIL experiment on CIFAR100 (Krizhevsky & Hinton, 2009) and ImageNet100 where", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 472, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 487 + ], + "score": 1.0, + "content": "we assume a memory budget of 2000 samples which are identical across all the baselines. Following", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "Rajasegaran et al. (2020) we divide CIFAR100 to 10 and 20 disjoint tasks with 10 and 5 classes at a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "time. Figures 3a and 3b show the classification accuracy upon learning each task on all seen classes.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 284, + 520 + ], + "score": 1.0, + "content": "We see a consistent average improvement of", + "type": "text" + }, + { + "bbox": [ + 285, + 506, + 315, + 517 + ], + "score": 0.91, + "content": "2 - 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 506, + 340, + 520 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 340, + 507, + 362, + 518 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "is added as an additional constraint", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "to preserve the model explanations across all methods, from the most naive memory-based method,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "experience replay (ER), to more sophisticated approaches which store a set of old class exemplars", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "along with meta-learning (iTAML), correct bias for new classes (BiC), and fine tune on the exemplar", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "set (EEIL). We also applied the RRR constraint on regularization-based methods such as EWC and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "LwF with no memory used as a replay buffer. The accuracy for both improves despite not benefiting", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "score": 1.0, + "content": "from revisiting the raw data. However, they fall behind all memory-based methods with or without", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 129, + 595 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 582, + 506, + 596 + ], + "score": 1.0, + "content": ". 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MethodsPG-ACC (%)PG-BWT (%)
ER54.0-17.4
ER+RRR58.5-15.6
TOPIC72.7-0.9
TOPIC+RRR74.2-2.1
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PrecisionRecall
MethodsPri,iPrT,iRei,iReT,i
ER80.068.964.165.1
ER+RRR82.170.364.266.8
TOPIC91.088.498.197.4
TOPIC+RRR92.889.199.699.2
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In particular, we want to evaluate how often", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "the model is “pointing” at the right evidence for its predictions, instead of focusing its maximum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 291 + ], + "score": 1.0, + "content": "attention on the background or other objects in the image. We use the Pointing Game experiment", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "(PG) (Zhang et al., 2018) for this evaluation, which was introduced to measure the discriminative-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "ness of a visualization method for target object localization. Here, we use ground truth segmentation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 447, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 447, + 326 + ], + "score": 1.0, + "content": "annotation labels provided with the CUB-200 dataset to define the true object region.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "First, we look into hits and misses defined by the PG experiment. When the location of the maximum", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "in a predicted saliency map falls inside the object, it is considered as a hit and otherwise it is a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "miss. Figure 5 shows an example from CUB200 where the segmentation annotation is used to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "determine whether the peak of the predicted saliency map (marked with red cross) falls inside the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "object region. This example is regarded as hit as the red cross is inside the segmentation mask", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "for the bird. PG localization accuracy is defined as the number of hits over the total number of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "predictions. We would like to measure both the overall PG performance of a continual learner as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "well as how much learning new tasks causes it to forget its ability to hit the target object. For", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 417, + 507, + 436 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 366, + 436 + ], + "score": 1.0, + "content": "these metrics, inspired by (Lopez-Paz et al., 2017), we define", + "type": "text" + }, + { + "bbox": [ + 366, + 417, + 476, + 432 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathrm { P G } { \\mathrm { - } } \\mathrm { A C C } = \\frac { 1 } { T } \\sum _ { i = 1 } ^ { \\bar { T } } \\dot { R _ { T , i } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 417, + 507, + 436 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "average PG localization accuracy computed over all prior tasks after training for each new task and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 439, + 507, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 257, + 456 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathrm { P G - B W T } = \\frac { 1 } { T - 1 } \\sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 439, + 507, + 460 + ], + "score": 1.0, + "content": "(backward transfer) which indicates how much learning new", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 403, + 468 + ], + "score": 1.0, + "content": "tasks has influenced the PG localization accuracy on previous tasks where", + "type": "text" + }, + { + "bbox": [ + 403, + 455, + 423, + 467 + ], + "score": 0.91, + "content": "R _ { n , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 455, + 478, + 468 + ], + "score": 1.0, + "content": "is the on task", + "type": "text" + }, + { + "bbox": [ + 478, + 456, + 483, + 465 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "after", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 156, + 480 + ], + "score": 1.0, + "content": "learning the", + "type": "text" + }, + { + "bbox": [ + 156, + 466, + 172, + 478 + ], + "score": 0.87, + "content": "n ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 466, + 376, + 480 + ], + "score": 1.0, + "content": "task. Results for ER and TOPIC with and without", + "type": "text" + }, + { + "bbox": [ + 377, + 468, + 399, + 479 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "on CUB200 are shown in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Table 1a. It shows how constraining different models to remember their initial evidence can lead to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 489, + 330, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 330, + 501 + ], + "score": 1.0, + "content": "better localization of the bird across learning new tasks.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "However, PG performance does not capture all of our desired properties for a continual learner.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "Ideally, we not only want a model to predict the object correctly if it is looking at the right evidence,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "but also we want it to not predict an object if it is not able to find the right evidence for it. To measure", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 429, + 552 + ], + "score": 1.0, + "content": "how close our baselines can get to this ideal model when they are combined with", + "type": "text" + }, + { + "bbox": [ + 429, + 540, + 451, + 551 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 538, + 506, + 552 + ], + "score": 1.0, + "content": ", we measure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 171, + 562 + ], + "score": 1.0, + "content": "the precision as", + "type": "text" + }, + { + "bbox": [ + 172, + 550, + 205, + 562 + ], + "score": 0.89, + "content": "\\mathrm { t p / ( t p / \\Delta \\mathfrak { p } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 551, + 263, + 562 + ], + "score": 1.0, + "content": ", and recall as", + "type": "text" + }, + { + "bbox": [ + 263, + 550, + 298, + 562 + ], + "score": 0.9, + "content": "\\mathrm { t p / ( t p / \\Delta \\mathfrak { t } f n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 551, + 506, + 562 + ], + "score": 1.0, + "content": ". 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The higher the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "precision for a model is, the less often it has made the right decision without looking at the right", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "evidence. On the other hand, the higher the recall, the less often it makes a wrong decision despite", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "looking at the correct evidence. We show our evaluation on these metrics in Table 1b for ER and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 209, + 640 + ], + "score": 1.0, + "content": "TOPIC with and without", + "type": "text" + }, + { + "bbox": [ + 209, + 627, + 232, + 638 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 626, + 312, + 640 + ], + "score": 1.0, + "content": "on CUB200 where", + "type": "text" + }, + { + "bbox": [ + 312, + 627, + 334, + 638 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "increases both precision and recall across", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "all methods, demonstrating that our approach continually makes better predictions because it finds", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 248, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 248, + 661 + ], + "score": 1.0, + "content": "the right evidence for its decisions.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 507, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 507, + 680 + ], + "score": 1.0, + "content": "In our final analysis, we would like to visualize the evolution of saliency maps across learning a", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "sequence of tasks. Figure 4 illustrates the evolution of saliency maps for an image from the test-set", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 281, + 700 + ], + "score": 1.0, + "content": "of the second task, which both ER without", + "type": "text" + }, + { + "bbox": [ + 281, + 688, + 304, + 699 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 687, + 383, + 700 + ], + "score": 1.0, + "content": "(top row) and with", + "type": "text" + }, + { + "bbox": [ + 384, + 688, + 406, + 699 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(bottom row) have seen", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "during training for the second task. We only visualize the generated saliencies after finishing tasks", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "#2, #5, #7, #9, and #11 for simplicity. We indicate the correctness of the prediction made by each", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "model with ‘correct’ or ‘incorrect’ written on top of their corresponding saliency map. Our goal is", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 51.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 135, + 149, + 279, + 197 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 114, + 135, + 299, + 145 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 113, + 134, + 300, + 146 + ], + "spans": [ + { + "bbox": [ + 113, + 134, + 300, + 146 + ], + "score": 1.0, + "content": "(a) PG localization accuracy and backward transfer", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 135, + 149, + 279, + 197 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 135, + 149, + 279, + 197 + ], + "spans": [ + { + "bbox": [ + 135, + 149, + 279, + 197 + ], + "score": 0.966, + "html": "
MethodsPG-ACC (%)PG-BWT (%)
ER54.0-17.4
ER+RRR58.5-15.6
TOPIC72.7-0.9
TOPIC+RRR74.2-2.1
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PrecisionRecall
MethodsPri,iPrT,iRei,iReT,i
ER80.068.964.165.1
ER+RRR82.170.364.266.8
TOPIC91.088.498.197.4
TOPIC+RRR92.889.199.699.2
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In particular, we want to evaluate how often", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "the model is “pointing” at the right evidence for its predictions, instead of focusing its maximum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 291 + ], + "score": 1.0, + "content": "attention on the background or other objects in the image. We use the Pointing Game experiment", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "(PG) (Zhang et al., 2018) for this evaluation, which was introduced to measure the discriminative-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "ness of a visualization method for target object localization. Here, we use ground truth segmentation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 447, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 447, + 326 + ], + "score": 1.0, + "content": "annotation labels provided with the CUB-200 dataset to define the true object region.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 247, + 505, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "First, we look into hits and misses defined by the PG experiment. When the location of the maximum", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 353 + ], + "score": 1.0, + "content": "in a predicted saliency map falls inside the object, it is considered as a hit and otherwise it is a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "miss. Figure 5 shows an example from CUB200 where the segmentation annotation is used to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "determine whether the peak of the predicted saliency map (marked with red cross) falls inside the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "object region. This example is regarded as hit as the red cross is inside the segmentation mask", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "for the bird. PG localization accuracy is defined as the number of hits over the total number of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "predictions. We would like to measure both the overall PG performance of a continual learner as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "well as how much learning new tasks causes it to forget its ability to hit the target object. For", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 417, + 507, + 436 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 366, + 436 + ], + "score": 1.0, + "content": "these metrics, inspired by (Lopez-Paz et al., 2017), we define", + "type": "text" + }, + { + "bbox": [ + 366, + 417, + 476, + 432 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathrm { P G } { \\mathrm { - } } \\mathrm { A C C } = \\frac { 1 } { T } \\sum _ { i = 1 } ^ { \\bar { T } } \\dot { R _ { T , i } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 417, + 507, + 436 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "average PG localization accuracy computed over all prior tasks after training for each new task and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 439, + 507, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 257, + 456 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathrm { P G - B W T } = \\frac { 1 } { T - 1 } \\sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 439, + 507, + 460 + ], + "score": 1.0, + "content": "(backward transfer) which indicates how much learning new", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 403, + 468 + ], + "score": 1.0, + "content": "tasks has influenced the PG localization accuracy on previous tasks where", + "type": "text" + }, + { + "bbox": [ + 403, + 455, + 423, + 467 + ], + "score": 0.91, + "content": "R _ { n , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 455, + 478, + 468 + ], + "score": 1.0, + "content": "is the on task", + "type": "text" + }, + { + "bbox": [ + 478, + 456, + 483, + 465 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "after", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 156, + 480 + ], + "score": 1.0, + "content": "learning the", + "type": "text" + }, + { + "bbox": [ + 156, + 466, + 172, + 478 + ], + "score": 0.87, + "content": "n ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 466, + 376, + 480 + ], + "score": 1.0, + "content": "task. Results for ER and TOPIC with and without", + "type": "text" + }, + { + "bbox": [ + 377, + 468, + 399, + 479 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "on CUB200 are shown in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Table 1a. It shows how constraining different models to remember their initial evidence can lead to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 489, + 330, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 330, + 501 + ], + "score": 1.0, + "content": "better localization of the bird across learning new tasks.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 329, + 507, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "However, PG performance does not capture all of our desired properties for a continual learner.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "Ideally, we not only want a model to predict the object correctly if it is looking at the right evidence,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "but also we want it to not predict an object if it is not able to find the right evidence for it. To measure", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 429, + 552 + ], + "score": 1.0, + "content": "how close our baselines can get to this ideal model when they are combined with", + "type": "text" + }, + { + "bbox": [ + 429, + 540, + 451, + 551 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 538, + 506, + 552 + ], + "score": 1.0, + "content": ", we measure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 171, + 562 + ], + "score": 1.0, + "content": "the precision as", + "type": "text" + }, + { + "bbox": [ + 172, + 550, + 205, + 562 + ], + "score": 0.89, + "content": "\\mathrm { t p / ( t p / \\Delta \\mathfrak { p } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 551, + 263, + 562 + ], + "score": 1.0, + "content": ", and recall as", + "type": "text" + }, + { + "bbox": [ + 263, + 550, + 298, + 562 + ], + "score": 0.9, + "content": "\\mathrm { t p / ( t p / \\Delta \\mathfrak { t } f n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 551, + 506, + 562 + ], + "score": 1.0, + "content": ". We evaluate these metrics once immediately after", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 233, + 575 + ], + "score": 1.0, + "content": "learning each task, denoted as", + "type": "text" + }, + { + "bbox": [ + 233, + 561, + 255, + 573 + ], + "score": 0.91, + "content": "P r _ { i , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 560, + 275, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 275, + 562, + 297, + 573 + ], + "score": 0.9, + "content": "R e _ { i , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 560, + 505, + 575 + ], + "score": 1.0, + "content": ", respectively, and again at the end of the learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 191, + 584 + ], + "score": 1.0, + "content": "process of final task", + "type": "text" + }, + { + "bbox": [ + 192, + 573, + 201, + 582 + ], + "score": 0.73, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 572, + 249, + 584 + ], + "score": 1.0, + "content": "denoted as", + "type": "text" + }, + { + "bbox": [ + 249, + 573, + 274, + 584 + ], + "score": 0.93, + "content": "P r _ { T , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 572, + 294, + 584 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 294, + 573, + 318, + 584 + ], + "score": 0.91, + "content": "R e _ { T , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 572, + 505, + 584 + ], + "score": 1.0, + "content": ", where the first subscript refers to the model", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "ID and the second subscript is the test dataset ID on which the model is evaluated. The higher the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "precision for a model is, the less often it has made the right decision without looking at the right", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "evidence. On the other hand, the higher the recall, the less often it makes a wrong decision despite", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "looking at the correct evidence. We show our evaluation on these metrics in Table 1b for ER and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 209, + 640 + ], + "score": 1.0, + "content": "TOPIC with and without", + "type": "text" + }, + { + "bbox": [ + 209, + 627, + 232, + 638 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 626, + 312, + 640 + ], + "score": 1.0, + "content": "on CUB200 where", + "type": "text" + }, + { + "bbox": [ + 312, + 627, + 334, + 638 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "increases both precision and recall across", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "all methods, demonstrating that our approach continually makes better predictions because it finds", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 248, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 248, + 661 + ], + "score": 1.0, + "content": "the right evidence for its decisions.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 505, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 507, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 507, + 680 + ], + "score": 1.0, + "content": "In our final analysis, we would like to visualize the evolution of saliency maps across learning a", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "sequence of tasks. Figure 4 illustrates the evolution of saliency maps for an image from the test-set", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 281, + 700 + ], + "score": 1.0, + "content": "of the second task, which both ER without", + "type": "text" + }, + { + "bbox": [ + 281, + 688, + 304, + 699 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 687, + 383, + 700 + ], + "score": 1.0, + "content": "(top row) and with", + "type": "text" + }, + { + "bbox": [ + 384, + 688, + 406, + 699 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "(bottom row) have seen", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "during training for the second task. We only visualize the generated saliencies after finishing tasks", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "#2, #5, #7, #9, and #11 for simplicity. We indicate the correctness of the prediction made by each", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "model with ‘correct’ or ‘incorrect’ written on top of their corresponding saliency map. Our goal is", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 245, + 304 + ], + "score": 1.0, + "content": "to visualize if adding the loss term", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 246, + 292, + 268, + 303 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 268, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "prevents the drifting of explanations. Given the same input", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 202, + 315 + ], + "score": 1.0, + "content": "image, the ER without", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 203, + 303, + 225, + 314 + ], + "score": 0.81, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 226, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "model makes an incorrect prediction after being continually trained", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "for 11 tasks while never recovering from its mistake. On the other hand, when it is combined with", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 129, + 336 + ], + "score": 0.79, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 129, + 324, + 505, + 336 + ], + "score": 1.0, + "content": ". it is able to recover from an early mistake after task 5. Considering the saliency map obtained", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "after finishing task one as a reference evidence, we can see that ER’s evidence drifts further from the", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "reference. On the bottom row, the region of focus of ER+RRR remains nearly identical to its initial", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 349, + 371 + ], + "score": 1.0, + "content": "evidence, apart from one incorrect prediction. As applying", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 350, + 357, + 373, + 369 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 373, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "corrects its saliency back to the", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "original, this prediction is corrected as well. 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Failure case", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 160, + 236 + ], + "score": 1.0, + "content": "for ER w.o.", + "type": "text" + }, + { + "bbox": [ + 160, + 223, + 183, + 234 + ], + "score": 0.73, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "(top row), where saliency drifts from the original and the prediction becomes", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 147, + 246 + ], + "score": 1.0, + "content": "incorrect.", + "type": "text" + }, + { + "bbox": [ + 148, + 235, + 188, + 245 + ], + "score": 0.55, + "content": "_ \\mathrm { E R + R R R }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "(bottom row) retains close to the original saliency as the model trains on more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 246, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 257 + ], + "score": 1.0, + "content": "tasks, with the exception of Task #5 which it is able to correct later on. Its performance is retained", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 257, + 311, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 311, + 268 + ], + "score": 1.0, + "content": "as well, for saliencies that are close to the original.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 245, + 304 + ], + "score": 1.0, + "content": "to visualize if adding the loss term", + "type": "text" + }, + { + "bbox": [ + 246, + 292, + 268, + 303 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "prevents the drifting of explanations. Given the same input", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 202, + 315 + ], + "score": 1.0, + "content": "image, the ER without", + "type": "text" + }, + { + "bbox": [ + 203, + 303, + 225, + 314 + ], + "score": 0.81, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "model makes an incorrect prediction after being continually trained", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "for 11 tasks while never recovering from its mistake. On the other hand, when it is combined with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 325, + 129, + 336 + ], + "score": 0.79, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 324, + 505, + 336 + ], + "score": 1.0, + "content": ". it is able to recover from an early mistake after task 5. Considering the saliency map obtained", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "after finishing task one as a reference evidence, we can see that ER’s evidence drifts further from the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "reference. On the bottom row, the region of focus of ER+RRR remains nearly identical to its initial", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 349, + 371 + ], + "score": 1.0, + "content": "evidence, apart from one incorrect prediction. As applying", + "type": "text" + }, + { + "bbox": [ + 350, + 357, + 373, + 369 + ], + "score": 0.85, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "corrects its saliency back to the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "original, this prediction is corrected as well. 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Right: the segmentation label where the red cross marks the peak saliency.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 520, + 227, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 229, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 229, + 536 + ], + "score": 1.0, + "content": "C TABULAR RESULTS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 504, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "In this section, we have tabulated results shown in Figure 2 and Figure 3 with means and standard", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 558, + 238, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 238, + 570 + ], + "score": 1.0, + "content": "deviations averaged over 3 runs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "table", + "bbox": [ + 106, + 623, + 505, + 717 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 580, + 505, + 615 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "Table 3: Classification accuracy of few-shot CIL learning of CUB200 at the end of 11 tasks for ER", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 592, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 176, + 604 + ], + "score": 1.0, + "content": "with and without", + "type": "text" + }, + { + "bbox": [ + 176, + 592, + 199, + 603 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 592, + 504, + 604 + ], + "score": 1.0, + "content": "using different backbone architectures and saliency map techniques. Results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "are averaged over 3 runs. Figure 2 (left) in the main paper is generated using numbers in this Table.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "table_body", + "bbox": [ + 106, + 623, + 505, + 717 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 623, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 717 + ], + "score": 0.978, + "html": "
234567891011
RN18-RRR-GCam67.8±0.853.5±0.745.6±0.6 39.6±0.735.3±0.932.3± 1.129.4±0.925.9±0.825.7±0.626.3±0.723.6±0.7
RN18-ER67.8±0.849.7 ±0.941.7 ±0.835.8 ±0.731.4 ± 0.928.5±0.825.5±0.822.1±0.821.8±0.822.5 ± 1.119.8± 0.9
RN18-RRR-Smooth67.8± 0.850.9±0.643.5 ± 0.937.0±0.833.0±0.729.5± 0.626.8±0.823.9 ±0.823.9±0.823.4± 0.821.5± 0.5
RN18-RRR-BP67.8±0.850.8±0.843.9 ±0.636.6 ±0.432.7±0.628.9±0.627.2± 0.523.8 ±0.623.8±0.624.0± 0.421.5± 0.6
RN18-Finetune67.8± 0.844.8 ± 0.632.2± 0.525.8±0.725.6 ± 0.725.2± 0.720.8±0.616.8 ± 0.718.8± 0.518.3 ± 0.417.1 ± 0.6
Alex-RRR-GCam56.7±0.746.6±0.543.9±0.741.3± 0.733.7 ± 0.527.4± 0.725.3±0.722.0±0.521.5±0.621.4± 0.621.2± 0.6
Alex-ER56.7± 0.744.6 ± 0.741.3 ± 0.738.7±0.731.1 ± 0.724.5± 0.722.6± 0.719.6 ± 0.619.1± 0.818.7 ± 0.819.1± 0.8
Alex-Finetune56.7±0.742.8 ± 0.839.6±0.836.9±0.829.5 ± 0.723.3±0.621.4± 0.817.9 ± 0.718.0 ±0.717.0 ± 0.516.9 ± 0.4
SQ-RRR-GCam46.8± 0.536.2 ±0.430.1±0.628.3±0.425.1 ± 0.523.4± 0.519.3± 0.619.0± 0.618.5± 0.518.4± 0.518.2 ±0.6
SQ-ER46.8 ± 0.533.2±0.5 27.1±0.625.3±0.622.1±0.5220.5±0.516.3± 0.416.0±0.615.5± 0.615.4 ± 0.615.2 ± 0.7
SQ-Finetune46.8 ± 0.532.0±0.7 25.2±0.723.9±0.720.2±0.8119.4 ± 0.414.9 ± 0.414.4 ± 0.513.8± 0.414.2 ± 0.513.7±0.6
", + "type": "table", + "image_path": "d91a0e49a90a15c0b2d1c86bd040faa5f4c70e3caf7ca5895edb811ccb835801.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 106, + 623, + 505, + 654.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 106, + 654.3333333333334, + 505, + 685.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 106, + 685.6666666666667, + 505, + 717.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "index": 27.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 182, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 261, + 119 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 262, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 262, + 120 + ], + "score": 1.0, + "content": "A.1 GRAD-CAM TARGET LAYERS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 128, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "Table 2 shows the target layer names used in Grad-CAM for different network architectures accord-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "ing to their standard PyTorch (Paszke et al., 2017) implementations. Saliency map size is equal to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 262, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 262, + 163 + ], + "score": 1.0, + "content": "the activation map of the target layers.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 128, + 505, + 163 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 205, + 505, + 266 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 173, + 504, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "Table 2: Target layer names and activation maps size for saliencies generated by different network", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 185, + 221, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 221, + 196 + ], + "score": 1.0, + "content": "architectures in Grad-CAM.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 205, + 505, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 205, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 266 + ], + "score": 0.978, + "html": "
Target layer name in PyTorch torchvision packageSaliency map size
SqueezeNet1_1features.0.12.expand3x313 ×13
AlexNetfeatures.0.1013 ×13
ResNet18features.7.1.conv27×7
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We used the segmentation anno-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "tation to verify whether the peak of the predicted saliency map (marked with red cross) falls inside", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 351, + 500, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 500, + 363 + ], + "score": 1.0, + "content": "the object region. It is regarded as hit as the red cross is inside the segmentation mask for the bird.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 328, + 505, + 363 + ] + }, + { + "type": "image", + "bbox": [ + 217, + 375, + 394, + 463 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 217, + 375, + 394, + 463 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 217, + 375, + 394, + 463 + ], + "spans": [ + { + "bbox": [ + 217, + 375, + 394, + 463 + ], + "score": 0.973, + "type": "image", + "image_path": "3e59b0dc386b86facc06d523b925d24c5ac95c5b1d5165a09da8cda4bb4f9375.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 375, + 394, + 389.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 217, + 389.6666666666667, + 394, + 404.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 217, + 404.33333333333337, + 394, + 419.00000000000006 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 217, + 419.00000000000006, + 394, + 433.66666666666674 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 217, + 433.66666666666674, + 394, + 448.3333333333334 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 217, + 448.3333333333334, + 394, + 463.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 473, + 505, + 493 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "Figure 5: An example of PG evaluation as hit for an image in CUB200. Left: image saliency map overlaid on", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 482, + 416, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 416, + 495 + ], + "score": 1.0, + "content": "the image. Right: the segmentation label where the red cross marks the peak saliency.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 520, + 227, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 229, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 229, + 536 + ], + "score": 1.0, + "content": "C TABULAR RESULTS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 504, + 569 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "In this section, we have tabulated results shown in Figure 2 and Figure 3 with means and standard", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 558, + 238, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 238, + 570 + ], + "score": 1.0, + "content": "deviations averaged over 3 runs.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 546, + 506, + 570 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 623, + 505, + 717 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 580, + 505, + 615 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 594 + ], + "score": 1.0, + "content": "Table 3: Classification accuracy of few-shot CIL learning of CUB200 at the end of 11 tasks for ER", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 592, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 176, + 604 + ], + "score": 1.0, + "content": "with and without", + "type": "text" + }, + { + "bbox": [ + 176, + 592, + 199, + 603 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 592, + 504, + 604 + ], + "score": 1.0, + "content": "using different backbone architectures and saliency map techniques. Results", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "are averaged over 3 runs. Figure 2 (left) in the main paper is generated using numbers in this Table.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "table_body", + "bbox": [ + 106, + 623, + 505, + 717 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 623, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 717 + ], + "score": 0.978, + "html": "
234567891011
RN18-RRR-GCam67.8±0.853.5±0.745.6±0.6 39.6±0.735.3±0.932.3± 1.129.4±0.925.9±0.825.7±0.626.3±0.723.6±0.7
RN18-ER67.8±0.849.7 ±0.941.7 ±0.835.8 ±0.731.4 ± 0.928.5±0.825.5±0.822.1±0.821.8±0.822.5 ± 1.119.8± 0.9
RN18-RRR-Smooth67.8± 0.850.9±0.643.5 ± 0.937.0±0.833.0±0.729.5± 0.626.8±0.823.9 ±0.823.9±0.823.4± 0.821.5± 0.5
RN18-RRR-BP67.8±0.850.8±0.843.9 ±0.636.6 ±0.432.7±0.628.9±0.627.2± 0.523.8 ±0.623.8±0.624.0± 0.421.5± 0.6
RN18-Finetune67.8± 0.844.8 ± 0.632.2± 0.525.8±0.725.6 ± 0.725.2± 0.720.8±0.616.8 ± 0.718.8± 0.518.3 ± 0.417.1 ± 0.6
Alex-RRR-GCam56.7±0.746.6±0.543.9±0.741.3± 0.733.7 ± 0.527.4± 0.725.3±0.722.0±0.521.5±0.621.4± 0.621.2± 0.6
Alex-ER56.7± 0.744.6 ± 0.741.3 ± 0.738.7±0.731.1 ± 0.724.5± 0.722.6± 0.719.6 ± 0.619.1± 0.818.7 ± 0.819.1± 0.8
Alex-Finetune56.7±0.742.8 ± 0.839.6±0.836.9±0.829.5 ± 0.723.3±0.621.4± 0.817.9 ± 0.718.0 ±0.717.0 ± 0.516.9 ± 0.4
SQ-RRR-GCam46.8± 0.536.2 ±0.430.1±0.628.3±0.425.1 ± 0.523.4± 0.519.3± 0.619.0± 0.618.5± 0.518.4± 0.518.2 ±0.6
SQ-ER46.8 ± 0.533.2±0.5 27.1±0.625.3±0.622.1±0.5220.5±0.516.3± 0.416.0±0.615.5± 0.615.4 ± 0.615.2 ± 0.7
SQ-Finetune46.8 ± 0.532.0±0.7 25.2±0.723.9±0.720.2±0.8119.4 ± 0.414.9 ± 0.414.4 ± 0.513.8± 0.414.2 ± 0.513.7±0.6
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1234567891011
EEIL68.6±0.453.6± 0.447.9 ± 0.344.2±0.836.3±0.927.4 ± 1.225.9±0.724.7±0.523.9±0.724.1± 0.722.1 ± 0.5
EEIL+RRR68.6±0.456.6±0.550.9±0.648.3± 0.539.7 ± 1.231.4± 0.728.3± 1.228.0±0.626.5±0.627.4± 0.625.2±0.9
iCaRL68.6±0.452.6±0.748.6± 1.244.1 ± 0.536.6±0.329.5±0.927.8± 0.426.2±0.524.0±0.623.8±0.621.1± 0.7
iCaRL+RRR68.6±0.455.6± 1.253.6±0.747.1 ± 0.839.6±0.532.5±0.831.8 ± 0.429.2±0.627.0±0.827.8±0.624.1 ± 0.3
TOPIC68.6 ± 0.462.4 ± 0.854.8 ± 0.449.9 ± 1.245.2 ±0.641.4± 0.338.3±0.835.3±0.632.2± 0.328.3±0.626.2 ± 1.2
TOPIC+RRR68.6±0.462.5 ± 0.956.8 ± 0.451.5 ± 0.548.2 ± 0.444.4 ± 0.442.3±0.738.3±0.635.2±0.932.3±0.929.2 ± 0.5
FT68.6±0.444.8 ± 0.532.2±0.825.8± 0.425.6 ± 1.125.2± 0.720.8± 1.116.7± 0.418.8 ± 1.118.2± 0.317.1 ± 0.8
ER67.8±0.849.7± 0.941.7 ± 0.835.8 ±0.731.4± 0.928.5±0.825.5±0.822.1± 0.821.8± 0.622.5 ± 1.119.8±0.9
RRR67.8±0.853.5± 0.745.6± 0.639.6± 0.735.3±0.932.3 ± 1.129.4± 0.925.9± 0.825.7±0.626.3±0.723.6±0.7
JT68.6±0.462.4± 0.457.2 ± 0.452.8±0.549.5 ± 0.946.1 ± 0.542.8 ± 1.140.1±0.838.7±0.737.1± 0.535.6±0.9
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12345678910
iTAML+RRR89.2± 0.592.3±0.789.5 ± 1.287.5 ± 1.284.1±0.883.5±0.983.9±0.781.2± 0.379.6 ± 0.979.7±0.5
iTAML89.2± 0.588.9±0.587.0 ± 1.185.7 ± 1.184.1 ± 1.181.8 ± 0.380.0±0.679.0± 0.378.6 ±0.877.8 ±0.6
BiC90.3±0.782.1 ± 0.775.1± 0.469.8 ± 1.265.3±0.861.3± 0.957.4± 0.754.9± 0.553.2 ±0.950.3±0.7
BiC+RRR90.3±0.784.9 ± 1.176.4± 0.669.3±0.365.1±0.963.3± 0.459.7 ± 1.155.4± 0.855.8 ± 0.752.1 ± 0.5
EEIL80.0±0.780.5± 1.275.5± 0.971.5± 0.468.0± 1.262.0±0.959.0± 0.755.1 ± 1.251.4 ± 0.848.7 ± 0.4
EEIL+RRR80.0±0.783.5± 0.378.7 ± 1.274.0 ± 1.271.7± 0.365.1 ± 0.461.2± 0.557.6± 0.554.1 ± 0.451.7± 0.3
LwF86.1 ± 1.269.0±0.755.0±0.345.8± 0.340.4± 0.536.7±0.930.8 ±0.728.6±0.526.1 ± 0.724.2 ±0.7
LwF+RRR86.1 ± 1.272.4± 0.857.0 ± 1.148.3 ± 0.343.2 ±0.839.3 ±0.534.1 ± 0.632.1 ± 1.129.8 ±0.727.1± 0.6
EWC86.1 ± 1.252.6 ± 0.448.6± 0.438.5±0.531.1 ± 0.926.5±0.321.7±0.620.0±0.718.9 ± 0.516.6 ±0.9
EWC+RRR86.1 ± 1.256.0± 0.453.9 ± 1.244.4 ± 0.935.1±0.528.6±0.625.1 ± 1.123.1± 0.518.8 ±0.919.0 ± 1.2
ER86.1 ± 1.274.5 ± 0.965.2±0.862.5± 0.856.7±0.750.5± 0.347.6 ± 0.443.4± 0.341.6 ± 0.938.1 ± 1.1
RRR86.1 ± 1.278.5± 0.969.2 ± 1.163.5 ± 1.258.7±0.853.5 ± 1.149.6± 0.744.4± 0.342.6 ± 1.239.1 ± 1.1
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iTAML99.4± 0.896.4±0.994.4 ± 0.993.0±0.392.4± 1.290.6±0.389.9 ±0.490.3±0.890.3 ±1.189.8±0.4
iTAML+RRR99.4± 0.897.3± 0.596.6±0.796.3 ± 1.195.3 ± 0.593.1 ± 0.592.1 ± 0.692.1±0.692.9±0.991.9 ± 0.4
EEIL99.5 ± 0.498.8 ± 1.195.9±0.993.0 ±0.488.3 ±1.186.7 ± 1.283.0 ±1.281.1± 0.578.2 ±0.775.4± 0.4
EEIL+RRR99.5± 0.498.1 ± 0.797.4 ± 1.196.7 ± 0.493.3 ± 0.589.4 ± 1.186.5±0.386.1 ± 1.181.8 ± 0.477.0± 0.3
BiC98.3±0.794.9 ±0.893.5±0.790.9 ± 1.289.0 ± 1.287.3± 0.685.2±0.783.2 ± 0.482.5±0.981.1 ± 1.1
BiC+RRR98.3±0.798.9 ±0.396.5±0.693.9 ± 0.492.0±0.789.3 ± 1.187.2 ±0.887.2 ± 1.185.5±0.984.1± 0.6
iCaRL99.3±0.497.2 ± 0.993.5±0.991.0 ± 0.387.5 ± 1.282.1 ± 1.277.1 ± 0.472.8± 0.667.1 ±0.863.5 ± 1.1
iCaRL+RRR99.3± 0.497.9 ± 1.294.1 ± 0.792.8 ±0.791.7 ± 0.985.7 ± 1.182.1 ± 0.674.4 ± 0.972.2 ± 0.868.1±0.9
LwF99.3±0.595.2 ± 0.985.9± 0.973.9 ± 1.163.7±0.854.8 ± 0.850.1 ± 0.644.5 ± 0.940.7 ± 0.536.7±0.3
LwF+RRR99.3±0.597.1 ± 1.289.3 ±0.679.1 ± 0.569.1 ± 1.159.4 ± 1.157.2 ± 0.748.2 ± 1.145.1 ± 0.641.5 ± 0.5
FT99.3± 0.549.4 ± 0.332.6±0.324.7 ± 0.620.0 ± 1.216.7 ± 0.313.9 ± 0.312.3 ± 0.711.1 ± 0.69.9 ±0.7
ER99.3± 0.595.2 ± 0.888.1±0.878.1± 0.972.5 ± 0.669.1± 0.867.1 ± 0.661.8 ±0.655.1± 0.350.1 ± 1.1
RRR99.3± 0.596.5 ± 0.393.4±0.884.8±0.778.7 ± 0.474.7 ± 0.473.1 ± 0.568.4±0.860.2±0.355.1 ±0.7
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1234567891011
EEIL68.6±0.453.6± 0.447.9 ± 0.344.2±0.836.3±0.927.4 ± 1.225.9±0.724.7±0.523.9±0.724.1± 0.722.1 ± 0.5
EEIL+RRR68.6±0.456.6±0.550.9±0.648.3± 0.539.7 ± 1.231.4± 0.728.3± 1.228.0±0.626.5±0.627.4± 0.625.2±0.9
iCaRL68.6±0.452.6±0.748.6± 1.244.1 ± 0.536.6±0.329.5±0.927.8± 0.426.2±0.524.0±0.623.8±0.621.1± 0.7
iCaRL+RRR68.6±0.455.6± 1.253.6±0.747.1 ± 0.839.6±0.532.5±0.831.8 ± 0.429.2±0.627.0±0.827.8±0.624.1 ± 0.3
TOPIC68.6 ± 0.462.4 ± 0.854.8 ± 0.449.9 ± 1.245.2 ±0.641.4± 0.338.3±0.835.3±0.632.2± 0.328.3±0.626.2 ± 1.2
TOPIC+RRR68.6±0.462.5 ± 0.956.8 ± 0.451.5 ± 0.548.2 ± 0.444.4 ± 0.442.3±0.738.3±0.635.2±0.932.3±0.929.2 ± 0.5
FT68.6±0.444.8 ± 0.532.2±0.825.8± 0.425.6 ± 1.125.2± 0.720.8± 1.116.7± 0.418.8 ± 1.118.2± 0.317.1 ± 0.8
ER67.8±0.849.7± 0.941.7 ± 0.835.8 ±0.731.4± 0.928.5±0.825.5±0.822.1± 0.821.8± 0.622.5 ± 1.119.8±0.9
RRR67.8±0.853.5± 0.745.6± 0.639.6± 0.735.3±0.932.3 ± 1.129.4± 0.925.9± 0.825.7±0.626.3±0.723.6±0.7
JT68.6±0.462.4± 0.457.2 ± 0.452.8±0.549.5 ± 0.946.1 ± 0.542.8 ± 1.140.1±0.838.7±0.737.1± 0.535.6±0.9
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iTAML+RRR89.2± 0.592.3±0.789.5 ± 1.287.5 ± 1.284.1±0.883.5±0.983.9±0.781.2± 0.379.6 ± 0.979.7±0.5
iTAML89.2± 0.588.9±0.587.0 ± 1.185.7 ± 1.184.1 ± 1.181.8 ± 0.380.0±0.679.0± 0.378.6 ±0.877.8 ±0.6
BiC90.3±0.782.1 ± 0.775.1± 0.469.8 ± 1.265.3±0.861.3± 0.957.4± 0.754.9± 0.553.2 ±0.950.3±0.7
BiC+RRR90.3±0.784.9 ± 1.176.4± 0.669.3±0.365.1±0.963.3± 0.459.7 ± 1.155.4± 0.855.8 ± 0.752.1 ± 0.5
EEIL80.0±0.780.5± 1.275.5± 0.971.5± 0.468.0± 1.262.0±0.959.0± 0.755.1 ± 1.251.4 ± 0.848.7 ± 0.4
EEIL+RRR80.0±0.783.5± 0.378.7 ± 1.274.0 ± 1.271.7± 0.365.1 ± 0.461.2± 0.557.6± 0.554.1 ± 0.451.7± 0.3
LwF86.1 ± 1.269.0±0.755.0±0.345.8± 0.340.4± 0.536.7±0.930.8 ±0.728.6±0.526.1 ± 0.724.2 ±0.7
LwF+RRR86.1 ± 1.272.4± 0.857.0 ± 1.148.3 ± 0.343.2 ±0.839.3 ±0.534.1 ± 0.632.1 ± 1.129.8 ±0.727.1± 0.6
EWC86.1 ± 1.252.6 ± 0.448.6± 0.438.5±0.531.1 ± 0.926.5±0.321.7±0.620.0±0.718.9 ± 0.516.6 ±0.9
EWC+RRR86.1 ± 1.256.0± 0.453.9 ± 1.244.4 ± 0.935.1±0.528.6±0.625.1 ± 1.123.1± 0.518.8 ±0.919.0 ± 1.2
ER86.1 ± 1.274.5 ± 0.965.2±0.862.5± 0.856.7±0.750.5± 0.347.6 ± 0.443.4± 0.341.6 ± 0.938.1 ± 1.1
RRR86.1 ± 1.278.5± 0.969.2 ± 1.163.5 ± 1.258.7±0.853.5 ± 1.149.6± 0.744.4± 0.342.6 ± 1.239.1 ± 1.1
", + "type": "table", + "image_path": "a36c84c93cbd9f6f56147d35481bdbe9a921307573394a0ff7cc54dd55d7ce76.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 106, + 292, + 505, + 325.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 106, + 325.3333333333333, + 505, + 358.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 106, + 358.66666666666663, + 505, + 391.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + } + ], + "index": 11.0 + }, + { + "type": "table", + "bbox": [ + 106, + 476, + 505, + 583 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 411, + 505, + 466 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 105, + 409, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 445, + 424 + ], + "score": 1.0, + "content": "Table 7: Performance of the state-of-the-art existing approaches with and without", + "type": "text" + }, + { + "bbox": [ + 446, + 411, + 468, + 422 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { R R R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 409, + 505, + 424 + ], + "score": 1.0, + "content": "on Ima-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "geNet100 in 10 tasks. Results for iTAML (Rajasegaran et al., 2020), BiC (Wu et al., 2019), and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "EEIL (Castro et al., 2018) are produced with their original implementation while EWC (Kirkpatrick", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "score": 1.0, + "content": "et al., 2017) and LwF (Li & Hoiem, 2016) are re-implemented by us. Results are averaged over 3", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 455, + 406, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 406, + 467 + ], + "score": 1.0, + "content": "runs. Figure 3c in the main paper is generated using numbers in this Table.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 106, + 476, + 505, + 583 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 583 + ], + "score": 0.982, + "html": "
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iTAML99.4± 0.896.4±0.994.4 ± 0.993.0±0.392.4± 1.290.6±0.389.9 ±0.490.3±0.890.3 ±1.189.8±0.4
iTAML+RRR99.4± 0.897.3± 0.596.6±0.796.3 ± 1.195.3 ± 0.593.1 ± 0.592.1 ± 0.692.1±0.692.9±0.991.9 ± 0.4
EEIL99.5 ± 0.498.8 ± 1.195.9±0.993.0 ±0.488.3 ±1.186.7 ± 1.283.0 ±1.281.1± 0.578.2 ±0.775.4± 0.4
EEIL+RRR99.5± 0.498.1 ± 0.797.4 ± 1.196.7 ± 0.493.3 ± 0.589.4 ± 1.186.5±0.386.1 ± 1.181.8 ± 0.477.0± 0.3
BiC98.3±0.794.9 ±0.893.5±0.790.9 ± 1.289.0 ± 1.287.3± 0.685.2±0.783.2 ± 0.482.5±0.981.1 ± 1.1
BiC+RRR98.3±0.798.9 ±0.396.5±0.693.9 ± 0.492.0±0.789.3 ± 1.187.2 ±0.887.2 ± 1.185.5±0.984.1± 0.6
iCaRL99.3±0.497.2 ± 0.993.5±0.991.0 ± 0.387.5 ± 1.282.1 ± 1.277.1 ± 0.472.8± 0.667.1 ±0.863.5 ± 1.1
iCaRL+RRR99.3± 0.497.9 ± 1.294.1 ± 0.792.8 ±0.791.7 ± 0.985.7 ± 1.182.1 ± 0.674.4 ± 0.972.2 ± 0.868.1±0.9
LwF99.3±0.595.2 ± 0.985.9± 0.973.9 ± 1.163.7±0.854.8 ± 0.850.1 ± 0.644.5 ± 0.940.7 ± 0.536.7±0.3
LwF+RRR99.3±0.597.1 ± 1.289.3 ±0.679.1 ± 0.569.1 ± 1.159.4 ± 1.157.2 ± 0.748.2 ± 1.145.1 ± 0.641.5 ± 0.5
FT99.3± 0.549.4 ± 0.332.6±0.324.7 ± 0.620.0 ± 1.216.7 ± 0.313.9 ± 0.312.3 ± 0.711.1 ± 0.69.9 ±0.7
ER99.3± 0.595.2 ± 0.888.1±0.878.1± 0.972.5 ± 0.669.1± 0.867.1 ± 0.661.8 ±0.655.1± 0.350.1 ± 1.1
RRR99.3± 0.596.5 ± 0.393.4±0.884.8±0.778.7 ± 0.474.7 ± 0.473.1 ± 0.568.4±0.860.2±0.355.1 ±0.7
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iTAML84.7±0.685.7 ±0.486.5±0.386.5±0.886.3±1.285.7±0.884.9 ± 1.182.6±0.380.8±0.782.4± 0.3
iTAML+RRR84.7±0.689.9± 0.589.2±0.989.2±0.689.0 ± 1.187.2 ±0.688.0±0.485.6 ± 1.186.6±0.385.4±0.3
BiC95.7±0.690.3±0.980.9±0.875.8 ± 0.873.5±0.671.5 ± 1.267.8±0.465.4±0.862.7 ± 1.261.9 ± 1.2
BiC+RRR95.7± 0.693.3±0.684.7 ± 1.177.5 ± 0.973.4±0.674.8 ± 0.669.6± 0.767.4± 0.365.7± 0.564.9±0.6
EEIL81.9 ± 0.586.3±0.384.9 ± 0.480.7±0.377.7 ±0.674.9± 0.370.9±0.767.4± 0.764.9 ± 0.562.4±0.3
EEIL+RRR81.9±0.588.4±0.887.6±0.782.6 ± 1.278.5 ± 0.676.9 ± 0.471.2 ±0.767.3± 0.467.0 ± 1.264.5±0.3
LwF85.1 ± 0.768.8 ±0.958.6 ±1.150.5 ± 1.243.5±0.937.5 ± 0.633.7±0.930.4±0.926.8 ± 1.124.9 ± 0.7
LwF+RRR85.1±0.771.6 ± 0.661.8±0.754.2 ± 0.546.2±0.940.7 ±0.736.7 ± 1.234.4± 0.429.8±0.727.2 ± 1.2
EWC85.1± 0.761.3± 0.547.4± 0.836.2±0.331.3±0.627.9± 0.523.7 ± 1.122.5± 0.420.8±0.818.9±0.7
EWC+RRR85.1±0.768.9 ±0.552.2 ±0.939.9 ±0.935.2±0.330.0±0.324.3 ± 0.824.0±0.623.7 ± 0.421.0 ± 1.1
ER85.1± 0.783.1± 0.981.8±0.774.9 ± 0.370.4± 0.361.5 ± 1.260.8 ± 1.157.0±0.754.3 ± 0.448.2±0.6
RRR85.1± 0.785.1 ± 0.983.8±0.477.9 ± 0.472.4 ± 1.264.5 ± 0.762.8±0.759.0±0.357.3±0.851.2 ± 1.1
", + "type": "table", + "image_path": "676956df21d0f00bae734752c33d3bc22a5457a67dca5308216eef3e529b5321.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 106, + 331, + 505, + 364.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 106, + 364.3333333333333, + 505, + 397.66666666666663 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 106, + 397.66666666666663, + 505, + 430.99999999999994 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 5 + }, + { + "type": "table", + "bbox": [ + 107, + 459, + 505, + 560 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 276, + 443, + 335, + 453 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 276, + 443, + 335, + 454 + ], + "spans": [ + { + "bbox": [ + 276, + 443, + 335, + 454 + ], + "score": 1.0, + "content": "(b) Tasks 11-20", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "table_body", + "bbox": [ + 107, + 459, + 505, + 560 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 459, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 505, + 560 + ], + "score": 0.974, + "html": "
11121314151617181920
iTAML80.0 ± 1.180.6±0.574.3 ± 0.870.7±0.671.3 ± 1.168.3±0.570.3 ±0.868.3±0.669.5 ± 0.366.0±0.6
iTAML+RRR85.5±0.585.2±0.879.7± 0.674.3 ± 0.474.0± 0.973.4 ± 1.174.8± 0.974.4 ± 0.473.9 ±0.571.8±0.9
BiC59.2 ±0.457.0± 0.656.1 ± 1.255.7± 0.653.8±0.552.4 ± 1.249.7 ± 0.649.2 ± 1.247.7 ± 1.146.7 ± 1.2
BiC+RRR62.2 ± 0.559.1 ±0.758.2±0.557.8±0.554.4 ± 1.256.6±0.953.9 ±0.752.4 ± 1.149.5 ± 0.849.4± 0.9
EEIL60.9±0.659.5 ± 0.657.8±0.655.1 ± 0.353.9±0.551.7± 0.350.1±0.849.4± 0.547.4± 0.646.9 ± 0.9
EEIL+RRR63.7±0.662.9 ± 0.459.7 ± 0.457.0±0.355.6±0.853.5± 0.453.5±0.352.7 ± 0.449.1 ± 0.347.8 ± 0.4
LwF23.9 ±0.721.4± 0.720.0±0.719.1 ± 0.918.7±0.817.1 ± 0.815.6 ±0.814.7 ± 0.814.0 ± 0.813.7 ± 1.1
LwF+RRR27.7± 0.726.9 ±0.925.7±0.724.5 ± 1.223.6±0.622.6±0.719.5 ± 0.318.6 ± 0.519.7 ± 0.818.4± 1.2
EWC17.2 ± 1.116.0 ± 0.515.0± 0.814.5 ± 0.813.4 ± 1.112.4 ± 0.412.3 ± 0.411.5 ± 0.811.2 ± 0.89.44± 0.5
EWC+RRR20.7±0.319.5 ± 0.418.4± 0.717.3 ± 0.516.2 ± 0.415.8 ± 0.515.0 ± 0.716.6± 0.914.3 ± 0.413.2± 0.3
ER45.8 ± 0.642.7± 0.741.6 ± 0.641.2 ± 0.636.5±0.436.5±0.633.8± 0.432.4± 1.231.4± 0.730.2±0.5
RRR48.8± 0.346.7 ± 0.943.6 ± 1.144.2 ± 0.739.5±0.338.5±0.935.8±0.333.4± 0.332.4±0.331.2±0.3
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12345678910
iTAML84.7±0.685.7 ±0.486.5±0.386.5±0.886.3±1.285.7±0.884.9 ± 1.182.6±0.380.8±0.782.4± 0.3
iTAML+RRR84.7±0.689.9± 0.589.2±0.989.2±0.689.0 ± 1.187.2 ±0.688.0±0.485.6 ± 1.186.6±0.385.4±0.3
BiC95.7±0.690.3±0.980.9±0.875.8 ± 0.873.5±0.671.5 ± 1.267.8±0.465.4±0.862.7 ± 1.261.9 ± 1.2
BiC+RRR95.7± 0.693.3±0.684.7 ± 1.177.5 ± 0.973.4±0.674.8 ± 0.669.6± 0.767.4± 0.365.7± 0.564.9±0.6
EEIL81.9 ± 0.586.3±0.384.9 ± 0.480.7±0.377.7 ±0.674.9± 0.370.9±0.767.4± 0.764.9 ± 0.562.4±0.3
EEIL+RRR81.9±0.588.4±0.887.6±0.782.6 ± 1.278.5 ± 0.676.9 ± 0.471.2 ±0.767.3± 0.467.0 ± 1.264.5±0.3
LwF85.1 ± 0.768.8 ±0.958.6 ±1.150.5 ± 1.243.5±0.937.5 ± 0.633.7±0.930.4±0.926.8 ± 1.124.9 ± 0.7
LwF+RRR85.1±0.771.6 ± 0.661.8±0.754.2 ± 0.546.2±0.940.7 ±0.736.7 ± 1.234.4± 0.429.8±0.727.2 ± 1.2
EWC85.1± 0.761.3± 0.547.4± 0.836.2±0.331.3±0.627.9± 0.523.7 ± 1.122.5± 0.420.8±0.818.9±0.7
EWC+RRR85.1±0.768.9 ±0.552.2 ±0.939.9 ±0.935.2±0.330.0±0.324.3 ± 0.824.0±0.623.7 ± 0.421.0 ± 1.1
ER85.1± 0.783.1± 0.981.8±0.774.9 ± 0.370.4± 0.361.5 ± 1.260.8 ± 1.157.0±0.754.3 ± 0.448.2±0.6
RRR85.1± 0.785.1 ± 0.983.8±0.477.9 ± 0.472.4 ± 1.264.5 ± 0.762.8±0.759.0±0.357.3±0.851.2 ± 1.1
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11121314151617181920
iTAML80.0 ± 1.180.6±0.574.3 ± 0.870.7±0.671.3 ± 1.168.3±0.570.3 ±0.868.3±0.669.5 ± 0.366.0±0.6
iTAML+RRR85.5±0.585.2±0.879.7± 0.674.3 ± 0.474.0± 0.973.4 ± 1.174.8± 0.974.4 ± 0.473.9 ±0.571.8±0.9
BiC59.2 ±0.457.0± 0.656.1 ± 1.255.7± 0.653.8±0.552.4 ± 1.249.7 ± 0.649.2 ± 1.247.7 ± 1.146.7 ± 1.2
BiC+RRR62.2 ± 0.559.1 ±0.758.2±0.557.8±0.554.4 ± 1.256.6±0.953.9 ±0.752.4 ± 1.149.5 ± 0.849.4± 0.9
EEIL60.9±0.659.5 ± 0.657.8±0.655.1 ± 0.353.9±0.551.7± 0.350.1±0.849.4± 0.547.4± 0.646.9 ± 0.9
EEIL+RRR63.7±0.662.9 ± 0.459.7 ± 0.457.0±0.355.6±0.853.5± 0.453.5±0.352.7 ± 0.449.1 ± 0.347.8 ± 0.4
LwF23.9 ±0.721.4± 0.720.0±0.719.1 ± 0.918.7±0.817.1 ± 0.815.6 ±0.814.7 ± 0.814.0 ± 0.813.7 ± 1.1
LwF+RRR27.7± 0.726.9 ±0.925.7±0.724.5 ± 1.223.6±0.622.6±0.719.5 ± 0.318.6 ± 0.519.7 ± 0.818.4± 1.2
EWC17.2 ± 1.116.0 ± 0.515.0± 0.814.5 ± 0.813.4 ± 1.112.4 ± 0.412.3 ± 0.411.5 ± 0.811.2 ± 0.89.44± 0.5
EWC+RRR20.7±0.319.5 ± 0.418.4± 0.717.3 ± 0.516.2 ± 0.415.8 ± 0.515.0 ± 0.716.6± 0.914.3 ± 0.413.2± 0.3
ER45.8 ± 0.642.7± 0.741.6 ± 0.641.2 ± 0.636.5±0.436.5±0.633.8± 0.432.4± 1.231.4± 0.730.2±0.5
RRR48.8± 0.346.7 ± 0.943.6 ± 1.144.2 ± 0.739.5±0.338.5±0.935.8±0.333.4± 0.332.4±0.331.2±0.3
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PrecisionRecall
MethodsPri,iPrT,iRei,iReT,i
ER80.068.964.165.1
ER+RRR82.170.364.266.8
TOPIC91.088.498.197.4
TOPIC+RRR92.889.199.699.2
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ER+RRR58.5-15.6
TOPIC72.7-0.9
TOPIC+RRR74.2-2.1
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Target layer name in PyTorch torchvision packageSaliency map size
SqueezeNet1_1features.0.12.expand3x313 ×13
AlexNetfeatures.0.1013 ×13
ResNet18features.7.1.conv27×7
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234567891011
RN18-RRR-GCam67.8±0.853.5±0.745.6±0.6 39.6±0.735.3±0.932.3± 1.129.4±0.925.9±0.825.7±0.626.3±0.723.6±0.7
RN18-ER67.8±0.849.7 ±0.941.7 ±0.835.8 ±0.731.4 ± 0.928.5±0.825.5±0.822.1±0.821.8±0.822.5 ± 1.119.8± 0.9
RN18-RRR-Smooth67.8± 0.850.9±0.643.5 ± 0.937.0±0.833.0±0.729.5± 0.626.8±0.823.9 ±0.823.9±0.823.4± 0.821.5± 0.5
RN18-RRR-BP67.8±0.850.8±0.843.9 ±0.636.6 ±0.432.7±0.628.9±0.627.2± 0.523.8 ±0.623.8±0.624.0± 0.421.5± 0.6
RN18-Finetune67.8± 0.844.8 ± 0.632.2± 0.525.8±0.725.6 ± 0.725.2± 0.720.8±0.616.8 ± 0.718.8± 0.518.3 ± 0.417.1 ± 0.6
Alex-RRR-GCam56.7±0.746.6±0.543.9±0.741.3± 0.733.7 ± 0.527.4± 0.725.3±0.722.0±0.521.5±0.621.4± 0.621.2± 0.6
Alex-ER56.7± 0.744.6 ± 0.741.3 ± 0.738.7±0.731.1 ± 0.724.5± 0.722.6± 0.719.6 ± 0.619.1± 0.818.7 ± 0.819.1± 0.8
Alex-Finetune56.7±0.742.8 ± 0.839.6±0.836.9±0.829.5 ± 0.723.3±0.621.4± 0.817.9 ± 0.718.0 ±0.717.0 ± 0.516.9 ± 0.4
SQ-RRR-GCam46.8± 0.536.2 ±0.430.1±0.628.3±0.425.1 ± 0.523.4± 0.519.3± 0.619.0± 0.618.5± 0.518.4± 0.518.2 ±0.6
SQ-ER46.8 ± 0.533.2±0.5 27.1±0.625.3±0.622.1±0.5220.5±0.516.3± 0.416.0±0.615.5± 0.615.4 ± 0.615.2 ± 0.7
SQ-Finetune46.8 ± 0.532.0±0.7 25.2±0.723.9±0.720.2±0.8119.4 ± 0.414.9 ± 0.414.4 ± 0.513.8± 0.414.2 ± 0.513.7±0.6
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iTAML99.4± 0.896.4±0.994.4 ± 0.993.0±0.392.4± 1.290.6±0.389.9 ±0.490.3±0.890.3 ±1.189.8±0.4
iTAML+RRR99.4± 0.897.3± 0.596.6±0.796.3 ± 1.195.3 ± 0.593.1 ± 0.592.1 ± 0.692.1±0.692.9±0.991.9 ± 0.4
EEIL99.5 ± 0.498.8 ± 1.195.9±0.993.0 ±0.488.3 ±1.186.7 ± 1.283.0 ±1.281.1± 0.578.2 ±0.775.4± 0.4
EEIL+RRR99.5± 0.498.1 ± 0.797.4 ± 1.196.7 ± 0.493.3 ± 0.589.4 ± 1.186.5±0.386.1 ± 1.181.8 ± 0.477.0± 0.3
BiC98.3±0.794.9 ±0.893.5±0.790.9 ± 1.289.0 ± 1.287.3± 0.685.2±0.783.2 ± 0.482.5±0.981.1 ± 1.1
BiC+RRR98.3±0.798.9 ±0.396.5±0.693.9 ± 0.492.0±0.789.3 ± 1.187.2 ±0.887.2 ± 1.185.5±0.984.1± 0.6
iCaRL99.3±0.497.2 ± 0.993.5±0.991.0 ± 0.387.5 ± 1.282.1 ± 1.277.1 ± 0.472.8± 0.667.1 ±0.863.5 ± 1.1
iCaRL+RRR99.3± 0.497.9 ± 1.294.1 ± 0.792.8 ±0.791.7 ± 0.985.7 ± 1.182.1 ± 0.674.4 ± 0.972.2 ± 0.868.1±0.9
LwF99.3±0.595.2 ± 0.985.9± 0.973.9 ± 1.163.7±0.854.8 ± 0.850.1 ± 0.644.5 ± 0.940.7 ± 0.536.7±0.3
LwF+RRR99.3±0.597.1 ± 1.289.3 ±0.679.1 ± 0.569.1 ± 1.159.4 ± 1.157.2 ± 0.748.2 ± 1.145.1 ± 0.641.5 ± 0.5
FT99.3± 0.549.4 ± 0.332.6±0.324.7 ± 0.620.0 ± 1.216.7 ± 0.313.9 ± 0.312.3 ± 0.711.1 ± 0.69.9 ±0.7
ER99.3± 0.595.2 ± 0.888.1±0.878.1± 0.972.5 ± 0.669.1± 0.867.1 ± 0.661.8 ±0.655.1± 0.350.1 ± 1.1
RRR99.3± 0.596.5 ± 0.393.4±0.884.8±0.778.7 ± 0.474.7 ± 0.473.1 ± 0.568.4±0.860.2±0.355.1 ±0.7
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iTAML+RRR89.2± 0.592.3±0.789.5 ± 1.287.5 ± 1.284.1±0.883.5±0.983.9±0.781.2± 0.379.6 ± 0.979.7±0.5
iTAML89.2± 0.588.9±0.587.0 ± 1.185.7 ± 1.184.1 ± 1.181.8 ± 0.380.0±0.679.0± 0.378.6 ±0.877.8 ±0.6
BiC90.3±0.782.1 ± 0.775.1± 0.469.8 ± 1.265.3±0.861.3± 0.957.4± 0.754.9± 0.553.2 ±0.950.3±0.7
BiC+RRR90.3±0.784.9 ± 1.176.4± 0.669.3±0.365.1±0.963.3± 0.459.7 ± 1.155.4± 0.855.8 ± 0.752.1 ± 0.5
EEIL80.0±0.780.5± 1.275.5± 0.971.5± 0.468.0± 1.262.0±0.959.0± 0.755.1 ± 1.251.4 ± 0.848.7 ± 0.4
EEIL+RRR80.0±0.783.5± 0.378.7 ± 1.274.0 ± 1.271.7± 0.365.1 ± 0.461.2± 0.557.6± 0.554.1 ± 0.451.7± 0.3
LwF86.1 ± 1.269.0±0.755.0±0.345.8± 0.340.4± 0.536.7±0.930.8 ±0.728.6±0.526.1 ± 0.724.2 ±0.7
LwF+RRR86.1 ± 1.272.4± 0.857.0 ± 1.148.3 ± 0.343.2 ±0.839.3 ±0.534.1 ± 0.632.1 ± 1.129.8 ±0.727.1± 0.6
EWC86.1 ± 1.252.6 ± 0.448.6± 0.438.5±0.531.1 ± 0.926.5±0.321.7±0.620.0±0.718.9 ± 0.516.6 ±0.9
EWC+RRR86.1 ± 1.256.0± 0.453.9 ± 1.244.4 ± 0.935.1±0.528.6±0.625.1 ± 1.123.1± 0.518.8 ±0.919.0 ± 1.2
ER86.1 ± 1.274.5 ± 0.965.2±0.862.5± 0.856.7±0.750.5± 0.347.6 ± 0.443.4± 0.341.6 ± 0.938.1 ± 1.1
RRR86.1 ± 1.278.5± 0.969.2 ± 1.163.5 ± 1.258.7±0.853.5 ± 1.149.6± 0.744.4± 0.342.6 ± 1.239.1 ± 1.1
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EEIL68.6±0.453.6± 0.447.9 ± 0.344.2±0.836.3±0.927.4 ± 1.225.9±0.724.7±0.523.9±0.724.1± 0.722.1 ± 0.5
EEIL+RRR68.6±0.456.6±0.550.9±0.648.3± 0.539.7 ± 1.231.4± 0.728.3± 1.228.0±0.626.5±0.627.4± 0.625.2±0.9
iCaRL68.6±0.452.6±0.748.6± 1.244.1 ± 0.536.6±0.329.5±0.927.8± 0.426.2±0.524.0±0.623.8±0.621.1± 0.7
iCaRL+RRR68.6±0.455.6± 1.253.6±0.747.1 ± 0.839.6±0.532.5±0.831.8 ± 0.429.2±0.627.0±0.827.8±0.624.1 ± 0.3
TOPIC68.6 ± 0.462.4 ± 0.854.8 ± 0.449.9 ± 1.245.2 ±0.641.4± 0.338.3±0.835.3±0.632.2± 0.328.3±0.626.2 ± 1.2
TOPIC+RRR68.6±0.462.5 ± 0.956.8 ± 0.451.5 ± 0.548.2 ± 0.444.4 ± 0.442.3±0.738.3±0.635.2±0.932.3±0.929.2 ± 0.5
FT68.6±0.444.8 ± 0.532.2±0.825.8± 0.425.6 ± 1.125.2± 0.720.8± 1.116.7± 0.418.8 ± 1.118.2± 0.317.1 ± 0.8
ER67.8±0.849.7± 0.941.7 ± 0.835.8 ±0.731.4± 0.928.5±0.825.5±0.822.1± 0.821.8± 0.622.5 ± 1.119.8±0.9
RRR67.8±0.853.5± 0.745.6± 0.639.6± 0.735.3±0.932.3 ± 1.129.4± 0.925.9± 0.825.7±0.626.3±0.723.6±0.7
JT68.6±0.462.4± 0.457.2 ± 0.452.8±0.549.5 ± 0.946.1 ± 0.542.8 ± 1.140.1±0.838.7±0.737.1± 0.535.6±0.9
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iTAML84.7±0.685.7 ±0.486.5±0.386.5±0.886.3±1.285.7±0.884.9 ± 1.182.6±0.380.8±0.782.4± 0.3
iTAML+RRR84.7±0.689.9± 0.589.2±0.989.2±0.689.0 ± 1.187.2 ±0.688.0±0.485.6 ± 1.186.6±0.385.4±0.3
BiC95.7±0.690.3±0.980.9±0.875.8 ± 0.873.5±0.671.5 ± 1.267.8±0.465.4±0.862.7 ± 1.261.9 ± 1.2
BiC+RRR95.7± 0.693.3±0.684.7 ± 1.177.5 ± 0.973.4±0.674.8 ± 0.669.6± 0.767.4± 0.365.7± 0.564.9±0.6
EEIL81.9 ± 0.586.3±0.384.9 ± 0.480.7±0.377.7 ±0.674.9± 0.370.9±0.767.4± 0.764.9 ± 0.562.4±0.3
EEIL+RRR81.9±0.588.4±0.887.6±0.782.6 ± 1.278.5 ± 0.676.9 ± 0.471.2 ±0.767.3± 0.467.0 ± 1.264.5±0.3
LwF85.1 ± 0.768.8 ±0.958.6 ±1.150.5 ± 1.243.5±0.937.5 ± 0.633.7±0.930.4±0.926.8 ± 1.124.9 ± 0.7
LwF+RRR85.1±0.771.6 ± 0.661.8±0.754.2 ± 0.546.2±0.940.7 ±0.736.7 ± 1.234.4± 0.429.8±0.727.2 ± 1.2
EWC85.1± 0.761.3± 0.547.4± 0.836.2±0.331.3±0.627.9± 0.523.7 ± 1.122.5± 0.420.8±0.818.9±0.7
EWC+RRR85.1±0.768.9 ±0.552.2 ±0.939.9 ±0.935.2±0.330.0±0.324.3 ± 0.824.0±0.623.7 ± 0.421.0 ± 1.1
ER85.1± 0.783.1± 0.981.8±0.774.9 ± 0.370.4± 0.361.5 ± 1.260.8 ± 1.157.0±0.754.3 ± 0.448.2±0.6
RRR85.1± 0.785.1 ± 0.983.8±0.477.9 ± 0.472.4 ± 1.264.5 ± 0.762.8±0.759.0±0.357.3±0.851.2 ± 1.1
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iTAML80.0 ± 1.180.6±0.574.3 ± 0.870.7±0.671.3 ± 1.168.3±0.570.3 ±0.868.3±0.669.5 ± 0.366.0±0.6
iTAML+RRR85.5±0.585.2±0.879.7± 0.674.3 ± 0.474.0± 0.973.4 ± 1.174.8± 0.974.4 ± 0.473.9 ±0.571.8±0.9
BiC59.2 ±0.457.0± 0.656.1 ± 1.255.7± 0.653.8±0.552.4 ± 1.249.7 ± 0.649.2 ± 1.247.7 ± 1.146.7 ± 1.2
BiC+RRR62.2 ± 0.559.1 ±0.758.2±0.557.8±0.554.4 ± 1.256.6±0.953.9 ±0.752.4 ± 1.149.5 ± 0.849.4± 0.9
EEIL60.9±0.659.5 ± 0.657.8±0.655.1 ± 0.353.9±0.551.7± 0.350.1±0.849.4± 0.547.4± 0.646.9 ± 0.9
EEIL+RRR63.7±0.662.9 ± 0.459.7 ± 0.457.0±0.355.6±0.853.5± 0.453.5±0.352.7 ± 0.449.1 ± 0.347.8 ± 0.4
LwF23.9 ±0.721.4± 0.720.0±0.719.1 ± 0.918.7±0.817.1 ± 0.815.6 ±0.814.7 ± 0.814.0 ± 0.813.7 ± 1.1
LwF+RRR27.7± 0.726.9 ±0.925.7±0.724.5 ± 1.223.6±0.622.6±0.719.5 ± 0.318.6 ± 0.519.7 ± 0.818.4± 1.2
EWC17.2 ± 1.116.0 ± 0.515.0± 0.814.5 ± 0.813.4 ± 1.112.4 ± 0.412.3 ± 0.411.5 ± 0.811.2 ± 0.89.44± 0.5
EWC+RRR20.7±0.319.5 ± 0.418.4± 0.717.3 ± 0.516.2 ± 0.415.8 ± 0.515.0 ± 0.716.6± 0.914.3 ± 0.413.2± 0.3
ER45.8 ± 0.642.7± 0.741.6 ± 0.641.2 ± 0.636.5±0.436.5±0.633.8± 0.432.4± 1.231.4± 0.730.2±0.5
RRR48.8± 0.346.7 ± 0.943.6 ± 1.144.2 ± 0.739.5±0.338.5±0.935.8±0.333.4± 0.332.4±0.331.2±0.3
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