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+# SNIPS: Solving Noisy Inverse Problems Stochastically
+
+Bahjat Kawar, Gregory Vaksman, Michael Elad Computer Science Department, Technion, Haifa, Israel {bahjat.kawar, grishav, elad}@cs.technion.ac.il
+
+# Abstract
+
+In this work we introduce a novel stochastic algorithm dubbed SNIPS, which draws samples from the posterior distribution of any linear inverse problem, where the observation is assumed to be contaminated by additive white Gaussian noise. Our solution incorporates ideas from Langevin dynamics and Newton’s method, and exploits a pre-trained minimum mean squared error (MMSE) Gaussian denoiser. The proposed approach relies on an intricate derivation of the posterior score function that includes a singular value decomposition (SVD) of the degradation operator, in order to obtain a tractable iterative algorithm for the desired sampling. Due to its stochasticity, the algorithm can produce multiple high perceptual quality samples for the same noisy observation. We demonstrate the abilities of the proposed paradigm for image deblurring, super-resolution, and compressive sensing. We show that the samples produced are sharp, detailed and consistent with the given measurements, and their diversity exposes the inherent uncertainty in the inverse problem being solved.
+
+# 1 Introduction
+
+Many problems in the field of image processing can be cast as noisy linear inverse problems. This family of tasks includes denoising, inpainting, deblurring, super resolution, compressive sensing, and many other image recovery problems. A general linear inverse problem is posed as
+
+$$
+\mathbf { y } = \mathbf { H } \mathbf { x } + \mathbf { z } ,
+$$
+
+where we aim to recover a signal $\mathbf { x }$ from its measurement $\mathbf { y }$ , given through a linear degradation operator $\mathbf { H }$ and a contaminating noise, being additive, white and Gaussian, $\mathbf { \overline { { z } } } \sim \mathcal { N } \left( 0 , \sigma _ { 0 } ^ { 2 } \mathbf { \check { I } } \right)$ . In this work we assume that both $\mathbf { H }$ and $\sigma _ { 0 }$ are known.
+
+Over the years, many strategies, algorithms and underlying statistical models were developed for handling image restoration problems. A key ingredient in many of the classic attempts is the prior that aims to regularize the inversion process and lead to visually pleasing results. Among the various options explored, we mention sparsity-inspired techniques [13, 55, 11], local Gaussian-mixture modeling [57, 63], and methods relying on non-local self-similarity [6, 9, 36, 51]. More recently, and with the emergence of deep learning techniques, a direct design of the recovery path from y to an estimate of $\mathbf { x }$ took the lead, yielding state-of-the-art results in various linear inverse problems, such as denoising [25, 59, 61, 52], deblurring [22, 48], super resolution [10, 17, 54] and other tasks [29, 28, 19, 16, 37, 58].
+
+Despite the evident success of the above techniques, many image restoration algorithms still have a critical shortcoming: In cases of severe degradation, most recovery algorithms tend to produce washed out reconstructions that lack details. Indeed, most image restoration techniques seek a reconstruction that minimizes the mean squared error between the restored image, ˆx, and the unknown original one, x. When the degradation is acute and information is irreversibly lost, image reconstruction becomes a highly ill-posed problem, implying that many possible clean images could explain the given measurements. The MMSE solution averages all these candidate solutions, being the conditional mean of the posterior of $\mathbf { x }$ given y, leading to an image with loss of fine details in the majority of practical cases. A recent work reported in [5] has shown that reconstruction algorithms necessarily suffer from a perception-distortion tradeoff, i.e., targeting a minimization of the error between $\hat { \bf x }$ and $\mathbf { x }$ (in any metric) is necessarily accompanied by a compromised perceptual quality. As a consequence, as long as we stick to the tendency to design recovery algorithms that aim for minimum MSE (or other distances), only a limited perceptual improvement can be expected.
+
+When perceptual quality becomes our prime objective, the strategy for solving inverse problems must necessarily change. More specifically, the solution should concentrate on producing a sample (or many of them) from the posterior distribution $p \left( \mathbf { x } | \mathbf { y } \right)$ instead of its conditional mean. Recently, two such approaches have been suggested – GAN-based and Langevin sampling. Generative Adversarial Networks (GANs) have shown impressive results in generating realistically looking images (e.g., [14, 35]). GANs can be utilized for solving inverse problems while producing high-quality images (see e.g. [2, 31, 34]). These solvers aim to produce a diverse set of output images that are consistent with the measurements, while also being aligned with the distribution of clean examples. A major disadvantage of GAN-based algorithms for inverse problems is their tendency (as practiced in [2, 31, 34]) to assume noiseless measurements, a condition seldom met in practice. An exception to this is the work reported in [33], which adapts a conditional GAN to become a stochastic denoiser.
+
+The second approach for sampling from the posterior, and the one we shall be focusing on in this paper, is based on Langevin dynamics. This core iterative technique enables sampling from a given distribution by leveraging the availability of the score function – the gradient of the log of the probability density function [38, 3]. The work reported in [44, 20, 46] utilizes the annealed Langevin dynamics method, both for image synthesis and for solving noiseless inverse problems.1 Their synthesis algorithm relies on an MMSE Gaussian denoiser (given as a neural network) for approximating a gradually blurred score function. In their treatment of inverse problems, the conditional score remains tractable and manageable due to the noiseless measurements assumption.
+
+The question addressed in this paper is the following: How can the above line of Langevin-based work be generalized for handling linear inverse problems, as in Equation 1, in which the measurements are noisy? A partial and limited answer to this question has already been given in [21] for the tasks of image denoising and inpainting. The present work generalizes these ([44, 20, 46, 21]) results, and introduces a systematic way for sampling from the posterior distribution of any given noisy linear inverse problem. As we carefully show, this extension is far from being trivial, due to two prime reasons: (i) The involvement of the degradation operator H, which poses a difficulty for establishing a relationship between the reconstructed image and the noisy observation; and (ii) The intricate connection between the measurements’ and the synthetic annealed Langevin noise. Our proposed remedy is a decorrelation of the measurements equation via a singular value decomposition (SVD) of the operator H, which decouples the dependencies between the measurements, enabling each to be addressed by an adapted iterative process. In addition, we define the annealing noise to be built as portions of the measurement noise itself, in a manner that facilitates a constructive derivation of the conditional score function.
+
+Following earlier work [44, 20, 46, 21], our algorithm is initialized with a random noise image, gradually converging to the reconstructed result, while following the direction of the log-posterior gradient, estimated using an MMSE denoiser. Via a careful construction of the gradual annealing noise sequence, from very high values to low ones, the entries in the derived score switch mode. Those referring to non-zero singular values start by being purely dependent on the measurements, and then transition to incorporate prior information based on the denoiser. As for entries referring to zero singular values, their corresponding entries undergo a pure synthesis process based on the prior-only score function. Note that the denoiser blends values in the evolving sample, thus intermixing the influence of the gradient entries. Our derivations include an analytical expression for a positiondependent step size vector, drawing inspiration from Newton’s method in optimization. This stabilizes the algorithm and is shown to be essential for its success.
+
+We refer hereafter to our algorithm as SNIPS (Solution of Noisy Inverse Problems Stochastically). Observe that as we target to sample from the posterior distribution $p \left( \mathbf { x } | \mathbf { y } \right)$ , different runs of SNIPS on the same input necessarily yield different results, all of which valid solutions to the given inverse problem. This should not come as a surprise, as ill-posedness implies that there are multiple viable solutions for the same data, as has already been suggested in the context of super resolution [31, 2, 34]. We demonstrate SNIPS on image deblurring, single image super resolution, and compressive sensing, all of which contain non-negligible noise, and emphasize the high perceptual quality of the results, their diversity, and their relation to the MMSE estimate.
+
+![](images/c82146786b5adb3c31fe1fe0e0cc72d1662774fbb84d513189a447eec3bd7cf3.jpg)  
+Figure 1: Deblurring results on CelebA [27] images (uniform $5 \times 5$ blur and an additive noise with $\sigma _ { 0 } = 0 . 1$ ). Here and in all other shown figures, the standard deviation image is scaled by 4 for better visual inspection.
+
+To summarize, this paper’s contributions are threefold:
+
+• We present an intricate derivation of the blurred posterior score function for general noisy inverse problems, where both the measurement and the target image contain delicately inter-connected additive white Gaussian noise. • We introduce a novel stochastic algorithm – SNIPS – that can sample from the posterior distribution of these problems. The algorithm relies on the availability of an MMSE denoiser. • We demonstrate impressive results of SNIPS on image deblurring, single image super resolution, and compressive sensing, all of which are highly noisy and ill-posed.
+
+Before diving into the details of this work, we should mention that using Gaussian denoisers iteratively for handling general linear inverse problems has been already proposed in the context of the Plugand-Play-Prior $( \mathrm { P n P } )$ method [53] and RED [39], and their many followup papers (e.g., [60, 30, 1, 49, 7, 50, 40, 4]). However, both PnP and RED are quite different from our work, as they do not target sampling from the posterior, but rather focus on MAP or MMSE estimation.
+
+# 2 Background
+
+The Langevin dynamics algorithm [3, 38] suggests sampling from a probability distribution $p \left( \mathbf { x } \right)$ using the iterative transition rule
+
+$$
+{ \bf x } _ { t + 1 } = { { \bf x } _ { t } } + \alpha \nabla _ { { \bf x } _ { t } } \log p \left( { { \bf x } _ { t } } \right) + \sqrt { 2 \alpha } { \bf z } _ { t } \mathrm { ~ , ~ }
+$$
+
+where $\mathbf { z } _ { t } \sim \mathcal { N } ( 0 , \mathbf { I } )$ and $\alpha$ is an appropriately chosen small constant. The added $\mathbf { z } _ { t }$ allows for stochastic sampling, avoiding a collapse to a maximum of the distribution. Initialized randomly, after a sufficiently large number of iterations, and under some mild conditions, this process converges to a sample from the desired distribution $p \left( \mathbf { x } \right)$ [38].
+
+The work reported in [44] extends the aforementioned algorithm into annealed Langevin dynamics. The annealing proposed replaces the score function in Equation 2 with a blurred version of it, $\nabla _ { \tilde { \mathbf { x } } _ { t } } \log { p \left( \tilde { \mathbf { x } } _ { t } \right) }$ , where $\tilde { \mathbf { x } } _ { \mathbf { t } } = \mathbf { x } _ { t } + \mathbf { n }$ and $\mathbf { n } \sim { \mathcal { N } } \left( 0 , \sigma ^ { 2 } \mathbf { I } \right)$ is a synthetically injected noise. The core idea is to start with a very high noise level $\sigma$ and gradually drop it to near-zero, all while using a step size $\alpha$ dependent on the noise level. These changes allow the algorithm to converge much faster and perform better, because it widens the basin of attraction of the sampling process. The work in [20] further develops this line of work by leveraging a brilliant relation attributed to Miyasawa [32] (also known as Stein’s integration by parts trick [47] or Tweedie’s identity [12]). It is given as
+
+$$
+{ \nabla } _ { \tilde { \mathbf { x } } _ { t } } \log p \left( \tilde { \mathbf { x } } _ { t } \right) = \frac { \mathbf { D } \left( \tilde { \mathbf { x } } _ { t } , \sigma \right) - \tilde { \mathbf { x } } _ { t } } { { \sigma } ^ { 2 } } ,
+$$
+
+where $\mathbf { D } \left( \tilde { \mathbf { x } } _ { t } , \sigma \right) = \mathbb { E } \left[ \mathbf { x } | \tilde { \mathbf { x } } _ { t } \right]$ is the minimizer of the MSE measure $\mathbb { E } \left[ \lVert \mathbf { x } - \mathbf { D } \left( \tilde { \mathbf { x } } _ { t } , \sigma \right) \rVert _ { 2 } ^ { 2 } \right]$ , which can be approximated using a denoising neural network. This facilitates the use of denoisers in Langevin dynamics as a replacement for the evasive score function.
+
+When turning to solve inverse problems, previous work suggests sampling from the posterior distribution $p \left( \mathbf { x } | \mathbf { y } \right)$ using annealed Langevin dynamics [20, 46, 21] or similar methods [15, 18, 42, 26], by replacing the score function used in the generation algorithm with a conditional one. As it turns out, if limiting assumptions can be posed on the measurements formation, the conditional score is tractable, and thus generalization of the annealed Langevin process to these problems is within reach. Indeed, in [44, 20, 46, 42, 26] the core assumption is $\mathbf { y } = \mathbf { H } \mathbf { x }$ for specific and simplified choices of $\mathbf { H }$ and with no noise in the measurements. The works in [15, 23] avoid these difficulties altogether by returning to the original (non-annealed) Langevin method, with the unavoidable cost of becoming extremely slow. In addition, their algorithms are demonstrated on inverse problems in which the additive noise is restricted to be very weak. The work in [21] is broader, allowing for an arbitrary additive white Gaussian noise, but limits $\mathbf { H }$ to the problems of denoising or inpainting. While all these works demonstrate high quality results, there is currently no clear way for deriving the blurred score function of a general linear inverse problem as posed in Equation 1. In the following, we present such a derivation.
+
+# 3 The Proposed Approach: Deriving the Conditional Score Function
+
+# 3.1 Problem Setting
+
+We consider the problem of recovering a signal $\mathbf { x } \in \mathbb { R } ^ { N }$ (where $\mathbf { x } \sim p \left( \mathbf { x } \right)$ and $p \left( \mathbf { x } \right)$ is unknown) from the observation $\mathbf { y } = \mathbf { H } \mathbf { x } + \mathbf { z }$ , where $\mathbf { y } \in \mathbb { R } ^ { M } , \mathbf { H } \in \mathbb { R } ^ { M \times N } , M \leq N , \mathbf { z } \sim \mathcal { N } \left( 0 , \sigma _ { 0 } ^ { 2 } \mathbf { I } \right)$ , and $\mathbf { H }$ and $\sigma _ { 0 }$ are known.2 Our ultimate goal is to sample from the posterior $p \left( \mathbf { x } | \mathbf { y } \right)$ . However, since access to the score function $\nabla _ { \mathbf { x } } \log p ( \mathbf { x } | \mathbf { y } )$ is not available, we retarget our goal, as explained above, to sampling from blurred posterior distributions, $p \left( \tilde { \mathbf { x } } | \mathbf { y } \right)$ , where $\tilde { \mathbf { x } } = \mathbf { x } + \mathbf { n }$ and $\mathbf { n } \sim { \mathcal { N } } \left( 0 , \sigma ^ { 2 } \mathbf { I } \right)$ , with noise levels $\sigma$ starting very high, and decreasing towards near-zero.
+
+As explained in the supplemental material, the sampling should be performed in the SVD domain in order to get a tractable derivation of the blurred score function. Thus, we consider the singular value decomposition (SVD) of $\mathbf { H }$ , given as $\mathbf { H } = \mathbf { U } \pmb { \Sigma } \mathbf { V } ^ { T }$ , where $\mathbf { U } \in \mathbb { R } ^ { M \times M }$ and $\mathbf { V } \in \mathbb { R } ^ { N \times N }$ are orthogonal matrices, and $\pmb { \Sigma } \in \mathbb { R } ^ { M \times N }$ is a rectangular diagonal matrix containing the singular values of H, denoted as {sj}Mj=1 i n descending order $\quad : s _ { 1 } > s _ { 2 } > \dots > s _ { M - 1 } > s _ { M } \geq 0 \quad$ ). For convenience of notations, we also define $s _ { j } = 0$ for $j = M + 1 , \dotsc , N$ . To that end, we notice that
+
+$$
+p \left( \tilde { \mathbf { x } } | \mathbf { y } \right) = p \left( \tilde { \mathbf { x } } | \mathbf { U } ^ { T } \mathbf { y } \right) = p \left( \mathbf { V } ^ { T } \tilde { \mathbf { x } } | \mathbf { U } ^ { T } \mathbf { y } \right) .
+$$
+
+The first equality holds because the multiplication of $\mathbf { y }$ by the orthogonal matrix $\mathbf { U } ^ { T }$ does not add or remove information, and the second equality holds because the multiplication of $\tilde { \bf x }$ by $\mathbf { V } ^ { T }$ does not change its probability [24]. Therefore, sampling from $p \left( \mathbf { V } ^ { T } \tilde { \mathbf { x } } | \mathbf { U } ^ { T } \mathbf { y } \right)$ and then multiplying the result by $\mathbf { V }$ will produce the desired sample from $p \left( \mathbf { \tilde { x } } | \mathbf { y } \right)$ . As we are using Langevin dynamics, we need to calculate the conditional score function $\nabla _ { \mathbf { V } ^ { T } \widetilde { \mathbf { x } } } \log p \left( \mathbf { V } ^ { T } \widetilde { \mathbf { x } } | \mathbf { U } ^ { T } \mathbf { y } \right)$ . For simplicity, we denote hereafter ${ \bf y } _ { T } = { \bf U } ^ { T } { \bf y } , { \bf z } _ { T } = { \bf U } ^ { T } { \bf z } , { \bf x } _ { T } = { \bf V } ^ { T } { \bf x } , { \bf n } _ { T } = \Sigma { \bf V } ^ { T } { \bf n }$ , and $\tilde { \mathbf { x } } _ { T } = \mathbf { V } ^ { T } \tilde { \mathbf { x } }$ . Observe that with these notations, the measurements equation becomes
+
+$$
+\mathbf { y } = \mathbf { H } \mathbf { x } + \mathbf { z } = \mathbf { U } \pmb { \Sigma } \mathbf { V } ^ { T } \mathbf { x } + \mathbf { z } ,
+$$
+
+and thus
+
+$$
+\mathbf { U } ^ { T } \mathbf { y } = \Sigma \mathbf { V } ^ { T } \mathbf { x } + \mathbf { U } ^ { T } \mathbf { z } = \Sigma \mathbf { V } ^ { T } ( \tilde { \mathbf { x } } - \mathbf { n } ) + \mathbf { U } ^ { T } \mathbf { z } = \Sigma \mathbf { V } ^ { T } \tilde { \mathbf { x } } - \Sigma \mathbf { V } ^ { T } \mathbf { n } + \mathbf { U } ^ { T } \mathbf { z } ,
+$$
+
+where we have relied on the relation $\tilde { \mathbf { x } } = \mathbf { x } + \mathbf { n }$ . This leads to
+
+$$
+{ \bf y } _ { T } = \pmb { \Sigma } \tilde { \bf x } _ { T } - { \bf n } _ { T } + { \bf z } _ { T } .
+$$
+
+In this formulation, which will aid in deriving the conditional score, our aim is to make design choices on ${ \bf n } _ { T }$ such that ${ \bf z } _ { T } - { \bf n } _ { T }$ has uncorrelated entries and is independent of $\tilde { \bf x } _ { T }$ . This brings us to the formation of the synthetic annealed noise, which is an intricate ingredient in our derivations.
+
+We base this formation on the definition of a sequence of noise levels $\{ \sigma _ { i } \} _ { i = 1 } ^ { L + 1 }$ such that $\sigma _ { 1 } > \sigma _ { 2 } > \cdot \cdot \cdot > \sigma _ { L } > \sigma _ { L + 1 } = 0$ , where $\sigma _ { 1 }$ is high (possibly $\sigma _ { 1 } > \| \mathbf { x } \| _ { \infty } )$ and $\sigma _ { L }$ is close to zero. We require that for every $j$ such that $s _ { j } \neq 0$ , there exists $i _ { j }$ such that $\sigma _ { i _ { j } } s _ { j } < \sigma _ { 0 }$ and $\sigma _ { i _ { j } - 1 } s _ { j } > \sigma _ { 0 }$ . This implies $\forall i : \sigma _ { i } s _ { j } \neq \sigma _ { 0 }$ , which helps ease notations. SNIPS works just as well for $\sigma _ { i } s _ { j } = \sigma _ { 0 }$ .
+
+Using $\{ \sigma _ { i } \} _ { i = 1 } ^ { L + 1 }$ , we would like to define $\left\{ \tilde { \mathbf { x } } _ { i } \right\} _ { i = 1 } ^ { L + 1 }$ , a sequence of noisy versions of $\mathbf { x }$ , where the noise level in $\tilde { \mathbf { x } } _ { i }$ is $\sigma _ { i }$ . One might be tempted to define these noise additions as independent of the measurement noise $\mathbf { z }$ . However, this option leads to a conditional score term that cannot be calculated analytically, as explained in the supplemental material. Therefore, we define these noise additions differently, as carved from $\mathbf { z }$ in a gradual fashion. To that end, we define $\tilde { \mathbf { x } } _ { L + 1 } = \mathbf { x }$ , and for every $i = L , L - 1 , \ldots , 1 \colon \tilde { \mathbf { x } } _ { i } = \tilde { \mathbf { x } } _ { i + 1 } + \pmb { \eta } _ { i }$ , where $\pmb { \eta } _ { i } \sim \mathcal { N } \left( 0 , \left( \sigma _ { i } ^ { 2 } - \sigma _ { i + 1 } ^ { 2 } \right) \mathbf { I } \right)$ . This results in $\tilde { \mathbf { x } } _ { i } = \mathbf { x } + \mathbf { n } _ { i }$ , where $\begin{array} { r } { \mathbf { n } _ { i } = \sum _ { k = i } ^ { L } \pmb { \eta } _ { k } \sim \mathcal { N } \left( 0 , \sigma _ { i } ^ { 2 } \mathbf { I } \right) } \end{array}$ .
+
+And now we turn to define the statistical dependencies between the measurements’ noise $\mathbf { z }$ and the artificial noise vectors $\eta _ { i }$ . Since $\eta _ { i }$ and $\mathbf { z }$ are each Gaussian with uncorrelated entries, so are the components of the vectors $\pmb { \Sigma } \mathbf { V } ^ { T } \pmb { \eta } _ { i }$ , $\pmb { \Sigma } \mathbf { V } ^ { T } \mathbf { n } _ { i }$ , and $\mathbf { z } _ { T }$ . In order to proceed while easing notations, let us focus on a single entry $j$ in these three vectors, for which $s _ { j } > 0$ , and omit this index. We denote these entries as $\eta _ { T , i } , n _ { T , i }$ and $z _ { T }$ , respectively. We construct $\eta _ { T , i }$ such that
+
+$$
+\mathbb { E } \left[ \eta _ { T , i } \cdot z _ { T } \right] = \left\{ \begin{array} { l l } { \mathbb { E } \left[ \eta _ { T , i } ^ { 2 } \right] } & { \mathrm { f o r } i \geq i _ { j } } \\ { \mathbb { E } \left[ \left( z _ { T } - n _ { T , i _ { j } } \right) ^ { 2 } \right] } & { \mathrm { f o r } i = i _ { j } - 1 } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} \right.
+$$
+
+This implies that the layers of noise $\eta _ { T , L + 1 } , \dots , \eta _ { T , i _ { j } }$ are all portions of $z _ { T }$ itself, with an additional portion being contained in $\eta _ { T , i _ { j } - 1 }$ . Afterwards, $\eta _ { T , i }$ become independent of $z _ { T }$ . In the case of $s _ { j } ~ = ~ 0$ , the above relations simplify to be $E [ \eta _ { T , i } \cdot z _ { T } ] = 0$ for all $i$ , implying no statistical dependency between the given and the synthetic noises. Consequently, it can be shown that the overall noise in Equation 5 satisfies
+
+$$
+\left( \Sigma \mathbf { V } ^ { T } \mathbf { n } _ { i } - \mathbf { z } _ { \mathbf { T } } \right) _ { j } = n _ { T , i } - z _ { T } \sim \left\{ \begin{array} { l l } { \mathcal { N } \left( 0 , s _ { j } ^ { 2 } \sigma _ { i } ^ { 2 } - \sigma _ { 0 } ^ { 2 } \right) } & { \mathrm { i f ~ } \sigma _ { i } s _ { j } > \sigma _ { 0 } } \\ { \mathcal { N } \left( 0 , \sigma _ { 0 } ^ { 2 } - s _ { j } ^ { 2 } \sigma _ { i } ^ { 2 } \right) } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
+$$
+
+The top option refers to high values of the annealed Langevin noise, in which, despite the possible decay caused by the singular value $s _ { j }$ , this noise is stronger than $z _ { T }$ . In this case, $n _ { T , i }$ contains all $z _ { T }$ and an additional independent portion of noise. The bottom part assumes that the annealed noise (with the influence of $s _ { j }$ ) is weaker than the measurements’ noise, and then it is fully immersed within $z _ { T }$ , with the difference being Gaussian and independent.
+
+# 3.2 Derivation of the Conditional Score Function
+
+The above derivations show that the noise in Equation 5 is zero-mean, Gaussian with uncorrelated entries and of known variance, and this noise is independent of $\widetilde { \mathbf { x } } _ { i }$ . Thus Equation 5 can be used conveniently for deriving the measurements part of the conditional score function. We denote $\tilde { \mathbf { x } } _ { T } = \mathbf { V } ^ { T } \tilde { \mathbf { x } } _ { i }$ , $\widetilde { \mathbf { x } } = \widetilde { \mathbf { x } } _ { i }$ , $\mathbf { n } = \mathbf { n } _ { i }$ for simplicity, and turn to calculate $\nabla _ { \tilde { \mathbf { x } } _ { T } } \log p \left( \tilde { \mathbf { x } } _ { T } | \mathbf { y } _ { T } \right)$ . We split $\tilde { \mathbf { x } } _ { T }$ into three parts: (i) $\tilde { \mathbf { x } } _ { T , 0 }$ refers to the entries $j$ for which $s _ { j } = 0$ ; (ii) $\tilde { \mathbf { x } } _ { T , < }$ corresponds to the entries $j$ for which $0 < \sigma _ { i } s _ { j } < \sigma _ { 0 }$ ; and (iii) $\tilde { \mathbf { x } } _ { T , > }$ includes the entries $j$ for which $\sigma _ { i } s _ { j } ~ > ~ \sigma _ { 0 }$ . Observe that this partition of the entries of $\tilde { \bf x } _ { T }$ is non-overlapping and fully covering. Similarly, we partition every vector $\mathbf { v } \in \mathbb { R } ^ { N }$ into $\mathbf { v } _ { 0 } , \mathbf { v } _ { < } , \mathbf { v } _ { > }$ , which are the entries of $\mathbf { v }$ corresponding to $\tilde { \bf x } _ { T , 0 } , \tilde { \bf x } _ { T , < } , \tilde { \bf x } _ { T , > }$ , respectively. Furthermore, we define $\mathbf { v } _ { \boldsymbol { \phi } } , \mathbf { v } _ { \mathcal { A } } , \mathbf { v } _ { \mathcal { P } }$ as all the entries of $\mathbf { v }$ except $\mathbf { v } _ { 0 } , \mathbf { v } _ { < } , \mathbf { v } _ { > }$ , respectively. With these definitions in place, the complete derivation of the score function is detailed in the supplemental material, and here we bring the final outcome. For $\tilde { \mathbf { x } } _ { T , 0 }$ , the score is independent of the measurements and given by
+
+$$
+\nabla _ { \tilde { \mathbf { x } } _ { T , 0 } } \log p \left( \tilde { \mathbf { x } } _ { T } | \mathbf { y } _ { T } \right) = \left( \mathbf { V } ^ { T } \nabla _ { \tilde { \mathbf { x } } } \log p \left( \tilde { \mathbf { x } } \right) \right) _ { 0 } .
+$$
+
+For the case of $\tilde { \mathbf { x } } _ { T , > }$ , the expression obtained is only measurements-dependent,
+
+$$
+\nabla _ { \tilde { \mathbf { x } } _ { T , > } } \log p \left( \tilde { \mathbf { x } } _ { T } | \mathbf { y } _ { T } \right) = \left( \Sigma ^ { T } \left( \sigma _ { i } ^ { 2 } \Sigma \Sigma ^ { T } - \sigma _ { 0 } ^ { 2 } \mathbf { I } \right) ^ { \dagger } \left( \mathbf { y } _ { T } - \Sigma \tilde { \mathbf { x } } _ { T } \right) \right) _ { > } .
+$$
+
+![](images/1bf7bf0fdb7b5ddbb342e405094746e2ae58288ea8fabffd3390f4076c6e4553.jpg)  
+Figure 2: Super resolution results on LSUN bedroom [56] images (downscaling $4 : 1$ by plain averaging and adding noise with $\sigma _ { 0 } = 0 . 0 4 \mathrm { , }$ ).
+
+Lastly, for the case of $\tilde { \mathbf { x } } _ { T , < }$ , the conditional score includes two terms – one referring to the plain (blurred) score, and the other depending on the measurements,
+
+$$
+\nabla _ { \mathbf { \tilde { x } } _ { T , < } } \log p \left( \mathbf { \tilde { x } } _ { T } | \mathbf { y } _ { T } \right) = \left( \Sigma ^ { T } \left( \sigma _ { 0 } ^ { 2 } \mathbf { I } - \sigma _ { i } ^ { 2 } \Sigma \Sigma ^ { T } \right) ^ { \dagger } \left( \mathbf { y } _ { T } - \Sigma \mathbf { \tilde { x } } _ { T } \right) \right) _ { < } + \left( \mathbf { V } ^ { T } \nabla _ { \mathbf { \tilde { x } } } \log p \left( \mathbf { \tilde { x } } \right) \right) _ { < } .
+$$
+
+As already mentioned, the full derivations of equations 7, 8, and 9 are detailed in the supplemental material. Aggregating all these results together, we obtain the following conditional score function:
+
+$$
+\begin{array} { r } { \nabla _ { \tilde { \mathbf { x } } _ { T } } \log p \left( \tilde { \mathbf { x } } _ { T } | \mathbf { y } _ { T } \right) = \Sigma ^ { T } \left| \sigma _ { 0 } ^ { 2 } \mathbf { I } - \sigma _ { i } ^ { 2 } \Sigma \Sigma ^ { T } \right| ^ { \frac { 1 } { \rho } } \left( \mathbf { y } _ { T } - \Sigma \tilde { \mathbf { x } } _ { T } \right) + \left. \left( \mathbf { V } ^ { T } \nabla _ { \tilde { \mathbf { x } } } \log p \left( \tilde { \mathbf { x } } \right) \right) \right| _ { \mathcal { X } } , } \end{array}
+$$
+
+where $( \mathbf { v } ) | _ { \ngtr }$ is the vector $\mathbf { v }$ , but with zeros in its entries that correspond to $\mathbf { v } _ { > }$ . Observe that the first term in Equation 10 contains zeros in the entries corresponding to $\tilde { \mathbf { x } } _ { T , 0 }$ , matching the above calculations. The vector $\nabla _ { \tilde { \mathbf { x } } } \log p \left( \tilde { \mathbf { x } } \right)$ can be estimated using a neural network as in [44], or using a pre-trained MMSE denoiser as in [20, 21]. All the other elements of this vector are given or can be easily obtained from $\mathbf { H }$ by calculating its SVD decomposition once at the beginning.
+
+# 4 The Proposed Algorithm
+
+Armed with the conditional score function in Equation 10, the Langevin dynamics algorithm can be run with a constant step size or an annealed step size as in [44], and this should converge to a sample from $p \left( \tilde { \mathbf { x } } _ { T } | \mathbf { y } _ { T } \right)$ . However, for this to perform well, one should use a very small step size, implying a devastatingly slow convergence behavior. This is mainly due to the fact that different entries of $\tilde { \bf x } _ { T }$ advance at different speeds, in accord with their corresponding singular values. As the added noise in each step has the same variance in every entry, this leads to an unbalanced signal-to-noise ratio, which considerably slows down the algorithm.
+
+In order to mitigate this problem, we suggest using a step size vector ${ \pmb { \alpha } } _ { i } \in \mathbb { R } ^ { N }$ . We denote $\mathbf { A } _ { i } = d i a g \left( \pmb { \alpha } _ { i } \right)$ , and obtain the following update formula for a Langevin dynamics algorithm:
+
+$$
+\mathbf { V } ^ { T } \tilde { \mathbf { x } } _ { i } = \mathbf { V } ^ { T } \tilde { \mathbf { x } } _ { i - 1 } + c \cdot \mathbf { A } _ { i } \cdot \nabla \mathbf { v } ^ { T } \tilde { \mathbf { x } } _ { i } \log p \left( \mathbf { V } ^ { T } \tilde { \mathbf { x } } _ { i } | \mathbf { y } _ { T } \right) + \sqrt { 2 \cdot c } \mathbf { A } _ { i } ^ { \frac { 1 } { 2 } } \cdot \mathbf { z } _ { i } ,
+$$
+
+where the conditional score function is estimated as described in subsection 3.2, and $c$ is some constant. For the choice of the step sizes in the diagonal of $\mathbf { A } _ { i }$ , we draw inspiration from Newton’s method in optimization, which is designed to speed up convergence to local maximum points. The update formula in Newton’s method is the same as Equation 11, but without the additional noise $\mathbf { z } _ { i }$ , and with $\mathbf { A } _ { i }$ being the negative inverse Hessian of $\log p \left( \mathbf { V } ^ { T } \tilde { \mathbf { x } } _ { i } | \mathbf { y } _ { T } \right)$ . We calculate a diagonal approximation of the Hessian, and set $\mathbf { A } _ { i }$ to be its negative inverse. We also estimate the conditional score function using Equation 10 and a neural network. Note that this mixture of Langevin dynamics and Newton’s method has been suggested in a slightly different context in [43], where the Hessian was approximated using a Quasi-Newton method. In our case, we analytically calculate a diagonal approximation of the negative inverse Hessian and obtain the following:
+
+$$
+\left( \alpha _ { i } \right) _ { j } = \left\{ \begin{array} { l l } { \sigma _ { i } ^ { 2 } , } & { s _ { j } = 0 } \\ { \sigma _ { i } ^ { 2 } - \frac { \sigma _ { 0 } ^ { 2 } } { s _ { j } ^ { 2 } } , } & { \sigma _ { i } s _ { j } > \sigma _ { 0 } } \\ { \sigma _ { i } ^ { 2 } \cdot \left( 1 - s _ { j } ^ { 2 } \frac { \sigma _ { i } ^ { 2 } } { \sigma _ { 0 } ^ { 2 } } \right) , } & { 0 < \sigma _ { i } s _ { j } < \sigma _ { 0 } . } \end{array} \right.
+$$
+
+![](images/f76bb7d49bb33b3a68815799a29373ea7f403eba8070bce309f48629298e43e9.jpg)  
+Figure 3: Compressive sensing results on a CelebA [27] image with an additive noise of $\sigma _ { 0 } = 0 . 1$
+
+The full derivations for each of the three cases are detailed in the supplemental material. Using these step sizes, the update formula in Equation 11, the conditional score function in Equation 10, and a neural network s $( \tilde { \mathbf { x } } , \sigma )$ that estimates the score function $\nabla _ { \tilde { \mathbf { x } } } \log p \left( \tilde { \mathbf { x } } \right)$ ,3 we obtain a tractable iterative algorithm for sampling from $p \left( \tilde { \mathbf { x } } _ { L } \mid \mathbf { y } \right)$ , where the noise in $ { \widetilde { \mathbf { x } } } _ { L }$ is sufficiently negligible to be considered as a sampling from the ideal image manifold.
+
+# Algorithm 1: SNIPS
+
+Input: $\left\{ \sigma _ { i } \right\} _ { i = 1 } ^ { L } , c , \tau , \mathbf { y } , \mathbf { H } , \sigma _ { 0 }$   
+$\mathbf { 1 } \ \mathbf { U } , \Sigma , \mathbf { V }  s { \bar { v } } d ( \mathbf { H } )$   
+2 Initialize $\mathbf { x _ { 0 } }$ with random noise $U \left[ 0 , 1 \right]$   
+3 for $i \gets 1$ to $L$ do   
+4 $( \mathbf { A } _ { i } ) _ { 0 }  \sigma _ { i } ^ { 2 } \mathbf { I }$   
+5 ( A i ) < ← σ 2i ·  I − σ 2iσ 2 Σ < Σ < T    
+6 $( \mathbf { A } _ { i } ) _ { > }  \sigma _ { i } ^ { 2 } \mathbf { I } - \sigma _ { 0 } ^ { 2 } \mathbf { \Sigma } \mathbf { \Sigma } _ { > } ^ { \dagger } \mathbf { \Sigma } \mathbf { \Sigma } \mathbf { \Sigma } _ { > } ^ { \dagger ^ { T } }$   
+7 for t ← 1 to τ do   
+8 Draw $\mathbf { z } _ { t } \sim \mathcal { N } \left( 0 , \mathbf { I } \right)$   
+9 $\begin{array} { r l } & { \mathbf { d } _ { t } \xleftarrow \Sigma ^ { T } \cdot \left| \sigma _ { 0 } ^ { 2 } \mathbf { I } - \sigma _ { i } ^ { 2 } \Sigma \Sigma ^ { T } \right| ^ { \dagger } \cdot \left( \mathbf { U } ^ { T } \mathbf { y } - \Sigma \mathbf { V } ^ { T } \mathbf { x } _ { t - 1 } \right) + \left( \mathbf { V } ^ { T } \cdot \mathbf { s } \left( \mathbf { x } _ { t - 1 } , \sigma _ { i } \right) \right) \big | _ { \ng } } \\ & { \mathbf { x } _ { t } \xleftarrow \mathbf { V } \cdot \left( \mathbf { V } ^ { T } \mathbf { x } _ { t - 1 } + c \mathbf { A } _ { i } \mathbf { d } _ { t } + \sqrt { 2 c } \mathbf { A } _ { i } ^ { \frac { 1 } { 2 } } \mathbf { z } _ { t } \right) } \end{array}$   
+10   
+11 end   
+12 $\mathbf { x } _ { 0 }  \mathbf { x } _ { \tau }$   
+13 end
+
+Note that when we set $\mathbf { H } = \mathbf { \Omega } 0$ and $\sigma _ { 0 } = 0$ , implying no measurements, the above algorithm degenerates to an image synthesis, exactly as in [44]. Two other special cases of this algorithm are obtained for $\mathbf { H } = \mathbf { I }$ or $\mathbf { H } = \mathbf { I }$ with some rows removed, the first referring to denoising and the second to noisy inpainting, both cases shown in [21]. Lastly, for the choices of $\mathbf { H }$ as in [20] or [44, 46] and with $\sigma _ { 0 } = 0$ , the above algorithm collapses to a close variant of their proposed iterative methods.
+
+# 5 Experimental Results
+
+In our experiments we use the NCSNv2 [45] network in order to estimate the score function of the prior distribution. Three different NCSNv2 models are used, each trained separately on the training sets of: (i) images of size $6 4 \times 6 4$ pixels from the CelebA dataset [27]; (ii) images of size $1 2 8 \times 1 2 8$ pixels from LSUN [56] bedrooms dataset; and (iii) LSUN $1 2 8 \times 1 2 8$ images of towers. We demonstrate SNIPS’ capabilities on the respective test sets for image deblurring, super resolution, and compressive sensing. In each of the experiments, we run our algorithm 8 times, producing 8 samples for each input. We examine both the samples themselves and their mean, which serves as an approximation of the MMSE solution, $\mathbb { E } \left[ \mathbf { x } | \mathbf { y } \right]$ .
+
+![](images/0fe681d53940c46fc3dcbf035756174ac25fbeaf06344ec3bce31a956ddec79b.jpg)  
+Figure 4: Super resolution results on CelebA [27] images (downscaling $4 : 1$ by plain averaging and adding noise with $\sigma _ { 0 } = 0 . 1$ ).
+
+![](images/6f296f6b2a2fc1162ce50cccb9a0e568e4e043a1cf1bb3e7155c6118f8d3aa75.jpg)  
+Figure 5: Super resolution results on CelebA [27] images (downscaling $2 : 1$ by plain averaging and adding noise with $\sigma _ { 0 } = 0 . 1$ ).
+
+For image deblurring, we use a uniform $5 \times 5$ blur kernel, and an additive white Gaussian noise with $\sigma _ { 0 } = 0 . 1$ (referring to pixel values in the range $[ 0 , 1 ] \rangle$ ). Figure 1 demonstrates the obtained results for several images taken from the CelebA dataset. As can be seen, SNIPS produces visually pleasing, diverse samples.
+
+For super resolution, the images are downscaled using a block averaging filter, i.e., each nonoverlapping block of pixels in the original image is averaged into one pixel in the low-resolution image. We use blocks of size $2 \times 2$ or $4 \times 4$ pixels, and assume the low-resolution image to include an additive white Gaussian noise. We showcase results on LSUN and CelebA in Figures 2, 4, and 5.
+
+For compressive sensing, we use three random projection matrices with singular values of 1, that compress the image by $2 5 \%$ , $1 2 . 5 \%$ , and $6 . 2 5 \%$ . As can be seen in Figure 3 and as expected, the more aggressive the compression, the more significant are the variations in reconstruction.
+
+We calculate the average PSNR (peak signal-to-noise ratio) of each of the 8 samples in our experiments, as well as the PSNR of their mean, as shown in Table 1. In all the experiments, the empirical conditional mean presents an improvement of around $2 . 4 \ : \mathrm { d B }$ in PSNR, even though it is less visually appealing compared to the samples. This is consistent with the theory in [5], which states that the difference in PSNR between posterior samples and the conditional mean (the MMSE estimator) should be $3 \mathrm { d B }$ , with the MMSE estimator having poorer perceptual quality but better PSNR.
+
+A comparison of our deblurring results to those obtained by RED [39] is detailed in the supplemental material. We show that SNIPS exhibits superior performance over RED, achieving more than $1 1 \%$ improvement in PSNR and more than $5 8 \%$ improvement in LPIPS [62], a perceptual quality metric.
+
+# 5.1 Assessing Faithfulness to the Measurements
+
+A valid solution to an inverse problem should satisfy two conditions: (i) It should be visually pleasing, consistent with the underlying prior distribution of images, and (ii) It should be faithful to the given measurement, maintaining the relationship as given in the problem setting. Since the prior distribution is unknown, we assess the first condition by visually observing the obtained solutions and their tendency to look realistic. As for the second condition, we perform the following computation: We degrade the obtained reconstruction $\hat { \bf x }$ by $\mathbf { H }$ , and calculate its difference from the given measurement $\mathbf { y }$ , obtaining $\mathbf { y } - \mathbf { H } \hat { \mathbf { x } }$ . According to the problem setting, this difference should be an additive white Gaussian noise vector with a standard deviation of $\sigma _ { 0 }$ . We examine this difference by calculating its empirical standard deviation, and performing the Pearson-D’Agostino [8] test of normality on it, accepting it as a Gaussian vector if the obtained p-value is greater than 0.05. We also calculate the Pearson correlation coefficient (denoted as $\rho$ ) among neighboring entries, accepting them as uncorrelated for coefficients smaller than 0.1 in absolute value. In all of our tests, the standard deviation matches $\sigma _ { 0 }$ almost exactly, the Pearson correlation coefficient satisfies $| \rho | < 0 . 1$ , and we obtain p-values greater than 0.05 in around $9 5 \%$ of the samples (across all experiments). These results empirically show that our algorithm produces valid solutions to the given inverse problems.
+
+Table 1: PSNR results for different inverse problems on 8 images from CelebA [27]. We ran SNIPS 8 times, and obtained 8 samples. The average PSNR for each of the samples is in the first column, while the average PSNR for the mean of the 8 samples for each image is in the second one.   
+
+<table><tr><td>Problem</td><td>Sample PSNR</td><td>Mean PSNR</td></tr><tr><td>Uniform deblurring</td><td>25.54</td><td>28.01</td></tr><tr><td>Super resolution (by 2)</td><td>25.58</td><td>28.03</td></tr><tr><td>Super resolution (by 4)</td><td>21.90</td><td>24.31</td></tr><tr><td>Compressive sensing (by 25%)</td><td>25.68</td><td>28.06</td></tr><tr><td>Compressive sensing (by 12.5%)</td><td>22.34</td><td>24.67</td></tr></table>
+
+![](images/31b80bec3255fef5b0e8eb8f928b222f9c9b72045dbebf8254989ebe23acad3b.jpg)  
+Figure 6: Compressive sensing results on LSUN [56] tower images (compression by $2 5 \%$ and adding noise with $\sigma _ { 0 } = 0 . 0 4 )$ .
+
+# 6 Conclusion and Future Work
+
+SNIPS, presented in this paper, is a novel stochastic algorithm for solving general noisy linear inverse problems. This method is based on annealed Langevin dynamics and Newton’s method, and relies on the availability of a pre-trained Gaussian MMSE denoiser. SNIPS produces a random variety of high quality samples from the posterior distribution of the unknown given the measurements, while guaranteeing their validity with respect to the given data. This algorithm’s derivation includes an intricate choice of the injected annealed noise in the Langevin update equations, and an SVD decomposition of the degradation operator for decoupling the measurements’ dependencies. We demonstrate SNIPS’ success on image deblurring, super resolution, and compressive sensing.
+
+Extensions of this work should focus on SNIPS’ limitations: (i) The need to deploy SVD decomposition of the degradation matrix requires a considerable amount of memory and computations, and hinders the algorithm’s scalability; (ii) The current version of SNIPS does not handle general content images, a fact that is related to the properties of the denoiser being used [41]; and (iii) SNIPS, as any other Langevin based method, requires (too) many iterations (e.g., in our super-resolution tests on CelebA, 2 minutes are required for producing 8 sample images), and means for its acceleration should be explored.
+
+# 7 Funding Transparency Statement
+
+This research was partially supported by the Israel Science Foundation (ISF) under Grant 335/18 and the Technion Hiroshi Fujiwara Cyber Security Research Center and the Israel Cyber Bureau. Bahjat Kawar’s scholarship was partially provided by Li Ka Shing Fellowships and the Planning and Budgeting Committee of the Israel Council for Higher Education.
+
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+        "type": "text",
+        "text": "SNIPS: Solving Noisy Inverse Problems Stochastically ",
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+        "text": "Bahjat Kawar, Gregory Vaksman, Michael Elad Computer Science Department, Technion, Haifa, Israel {bahjat.kawar, grishav, elad}@cs.technion.ac.il ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "In this work we introduce a novel stochastic algorithm dubbed SNIPS, which draws samples from the posterior distribution of any linear inverse problem, where the observation is assumed to be contaminated by additive white Gaussian noise. Our solution incorporates ideas from Langevin dynamics and Newton’s method, and exploits a pre-trained minimum mean squared error (MMSE) Gaussian denoiser. The proposed approach relies on an intricate derivation of the posterior score function that includes a singular value decomposition (SVD) of the degradation operator, in order to obtain a tractable iterative algorithm for the desired sampling. Due to its stochasticity, the algorithm can produce multiple high perceptual quality samples for the same noisy observation. We demonstrate the abilities of the proposed paradigm for image deblurring, super-resolution, and compressive sensing. We show that the samples produced are sharp, detailed and consistent with the given measurements, and their diversity exposes the inherent uncertainty in the inverse problem being solved. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Many problems in the field of image processing can be cast as noisy linear inverse problems. This family of tasks includes denoising, inpainting, deblurring, super resolution, compressive sensing, and many other image recovery problems. A general linear inverse problem is posed as ",
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+    {
+        "type": "equation",
+        "img_path": "images/c41395ed0dd6930514e9332a785616ea4af1c7e258a5b7f5cd52905e5f8573c0.jpg",
+        "text": "$$\n\\mathbf { y } = \\mathbf { H } \\mathbf { x } + \\mathbf { z } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where we aim to recover a signal $\\mathbf { x }$ from its measurement $\\mathbf { y }$ , given through a linear degradation operator $\\mathbf { H }$ and a contaminating noise, being additive, white and Gaussian, $\\mathbf { \\overline { { z } } } \\sim \\mathcal { N } \\left( 0 , \\sigma _ { 0 } ^ { 2 } \\mathbf { \\check { I } } \\right)$ . In this work we assume that both $\\mathbf { H }$ and $\\sigma _ { 0 }$ are known. ",
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+        "text": "Over the years, many strategies, algorithms and underlying statistical models were developed for handling image restoration problems. A key ingredient in many of the classic attempts is the prior that aims to regularize the inversion process and lead to visually pleasing results. Among the various options explored, we mention sparsity-inspired techniques [13, 55, 11], local Gaussian-mixture modeling [57, 63], and methods relying on non-local self-similarity [6, 9, 36, 51]. More recently, and with the emergence of deep learning techniques, a direct design of the recovery path from y to an estimate of $\\mathbf { x }$ took the lead, yielding state-of-the-art results in various linear inverse problems, such as denoising [25, 59, 61, 52], deblurring [22, 48], super resolution [10, 17, 54] and other tasks [29, 28, 19, 16, 37, 58]. ",
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+    {
+        "type": "text",
+        "text": "Despite the evident success of the above techniques, many image restoration algorithms still have a critical shortcoming: In cases of severe degradation, most recovery algorithms tend to produce washed out reconstructions that lack details. Indeed, most image restoration techniques seek a reconstruction that minimizes the mean squared error between the restored image, ˆx, and the unknown original one, x. When the degradation is acute and information is irreversibly lost, image reconstruction becomes a highly ill-posed problem, implying that many possible clean images could explain the given measurements. The MMSE solution averages all these candidate solutions, being the conditional mean of the posterior of $\\mathbf { x }$ given y, leading to an image with loss of fine details in the majority of practical cases. A recent work reported in [5] has shown that reconstruction algorithms necessarily suffer from a perception-distortion tradeoff, i.e., targeting a minimization of the error between $\\hat { \\bf x }$ and $\\mathbf { x }$ (in any metric) is necessarily accompanied by a compromised perceptual quality. As a consequence, as long as we stick to the tendency to design recovery algorithms that aim for minimum MSE (or other distances), only a limited perceptual improvement can be expected. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "When perceptual quality becomes our prime objective, the strategy for solving inverse problems must necessarily change. More specifically, the solution should concentrate on producing a sample (or many of them) from the posterior distribution $p \\left( \\mathbf { x } | \\mathbf { y } \\right)$ instead of its conditional mean. Recently, two such approaches have been suggested – GAN-based and Langevin sampling. Generative Adversarial Networks (GANs) have shown impressive results in generating realistically looking images (e.g., [14, 35]). GANs can be utilized for solving inverse problems while producing high-quality images (see e.g. [2, 31, 34]). These solvers aim to produce a diverse set of output images that are consistent with the measurements, while also being aligned with the distribution of clean examples. A major disadvantage of GAN-based algorithms for inverse problems is their tendency (as practiced in [2, 31, 34]) to assume noiseless measurements, a condition seldom met in practice. An exception to this is the work reported in [33], which adapts a conditional GAN to become a stochastic denoiser. ",
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+    {
+        "type": "text",
+        "text": "The second approach for sampling from the posterior, and the one we shall be focusing on in this paper, is based on Langevin dynamics. This core iterative technique enables sampling from a given distribution by leveraging the availability of the score function – the gradient of the log of the probability density function [38, 3]. The work reported in [44, 20, 46] utilizes the annealed Langevin dynamics method, both for image synthesis and for solving noiseless inverse problems.1 Their synthesis algorithm relies on an MMSE Gaussian denoiser (given as a neural network) for approximating a gradually blurred score function. In their treatment of inverse problems, the conditional score remains tractable and manageable due to the noiseless measurements assumption. ",
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+        "text": "The question addressed in this paper is the following: How can the above line of Langevin-based work be generalized for handling linear inverse problems, as in Equation 1, in which the measurements are noisy? A partial and limited answer to this question has already been given in [21] for the tasks of image denoising and inpainting. The present work generalizes these ([44, 20, 46, 21]) results, and introduces a systematic way for sampling from the posterior distribution of any given noisy linear inverse problem. As we carefully show, this extension is far from being trivial, due to two prime reasons: (i) The involvement of the degradation operator H, which poses a difficulty for establishing a relationship between the reconstructed image and the noisy observation; and (ii) The intricate connection between the measurements’ and the synthetic annealed Langevin noise. Our proposed remedy is a decorrelation of the measurements equation via a singular value decomposition (SVD) of the operator H, which decouples the dependencies between the measurements, enabling each to be addressed by an adapted iterative process. In addition, we define the annealing noise to be built as portions of the measurement noise itself, in a manner that facilitates a constructive derivation of the conditional score function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following earlier work [44, 20, 46, 21], our algorithm is initialized with a random noise image, gradually converging to the reconstructed result, while following the direction of the log-posterior gradient, estimated using an MMSE denoiser. Via a careful construction of the gradual annealing noise sequence, from very high values to low ones, the entries in the derived score switch mode. Those referring to non-zero singular values start by being purely dependent on the measurements, and then transition to incorporate prior information based on the denoiser. As for entries referring to zero singular values, their corresponding entries undergo a pure synthesis process based on the prior-only score function. Note that the denoiser blends values in the evolving sample, thus intermixing the influence of the gradient entries. Our derivations include an analytical expression for a positiondependent step size vector, drawing inspiration from Newton’s method in optimization. This stabilizes the algorithm and is shown to be essential for its success. ",
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+        "type": "text",
+        "text": "We refer hereafter to our algorithm as SNIPS (Solution of Noisy Inverse Problems Stochastically). Observe that as we target to sample from the posterior distribution $p \\left( \\mathbf { x } | \\mathbf { y } \\right)$ , different runs of SNIPS on the same input necessarily yield different results, all of which valid solutions to the given inverse problem. This should not come as a surprise, as ill-posedness implies that there are multiple viable solutions for the same data, as has already been suggested in the context of super resolution [31, 2, 34]. We demonstrate SNIPS on image deblurring, single image super resolution, and compressive sensing, all of which contain non-negligible noise, and emphasize the high perceptual quality of the results, their diversity, and their relation to the MMSE estimate. ",
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+        "img_path": "images/c82146786b5adb3c31fe1fe0e0cc72d1662774fbb84d513189a447eec3bd7cf3.jpg",
+        "image_caption": [
+            "Figure 1: Deblurring results on CelebA [27] images (uniform $5 \\times 5$ blur and an additive noise with $\\sigma _ { 0 } = 0 . 1$ ). Here and in all other shown figures, the standard deviation image is scaled by 4 for better visual inspection. "
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+    {
+        "type": "text",
+        "text": "To summarize, this paper’s contributions are threefold: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We present an intricate derivation of the blurred posterior score function for general noisy inverse problems, where both the measurement and the target image contain delicately inter-connected additive white Gaussian noise. • We introduce a novel stochastic algorithm – SNIPS – that can sample from the posterior distribution of these problems. The algorithm relies on the availability of an MMSE denoiser. • We demonstrate impressive results of SNIPS on image deblurring, single image super resolution, and compressive sensing, all of which are highly noisy and ill-posed. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Before diving into the details of this work, we should mention that using Gaussian denoisers iteratively for handling general linear inverse problems has been already proposed in the context of the Plugand-Play-Prior $( \\mathrm { P n P } )$ method [53] and RED [39], and their many followup papers (e.g., [60, 30, 1, 49, 7, 50, 40, 4]). However, both PnP and RED are quite different from our work, as they do not target sampling from the posterior, but rather focus on MAP or MMSE estimation. ",
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+        "type": "text",
+        "text": "2 Background ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The Langevin dynamics algorithm [3, 38] suggests sampling from a probability distribution $p \\left( \\mathbf { x } \\right)$ using the iterative transition rule ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/65fe25b37c181016c6e26fe3b3166fadef13175e5afdcb074d833e13023e5d2e.jpg",
+        "text": "$$\n{ \\bf x } _ { t + 1 } = { { \\bf x } _ { t } } + \\alpha \\nabla _ { { \\bf x } _ { t } } \\log p \\left( { { \\bf x } _ { t } } \\right) + \\sqrt { 2 \\alpha } { \\bf z } _ { t } \\mathrm { ~ , ~ }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { z } _ { t } \\sim \\mathcal { N } ( 0 , \\mathbf { I } )$ and $\\alpha$ is an appropriately chosen small constant. The added $\\mathbf { z } _ { t }$ allows for stochastic sampling, avoiding a collapse to a maximum of the distribution. Initialized randomly, after a sufficiently large number of iterations, and under some mild conditions, this process converges to a sample from the desired distribution $p \\left( \\mathbf { x } \\right)$ [38]. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The work reported in [44] extends the aforementioned algorithm into annealed Langevin dynamics. The annealing proposed replaces the score function in Equation 2 with a blurred version of it, $\\nabla _ { \\tilde { \\mathbf { x } } _ { t } } \\log { p \\left( \\tilde { \\mathbf { x } } _ { t } \\right) }$ , where $\\tilde { \\mathbf { x } } _ { \\mathbf { t } } = \\mathbf { x } _ { t } + \\mathbf { n }$ and $\\mathbf { n } \\sim { \\mathcal { N } } \\left( 0 , \\sigma ^ { 2 } \\mathbf { I } \\right)$ is a synthetically injected noise. The core idea is to start with a very high noise level $\\sigma$ and gradually drop it to near-zero, all while using a step size $\\alpha$ dependent on the noise level. These changes allow the algorithm to converge much faster and perform better, because it widens the basin of attraction of the sampling process. The work in [20] further develops this line of work by leveraging a brilliant relation attributed to Miyasawa [32] (also known as Stein’s integration by parts trick [47] or Tweedie’s identity [12]). It is given as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8e4563b900dd22b4a378f4305ed6f38a4a6bac2415dc5b293ef59bccbd807e8e.jpg",
+        "text": "$$\n{ \\nabla } _ { \\tilde { \\mathbf { x } } _ { t } } \\log p \\left( \\tilde { \\mathbf { x } } _ { t } \\right) = \\frac { \\mathbf { D } \\left( \\tilde { \\mathbf { x } } _ { t } , \\sigma \\right) - \\tilde { \\mathbf { x } } _ { t } } { { \\sigma } ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { D } \\left( \\tilde { \\mathbf { x } } _ { t } , \\sigma \\right) = \\mathbb { E } \\left[ \\mathbf { x } | \\tilde { \\mathbf { x } } _ { t } \\right]$ is the minimizer of the MSE measure $\\mathbb { E } \\left[ \\lVert \\mathbf { x } - \\mathbf { D } \\left( \\tilde { \\mathbf { x } } _ { t } , \\sigma \\right) \\rVert _ { 2 } ^ { 2 } \\right]$ , which can be approximated using a denoising neural network. This facilitates the use of denoisers in Langevin dynamics as a replacement for the evasive score function. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "When turning to solve inverse problems, previous work suggests sampling from the posterior distribution $p \\left( \\mathbf { x } | \\mathbf { y } \\right)$ using annealed Langevin dynamics [20, 46, 21] or similar methods [15, 18, 42, 26], by replacing the score function used in the generation algorithm with a conditional one. As it turns out, if limiting assumptions can be posed on the measurements formation, the conditional score is tractable, and thus generalization of the annealed Langevin process to these problems is within reach. Indeed, in [44, 20, 46, 42, 26] the core assumption is $\\mathbf { y } = \\mathbf { H } \\mathbf { x }$ for specific and simplified choices of $\\mathbf { H }$ and with no noise in the measurements. The works in [15, 23] avoid these difficulties altogether by returning to the original (non-annealed) Langevin method, with the unavoidable cost of becoming extremely slow. In addition, their algorithms are demonstrated on inverse problems in which the additive noise is restricted to be very weak. The work in [21] is broader, allowing for an arbitrary additive white Gaussian noise, but limits $\\mathbf { H }$ to the problems of denoising or inpainting. While all these works demonstrate high quality results, there is currently no clear way for deriving the blurred score function of a general linear inverse problem as posed in Equation 1. In the following, we present such a derivation. ",
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+    {
+        "type": "text",
+        "text": "3 The Proposed Approach: Deriving the Conditional Score Function ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Problem Setting ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We consider the problem of recovering a signal $\\mathbf { x } \\in \\mathbb { R } ^ { N }$ (where $\\mathbf { x } \\sim p \\left( \\mathbf { x } \\right)$ and $p \\left( \\mathbf { x } \\right)$ is unknown) from the observation $\\mathbf { y } = \\mathbf { H } \\mathbf { x } + \\mathbf { z }$ , where $\\mathbf { y } \\in \\mathbb { R } ^ { M } , \\mathbf { H } \\in \\mathbb { R } ^ { M \\times N } , M \\leq N , \\mathbf { z } \\sim \\mathcal { N } \\left( 0 , \\sigma _ { 0 } ^ { 2 } \\mathbf { I } \\right)$ , and $\\mathbf { H }$ and $\\sigma _ { 0 }$ are known.2 Our ultimate goal is to sample from the posterior $p \\left( \\mathbf { x } | \\mathbf { y } \\right)$ . However, since access to the score function $\\nabla _ { \\mathbf { x } } \\log p ( \\mathbf { x } | \\mathbf { y } )$ is not available, we retarget our goal, as explained above, to sampling from blurred posterior distributions, $p \\left( \\tilde { \\mathbf { x } } | \\mathbf { y } \\right)$ , where $\\tilde { \\mathbf { x } } = \\mathbf { x } + \\mathbf { n }$ and $\\mathbf { n } \\sim { \\mathcal { N } } \\left( 0 , \\sigma ^ { 2 } \\mathbf { I } \\right)$ , with noise levels $\\sigma$ starting very high, and decreasing towards near-zero. ",
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+    {
+        "type": "text",
+        "text": "As explained in the supplemental material, the sampling should be performed in the SVD domain in order to get a tractable derivation of the blurred score function. Thus, we consider the singular value decomposition (SVD) of $\\mathbf { H }$ , given as $\\mathbf { H } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { T }$ , where $\\mathbf { U } \\in \\mathbb { R } ^ { M \\times M }$ and $\\mathbf { V } \\in \\mathbb { R } ^ { N \\times N }$ are orthogonal matrices, and $\\pmb { \\Sigma } \\in \\mathbb { R } ^ { M \\times N }$ is a rectangular diagonal matrix containing the singular values of H, denoted as {sj}Mj=1 i n descending order $\\quad : s _ { 1 } > s _ { 2 } > \\dots > s _ { M - 1 } > s _ { M } \\geq 0 \\quad$ ). For convenience of notations, we also define $s _ { j } = 0$ for $j = M + 1 , \\dotsc , N$ . To that end, we notice that ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/863ea856b7cce1c3b58de72c2976483b4aca738e0aec67aa4fe0802307840972.jpg",
+        "text": "$$\np \\left( \\tilde { \\mathbf { x } } | \\mathbf { y } \\right) = p \\left( \\tilde { \\mathbf { x } } | \\mathbf { U } ^ { T } \\mathbf { y } \\right) = p \\left( \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } | \\mathbf { U } ^ { T } \\mathbf { y } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The first equality holds because the multiplication of $\\mathbf { y }$ by the orthogonal matrix $\\mathbf { U } ^ { T }$ does not add or remove information, and the second equality holds because the multiplication of $\\tilde { \\bf x }$ by $\\mathbf { V } ^ { T }$ does not change its probability [24]. Therefore, sampling from $p \\left( \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } | \\mathbf { U } ^ { T } \\mathbf { y } \\right)$ and then multiplying the result by $\\mathbf { V }$ will produce the desired sample from $p \\left( \\mathbf { \\tilde { x } } | \\mathbf { y } \\right)$ . As we are using Langevin dynamics, we need to calculate the conditional score function $\\nabla _ { \\mathbf { V } ^ { T } \\widetilde { \\mathbf { x } } } \\log p \\left( \\mathbf { V } ^ { T } \\widetilde { \\mathbf { x } } | \\mathbf { U } ^ { T } \\mathbf { y } \\right)$ . For simplicity, we denote hereafter ${ \\bf y } _ { T } = { \\bf U } ^ { T } { \\bf y } , { \\bf z } _ { T } = { \\bf U } ^ { T } { \\bf z } , { \\bf x } _ { T } = { \\bf V } ^ { T } { \\bf x } , { \\bf n } _ { T } = \\Sigma { \\bf V } ^ { T } { \\bf n }$ , and $\\tilde { \\mathbf { x } } _ { T } = \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } }$ . Observe that with these notations, the measurements equation becomes ",
+        "bbox": [
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+        "page_idx": 3
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+        "type": "equation",
+        "img_path": "images/4159b80da532b10c585424ee53f1cacc669df2fb68d4a1d7165dd8b2d3153908.jpg",
+        "text": "$$\n\\mathbf { y } = \\mathbf { H } \\mathbf { x } + \\mathbf { z } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { T } \\mathbf { x } + \\mathbf { z } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "and thus ",
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+    {
+        "type": "equation",
+        "img_path": "images/e2bc92fec592a5fe51b5e71d5a54825861dd4e27b6fc667d33e166ddcfcdb2b6.jpg",
+        "text": "$$\n\\mathbf { U } ^ { T } \\mathbf { y } = \\Sigma \\mathbf { V } ^ { T } \\mathbf { x } + \\mathbf { U } ^ { T } \\mathbf { z } = \\Sigma \\mathbf { V } ^ { T } ( \\tilde { \\mathbf { x } } - \\mathbf { n } ) + \\mathbf { U } ^ { T } \\mathbf { z } = \\Sigma \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } - \\Sigma \\mathbf { V } ^ { T } \\mathbf { n } + \\mathbf { U } ^ { T } \\mathbf { z } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where we have relied on the relation $\\tilde { \\mathbf { x } } = \\mathbf { x } + \\mathbf { n }$ . This leads to ",
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+            586,
+            801
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e55a43e28f9c7f94ced77a84b4003c8cc93fcc1582d821d290152ac8f83728c.jpg",
+        "text": "$$\n{ \\bf y } _ { T } = \\pmb { \\Sigma } \\tilde { \\bf x } _ { T } - { \\bf n } _ { T } + { \\bf z } _ { T } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            806,
+            581,
+            823
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this formulation, which will aid in deriving the conditional score, our aim is to make design choices on ${ \\bf n } _ { T }$ such that ${ \\bf z } _ { T } - { \\bf n } _ { T }$ has uncorrelated entries and is independent of $\\tilde { \\bf x } _ { T }$ . This brings us to the formation of the synthetic annealed noise, which is an intricate ingredient in our derivations. ",
+        "bbox": [
+            174,
+            833,
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+            876
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We base this formation on the definition of a sequence of noise levels $\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { L + 1 }$ such that $\\sigma _ { 1 } > \\sigma _ { 2 } > \\cdot \\cdot \\cdot > \\sigma _ { L } > \\sigma _ { L + 1 } = 0$ , where $\\sigma _ { 1 }$ is high (possibly $\\sigma _ { 1 } > \\| \\mathbf { x } \\| _ { \\infty } )$ and $\\sigma _ { L }$ is close to zero. We require that for every $j$ such that $s _ { j } \\neq 0$ , there exists $i _ { j }$ such that $\\sigma _ { i _ { j } } s _ { j } < \\sigma _ { 0 }$ and $\\sigma _ { i _ { j } - 1 } s _ { j } > \\sigma _ { 0 }$ . This implies $\\forall i : \\sigma _ { i } s _ { j } \\neq \\sigma _ { 0 }$ , which helps ease notations. SNIPS works just as well for $\\sigma _ { i } s _ { j } = \\sigma _ { 0 }$ . ",
+        "bbox": [
+            173,
+            89,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using $\\{ \\sigma _ { i } \\} _ { i = 1 } ^ { L + 1 }$ , we would like to define $\\left\\{ \\tilde { \\mathbf { x } } _ { i } \\right\\} _ { i = 1 } ^ { L + 1 }$ , a sequence of noisy versions of $\\mathbf { x }$ , where the noise level in $\\tilde { \\mathbf { x } } _ { i }$ is $\\sigma _ { i }$ . One might be tempted to define these noise additions as independent of the measurement noise $\\mathbf { z }$ . However, this option leads to a conditional score term that cannot be calculated analytically, as explained in the supplemental material. Therefore, we define these noise additions differently, as carved from $\\mathbf { z }$ in a gradual fashion. To that end, we define $\\tilde { \\mathbf { x } } _ { L + 1 } = \\mathbf { x }$ , and for every $i = L , L - 1 , \\ldots , 1 \\colon \\tilde { \\mathbf { x } } _ { i } = \\tilde { \\mathbf { x } } _ { i + 1 } + \\pmb { \\eta } _ { i }$ , where $\\pmb { \\eta } _ { i } \\sim \\mathcal { N } \\left( 0 , \\left( \\sigma _ { i } ^ { 2 } - \\sigma _ { i + 1 } ^ { 2 } \\right) \\mathbf { I } \\right)$ . This results in $\\tilde { \\mathbf { x } } _ { i } = \\mathbf { x } + \\mathbf { n } _ { i }$ , where $\\begin{array} { r } { \\mathbf { n } _ { i } = \\sum _ { k = i } ^ { L } \\pmb { \\eta } _ { k } \\sim \\mathcal { N } \\left( 0 , \\sigma _ { i } ^ { 2 } \\mathbf { I } \\right) } \\end{array}$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "And now we turn to define the statistical dependencies between the measurements’ noise $\\mathbf { z }$ and the artificial noise vectors $\\eta _ { i }$ . Since $\\eta _ { i }$ and $\\mathbf { z }$ are each Gaussian with uncorrelated entries, so are the components of the vectors $\\pmb { \\Sigma } \\mathbf { V } ^ { T } \\pmb { \\eta } _ { i }$ , $\\pmb { \\Sigma } \\mathbf { V } ^ { T } \\mathbf { n } _ { i }$ , and $\\mathbf { z } _ { T }$ . In order to proceed while easing notations, let us focus on a single entry $j$ in these three vectors, for which $s _ { j } > 0$ , and omit this index. We denote these entries as $\\eta _ { T , i } , n _ { T , i }$ and $z _ { T }$ , respectively. We construct $\\eta _ { T , i }$ such that ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca598f71ce34de7d18b0e938cffeed2cac4d504ce7554612e0b5083a4359bf45.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\eta _ { T , i } \\cdot z _ { T } \\right] = \\left\\{ \\begin{array} { l l } { \\mathbb { E } \\left[ \\eta _ { T , i } ^ { 2 } \\right] } & { \\mathrm { f o r } i \\geq i _ { j } } \\\\ { \\mathbb { E } \\left[ \\left( z _ { T } - n _ { T , i _ { j } } \\right) ^ { 2 } \\right] } & { \\mathrm { f o r } i = i _ { j } - 1 } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            343,
+            673,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This implies that the layers of noise $\\eta _ { T , L + 1 } , \\dots , \\eta _ { T , i _ { j } }$ are all portions of $z _ { T }$ itself, with an additional portion being contained in $\\eta _ { T , i _ { j } - 1 }$ . Afterwards, $\\eta _ { T , i }$ become independent of $z _ { T }$ . In the case of $s _ { j } ~ = ~ 0$ , the above relations simplify to be $E [ \\eta _ { T , i } \\cdot z _ { T } ] = 0$ for all $i$ , implying no statistical dependency between the given and the synthetic noises. Consequently, it can be shown that the overall noise in Equation 5 satisfies ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6dfbaf421c34fffe10f0fd9447266571ea61a2d747c32fd14a02d51d83547116.jpg",
+        "text": "$$\n\\left( \\Sigma \\mathbf { V } ^ { T } \\mathbf { n } _ { i } - \\mathbf { z } _ { \\mathbf { T } } \\right) _ { j } = n _ { T , i } - z _ { T } \\sim \\left\\{ \\begin{array} { l l } { \\mathcal { N } \\left( 0 , s _ { j } ^ { 2 } \\sigma _ { i } ^ { 2 } - \\sigma _ { 0 } ^ { 2 } \\right) } & { \\mathrm { i f ~ } \\sigma _ { i } s _ { j } > \\sigma _ { 0 } } \\\\ { \\mathcal { N } \\left( 0 , \\sigma _ { 0 } ^ { 2 } - s _ { j } ^ { 2 } \\sigma _ { i } ^ { 2 } \\right) } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            489,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The top option refers to high values of the annealed Langevin noise, in which, despite the possible decay caused by the singular value $s _ { j }$ , this noise is stronger than $z _ { T }$ . In this case, $n _ { T , i }$ contains all $z _ { T }$ and an additional independent portion of noise. The bottom part assumes that the annealed noise (with the influence of $s _ { j }$ ) is weaker than the measurements’ noise, and then it is fully immersed within $z _ { T }$ , with the difference being Gaussian and independent. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Derivation of the Conditional Score Function ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            621,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above derivations show that the noise in Equation 5 is zero-mean, Gaussian with uncorrelated entries and of known variance, and this noise is independent of $\\widetilde { \\mathbf { x } } _ { i }$ . Thus Equation 5 can be used conveniently for deriving the measurements part of the conditional score function. We denote $\\tilde { \\mathbf { x } } _ { T } = \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } _ { i }$ , $\\widetilde { \\mathbf { x } } = \\widetilde { \\mathbf { x } } _ { i }$ , $\\mathbf { n } = \\mathbf { n } _ { i }$ for simplicity, and turn to calculate $\\nabla _ { \\tilde { \\mathbf { x } } _ { T } } \\log p \\left( \\tilde { \\mathbf { x } } _ { T } | \\mathbf { y } _ { T } \\right)$ . We split $\\tilde { \\mathbf { x } } _ { T }$ into three parts: (i) $\\tilde { \\mathbf { x } } _ { T , 0 }$ refers to the entries $j$ for which $s _ { j } = 0$ ; (ii) $\\tilde { \\mathbf { x } } _ { T , < }$ corresponds to the entries $j$ for which $0 < \\sigma _ { i } s _ { j } < \\sigma _ { 0 }$ ; and (iii) $\\tilde { \\mathbf { x } } _ { T , > }$ includes the entries $j$ for which $\\sigma _ { i } s _ { j } ~ > ~ \\sigma _ { 0 }$ . Observe that this partition of the entries of $\\tilde { \\bf x } _ { T }$ is non-overlapping and fully covering. Similarly, we partition every vector $\\mathbf { v } \\in \\mathbb { R } ^ { N }$ into $\\mathbf { v } _ { 0 } , \\mathbf { v } _ { < } , \\mathbf { v } _ { > }$ , which are the entries of $\\mathbf { v }$ corresponding to $\\tilde { \\bf x } _ { T , 0 } , \\tilde { \\bf x } _ { T , < } , \\tilde { \\bf x } _ { T , > }$ , respectively. Furthermore, we define $\\mathbf { v } _ { \\boldsymbol { \\phi } } , \\mathbf { v } _ { \\mathcal { A } } , \\mathbf { v } _ { \\mathcal { P } }$ as all the entries of $\\mathbf { v }$ except $\\mathbf { v } _ { 0 } , \\mathbf { v } _ { < } , \\mathbf { v } _ { > }$ , respectively. With these definitions in place, the complete derivation of the score function is detailed in the supplemental material, and here we bring the final outcome. For $\\tilde { \\mathbf { x } } _ { T , 0 }$ , the score is independent of the measurements and given by ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce2641f7f2d032f69bd277e4dcc9529ab3ec60a8e3de0a242cf7a9c5ae304992.jpg",
+        "text": "$$\n\\nabla _ { \\tilde { \\mathbf { x } } _ { T , 0 } } \\log p \\left( \\tilde { \\mathbf { x } } _ { T } | \\mathbf { y } _ { T } \\right) = \\left( \\mathbf { V } ^ { T } \\nabla _ { \\tilde { \\mathbf { x } } } \\log p \\left( \\tilde { \\mathbf { x } } \\right) \\right) _ { 0 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            821,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the case of $\\tilde { \\mathbf { x } } _ { T , > }$ , the expression obtained is only measurements-dependent, ",
+        "bbox": [
+            173,
+            856,
+            692,
+            872
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/75073e86807d6da436683e8d4488bc04bc9f364ef632d947b298e64aba7955aa.jpg",
+        "text": "$$\n\\nabla _ { \\tilde { \\mathbf { x } } _ { T , > } } \\log p \\left( \\tilde { \\mathbf { x } } _ { T } | \\mathbf { y } _ { T } \\right) = \\left( \\Sigma ^ { T } \\left( \\sigma _ { i } ^ { 2 } \\Sigma \\Sigma ^ { T } - \\sigma _ { 0 } ^ { 2 } \\mathbf { I } \\right) ^ { \\dagger } \\left( \\mathbf { y } _ { T } - \\Sigma \\tilde { \\mathbf { x } } _ { T } \\right) \\right) _ { > } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            880,
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+            916
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/1bf7bf0fdb7b5ddbb342e405094746e2ae58288ea8fabffd3390f4076c6e4553.jpg",
+        "image_caption": [
+            "Figure 2: Super resolution results on LSUN bedroom [56] images (downscaling $4 : 1$ by plain averaging and adding noise with $\\sigma _ { 0 } = 0 . 0 4 \\mathrm { , }$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            88,
+            816,
+            229
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lastly, for the case of $\\tilde { \\mathbf { x } } _ { T , < }$ , the conditional score includes two terms – one referring to the plain (blurred) score, and the other depending on the measurements, ",
+        "bbox": [
+            173,
+            286,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15dfe9ad0010cd227855bf423b32c69ed21b0e109d7a57938953758a4c0d7bcc.jpg",
+        "text": "$$\n\\nabla _ { \\mathbf { \\tilde { x } } _ { T , < } } \\log p \\left( \\mathbf { \\tilde { x } } _ { T } | \\mathbf { y } _ { T } \\right) = \\left( \\Sigma ^ { T } \\left( \\sigma _ { 0 } ^ { 2 } \\mathbf { I } - \\sigma _ { i } ^ { 2 } \\Sigma \\Sigma ^ { T } \\right) ^ { \\dagger } \\left( \\mathbf { y } _ { T } - \\Sigma \\mathbf { \\tilde { x } } _ { T } \\right) \\right) _ { < } + \\left( \\mathbf { V } ^ { T } \\nabla _ { \\mathbf { \\tilde { x } } } \\log p \\left( \\mathbf { \\tilde { x } } \\right) \\right) _ { < } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            318,
+            789,
+            354
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As already mentioned, the full derivations of equations 7, 8, and 9 are detailed in the supplemental material. Aggregating all these results together, we obtain the following conditional score function: ",
+        "bbox": [
+            173,
+            363,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1e82236f44445c72e4e55abcb1b1603367ae9a77c9049541095a2831b1af43a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\nabla _ { \\tilde { \\mathbf { x } } _ { T } } \\log p \\left( \\tilde { \\mathbf { x } } _ { T } | \\mathbf { y } _ { T } \\right) = \\Sigma ^ { T } \\left| \\sigma _ { 0 } ^ { 2 } \\mathbf { I } - \\sigma _ { i } ^ { 2 } \\Sigma \\Sigma ^ { T } \\right| ^ { \\frac { 1 } { \\rho } } \\left( \\mathbf { y } _ { T } - \\Sigma \\tilde { \\mathbf { x } } _ { T } \\right) + \\left. \\left( \\mathbf { V } ^ { T } \\nabla _ { \\tilde { \\mathbf { x } } } \\log p \\left( \\tilde { \\mathbf { x } } \\right) \\right) \\right| _ { \\mathcal { X } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            393,
+            761,
+            424
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $( \\mathbf { v } ) | _ { \\ngtr }$ is the vector $\\mathbf { v }$ , but with zeros in its entries that correspond to $\\mathbf { v } _ { > }$ . Observe that the first term in Equation 10 contains zeros in the entries corresponding to $\\tilde { \\mathbf { x } } _ { T , 0 }$ , matching the above calculations. The vector $\\nabla _ { \\tilde { \\mathbf { x } } } \\log p \\left( \\tilde { \\mathbf { x } } \\right)$ can be estimated using a neural network as in [44], or using a pre-trained MMSE denoiser as in [20, 21]. All the other elements of this vector are given or can be easily obtained from $\\mathbf { H }$ by calculating its SVD decomposition once at the beginning. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 The Proposed Algorithm ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            516,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Armed with the conditional score function in Equation 10, the Langevin dynamics algorithm can be run with a constant step size or an annealed step size as in [44], and this should converge to a sample from $p \\left( \\tilde { \\mathbf { x } } _ { T } | \\mathbf { y } _ { T } \\right)$ . However, for this to perform well, one should use a very small step size, implying a devastatingly slow convergence behavior. This is mainly due to the fact that different entries of $\\tilde { \\bf x } _ { T }$ advance at different speeds, in accord with their corresponding singular values. As the added noise in each step has the same variance in every entry, this leads to an unbalanced signal-to-noise ratio, which considerably slows down the algorithm. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to mitigate this problem, we suggest using a step size vector ${ \\pmb { \\alpha } } _ { i } \\in \\mathbb { R } ^ { N }$ . We denote $\\mathbf { A } _ { i } = d i a g \\left( \\pmb { \\alpha } _ { i } \\right)$ , and obtain the following update formula for a Langevin dynamics algorithm: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf9d7ee13018f11e6687f32b2e40b3a7e3696f7589edc99ae00688480e279db1.jpg",
+        "text": "$$\n\\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } _ { i } = \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } _ { i - 1 } + c \\cdot \\mathbf { A } _ { i } \\cdot \\nabla \\mathbf { v } ^ { T } \\tilde { \\mathbf { x } } _ { i } \\log p \\left( \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } _ { i } | \\mathbf { y } _ { T } \\right) + \\sqrt { 2 \\cdot c } \\mathbf { A } _ { i } ^ { \\frac { 1 } { 2 } } \\cdot \\mathbf { z } _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            683,
+            735,
+            707
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the conditional score function is estimated as described in subsection 3.2, and $c$ is some constant. For the choice of the step sizes in the diagonal of $\\mathbf { A } _ { i }$ , we draw inspiration from Newton’s method in optimization, which is designed to speed up convergence to local maximum points. The update formula in Newton’s method is the same as Equation 11, but without the additional noise $\\mathbf { z } _ { i }$ , and with $\\mathbf { A } _ { i }$ being the negative inverse Hessian of $\\log p \\left( \\mathbf { V } ^ { T } \\tilde { \\mathbf { x } } _ { i } | \\mathbf { y } _ { T } \\right)$ . We calculate a diagonal approximation of the Hessian, and set $\\mathbf { A } _ { i }$ to be its negative inverse. We also estimate the conditional score function using Equation 10 and a neural network. Note that this mixture of Langevin dynamics and Newton’s method has been suggested in a slightly different context in [43], where the Hessian was approximated using a Quasi-Newton method. In our case, we analytically calculate a diagonal approximation of the negative inverse Hessian and obtain the following: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a33a533a03625d938bb3b08392eb96aa6484939f33077a88b6adf23ff1a151ce.jpg",
+        "text": "$$\n\\left( \\alpha _ { i } \\right) _ { j } = \\left\\{ \\begin{array} { l l } { \\sigma _ { i } ^ { 2 } , } & { s _ { j } = 0 } \\\\ { \\sigma _ { i } ^ { 2 } - \\frac { \\sigma _ { 0 } ^ { 2 } } { s _ { j } ^ { 2 } } , } & { \\sigma _ { i } s _ { j } > \\sigma _ { 0 } } \\\\ { \\sigma _ { i } ^ { 2 } \\cdot \\left( 1 - s _ { j } ^ { 2 } \\frac { \\sigma _ { i } ^ { 2 } } { \\sigma _ { 0 } ^ { 2 } } \\right) , } & { 0 < \\sigma _ { i } s _ { j } < \\sigma _ { 0 } . } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/f76bb7d49bb33b3a68815799a29373ea7f403eba8070bce309f48629298e43e9.jpg",
+        "image_caption": [
+            "Figure 3: Compressive sensing results on a CelebA [27] image with an additive noise of $\\sigma _ { 0 } = 0 . 1$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            349,
+            88,
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+            229
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The full derivations for each of the three cases are detailed in the supplemental material. Using these step sizes, the update formula in Equation 11, the conditional score function in Equation 10, and a neural network s $( \\tilde { \\mathbf { x } } , \\sigma )$ that estimates the score function $\\nabla _ { \\tilde { \\mathbf { x } } } \\log p \\left( \\tilde { \\mathbf { x } } \\right)$ ,3 we obtain a tractable iterative algorithm for sampling from $p \\left( \\tilde { \\mathbf { x } } _ { L } \\mid \\mathbf { y } \\right)$ , where the noise in $ { \\widetilde { \\mathbf { x } } } _ { L }$ is sufficiently negligible to be considered as a sampling from the ideal image manifold. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: SNIPS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Input: $\\left\\{ \\sigma _ { i } \\right\\} _ { i = 1 } ^ { L } , c , \\tau , \\mathbf { y } , \\mathbf { H } , \\sigma _ { 0 }$   \n$\\mathbf { 1 } \\ \\mathbf { U } , \\Sigma , \\mathbf { V }  s { \\bar { v } } d ( \\mathbf { H } )$   \n2 Initialize $\\mathbf { x _ { 0 } }$ with random noise $U \\left[ 0 , 1 \\right]$   \n3 for $i \\gets 1$ to $L$ do   \n4 $( \\mathbf { A } _ { i } ) _ { 0 }  \\sigma _ { i } ^ { 2 } \\mathbf { I }$   \n5 ( A i ) < ← σ 2i · \u0010 I − σ 2iσ 2 Σ < Σ < T \u0011   \n6 $( \\mathbf { A } _ { i } ) _ { > }  \\sigma _ { i } ^ { 2 } \\mathbf { I } - \\sigma _ { 0 } ^ { 2 } \\mathbf { \\Sigma } \\mathbf { \\Sigma } _ { > } ^ { \\dagger } \\mathbf { \\Sigma } \\mathbf { \\Sigma } \\mathbf { \\Sigma } _ { > } ^ { \\dagger ^ { T } }$   \n7 for t ← 1 to τ do   \n8 Draw $\\mathbf { z } _ { t } \\sim \\mathcal { N } \\left( 0 , \\mathbf { I } \\right)$   \n9 $\\begin{array} { r l } & { \\mathbf { d } _ { t } \\xleftarrow \\Sigma ^ { T } \\cdot \\left| \\sigma _ { 0 } ^ { 2 } \\mathbf { I } - \\sigma _ { i } ^ { 2 } \\Sigma \\Sigma ^ { T } \\right| ^ { \\dagger } \\cdot \\left( \\mathbf { U } ^ { T } \\mathbf { y } - \\Sigma \\mathbf { V } ^ { T } \\mathbf { x } _ { t - 1 } \\right) + \\left( \\mathbf { V } ^ { T } \\cdot \\mathbf { s } \\left( \\mathbf { x } _ { t - 1 } , \\sigma _ { i } \\right) \\right) \\big | _ { \\ng } } \\\\ & { \\mathbf { x } _ { t } \\xleftarrow \\mathbf { V } \\cdot \\left( \\mathbf { V } ^ { T } \\mathbf { x } _ { t - 1 } + c \\mathbf { A } _ { i } \\mathbf { d } _ { t } + \\sqrt { 2 c } \\mathbf { A } _ { i } ^ { \\frac { 1 } { 2 } } \\mathbf { z } _ { t } \\right) } \\end{array}$   \n10   \n11 end   \n12 $\\mathbf { x } _ { 0 }  \\mathbf { x } _ { \\tau }$   \n13 end ",
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+    {
+        "type": "text",
+        "text": "Note that when we set $\\mathbf { H } = \\mathbf { \\Omega } 0$ and $\\sigma _ { 0 } = 0$ , implying no measurements, the above algorithm degenerates to an image synthesis, exactly as in [44]. Two other special cases of this algorithm are obtained for $\\mathbf { H } = \\mathbf { I }$ or $\\mathbf { H } = \\mathbf { I }$ with some rows removed, the first referring to denoising and the second to noisy inpainting, both cases shown in [21]. Lastly, for the choices of $\\mathbf { H }$ as in [20] or [44, 46] and with $\\sigma _ { 0 } = 0$ , the above algorithm collapses to a close variant of their proposed iterative methods. ",
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+        "type": "text",
+        "text": "5 Experimental Results ",
+        "text_level": 1,
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+        "text": "In our experiments we use the NCSNv2 [45] network in order to estimate the score function of the prior distribution. Three different NCSNv2 models are used, each trained separately on the training sets of: (i) images of size $6 4 \\times 6 4$ pixels from the CelebA dataset [27]; (ii) images of size $1 2 8 \\times 1 2 8$ pixels from LSUN [56] bedrooms dataset; and (iii) LSUN $1 2 8 \\times 1 2 8$ images of towers. We demonstrate SNIPS’ capabilities on the respective test sets for image deblurring, super resolution, and compressive sensing. In each of the experiments, we run our algorithm 8 times, producing 8 samples for each input. We examine both the samples themselves and their mean, which serves as an approximation of the MMSE solution, $\\mathbb { E } \\left[ \\mathbf { x } | \\mathbf { y } \\right]$ . ",
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+        "type": "image",
+        "img_path": "images/0fe681d53940c46fc3dcbf035756174ac25fbeaf06344ec3bce31a956ddec79b.jpg",
+        "image_caption": [
+            "Figure 4: Super resolution results on CelebA [27] images (downscaling $4 : 1$ by plain averaging and adding noise with $\\sigma _ { 0 } = 0 . 1$ ). "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/6f296f6b2a2fc1162ce50cccb9a0e568e4e043a1cf1bb3e7155c6118f8d3aa75.jpg",
+        "image_caption": [
+            "Figure 5: Super resolution results on CelebA [27] images (downscaling $2 : 1$ by plain averaging and adding noise with $\\sigma _ { 0 } = 0 . 1$ ). "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "For image deblurring, we use a uniform $5 \\times 5$ blur kernel, and an additive white Gaussian noise with $\\sigma _ { 0 } = 0 . 1$ (referring to pixel values in the range $[ 0 , 1 ] \\rangle$ ). Figure 1 demonstrates the obtained results for several images taken from the CelebA dataset. As can be seen, SNIPS produces visually pleasing, diverse samples. ",
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+        "type": "text",
+        "text": "For super resolution, the images are downscaled using a block averaging filter, i.e., each nonoverlapping block of pixels in the original image is averaged into one pixel in the low-resolution image. We use blocks of size $2 \\times 2$ or $4 \\times 4$ pixels, and assume the low-resolution image to include an additive white Gaussian noise. We showcase results on LSUN and CelebA in Figures 2, 4, and 5. ",
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+        "type": "text",
+        "text": "For compressive sensing, we use three random projection matrices with singular values of 1, that compress the image by $2 5 \\%$ , $1 2 . 5 \\%$ , and $6 . 2 5 \\%$ . As can be seen in Figure 3 and as expected, the more aggressive the compression, the more significant are the variations in reconstruction. ",
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+        "type": "text",
+        "text": "We calculate the average PSNR (peak signal-to-noise ratio) of each of the 8 samples in our experiments, as well as the PSNR of their mean, as shown in Table 1. In all the experiments, the empirical conditional mean presents an improvement of around $2 . 4 \\ : \\mathrm { d B }$ in PSNR, even though it is less visually appealing compared to the samples. This is consistent with the theory in [5], which states that the difference in PSNR between posterior samples and the conditional mean (the MMSE estimator) should be $3 \\mathrm { d B }$ , with the MMSE estimator having poorer perceptual quality but better PSNR. ",
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+        "type": "text",
+        "text": "A comparison of our deblurring results to those obtained by RED [39] is detailed in the supplemental material. We show that SNIPS exhibits superior performance over RED, achieving more than $1 1 \\%$ improvement in PSNR and more than $5 8 \\%$ improvement in LPIPS [62], a perceptual quality metric. ",
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+    {
+        "type": "text",
+        "text": "5.1 Assessing Faithfulness to the Measurements ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A valid solution to an inverse problem should satisfy two conditions: (i) It should be visually pleasing, consistent with the underlying prior distribution of images, and (ii) It should be faithful to the given measurement, maintaining the relationship as given in the problem setting. Since the prior distribution is unknown, we assess the first condition by visually observing the obtained solutions and their tendency to look realistic. As for the second condition, we perform the following computation: We degrade the obtained reconstruction $\\hat { \\bf x }$ by $\\mathbf { H }$ , and calculate its difference from the given measurement $\\mathbf { y }$ , obtaining $\\mathbf { y } - \\mathbf { H } \\hat { \\mathbf { x } }$ . According to the problem setting, this difference should be an additive white Gaussian noise vector with a standard deviation of $\\sigma _ { 0 }$ . We examine this difference by calculating its empirical standard deviation, and performing the Pearson-D’Agostino [8] test of normality on it, accepting it as a Gaussian vector if the obtained p-value is greater than 0.05. We also calculate the Pearson correlation coefficient (denoted as $\\rho$ ) among neighboring entries, accepting them as uncorrelated for coefficients smaller than 0.1 in absolute value. In all of our tests, the standard deviation matches $\\sigma _ { 0 }$ almost exactly, the Pearson correlation coefficient satisfies $| \\rho | < 0 . 1$ , and we obtain p-values greater than 0.05 in around $9 5 \\%$ of the samples (across all experiments). These results empirically show that our algorithm produces valid solutions to the given inverse problems. ",
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+    {
+        "type": "table",
+        "img_path": "images/cb248dd8876a0120bd0ee7cf6851a981242a74a7f9f9b9391e6cbe6743fe8568.jpg",
+        "table_caption": [
+            "Table 1: PSNR results for different inverse problems on 8 images from CelebA [27]. We ran SNIPS 8 times, and obtained 8 samples. The average PSNR for each of the samples is in the first column, while the average PSNR for the mean of the 8 samples for each image is in the second one. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Problem</td><td>Sample PSNR</td><td>Mean PSNR</td></tr><tr><td>Uniform deblurring</td><td>25.54</td><td>28.01</td></tr><tr><td>Super resolution (by 2)</td><td>25.58</td><td>28.03</td></tr><tr><td>Super resolution (by 4)</td><td>21.90</td><td>24.31</td></tr><tr><td>Compressive sensing (by 25%)</td><td>25.68</td><td>28.06</td></tr><tr><td>Compressive sensing (by 12.5%)</td><td>22.34</td><td>24.67</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/31b80bec3255fef5b0e8eb8f928b222f9c9b72045dbebf8254989ebe23acad3b.jpg",
+        "image_caption": [
+            "Figure 6: Compressive sensing results on LSUN [56] tower images (compression by $2 5 \\%$ and adding noise with $\\sigma _ { 0 } = 0 . 0 4 )$ . "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 Conclusion and Future Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "SNIPS, presented in this paper, is a novel stochastic algorithm for solving general noisy linear inverse problems. This method is based on annealed Langevin dynamics and Newton’s method, and relies on the availability of a pre-trained Gaussian MMSE denoiser. SNIPS produces a random variety of high quality samples from the posterior distribution of the unknown given the measurements, while guaranteeing their validity with respect to the given data. This algorithm’s derivation includes an intricate choice of the injected annealed noise in the Langevin update equations, and an SVD decomposition of the degradation operator for decoupling the measurements’ dependencies. We demonstrate SNIPS’ success on image deblurring, super resolution, and compressive sensing. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Extensions of this work should focus on SNIPS’ limitations: (i) The need to deploy SVD decomposition of the degradation matrix requires a considerable amount of memory and computations, and hinders the algorithm’s scalability; (ii) The current version of SNIPS does not handle general content images, a fact that is related to the properties of the denoiser being used [41]; and (iii) SNIPS, as any other Langevin based method, requires (too) many iterations (e.g., in our super-resolution tests on CelebA, 2 minutes are required for producing 8 sample images), and means for its acceleration should be explored. ",
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+    {
+        "type": "text",
+        "text": "7 Funding Transparency Statement ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This research was partially supported by the Israel Science Foundation (ISF) under Grant 335/18 and the Technion Hiroshi Fujiwara Cyber Security Research Center and the Israel Cyber Bureau. Bahjat Kawar’s scholarship was partially provided by Li Ka Shing Fellowships and the Planning and Budgeting Committee of the Israel Council for Higher Education. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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ISTA-Net: interpretable optimization-inspired deep network for image compressive sensing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1828–1837, 2018.   \n[59] K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang. Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising. IEEE Transactions on Image Processing, 26(7): 3142–3155, 2017.   \n[60] K. Zhang, W. Zuo, S. Gu, and L. Zhang. Learning deep CNN denoiser prior for image restoration. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3929–3938, 2017.   \n[61] K. Zhang, W. Zuo, and L. Zhang. FFDNet: toward a fast and flexible solution for CNN-based image denoising. IEEE Transactions on Image Processing, 27(9):4608–4622, 2018.   \n[62] R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 586–595, 2018.   \n[63] D. Zoran and Y. Weiss. From learning models of natural image patches to whole image restoration. In 2011 International Conference on Computer Vision, pages 479–486. IEEE, 2011. ",
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parse/train/B12Js_yRb/B12Js_yRb.md

@@ -0,0 +1,342 @@
+# LEARNING TO COUNT OBJECTS IN NATURAL IMAGES FOR VISUAL QUESTION ANSWERING
+
+Yan Zhang & Jonathon Hare & Adam Prugel-Bennett ¨
+
+Department of Electronics and Computer Science University of Southampton {yz5n12,jsh2,apb}@ecs.soton.ac.uk
+
+# ABSTRACT
+
+Visual Question Answering (VQA) models have struggled with counting objects in natural images so far. We identify a fundamental problem due to soft attention in these models as a cause. To circumvent this problem, we propose a neural network component that allows robust counting from object proposals. Experiments on a toy task show the effectiveness of this component and we obtain state-of-theart accuracy on the number category of the VQA v2 dataset without negatively affecting other categories, even outperforming ensemble models with our single model. On a difficult balanced pair metric, the component gives a substantial improvement in counting over a strong baseline by $6 . 6 \%$ .
+
+# 1 INTRODUCTION
+
+Consider the problem of counting how many cats there are in Figure 1. Solving this involves several rough steps: understanding what instances of that type can look like, finding them in the image, and adding them up. This is a common task in Visual Question Answering (VQA) – answering questions about images – and is rated as among the tasks requiring the lowest human age to be able to answer (Antol et al., 2015). However, current models for VQA on natural images struggle to answer any counting questions successfully outside of dataset biases (Jabri et al., 2016).
+
+One reason for this is the presence of a fundamental problem with counting in the widely-used soft attention mechanisms (section 3). Another reason is that unlike standard counting tasks, there is no ground truth labeling of where the objects to count are. Coupled with the fact that models need to be able to count a large variety of objects and that, ideally, performance on non-counting questions should not be compromised, the task of counting in VQA seems very challenging.
+
+To make this task easier, we can use object proposals – pairs of a bounding box and object features – from object detection networks as input instead of learning from pixels directly. In any moderately complex scene, this runs into the issue of double-counting overlapping object proposals. This is a problem present in many natural images, which leads to inaccurate counting in real-world scenarios.
+
+Our main contribution is a differentiable neural network component that tackles this problem and consequently can learn to count (section 4). Used alongside an attention mechanism, this component avoids a fundamental limitation of soft attention while producing strong counting features. We provide experimental evidence of the effectiveness of this component (section 5). On a toy dataset, we demonstrate that this component enables robust counting in a variety of scenarios. On the number category of the VQA v2 Open-Ended dataset (Goyal et al., 2017), a relatively simple baseline model using the counting component outperforms all previous models – including large ensembles of state-of-the-art methods – without degrading performance on other categories.
+
+# 2 RELATED WORK
+
+Usually, greedy non-maximum suppression (NMS) is used to eliminate duplicate bounding boxes. The main problem with using it as part of a model is that its gradient is piecewise constant. Various differentiable variants such as by Azadi et al. (2017), Hosang et al. (2017), and Henderson & Ferrari (2017) exist. The main difference is that, since we are interested in counting, our component does not need to make discrete decisions about which bounding boxes to keep; it outputs counting features, not a smaller set of bounding boxes. Our component is also easily integrated into standard VQA models that utilize soft attention without any need for other network architecture changes and can be used without using true bounding boxes for supervision.
+
+On the VQA v2 dataset (Goyal et al., 2017) that we apply our method on, only few advances on counting questions have been made. The main improvement in accuracy is due to the use of object proposals in the visual processing pipeline, proposed by Anderson et al. (2017). Their object proposal network is trained with classes in singular and plural forms, for example “tree” versus “trees”, which only allows primitive counting information to be present in the object features after region-of-interest pooling. Our approach differs in the way that instead of relying on counting features being present in the input, we create counting features using information present in the attention map over object proposals. This has the benefit of being able to count anything that the attention mechanism can discriminate instead of only objects that belong to the predetermined set of classes that had plural forms.
+
+Using these object proposals, Trott et al. (2018) train a sequential counting mechanism with a reinforcement learning loss on the counting question subsets of VQA v2 and Visual Genome. They achieve a small increase in accuracy and can obtain an interpretable set of objects that their model counted, but it is unclear whether their method can be integrated into traditional VQA models due to their loss not applying to non-counting questions. Since they evaluate on their own dataset, their results can not be easily compared to existing results in VQA.
+
+Methods such as by Santoro et al. (2017) and Perez et al. (2017) can count on the synthetic CLEVR VQA dataset (Johnson et al., 2017) successfully without bounding boxes and supervision of where the objects to count are. They also use more training data ${ \sim } 2 5 0 { , } 0 0 0$ counting questions in the CLEVR training set versus $\sim 5 0 { , } 0 0 0$ counting questions in the VQA v2 training set), much simpler objects, and synthetic question structures.
+
+More traditional approaches based on Lempitsky & Zisserman (2010) learn to produce a target density map, from which a count is computed by integrating over it. In this setting, Cohen et al. (2017) make use of overlaps of convolutional receptive fields to improve counting performance. Chattopadhyay et al. (2017) use an approach that divides the image into smaller non-overlapping chunks, each of which is counted individually and combined together at the end. In both of these contexts, the convolutional receptive fields or chunks can be seen as sets of bounding boxes with a fixed structure in their positioning. Note that while Chattopadhyay et al. (2017) evaluate their models on a small subset of counting questions in VQA, major differences in training setup make their results not comparable to our work.
+
+# 3 PROBLEMS WITH SOFT ATTENTION
+
+The main message in this section is that using the feature vectors obtained after the attention mechanism is not enough to be able to count; the attention maps themselves should be used, which is what we do in our counting component.
+
+Models in VQA have consistently benefited from the use of soft attention (Mnih et al., 2014; Bahdanau et al., 2015) on the image, commonly implemented with a shallow convolutional network. It learns to output a weight for the feature vector at each spatial position in the feature map, which is first normalized and then used for performing a weighted sum over the spatial positions to produce a single feature vector. However, soft spatial attention severely limits the ability for a model to count.
+
+Consider the task of counting the number of cats for two images: an image showing a single cat on a clean background and an image that consists of two side-by-side copies of the first image. What we will describe applies to both spatial feature maps and sets of object proposals as input, but we focus on the latter case for simplicity. With an object detection network, we detect one cat in the first image and two cats in the second image, producing the same feature vector for all three detections. The attention mechanism then assigns all three instances of the same cat the same weight.
+
+![](images/01c2c43239e5530d6490c14c6c93b92ad5fef0ff207f0d931d1ff4b6596146b9.jpg)  
+Figure 1: Simplified example about counting the number of cats. The light-colored cat is detected twice and results in a duplicate proposal. This shows the conversion from the attention weights a to a graph representation A and the eventual goal of this component with exactly one proposal per true object. There are 4 proposals (vertices) capturing 3 underlying objects (groups in dotted lines). There are 3 relevant proposals (black with weight 1) and 1 irrelevant proposal (white with weight 0). Red edges mark intra-object edges between duplicate proposals and blue edges mark the main inter-object duplicate edges. In graph form, the object groups, coloring of edges, and shading of vertices serve illustration purposes only; the model does not have these access to these directly.
+
+The usual normalization used for the attention weights is the softmax function, which normalizes the weights to sum to 1. Herein lies the problem: the cat in the first image receives a normalized weight of 1, but the two cats in the second image now each receive a weight of 0.5. After the weighted sum, we are effectively averaging the two cats in the second image back to a single cat. As a consequence, the feature vector obtained after the weighted sum is exactly the same between the two images and we have lost all information about a possible count from the attention map. Any method that normalizes the weights to sum to 1 suffers from this issue.
+
+Multiple glimpses (Larochelle & Hinton, 2010) – sets of attention weights that the attention mechanism outputs – or several steps of attention (Yang et al., 2016; Lu et al., 2016) do not circumvent this problem. Each glimpse or step can not separate out an object each, since the attention weight given to one feature vector does not depend on the other feature vectors to be attended over. Hard attention (Ba et al., 2015; Mnih et al., 2014) and structured attention (Kim et al., 2017) may be possible solutions to this, though no significant improvement in counting ability has been found for the latter so far (Zhu et al., 2017). Ren & Zemel (2017) circumvent the problem by limiting attention to only work within one bounding box at a time, remotely similar to our approach of using object proposal features.
+
+Without normalization of weights to sum to one, the scale of the output features depends on the number of objects detected. In an image with 10 cats, the output feature vector is scaled up by 10. Since deep neural networks are typically very scale-sensitive – the scale of weight initializations and activations is generally considered quite important (Mishkin & Matas, 2016) – and the classifier would have to learn that joint scaling of all features is somehow related to count, this approach is not reasonable for counting objects. This is evidenced in Teney et al. (2017) where they provide evidence that sigmoid normalization not only degrades accuracy on non-number questions slightly, but also does not help with counting.
+
+# 4 COUNTING COMPONENT
+
+In this section, we describe a differentiable mechanism for counting from attention weights, while also dealing with the problem of overlapping object proposals to reduce double-counting of objects. This involves some nontrivial details to produce counts that are as accurate as possible. The main idea is illustrated in Figure 1 with the two main steps shown in Figure 2 and Figure 3. The use of this component allows a model to count while still being able to exploit the benefits of soft attention.
+
+Our key idea for dealing with overlapping object proposals is to turn these object proposals into a graph that is based on how they overlap. We then remove and scale edges in a specific way such that an estimate of the number of underlying objects is recovered.
+
+Our general strategy is to primarily design the component for the unrealistic extreme cases of perfect attention maps and bounding boxes that are either fully overlapping or fully distinct. By introducing some parameters and only using differentiable operations, we give the ability for the module to interpolate between the correct behaviours for these extreme cases to handle the more realistic cases.
+
+These parameters are responsible for handling variations in attention weights and partial bounding box overlaps in a manner suitable for a given dataset.
+
+To achieve this, we use several piecewise linear functions $f _ { 1 } , \ldots , f _ { 8 }$ as activation functions (defined in Appendix A), approximating arbitrary functions with domain and range [0, 1]. The shapes of these functions are learned to handle the specific nonlinear interactions necessary for dealing with overlapping proposals. Through their parametrization we enforce that $f _ { k } ( 0 ) = 0$ , $f _ { k } ( 1 ) = 1$ , and that they are monotonically increasing. The first two properties are required so that the extreme cases that we explicitly handle are left unchanged. In those cases, $f _ { k }$ is only applied to values of 0 or 1, so the activation functions can be safely ignored for understanding how the component handles them. By enforcing monotonicity, we can make sure that, for example, an increased value in an attention map should never result in the prediction of the count to decrease.
+
+# 4.1 INPUT
+
+Given a set of features from object proposals, an attention mechanism produces a weight for each proposal based on the question. The counting component takes as input the $n$ largest attention weights $\mathbf { \bar { a } } = [ a _ { 1 } , \ldots , a _ { n } ] ^ { \mathsf { T } }$ and their corresponding bounding boxes $\mathbf { b } = [ b _ { 1 } , \ldots , b _ { n } ] ^ { \mathsf { T } }$ . We assume that the weights lie in the interval $[ 0 , 1 ]$ , which can easily be achieved by applying a logistic function.
+
+In the extreme cases that we explicitly handle, we assume that the attention mechanism assigns a value of 1 to $a _ { i }$ whenever the ith proposal contains a relevant object and a value of 0 whenever it does not. This is in line with what usual soft attention mechanisms learn, as they produce higher weights for relevant inputs. We also assume that either two object proposals fully overlap (in which case they must be showing the same object and thus receive the same attention weight) or that they are fully distinct (in which case they show different objects). Keep in mind that while we make these assumptions to make reasoning about the behaviour easier, the learned parameters in the activation functions are intended to handle the more realistic scenarios when the assumptions do not apply.
+
+Instead of partially overlapping proposals, the problem now becomes the handling of exact duplicate proposals of underlying objects in a differentiable manner.
+
+# 4.2 DEDUPLICATION
+
+We start by changing the vector of attention weights a into a graph representation in which bounding boxes can be utilized more easily. Hence, we compute the outer product of the attention weights to obtain an attention matrix.
+
+$$
+\mathbf { A } = \mathbf { a } \mathbf { a } ^ { \mathsf { T } }
+$$
+
+$\mathbf { A } \in \mathbb { R } ^ { n \times n }$ can be interpreted as an adjacency matrix for a weighted directed graph. In this graph, the ith vertex represents the object proposal associated with $a _ { i }$ and the edge between any pair of vertices $( i , j )$ has weight $a _ { i } a _ { j }$ . In the extreme case where $a _ { i }$ is virtually 0 or 1, products are equivalent to logical AND operators. It follows that the subgraph containing only the vertices satisfying $a _ { i } = 1$ is a complete digraph with self-loops.
+
+In this representation, our objective is to eliminate edges in such a way that, conceptually, the underlying true objects – instead of proposals thereof – are the vertices of that complete subgraph. In order to then turn that graph into a count, recall that the number of edges $| E |$ in a complete digraph with self-loops relates to the number of vertices $| V |$ through $| E | = | V | ^ { 2 }$ . $| E |$ can be computed by summing over the entries in an adjacency matrix and $| V |$ is then the count. Notice how when $| E |$ is set to the sum over A, ${ \sqrt { \textstyle | E | } } = \sum _ { i } a _ { i }$ holds. This convenient property implies that when all proposals are fully distinct, the component can output the same as simply summing over the original attention weights by default.
+
+There are two types of duplicate edges to eliminate to achieve our objective: intra-object edges and inter-object edges.
+
+# 4.2.1 INTRA-OBJECT EDGES
+
+First, we eliminate intra-object edges between duplicate proposals of a single underlying object.
+
+![](images/ebf111b774761b74e228aac07c7d7d09a9fdfade0a9ad743d7521aa694abb69f.jpg)
+
+![](images/afc38c727db1f3d829ce09537e6ec8c8c604979b7eca9b190e7296a9afa5152c.jpg)  
+Figure 2: Removal of intra-object edges by masking the edges of the attention matrix A with the distance matrix D. The black vertices now form a graph without self-loops. The self-loops need to be added back in later.   
+Figure 3: Removal of duplicate inter-object edges by computing a scaling factor for each vertex and scaling $\tilde { \mathbf { A } } ^ { \prime }$ accordingly. $\bar { \mathbf { A } } ^ { \prime }$ is $\tilde { \mathbf { A } }$ with self-loops already added back in. The scaling factor for one vertex is computed by counting how many vertices have outgoing edges to the same set of vertices; all edges of the two proposals on the right are scaled by 0.5. This can be seen as averaging proposals within each object and is equivalent to removing duplicate proposals altogether under a sum.
+
+To compare two bounding boxes, we use the usual intersection-over-union (IoU) metric. We define the distance matrix $\mathbf { D } \in \bar { \mathbb { R } } ^ { n \times n }$ to be
+
+$$
+D _ { i j } = 1 - \mathrm { I o U } ( b _ { i } , b _ { j } )
+$$
+
+$\mathbf { D }$ can also be interpreted as an adjacency matrix. It represents a graph that has edges everywhere except when the two bounding boxes that an edge connects would overlap.
+
+Intra-object edges are removed by elementwise multiplying $( \odot )$ the distance matrix with the attention matrix (Figure 2).
+
+$$
+\tilde { \mathbf { A } } = f _ { 1 } ( \mathbf { A } ) \odot f _ { 2 } ( \mathbf { D } )
+$$
+
+$\tilde { \mathbf { A } }$ no longer has self-loops, so we need to add them back in at a later point to still satisfy $| E | = | V | ^ { 2 }$ Notice that we start making use of the activation functions mentioned earlier to handle intermediate values in the interval $( 0 , 1 )$ for both A and $\mathbf { D }$ . They regulate the influence of attention weights that are not close to 0 or 1 and the influence of partial overlaps.
+
+# 4.2.2 INTER-OBJECT EDGES
+
+Second, we eliminate inter-object edges between duplicate proposals of different underlying objects.
+
+The main idea (depicted in Figure 3) is to count the number of proposals associated to each invidual object, then scale down the weight of their associated edges by that number. If there are two proposals of a single object, the edges involving those proposals should be scaled by 0.5. In essence, this averages over the proposals within each underlying object because we only use the sum over the edge weights to compute the count at the end. Conceptually, this reduces multiple proposals of an object down to one as desired. Since we do not know how many proposals belong to an object, we have to estimate this. We do this by using the fact that proposals of the same object are similar.
+
+Keep in mind that $\tilde { \mathbf { A } }$ has no self-loops nor edges between proposals of the same object. As a consequence, two nonzero rows in $\tilde { \mathbf { A } }$ are the same if and only if the proposals are the same. If the two rows differ in at least one entry, then one proposal overlaps a proposal that the other proposal does not overlap, so they must be different proposals. This means for comparing rows, we need a similarity function that satisfies the criteria of taking the value 1 when they differ in no places and 0 if they differ in at least one place. We define a differentiable similarity between proposals $i$ and $j$ as
+
+$$
+\mathrm { S i m } _ { i j } = f _ { 3 } ( 1 - | a _ { i } - a _ { j } | ) \prod _ { k } f _ { 3 } ( 1 - | X _ { i k } - X _ { j k } | )
+$$
+
+where $\mathbf { X } = f _ { 4 } ( \mathbf { A } ) \odot f _ { 5 } ( \mathbf { D } )$ is the same as $\tilde { \mathbf { A } }$ except with different activation functions. The $\prod$ term compares the rows of proposals $i$ and $j$ . Using this term instead of $f _ { 4 } ( 1 - D _ { i j } )$ was more robust to inaccurate bounding boxes in initial experiments.
+
+Note that the $f _ { 3 } ( 1 - | a _ { i } - a _ { j } | )$ term handles the edge case when there is only one proposal to count. Since $\mathbf { X }$ does not have self-loops, $\mathbf { X }$ contains only zeros in that case, which causes the row corresponding to $a _ { i } = 1$ to be incorrectly similar to the rows where $a _ { j \neq i } = 0$ . By comparing the attention weights through that term as well, this issue is avoided.
+
+Now that we can check how similar two proposals are, we count the number of times any row is the same as any other row and compute a scaling factor $s _ { i }$ for each vertex $i$ .
+
+$$
+s _ { i } = 1 / \sum _ { j } \mathrm { S i m } _ { i j }
+$$
+
+The time complexity of computing $\mathbf { s } = [ s _ { 1 } , \ldots , s _ { n } ] ^ { \mathsf { T } }$ is $\Theta ( n ^ { 3 } )$ as there are $n ^ { 2 }$ pairs of rows and $\Theta ( n )$ operations to compute the similarity of any pair of rows.
+
+Since these scaling factors apply to each vertex, we have to expand s into a matrix using the outer product in order to scale both incoming and outgoing edges of each vertex. We can also add self-loops back in, which need to be scaled by s as well. Then, the count matrix $\mathbf { C }$ is
+
+$$
+\mathbf { C } = \tilde { \mathbf { A } } \odot \mathbf { s s } ^ { \mathsf { T } } + \mathrm { { d i a g } } ( \mathbf { s } \odot f _ { 1 } ( \mathbf { a } \odot \mathbf { a } ) )
+$$
+
+where $\mathrm { d i a g ( \cdot ) }$ expands a vector into a diagonal matrix with the vector on the diagonal.
+
+The scaling of self-loops involves a non-obvious detail. Recall that the diagonal that was removed when going from $\mathbf { A }$ to $\tilde { \mathbf { A } }$ contains the entries $f _ { 1 } ( \mathbf { a } \odot \mathbf { a } )$ . Notice however that we are scaling this diagonal by s and not s $\odot$ s. This is because the number of inter-object edges scales quadratically with respect to the number of proposals per object, but the number of self-loops only scales linearly.
+
+# 4.3 OUTPUT
+
+Under a sum, $\mathbf { C }$ is now equivalent to a complete graph with self-loops that involves all relevant objects instead of relevant proposals as originally desired.
+
+To turn $\mathbf { C }$ into a count $c$ , we set $\begin{array} { r } { | E | = \sum _ { i , j } C _ { i j } } \end{array}$ as mentioned and
+
+$$
+c = | V | = \sqrt { | E | }
+$$
+
+We verified experimentally that when our extreme case assumptions hold, $c$ is always an integer and equal to the correct count, regardless of the number of duplicate object proposals.
+
+To avoid issues with scale when the number of objects is large, we turn this single feature into several classes, one for each possible number. Since we only used the object proposals with the largest $n$ weights, the predicted count $c$ can be at most $n$ . We define the output $\mathbf { o } ^ { \mathsf { ^ { - } } } = [ o _ { 0 } , o _ { 1 } , \ldots , o _ { n } ] ^ { \mathsf { T } }$ to be
+
+$$
+o _ { i } = \operatorname* { m a x } ( 0 , 1 - | c - i | )
+$$
+
+This results in a vector that is 1 at the index of the count and 0 everywhere else when $c$ is exactly an integer, and a linear interpolation between the two corresponding one-hot vectors when the count falls inbetween two integers.
+
+# 4.3.1 OUTPUT CONFIDENCE
+
+Finally, we might consider a prediction made from values of a and $\mathbf { D }$ that are either close to 0 or close to 1 to be more reliable – we explicitly handle these after all – than when many values are close to 0.5. To incorporate this idea, we scale $\mathbf { o }$ by a confidence value in the interval $[ 0 , 1 ]$ .
+
+We define $p _ { \mathbf { a } }$ and $p _ { \mathbf { D } }$ to be the average distances to 0.5. The choice of 0.5 is not important, because the module can learn to change it by changing where $f _ { 6 } ( x ) = 0 . 5$ and $f _ { 7 } ( x ) = 0 . 5$ .
+
+$$
+\begin{array} { l } { { \displaystyle p _ { \mathbf { a } } = \frac { 1 } { n } \sum _ { i } \left. f _ { 6 } ( a _ { i } ) - 0 . 5 \right. } } \\ { { \displaystyle p _ { \mathbf { D } } = \frac { 1 } { n ^ { 2 } } \sum _ { i , j } \left. f _ { 7 } ( D _ { i j } ) - 0 . 5 \right. } } \end{array}
+$$
+
+Then, the output of the component with confidence scaling is
+
+$$
+\tilde { \mathbf { o } } = f _ { 8 } ( p _ { \mathbf { a } } + p _ { \mathbf { D } } ) \cdot \mathbf { o }
+$$
+
+In summary, we only used diffentiable operations to deduplicate object proposals and obtain a feature vector that represents the predicted count. This allows easy integration into any model with soft attention, enabling a model to count from an attention map.
+
+# 5 EXPERIMENTS
+
+# 5.1 TOY TASK
+
+First, we design a simple toy task to evaluate counting ability. This dataset is intended to only evaluate the performance of counting; thus, we skip any processing steps that are not directly related such as the processing of an input image. Samples from this dataset are given in Appendix D
+
+The classification task is to predict an integer count $\hat { c }$ of true objects, uniformly drawn from 0 to 10 inclusive, from a set of bounding boxes and the associated attention weights. 10 square bounding boxes with side length $l \in ( 0 , 1 \bar { ] }$ are placed in a square image with unit side length. The $\mathbf { X }$ and y coordinates of their top left corners are uniformly drawn from $U ( 0 , 1 - l )$ so that the boxes do not extend beyond the image border. $l$ is used to control the overlapping of bounding boxes: a larger $l$ leads to the fixed number of objects to be more tightly packed, increasing the chance of overlaps. $\hat { c }$ number of these boxes are randomly chosen to be true bounding boxes. The score of a bounding box is the maximum IoU overlap of it with any true bounding box. Then, the attention weight is a linear interpolation between the score and a noise value drawn from $U ( 0 , 1 )$ , with $q \in [ 0 , 1 ]$ controlling this trade-off. $q$ is the attention noise parameter: when $q$ is 0, there is no noise and when $q$ is 1, there is no signal. Increasing $q$ also indirectly simulates imprecise placements of bounding boxes.
+
+We compare the counting component against a simple baseline that simply sums the attention weights and turns the sum into a feature vector with Equation 8. Both models are followed by a linear projection to the classes 0 to 10 inclusive and a softmax activation. They are trained with crossentropy loss for 1000 iterations using Adam (Kingma & Ba, 2015) with a learning rate of 0.01 and a batch size of 1024.
+
+# 5.1.1 RESULTS
+
+The results of varying $l$ while keeping $q$ fixed at various values and vice versa are shown in Figure 4. Regardless of $l$ and $q$ , the counting component performs better than the baseline in most cases, often significantly so. Particularly when the noise is low, the component can deal with high values for $l$ very successfully, showing that it accomplishes the goal of increased robustness to overlapping proposals. The component also handles moderate noise levels decently as long as the overlaps are limited. The performance when both $l$ and $q$ are high is closely matched by the baseline, likely due to the high difficulty of those parametrizations leaving little information to extract in the first place.
+
+We can also look at the shape of the activation functions themselves, shown in Figure 5 and Appendix C, to understand how the behaviour changes with varying dataset parameters. For simplicity, we limit our description to the two easiest-to-interpret functions: $f _ { 1 }$ for the attention weights and $f _ { 2 }$ for the bounding box distances.
+
+![](images/3fb48560d3ebed9fd32e0c830572ff36757f872f81d3938700678967ef165a4e.jpg)  
+Figure 4: Accuracies on the toy task as side length $l$ and noise $q$ are varied in 0.01 step sizes.
+
+![](images/21e53377d411fcb31ec0d56dc4ccce342dbc7c34ca06a323a7b0659898b842de.jpg)  
+Figure 5: Shapes of trained activation functions $f _ { 1 }$ (attention weights) and $f _ { 2 }$ (bounding box distances) for varying bounding box side lengths (left) or the noise (right) in the dataset, varied in 0.01 step sizes. Best viewed in color.
+
+When increasing the side length, the height of the “step” in $f _ { 1 }$ decreases to compensate for the generally greater degree of overlapping bounding boxes. A similar effect is seen with $f _ { 2 }$ : it varies over requiring a high pairwise distance when $l$ is low – when partial overlaps are most likely spurious – and considering small distances enough for proposals to be considered different when $l$ is high. At the highest values for $l$ , there is little signal in the overlaps left since everything overlaps with everything, which explains why $f _ { 2 }$ returns to its default linear initialization for those parameters.
+
+When varying the amount of noise, without noise $f _ { 1 }$ resembles a step function where the step starts close to $x = 1$ and takes a value of close to 1 after the step. Since a true proposal will always have a weight of 1 when there is no noise, anything below this can be safely zeroed out. With increasing noise, this step moves away from 1 for both $x$ and $f _ { 1 } ( x )$ , capturing the uncertainty when a bounding box belongs to a true object. With lower $q$ , $f _ { 2 }$ considers a pair of proposals to be distinct for lower distances, whereas with higher $q$ , $f _ { 2 }$ follows a more sigmoidal shape. This can be explained by the model taking the increased uncertainty of the precise bounding box placements into account by requiring higher distances for proposals to be considered completely different.
+
+# 5.2 VQA
+
+VQA v2 (Goyal et al., 2017) is the updated version of the VQA v1 dataset (Antol et al., 2015) where greater care has been taken to reduce dataset biases through balanced pairs: for each question, a pair of images is identified where the answer to that question differs. The standard accuracy metric on this dataset accounts for disagreements in human answers by averaging $\mathrm { m i n } ( \textstyle { \frac { 1 } { 3 } }$ agreeing, 1) over all 10-choose-9 subsets of human answers, where agreeing is the number of human answers that agree with the given answer. This can be shown to be equal to $\operatorname* { m i n } ( 0 . 3 a g r e e i n g , 1 )$ without averaging.
+
+We use an improved version of the strong VQA baseline by Kazemi & Elqursh (2017) as baseline model (details in Appendix B). We have not performed any tuning of this baseline to maximize the performance difference between it and the baseline with counting module. To augment this model with the counting component, we extract the attention weights of the first attention glimpse (there are two in the baseline) before softmax normalization, and feed them into the counting component after applying a logistic function. Since object proposal features from Anderson et al. (2017) vary from 10 to 100 per image, a natural choice for the number of top- $^ n$ proposals to use is 10. The output of the component is linearly projected into the same space as the hidden layer of the classifier, followed by ReLU activation, batch normalization, and addition with the features in the hidden layer.
+
+Table 1: Results on VQA v2 of the top models along with our results. Entries marked with (Ens.) are ensembles of models. At the time of writing, our model with the counting module places third among all entries. All models listed here use object proposal features and are trained on the training and validation sets. The top-performing ensemble models use additional pre-trained word embeddings, which we do not use.   
+
+<table><tr><td></td><td colspan="4">VQA v2 test-dev</td><td colspan="4">VQA v2 test</td></tr><tr><td>Model</td><td>Yes/No</td><td>Number</td><td>Other</td><td>All</td><td>Yes/No</td><td>Number</td><td>Other</td><td>All</td></tr><tr><td>Teney et al. (2017)</td><td>81.82</td><td>44.21</td><td>56.05</td><td>65.32</td><td>82.20</td><td>43.90</td><td>56.26</td><td>65.67</td></tr><tr><td>Teney et al. (2017) (Ens.)</td><td>86.08</td><td>48.99</td><td>60.80</td><td>69.87</td><td>86.60</td><td>48.64</td><td>61.15</td><td>70.34</td></tr><tr><td>Zhou et al. (2017)</td><td>84.27</td><td>49.56</td><td>59.89</td><td>68.76</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Zhou et al. (2017) (Ens.)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>86.65</td><td>51.13</td><td>61.75</td><td>70.92</td></tr><tr><td>Baseline</td><td>82.98</td><td>46.88</td><td>58.99</td><td>67.50</td><td>83.21</td><td>46.60</td><td>59.20</td><td>67.78</td></tr><tr><td>+ counting module</td><td>83.14</td><td>51.62</td><td>58.97</td><td>68.09</td><td>83.56</td><td>51.39</td><td>59.11</td><td>68.41</td></tr></table>
+
+Table 2: Results on the VQA v2 validation set with models trained only on the training set. Reported are the mean accuracies and sample standard deviations $( \pm )$ over 4 random initializations.   
+
+<table><tr><td></td><td colspan="3">VQA accuracy</td><td colspan="3">Balanced pair accuracy</td></tr><tr><td>Model</td><td>Number</td><td>Count</td><td>All</td><td>Number</td><td>Count</td><td>All</td></tr><tr><td>Baseline</td><td>44.83±0.2</td><td>51.69±0.2</td><td>64.80±0.0</td><td>17.34±0.2</td><td>20.02±0.2</td><td>36.44±0.1</td></tr><tr><td>+ NMS</td><td>44.60±0.1</td><td>51.41±0.1</td><td>64.80±0.1</td><td>17.06±0.1</td><td>19.72±0.1</td><td>36.44±0.2</td></tr><tr><td> + counting module</td><td>49.36±0.1</td><td>57.03±0.0</td><td>65.42±0.1</td><td>23.10±0.2</td><td>26.63±0.2</td><td>37.19±0.1</td></tr></table>
+
+# 5.2.1 RESULTS
+
+Table 1 shows the results on the official VQA v2 leaderboard. The baseline with our component has a significantly higher accuracy on number questions without compromising accuracy on other categories compared to the baseline result. Despite our single-model baseline being substantially worse than the state-of-the-art, by simply adding the counting component we outperform even the 8-model ensemble in Zhou et al. (2017) on the number category. We expect further improvements in number accuracy when incorporating their techniques to improve the quality of attention weights, especially since the current state-of-the-art models suffer from the problems with counting that we mention in section 3. Some qualitative examples of inputs and activations within the counting component are shown in Appendix E.
+
+We also evaluate our models on the validation set of VQA v2, shown in Table 2. This allows us to consider only the counting questions within number questions, since number questions include questions such as ”what time is it?” as well. We treat any question starting with the words ”how many” as a counting question. As we expect, the benefit of using the counting module on the counting question subset is higher than on number questions in general. Additionally, we try an approach where we simply replace the counting module with NMS, using the average of the attention glimpses as scoring, and one-hot encoding the number of proposals left. The NMS-based approach, using an IoU threshold of 0.5 and no score thresholding based on validation set performance, does not improve on the baseline, which suggests that the piecewise gradient of NMS is a major problem for learning to count in VQA and that conversely, there is a substantial benefit to being able to differentiate through the counting module.
+
+Additionally, we can evaluate the accuracy over balanced pairs as proposed by Teney et al. (2017): the ratio of balanced pairs on which the VQA accuracy for both questions is 1.0. This is a much more difficult metric, since it requires the model to find the subtle details between images instead of being able to rely on question biases in the dataset. First, notice how all balanced pair accuracies are greatly reduced compared to their respective VQA accuracy. More importantly, the absolute accuracy improvement of the counting module is still fully present with the more challenging metric, which is further evidence that the component can properly count rather than simply fitting better to dataset biases.
+
+When looking at the activation functions of the trained model, shown in Figure 9, we find that some characteristics of them are shared with high-noise parametrizations of the toy dataset. This suggests that the current attention mechanisms and object proposal network are still very inaccurate, which explains the perhaps small-seeming increase in counting performance. This provides further evidence that the balanced pair accuracy is maybe a more reflective measure of how well current VQA models perform than the overall VQA accuracies of over $70 \%$ of the current top models.
+
+# 6 CONCLUSION
+
+After understanding why VQA models struggle to count, we designed a counting component that alleviates this problem through differentiable bounding box deduplication. The component can readily be used alongside any future improvements in VQA models, as long as they still use soft attention as all current top models on VQA v2 do. It has uses outside of VQA as well: for many counting tasks, it can allow an object-proposal-based approach to work without ground-truth objects available as long as there is a – possibly learned – per-proposal scoring (for example using a classification score) and a notion of how dissimilar a pair of proposals are. Since each step in the component has a clear purpose and interpretation, the learned weights of the activation functions are also interpretable. The design of the counting component is an example showing how by encoding inductive biases into a deep learning model, challenging problems such as counting of arbitrary objects can be approached when only relatively little supervisory information is available.
+
+For future research, it should be kept in mind that VQA v2 requires a versatile skill set that current models do not have. To make progress on this dataset, we advocate focusing on understanding of what the current shortcomings of models are and finding ways to mitigate them.
+
+# REFERENCES
+
+Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang. Bottom-up and top-down attention for image captioning and VQA. CoRR, arXiv:1707.07998, 2017.   
+Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C. Lawrence Zitnick, and Devi Parikh. VQA: Visual Question Answering. In ICCV, 2015.   
+Samaneh Azadi, Jiashi Feng, and Trevor Darrell. Learning detection with diverse proposals. In CVPR, 2017.   
+Jimmy Ba, Volodymyr Mnih, and Koray Kavukcuoglu. Multiple object recognition with visual attention. In ICLR, 2015.   
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In ICLR, 2015.   
+Prithvijit Chattopadhyay, Ramakrishna Vedantam, Ramprasaath R. Selvaraju, Dhruv Batra, and Devi Parikh. Counting everyday objects in everyday scenes. In CVPR, 2017.   
+Kyunghyun Cho, B van Merrienboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties of neural machine translation: Encoder-decoder approaches. In SSST@EMNLP, 2014.   
+Joseph Paul Cohen, Henry Z. Lo, and Yoshua Bengio. Count-ception: Counting by fully convolutional redundant counting. CoRR, arXiv:1703.08710, 2017.   
+Yash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. Making the V in VQA matter: Elevating the role of image understanding in Visual Question Answering. In CVPR, 2017.
+
+Paul Henderson and Vittorio Ferrari. End-to-end training of object class detectors for mean average precision. In ACCV, 2017.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural Computation, 1997.
+
+Jan Hosang, Rodrigo Benenson, and Bernt Schiele. Learning non-maximum suppression. In CVPR, 2017.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, 2015.
+
+Allan Jabri, Armand Joulin, and Laurens van der Maaten. Revisiting visual question answering baselines. In ECCV, 2016.
+
+Max Jaderberg, Karen Simonyan, Andrew Zisserman, and Koray Kavukcuoglu. Spatial transformer networks. In NIPS, 2015.
+
+Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C. Lawrence Zitnick, and Ross Girshick. CLEVR: A diagnostic dataset for compositional language and elementary visual reasoning. In CVPR, 2017.
+
+Vahid Kazemi and Ali Elqursh. Show, ask, attend, and answer: A strong baseline for visual question answering. CoRR, arXiv:1704.03162, 2017.
+
+Yoon Kim, Carl Denton, Luong Hoang, and Alexander M. Rush. Structured attention networks. In ICLR, 2017.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015.
+
+Hugo Larochelle and Geoffrey E Hinton. Learning to combine foveal glimpses with a third-order boltzmann machine. In NIPS, 2010.
+
+Victor Lempitsky and Andrew Zisserman. Learning to count objects in images. In NIPS, 2010.
+
+Jiasen Lu, Jianwei Yang, Dhruv Batra, and Devi Parikh. Hierarchical question-image co-attention for visual question answering. In NIPS, 2016.
+
+Dmytro Mishkin and Jiri Matas. All you need is a good init. In ICLR, 2016.
+
+Volodymyr Mnih, Nicolas Heess, Alex Graves, and Koray Kavukcuoglu. Recurrent models of visual attention. In NIPS, 2014.
+
+Ethan Perez, Florian Strub, Harm de Vries, Vincent Dumoulin, and Aaron Courville. FiLM: Visual reasoning with a general conditioning layer. CoRR, arXiv:1709.07871, 2017.
+
+Mengye Ren and Richard S. Zemel. End-to-end instance segmentation with recurrent attention. In CVPR, 2017.
+
+Adam Santoro, David Raposo, David G. T. Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Timothy P. Lillicrap. A simple neural network module for relational reasoning. In NIPS, 2017.
+
+Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. JMLR, 2014.
+
+Damien Teney, Peter Anderson, Xiaodong He, and Anton van den Hengel. Tips and tricks for visual question answering: Learnings from the 2017 challenge. CoRR, arXiv:1708.02711, 2017.
+
+Alexander Trott, Caiming Xiong, and Richard Socher. Interpretable counting for visual question answering. In ICLR, 2018.
+
+Zichao Yang, Xiaodong He, Jianfeng Gao, Li Deng, and Alexander J. Smola. Stacked attention networks for image question answering. In CVPR, 2016.
+
+Seungil You, David Ding, Kevin Canini, Jan Pfeifer, and Maya Gupta. Deep Lattice Networks and Partial Monotonic Functions. In NIPS, 2017.
+
+Yu Zhou, Yu Jun, Xiang Chenchao, Fan Jianping, and Tao Dacheng. Beyond bilinear: Generalized multi-modal factorized high-order pooling for visual question answering. CoRR, arXiv:1708.03619, 2017.
+
+Chen Zhu, Yanpeng Zhao, Shuaiyi Huang, Kewei Tu, and Yi Ma. Structured attentions for visual question answering. CoRR, arXiv:1708.02071, 2017.
+
+# A PIECEWISE LINEAR ACTIVATION FUNCTION
+
+Intuitively, the interval $[ 0 , 1 ]$ is split into $d$ equal size intervals. Each contains a line segment that is connected to the neighboring line segments at the boundaries of the intervals. These line segments form the shape of the activation function.
+
+For each function $f _ { k }$ , there are $d$ weights $w _ { k 1 } , \ldots , w _ { k d }$ , where the weight $w _ { k i }$ is the gradient for the interval $[ \textstyle { \frac { i - 1 } { d } } , \textstyle { \frac { i } { d } } )$ . We arbitrarily fix $d$ to be 16 in this paper, observing no significant difference when changing it to 8 and 32 in preliminary experiments. All $w _ { k i }$ are enforced to be non-negative by always using the absolute value of them, which yields the monotonicity property. Dividing the weights by $\Sigma _ { m } ^ { d } \mid w _ { k m } \mid$ yields the property that $f ( 1 ) = 1$ . The function can be written as
+
+$$
+f _ { k } ( x ) = \sum _ { i = 1 } ^ { d } \operatorname* { m a x } ( 0 , 1 - | d x - i | ) \frac { \sum _ { j = 1 } ^ { i } | w _ { k j } | } { \sum _ { m = 1 } ^ { d } | w _ { k m } | }
+$$
+
+In essence, the max term selects the two nearest boundary values of an interval, which are normalized cumulative sums over the $w _ { k }$ weights, and linearly interpolates between the two. This approach is similar to the subgradient approach by Jaderberg et al. (2015) to make sampling from indices differentiable. All $w _ { k i }$ are initialized to 1, which makes the functions linear on initialization. When applying $f _ { k } ( { \bf x } )$ to a vector-valued input $\mathbf { x }$ , it is assumed to be applied elementwise. By caching the normalized cumulative sum $\begin{array} { r } { \sum _ { j } ^ { i } | w _ { k j } | / \sum _ { m } ^ { d } | w _ { k m } | } \end{array}$ , this function has linear time complexity with respect to $d$ and is efficiently implementable on GPUs.
+
+Extensions to this are possible through Deep Lattice Networks (You et al., 2017), which preserve monotonicity across several nonlinear neural network layers. They would allow A and $\mathbf { D }$ to be combined in more sophisticated ways beyond an elementwise product, possibly improving counting performance as long as the property of the range lying within $[ 0 , 1 ]$ is still enforced in some way.
+
+# B BASELINE ARCHITECTURE
+
+This model is based on the work of Kazemi & Elqursh (2017), who outperformed most previous VQA models on the VQA v1 dataset with a simple baseline architecture. We adapt the model to the VQA v2 dataset and make various tweaks that improve validation accuracy slightly. The architecture is illustrated in Figure 6. Details not mentioned here can be assumed to be the same as in their paper.
+
+The most significant change that we make is the use of object proposal features by Anderson et al. (2017) as previously mentioned. The following tweaks were made without considering the performance impact on the counting component; only the validation accuracy of the baseline was optimized.
+
+To fuse vision features $\mathbf { x }$ and question features y, the baseline concatenates and linearly projects them, followed by a ReLU activation. This is equivalent to ReLU $( \mathbf { W } _ { x } \mathbf { x } + \mathbf { W } _ { y } \mathbf { y } )$ . We include an additional term that measures how different the projected $\mathbf { x }$ is from the projected y, changing the fusion mechanism to $\mathbf { x } \odot \mathbf { y } = \operatorname { R e L U } ( \mathbf { W } _ { x } \mathbf { x } + \mathbf { W } _ { y } \mathbf { y } ) - ( \mathbf { W } _ { x } \mathbf { x } - \mathbf { W } _ { y } \mathbf { y } ) ^ { 2 }$ .
+
+The LSTM (Hochreiter & Schmidhuber, 1997) for question encoding is replaced with a GRU (Cho et al., 2014) with the same hidden size with dynamic per-example unrolling instead of a fixed 14 words per question. We apply batch normalization (Ioffe & Szegedy, 2015) before the last linear projection in the classifier to the 3000 classes. The learning rate is increased from 0.001 to 0.0015 and the batch size is doubled to 256. The model is trained for 100 epochs (1697 iterations per epoch to train on the training set, 2517 iterations per epoch to train on both training and validation sets) instead of 100,000 iterations, roughly in line with the doubling of dataset size when going from VQA v1 to VQA v2.
+
+Note that this single-model baseline is regularized with dropout (Srivastava et al., 2014), while the other current top models skip this and rely on ensembling to reduce overfitting. This explains why our single-model baseline outperforms most single-model results of the state-of-the-art models. We found ensembling of the regularized baseline to provide a much smaller benefit in preliminary experiments compared to the results of ensembling unregularized networks reported in Teney et al. (2017).
+
+![](images/490c1f6c072e0a062d39ff4d6cc97980d0cf6261d8fcaf1d9d6a6f9cad6001a1.jpg)  
+Figure 6: Schematic view of a model using our counting component. The modifications made to the baseline model when including the counting component are marked in red. Blue blocks mark components with trainable parameters, gray blocks mark components without trainable parameters. White $\textsuperscript { \textregistered }$ mark linear layers, either linear projections or convolutions with a spatial size of 1 depending on the context. Dropout with drop probability 0.5 is applied before the GRU and every $\textsuperscript { \textregistered }$ , except before the $\textsuperscript { \textregistered }$ after the counting component. $\diamond$ stands for the fusion function we define in Appendix B, BN stands for batch normalization, $\sigma$ stands for a logistic, and Embedding is a word embedding that has been fed through a tanh function. The two glimpses of the attention mechanism are represented with the two lines exiting the $\textsuperscript { \textregistered }$ . Note that one of the two glimpses is shared with the counting component.
+
+![](images/07d2e4d6eedacaf38bf7ab5fb51fab2d11a25b373c83069db69f688b7b40cae0.jpg)  
+Figure 7: Shape of activation functions as $l$ is varied for $q = 0 . 5$ on the toy dataset. Each line shows the shape of the activation function when $l$ is set to the value associated to its color. Best viewed in color.
+
+![](images/e630ceffcd3a5654e3eb6233016450532e537c9e33e24a644459ee0f3beb38ae.jpg)  
+Figure 8: Shape of activation functions as $q$ is varied for $l = 0 . 5$ on the toy dataset. Each line shows the shape of the activation function when $q$ is set to the value associated to its color. Best viewed in color.
+
+![](images/0c0fbd90d5fdb814b3a5f433d4b3e097fa2287a734697d8678539536c723c7d2.jpg)  
+Figure 9: Shape of activation functions for a model trained on the train and validation sets of VQA v2 (thick black), compared against the shapes when parametrizing the toy dataset with $q$ around 0.4 (green), 0.7 (orange), or 1.0 (red) with fixed $l = 0 . 2$ . Best viewed in color.
+
+![](images/82a95f366fb3f29067d4526bec9b7457d9cdecee732615090ede2cab04ee24a8.jpg)  
+Figure 10: Example toy dataset data for varying bounding box side lengths $l$ and noise $q$ . The ground truth column shows bounding boxes of randomly placed true objects (blue) and of irrelevant objects (red). The data column visualizes the samples that are actually used as input (dark blues represent weights close to 1, dark reds represent weights close to 0, lighter colors represent weights closer to 0.5). The weight of the ith bounding box $b _ { i }$ is defined as $a _ { i } = ( 1 - q )$ score $+ q z$ where the score is the maximum overlap of $b _ { i }$ with any true bounding box or 0 if there are no true bounding boxes and $z$ is drawn from $U ( 0 , 1 )$ . Note how this turns red bounding boxes that overlap a lot with a blue bounding box in the ground truth column into a blue bounding box in the data column, which simulates the duplicate proposal that we have to deal with. Best viewed in color.
+
+![](images/2f649731cb7ae201b34fb60357ccbb8c13f0f0e0a6906d1cd4d0372f1be0a82e.jpg)  
+Figure 11: Selection of validation images with overlaid bounding boxes, values of the attention matrix A, distance matrix D, and the resulting count matrix C. White entries represent values close to 1, black entries represent values close to 0. The count $c$ is the usual square root of the sum over the elements of C. Notice how particularly in the third example, A clearly contains more rows/columns with high activations than there are actual objects (a sign of overlapping bounding boxes) and the counting module successfully removes intra- and inter-object edges to arrive at the correct prediction regardless. The prediction is not necessarily – though often is – the rounded value of $c$ .

parse/train/B12Js_yRb/B12Js_yRb_content_list.json

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+    {
+        "type": "text",
+        "text": "LEARNING TO COUNT OBJECTS IN NATURAL IMAGES FOR VISUAL QUESTION ANSWERING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yan Zhang & Jonathon Hare & Adam Prugel-Bennett ¨ ",
+        "bbox": [
+            183,
+            172,
+            562,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Electronics and Computer Science University of Southampton {yz5n12,jsh2,apb}@ecs.soton.ac.uk ",
+        "bbox": [
+            184,
+            189,
+            508,
+            229
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            266,
+            544,
+            281
+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "Visual Question Answering (VQA) models have struggled with counting objects in natural images so far. We identify a fundamental problem due to soft attention in these models as a cause. To circumvent this problem, we propose a neural network component that allows robust counting from object proposals. Experiments on a toy task show the effectiveness of this component and we obtain state-of-theart accuracy on the number category of the VQA v2 dataset without negatively affecting other categories, even outperforming ensemble models with our single model. On a difficult balanced pair metric, the component gives a substantial improvement in counting over a strong baseline by $6 . 6 \\%$ . ",
+        "bbox": [
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+            296,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            446,
+            336,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Consider the problem of counting how many cats there are in Figure 1. Solving this involves several rough steps: understanding what instances of that type can look like, finding them in the image, and adding them up. This is a common task in Visual Question Answering (VQA) – answering questions about images – and is rated as among the tasks requiring the lowest human age to be able to answer (Antol et al., 2015). However, current models for VQA on natural images struggle to answer any counting questions successfully outside of dataset biases (Jabri et al., 2016). ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One reason for this is the presence of a fundamental problem with counting in the widely-used soft attention mechanisms (section 3). Another reason is that unlike standard counting tasks, there is no ground truth labeling of where the objects to count are. Coupled with the fact that models need to be able to count a large variety of objects and that, ideally, performance on non-counting questions should not be compromised, the task of counting in VQA seems very challenging. ",
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+    {
+        "type": "text",
+        "text": "To make this task easier, we can use object proposals – pairs of a bounding box and object features – from object detection networks as input instead of learning from pixels directly. In any moderately complex scene, this runs into the issue of double-counting overlapping object proposals. This is a problem present in many natural images, which leads to inaccurate counting in real-world scenarios. ",
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+        "text": "Our main contribution is a differentiable neural network component that tackles this problem and consequently can learn to count (section 4). Used alongside an attention mechanism, this component avoids a fundamental limitation of soft attention while producing strong counting features. We provide experimental evidence of the effectiveness of this component (section 5). On a toy dataset, we demonstrate that this component enables robust counting in a variety of scenarios. On the number category of the VQA v2 Open-Ended dataset (Goyal et al., 2017), a relatively simple baseline model using the counting component outperforms all previous models – including large ensembles of state-of-the-art methods – without degrading performance on other categories. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Usually, greedy non-maximum suppression (NMS) is used to eliminate duplicate bounding boxes. The main problem with using it as part of a model is that its gradient is piecewise constant. Various differentiable variants such as by Azadi et al. (2017), Hosang et al. (2017), and Henderson & Ferrari (2017) exist. The main difference is that, since we are interested in counting, our component does not need to make discrete decisions about which bounding boxes to keep; it outputs counting features, not a smaller set of bounding boxes. Our component is also easily integrated into standard VQA models that utilize soft attention without any need for other network architecture changes and can be used without using true bounding boxes for supervision. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "On the VQA v2 dataset (Goyal et al., 2017) that we apply our method on, only few advances on counting questions have been made. The main improvement in accuracy is due to the use of object proposals in the visual processing pipeline, proposed by Anderson et al. (2017). Their object proposal network is trained with classes in singular and plural forms, for example “tree” versus “trees”, which only allows primitive counting information to be present in the object features after region-of-interest pooling. Our approach differs in the way that instead of relying on counting features being present in the input, we create counting features using information present in the attention map over object proposals. This has the benefit of being able to count anything that the attention mechanism can discriminate instead of only objects that belong to the predetermined set of classes that had plural forms. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Using these object proposals, Trott et al. (2018) train a sequential counting mechanism with a reinforcement learning loss on the counting question subsets of VQA v2 and Visual Genome. They achieve a small increase in accuracy and can obtain an interpretable set of objects that their model counted, but it is unclear whether their method can be integrated into traditional VQA models due to their loss not applying to non-counting questions. Since they evaluate on their own dataset, their results can not be easily compared to existing results in VQA. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Methods such as by Santoro et al. (2017) and Perez et al. (2017) can count on the synthetic CLEVR VQA dataset (Johnson et al., 2017) successfully without bounding boxes and supervision of where the objects to count are. They also use more training data ${ \\sim } 2 5 0 { , } 0 0 0$ counting questions in the CLEVR training set versus $\\sim 5 0 { , } 0 0 0$ counting questions in the VQA v2 training set), much simpler objects, and synthetic question structures. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "More traditional approaches based on Lempitsky & Zisserman (2010) learn to produce a target density map, from which a count is computed by integrating over it. In this setting, Cohen et al. (2017) make use of overlaps of convolutional receptive fields to improve counting performance. Chattopadhyay et al. (2017) use an approach that divides the image into smaller non-overlapping chunks, each of which is counted individually and combined together at the end. In both of these contexts, the convolutional receptive fields or chunks can be seen as sets of bounding boxes with a fixed structure in their positioning. Note that while Chattopadhyay et al. (2017) evaluate their models on a small subset of counting questions in VQA, major differences in training setup make their results not comparable to our work. ",
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+        "type": "text",
+        "text": "3 PROBLEMS WITH SOFT ATTENTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The main message in this section is that using the feature vectors obtained after the attention mechanism is not enough to be able to count; the attention maps themselves should be used, which is what we do in our counting component. ",
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+        "type": "text",
+        "text": "Models in VQA have consistently benefited from the use of soft attention (Mnih et al., 2014; Bahdanau et al., 2015) on the image, commonly implemented with a shallow convolutional network. It learns to output a weight for the feature vector at each spatial position in the feature map, which is first normalized and then used for performing a weighted sum over the spatial positions to produce a single feature vector. However, soft spatial attention severely limits the ability for a model to count. ",
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+    {
+        "type": "text",
+        "text": "Consider the task of counting the number of cats for two images: an image showing a single cat on a clean background and an image that consists of two side-by-side copies of the first image. What we will describe applies to both spatial feature maps and sets of object proposals as input, but we focus on the latter case for simplicity. With an object detection network, we detect one cat in the first image and two cats in the second image, producing the same feature vector for all three detections. The attention mechanism then assigns all three instances of the same cat the same weight. ",
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+    {
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+        "img_path": "images/01c2c43239e5530d6490c14c6c93b92ad5fef0ff207f0d931d1ff4b6596146b9.jpg",
+        "image_caption": [
+            "Figure 1: Simplified example about counting the number of cats. The light-colored cat is detected twice and results in a duplicate proposal. This shows the conversion from the attention weights a to a graph representation A and the eventual goal of this component with exactly one proposal per true object. There are 4 proposals (vertices) capturing 3 underlying objects (groups in dotted lines). There are 3 relevant proposals (black with weight 1) and 1 irrelevant proposal (white with weight 0). Red edges mark intra-object edges between duplicate proposals and blue edges mark the main inter-object duplicate edges. In graph form, the object groups, coloring of edges, and shading of vertices serve illustration purposes only; the model does not have these access to these directly. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The usual normalization used for the attention weights is the softmax function, which normalizes the weights to sum to 1. Herein lies the problem: the cat in the first image receives a normalized weight of 1, but the two cats in the second image now each receive a weight of 0.5. After the weighted sum, we are effectively averaging the two cats in the second image back to a single cat. As a consequence, the feature vector obtained after the weighted sum is exactly the same between the two images and we have lost all information about a possible count from the attention map. Any method that normalizes the weights to sum to 1 suffers from this issue. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multiple glimpses (Larochelle & Hinton, 2010) – sets of attention weights that the attention mechanism outputs – or several steps of attention (Yang et al., 2016; Lu et al., 2016) do not circumvent this problem. Each glimpse or step can not separate out an object each, since the attention weight given to one feature vector does not depend on the other feature vectors to be attended over. Hard attention (Ba et al., 2015; Mnih et al., 2014) and structured attention (Kim et al., 2017) may be possible solutions to this, though no significant improvement in counting ability has been found for the latter so far (Zhu et al., 2017). Ren & Zemel (2017) circumvent the problem by limiting attention to only work within one bounding box at a time, remotely similar to our approach of using object proposal features. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Without normalization of weights to sum to one, the scale of the output features depends on the number of objects detected. In an image with 10 cats, the output feature vector is scaled up by 10. Since deep neural networks are typically very scale-sensitive – the scale of weight initializations and activations is generally considered quite important (Mishkin & Matas, 2016) – and the classifier would have to learn that joint scaling of all features is somehow related to count, this approach is not reasonable for counting objects. This is evidenced in Teney et al. (2017) where they provide evidence that sigmoid normalization not only degrades accuracy on non-number questions slightly, but also does not help with counting. ",
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+    {
+        "type": "text",
+        "text": "4 COUNTING COMPONENT ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we describe a differentiable mechanism for counting from attention weights, while also dealing with the problem of overlapping object proposals to reduce double-counting of objects. This involves some nontrivial details to produce counts that are as accurate as possible. The main idea is illustrated in Figure 1 with the two main steps shown in Figure 2 and Figure 3. The use of this component allows a model to count while still being able to exploit the benefits of soft attention. ",
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+        "type": "text",
+        "text": "Our key idea for dealing with overlapping object proposals is to turn these object proposals into a graph that is based on how they overlap. We then remove and scale edges in a specific way such that an estimate of the number of underlying objects is recovered. ",
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+        "type": "text",
+        "text": "Our general strategy is to primarily design the component for the unrealistic extreme cases of perfect attention maps and bounding boxes that are either fully overlapping or fully distinct. By introducing some parameters and only using differentiable operations, we give the ability for the module to interpolate between the correct behaviours for these extreme cases to handle the more realistic cases. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These parameters are responsible for handling variations in attention weights and partial bounding box overlaps in a manner suitable for a given dataset. ",
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+    {
+        "type": "text",
+        "text": "To achieve this, we use several piecewise linear functions $f _ { 1 } , \\ldots , f _ { 8 }$ as activation functions (defined in Appendix A), approximating arbitrary functions with domain and range [0, 1]. The shapes of these functions are learned to handle the specific nonlinear interactions necessary for dealing with overlapping proposals. Through their parametrization we enforce that $f _ { k } ( 0 ) = 0$ , $f _ { k } ( 1 ) = 1$ , and that they are monotonically increasing. The first two properties are required so that the extreme cases that we explicitly handle are left unchanged. In those cases, $f _ { k }$ is only applied to values of 0 or 1, so the activation functions can be safely ignored for understanding how the component handles them. By enforcing monotonicity, we can make sure that, for example, an increased value in an attention map should never result in the prediction of the count to decrease. ",
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+    {
+        "type": "text",
+        "text": "4.1 INPUT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given a set of features from object proposals, an attention mechanism produces a weight for each proposal based on the question. The counting component takes as input the $n$ largest attention weights $\\mathbf { \\bar { a } } = [ a _ { 1 } , \\ldots , a _ { n } ] ^ { \\mathsf { T } }$ and their corresponding bounding boxes $\\mathbf { b } = [ b _ { 1 } , \\ldots , b _ { n } ] ^ { \\mathsf { T } }$ . We assume that the weights lie in the interval $[ 0 , 1 ]$ , which can easily be achieved by applying a logistic function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the extreme cases that we explicitly handle, we assume that the attention mechanism assigns a value of 1 to $a _ { i }$ whenever the ith proposal contains a relevant object and a value of 0 whenever it does not. This is in line with what usual soft attention mechanisms learn, as they produce higher weights for relevant inputs. We also assume that either two object proposals fully overlap (in which case they must be showing the same object and thus receive the same attention weight) or that they are fully distinct (in which case they show different objects). Keep in mind that while we make these assumptions to make reasoning about the behaviour easier, the learned parameters in the activation functions are intended to handle the more realistic scenarios when the assumptions do not apply. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Instead of partially overlapping proposals, the problem now becomes the handling of exact duplicate proposals of underlying objects in a differentiable manner. ",
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+    {
+        "type": "text",
+        "text": "4.2 DEDUPLICATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We start by changing the vector of attention weights a into a graph representation in which bounding boxes can be utilized more easily. Hence, we compute the outer product of the attention weights to obtain an attention matrix. ",
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+    {
+        "type": "equation",
+        "img_path": "images/45d21d32766cf3e79ac4874ade1c995e6282fbe88bba109fb973d8aace126ef1.jpg",
+        "text": "$$\n\\mathbf { A } = \\mathbf { a } \\mathbf { a } ^ { \\mathsf { T } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { A } \\in \\mathbb { R } ^ { n \\times n }$ can be interpreted as an adjacency matrix for a weighted directed graph. In this graph, the ith vertex represents the object proposal associated with $a _ { i }$ and the edge between any pair of vertices $( i , j )$ has weight $a _ { i } a _ { j }$ . In the extreme case where $a _ { i }$ is virtually 0 or 1, products are equivalent to logical AND operators. It follows that the subgraph containing only the vertices satisfying $a _ { i } = 1$ is a complete digraph with self-loops. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this representation, our objective is to eliminate edges in such a way that, conceptually, the underlying true objects – instead of proposals thereof – are the vertices of that complete subgraph. In order to then turn that graph into a count, recall that the number of edges $| E |$ in a complete digraph with self-loops relates to the number of vertices $| V |$ through $| E | = | V | ^ { 2 }$ . $| E |$ can be computed by summing over the entries in an adjacency matrix and $| V |$ is then the count. Notice how when $| E |$ is set to the sum over A, ${ \\sqrt { \\textstyle | E | } } = \\sum _ { i } a _ { i }$ holds. This convenient property implies that when all proposals are fully distinct, the component can output the same as simply summing over the original attention weights by default. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are two types of duplicate edges to eliminate to achieve our objective: intra-object edges and inter-object edges. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 INTRA-OBJECT EDGES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First, we eliminate intra-object edges between duplicate proposals of a single underlying object. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/ebf111b774761b74e228aac07c7d7d09a9fdfade0a9ad743d7521aa694abb69f.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/afc38c727db1f3d829ce09537e6ec8c8c604979b7eca9b190e7296a9afa5152c.jpg",
+        "image_caption": [
+            "Figure 2: Removal of intra-object edges by masking the edges of the attention matrix A with the distance matrix D. The black vertices now form a graph without self-loops. The self-loops need to be added back in later. ",
+            "Figure 3: Removal of duplicate inter-object edges by computing a scaling factor for each vertex and scaling $\\tilde { \\mathbf { A } } ^ { \\prime }$ accordingly. $\\bar { \\mathbf { A } } ^ { \\prime }$ is $\\tilde { \\mathbf { A } }$ with self-loops already added back in. The scaling factor for one vertex is computed by counting how many vertices have outgoing edges to the same set of vertices; all edges of the two proposals on the right are scaled by 0.5. This can be seen as averaging proposals within each object and is equivalent to removing duplicate proposals altogether under a sum. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To compare two bounding boxes, we use the usual intersection-over-union (IoU) metric. We define the distance matrix $\\mathbf { D } \\in \\bar { \\mathbb { R } } ^ { n \\times n }$ to be ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e17173feaf5840ff4ccb3f0036fe26270a9b13dcff514013b1adda23f4241173.jpg",
+        "text": "$$\nD _ { i j } = 1 - \\mathrm { I o U } ( b _ { i } , b _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { D }$ can also be interpreted as an adjacency matrix. It represents a graph that has edges everywhere except when the two bounding boxes that an edge connects would overlap. ",
+        "bbox": [
+            171,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Intra-object edges are removed by elementwise multiplying $( \\odot )$ the distance matrix with the attention matrix (Figure 2). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c2d701d5864f3e52114d9964ac794164b7ab3bd62fc612ee934a0a81b728e32.jpg",
+        "text": "$$\n\\tilde { \\mathbf { A } } = f _ { 1 } ( \\mathbf { A } ) \\odot f _ { 2 } ( \\mathbf { D } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\tilde { \\mathbf { A } }$ no longer has self-loops, so we need to add them back in at a later point to still satisfy $| E | = | V | ^ { 2 }$ Notice that we start making use of the activation functions mentioned earlier to handle intermediate values in the interval $( 0 , 1 )$ for both A and $\\mathbf { D }$ . They regulate the influence of attention weights that are not close to 0 or 1 and the influence of partial overlaps. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 INTER-OBJECT EDGES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Second, we eliminate inter-object edges between duplicate proposals of different underlying objects. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The main idea (depicted in Figure 3) is to count the number of proposals associated to each invidual object, then scale down the weight of their associated edges by that number. If there are two proposals of a single object, the edges involving those proposals should be scaled by 0.5. In essence, this averages over the proposals within each underlying object because we only use the sum over the edge weights to compute the count at the end. Conceptually, this reduces multiple proposals of an object down to one as desired. Since we do not know how many proposals belong to an object, we have to estimate this. We do this by using the fact that proposals of the same object are similar. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Keep in mind that $\\tilde { \\mathbf { A } }$ has no self-loops nor edges between proposals of the same object. As a consequence, two nonzero rows in $\\tilde { \\mathbf { A } }$ are the same if and only if the proposals are the same. If the two rows differ in at least one entry, then one proposal overlaps a proposal that the other proposal does not overlap, so they must be different proposals. This means for comparing rows, we need a similarity function that satisfies the criteria of taking the value 1 when they differ in no places and 0 if they differ in at least one place. We define a differentiable similarity between proposals $i$ and $j$ as ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df2beb019f6e78886f4c0f3253d9ce6a6d36b7087d2fbfc8426323e8796c19a7.jpg",
+        "text": "$$\n\\mathrm { S i m } _ { i j } = f _ { 3 } ( 1 - | a _ { i } - a _ { j } | ) \\prod _ { k } f _ { 3 } ( 1 - | X _ { i k } - X _ { j k } | )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { X } = f _ { 4 } ( \\mathbf { A } ) \\odot f _ { 5 } ( \\mathbf { D } )$ is the same as $\\tilde { \\mathbf { A } }$ except with different activation functions. The $\\prod$ term compares the rows of proposals $i$ and $j$ . Using this term instead of $f _ { 4 } ( 1 - D _ { i j } )$ was more robust to inaccurate bounding boxes in initial experiments. ",
+        "bbox": [
+            173,
+            193,
+            825,
+            237
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the $f _ { 3 } ( 1 - | a _ { i } - a _ { j } | )$ term handles the edge case when there is only one proposal to count. Since $\\mathbf { X }$ does not have self-loops, $\\mathbf { X }$ contains only zeros in that case, which causes the row corresponding to $a _ { i } = 1$ to be incorrectly similar to the rows where $a _ { j \\neq i } = 0$ . By comparing the attention weights through that term as well, this issue is avoided. ",
+        "bbox": [
+            173,
+            242,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Now that we can check how similar two proposals are, we count the number of times any row is the same as any other row and compute a scaling factor $s _ { i }$ for each vertex $i$ . ",
+        "bbox": [
+            171,
+            306,
+            823,
+            335
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efbc47996a76c7bfd2142cf8cdde1c5a64c1cfa1f607e7ef1bf94de3ee8043b5.jpg",
+        "text": "$$\ns _ { i } = 1 / \\sum _ { j } \\mathrm { S i m } _ { i j }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            353,
+            560,
+            388
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The time complexity of computing $\\mathbf { s } = [ s _ { 1 } , \\ldots , s _ { n } ] ^ { \\mathsf { T } }$ is $\\Theta ( n ^ { 3 } )$ as there are $n ^ { 2 }$ pairs of rows and $\\Theta ( n )$ operations to compute the similarity of any pair of rows. ",
+        "bbox": [
+            171,
+            404,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since these scaling factors apply to each vertex, we have to expand s into a matrix using the outer product in order to scale both incoming and outgoing edges of each vertex. We can also add self-loops back in, which need to be scaled by s as well. Then, the count matrix $\\mathbf { C }$ is ",
+        "bbox": [
+            173,
+            439,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a8773de135a8cbfd0753c09ba2918aff0ae26c0010d20af34943639f7db57f6b.jpg",
+        "text": "$$\n\\mathbf { C } = \\tilde { \\mathbf { A } } \\odot \\mathbf { s s } ^ { \\mathsf { T } } + \\mathrm { { d i a g } } ( \\mathbf { s } \\odot f _ { 1 } ( \\mathbf { a } \\odot \\mathbf { a } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            502,
+            624,
+            523
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { d i a g ( \\cdot ) }$ expands a vector into a diagonal matrix with the vector on the diagonal. ",
+        "bbox": [
+            173,
+            536,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The scaling of self-loops involves a non-obvious detail. Recall that the diagonal that was removed when going from $\\mathbf { A }$ to $\\tilde { \\mathbf { A } }$ contains the entries $f _ { 1 } ( \\mathbf { a } \\odot \\mathbf { a } )$ . Notice however that we are scaling this diagonal by s and not s $\\odot$ s. This is because the number of inter-object edges scales quadratically with respect to the number of proposals per object, but the number of self-loops only scales linearly. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 OUTPUT ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            636,
+            274,
+            650
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Under a sum, $\\mathbf { C }$ is now equivalent to a complete graph with self-loops that involves all relevant objects instead of relevant proposals as originally desired. ",
+        "bbox": [
+            173,
+            662,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To turn $\\mathbf { C }$ into a count $c$ , we set $\\begin{array} { r } { | E | = \\sum _ { i , j } C _ { i j } } \\end{array}$ as mentioned and ",
+        "bbox": [
+            173,
+            696,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/642d05cd0b486f94409c6bd6c12f032c0238a4fd56a467acc8fb930fe3c9000b.jpg",
+        "text": "$$\nc = | V | = \\sqrt { | E | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            724,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We verified experimentally that when our extreme case assumptions hold, $c$ is always an integer and equal to the correct count, regardless of the number of duplicate object proposals. ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To avoid issues with scale when the number of objects is large, we turn this single feature into several classes, one for each possible number. Since we only used the object proposals with the largest $n$ weights, the predicted count $c$ can be at most $n$ . We define the output $\\mathbf { o } ^ { \\mathsf { ^ { - } } } = [ o _ { 0 } , o _ { 1 } , \\ldots , o _ { n } ] ^ { \\mathsf { T } }$ to be ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce3db8dd9abdf02d670c56ed54df93933685a0172f09abb3c9f5ab60cc58665b.jpg",
+        "text": "$$\no _ { i } = \\operatorname* { m a x } ( 0 , 1 - | c - i | )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This results in a vector that is 1 at the index of the count and 0 everywhere else when $c$ is exactly an integer, and a linear interpolation between the two corresponding one-hot vectors when the count falls inbetween two integers. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3.1 OUTPUT CONFIDENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            382,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we might consider a prediction made from values of a and $\\mathbf { D }$ that are either close to 0 or close to 1 to be more reliable – we explicitly handle these after all – than when many values are close to 0.5. To incorporate this idea, we scale $\\mathbf { o }$ by a confidence value in the interval $[ 0 , 1 ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We define $p _ { \\mathbf { a } }$ and $p _ { \\mathbf { D } }$ to be the average distances to 0.5. The choice of 0.5 is not important, because the module can learn to change it by changing where $f _ { 6 } ( x ) = 0 . 5$ and $f _ { 7 } ( x ) = 0 . 5$ . ",
+        "bbox": [
+            169,
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/07e1cfab90748045f79d879721cbde8d479163716e70cd7fd17c744ddb9ade28.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle p _ { \\mathbf { a } } = \\frac { 1 } { n } \\sum _ { i } \\left. f _ { 6 } ( a _ { i } ) - 0 . 5 \\right. } } \\\\ { { \\displaystyle p _ { \\mathbf { D } } = \\frac { 1 } { n ^ { 2 } } \\sum _ { i , j } \\left. f _ { 7 } ( D _ { i j } ) - 0 . 5 \\right. } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Then, the output of the component with confidence scaling is ",
+        "bbox": [
+            173,
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+            575,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/53af49ee534a302b9679dcb5bea5e590645d46ffb83dc1cf2115eda86e98d6ae.jpg",
+        "text": "$$\n\\tilde { \\mathbf { o } } = f _ { 8 } ( p _ { \\mathbf { a } } + p _ { \\mathbf { D } } ) \\cdot \\mathbf { o }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In summary, we only used diffentiable operations to deduplicate object proposals and obtain a feature vector that represents the predicted count. This allows easy integration into any model with soft attention, enabling a model to count from an attention map. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 TOY TASK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "First, we design a simple toy task to evaluate counting ability. This dataset is intended to only evaluate the performance of counting; thus, we skip any processing steps that are not directly related such as the processing of an input image. Samples from this dataset are given in Appendix D ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The classification task is to predict an integer count $\\hat { c }$ of true objects, uniformly drawn from 0 to 10 inclusive, from a set of bounding boxes and the associated attention weights. 10 square bounding boxes with side length $l \\in ( 0 , 1 \\bar { ] }$ are placed in a square image with unit side length. The $\\mathbf { X }$ and y coordinates of their top left corners are uniformly drawn from $U ( 0 , 1 - l )$ so that the boxes do not extend beyond the image border. $l$ is used to control the overlapping of bounding boxes: a larger $l$ leads to the fixed number of objects to be more tightly packed, increasing the chance of overlaps. $\\hat { c }$ number of these boxes are randomly chosen to be true bounding boxes. The score of a bounding box is the maximum IoU overlap of it with any true bounding box. Then, the attention weight is a linear interpolation between the score and a noise value drawn from $U ( 0 , 1 )$ , with $q \\in [ 0 , 1 ]$ controlling this trade-off. $q$ is the attention noise parameter: when $q$ is 0, there is no noise and when $q$ is 1, there is no signal. Increasing $q$ also indirectly simulates imprecise placements of bounding boxes. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare the counting component against a simple baseline that simply sums the attention weights and turns the sum into a feature vector with Equation 8. Both models are followed by a linear projection to the classes 0 to 10 inclusive and a softmax activation. They are trained with crossentropy loss for 1000 iterations using Adam (Kingma & Ba, 2015) with a learning rate of 0.01 and a batch size of 1024. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1.1 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            766,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results of varying $l$ while keeping $q$ fixed at various values and vice versa are shown in Figure 4. Regardless of $l$ and $q$ , the counting component performs better than the baseline in most cases, often significantly so. Particularly when the noise is low, the component can deal with high values for $l$ very successfully, showing that it accomplishes the goal of increased robustness to overlapping proposals. The component also handles moderate noise levels decently as long as the overlaps are limited. The performance when both $l$ and $q$ are high is closely matched by the baseline, likely due to the high difficulty of those parametrizations leaving little information to extract in the first place. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We can also look at the shape of the activation functions themselves, shown in Figure 5 and Appendix C, to understand how the behaviour changes with varying dataset parameters. For simplicity, we limit our description to the two easiest-to-interpret functions: $f _ { 1 }$ for the attention weights and $f _ { 2 }$ for the bounding box distances. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3fb48560d3ebed9fd32e0c830572ff36757f872f81d3938700678967ef165a4e.jpg",
+        "image_caption": [
+            "Figure 4: Accuracies on the toy task as side length $l$ and noise $q$ are varied in 0.01 step sizes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/21e53377d411fcb31ec0d56dc4ccce342dbc7c34ca06a323a7b0659898b842de.jpg",
+        "image_caption": [
+            "Figure 5: Shapes of trained activation functions $f _ { 1 }$ (attention weights) and $f _ { 2 }$ (bounding box distances) for varying bounding box side lengths (left) or the noise (right) in the dataset, varied in 0.01 step sizes. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When increasing the side length, the height of the “step” in $f _ { 1 }$ decreases to compensate for the generally greater degree of overlapping bounding boxes. A similar effect is seen with $f _ { 2 }$ : it varies over requiring a high pairwise distance when $l$ is low – when partial overlaps are most likely spurious – and considering small distances enough for proposals to be considered different when $l$ is high. At the highest values for $l$ , there is little signal in the overlaps left since everything overlaps with everything, which explains why $f _ { 2 }$ returns to its default linear initialization for those parameters. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When varying the amount of noise, without noise $f _ { 1 }$ resembles a step function where the step starts close to $x = 1$ and takes a value of close to 1 after the step. Since a true proposal will always have a weight of 1 when there is no noise, anything below this can be safely zeroed out. With increasing noise, this step moves away from 1 for both $x$ and $f _ { 1 } ( x )$ , capturing the uncertainty when a bounding box belongs to a true object. With lower $q$ , $f _ { 2 }$ considers a pair of proposals to be distinct for lower distances, whereas with higher $q$ , $f _ { 2 }$ follows a more sigmoidal shape. This can be explained by the model taking the increased uncertainty of the precise bounding box placements into account by requiring higher distances for proposals to be considered completely different. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 VQA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            736,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "VQA v2 (Goyal et al., 2017) is the updated version of the VQA v1 dataset (Antol et al., 2015) where greater care has been taken to reduce dataset biases through balanced pairs: for each question, a pair of images is identified where the answer to that question differs. The standard accuracy metric on this dataset accounts for disagreements in human answers by averaging $\\mathrm { m i n } ( \\textstyle { \\frac { 1 } { 3 } }$ agreeing, 1) over all 10-choose-9 subsets of human answers, where agreeing is the number of human answers that agree with the given answer. This can be shown to be equal to $\\operatorname* { m i n } ( 0 . 3 a g r e e i n g , 1 )$ without averaging. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use an improved version of the strong VQA baseline by Kazemi & Elqursh (2017) as baseline model (details in Appendix B). We have not performed any tuning of this baseline to maximize the performance difference between it and the baseline with counting module. To augment this model with the counting component, we extract the attention weights of the first attention glimpse (there are two in the baseline) before softmax normalization, and feed them into the counting component after applying a logistic function. Since object proposal features from Anderson et al. (2017) vary from 10 to 100 per image, a natural choice for the number of top- $^ n$ proposals to use is 10. The output of the component is linearly projected into the same space as the hidden layer of the classifier, followed by ReLU activation, batch normalization, and addition with the features in the hidden layer. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c12ac0860ac18470491eb64dfeb797a85057704bc04f85b9dc1e0148c4e3864d.jpg",
+        "table_caption": [
+            "Table 1: Results on VQA v2 of the top models along with our results. Entries marked with (Ens.) are ensembles of models. At the time of writing, our model with the counting module places third among all entries. All models listed here use object proposal features and are trained on the training and validation sets. The top-performing ensemble models use additional pre-trained word embeddings, which we do not use. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">VQA v2 test-dev</td><td colspan=\"4\">VQA v2 test</td></tr><tr><td>Model</td><td>Yes/No</td><td>Number</td><td>Other</td><td>All</td><td>Yes/No</td><td>Number</td><td>Other</td><td>All</td></tr><tr><td>Teney et al. (2017)</td><td>81.82</td><td>44.21</td><td>56.05</td><td>65.32</td><td>82.20</td><td>43.90</td><td>56.26</td><td>65.67</td></tr><tr><td>Teney et al. (2017) (Ens.)</td><td>86.08</td><td>48.99</td><td>60.80</td><td>69.87</td><td>86.60</td><td>48.64</td><td>61.15</td><td>70.34</td></tr><tr><td>Zhou et al. (2017)</td><td>84.27</td><td>49.56</td><td>59.89</td><td>68.76</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Zhou et al. (2017) (Ens.)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>86.65</td><td>51.13</td><td>61.75</td><td>70.92</td></tr><tr><td>Baseline</td><td>82.98</td><td>46.88</td><td>58.99</td><td>67.50</td><td>83.21</td><td>46.60</td><td>59.20</td><td>67.78</td></tr><tr><td>+ counting module</td><td>83.14</td><td>51.62</td><td>58.97</td><td>68.09</td><td>83.56</td><td>51.39</td><td>59.11</td><td>68.41</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a485d8326b1c17cd1b9585e152972b482af0f4769cb7d35c1191a8d47936aef.jpg",
+        "table_caption": [
+            "Table 2: Results on the VQA v2 validation set with models trained only on the training set. Reported are the mean accuracies and sample standard deviations $( \\pm )$ over 4 random initializations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">VQA accuracy</td><td colspan=\"3\">Balanced pair accuracy</td></tr><tr><td>Model</td><td>Number</td><td>Count</td><td>All</td><td>Number</td><td>Count</td><td>All</td></tr><tr><td>Baseline</td><td>44.83±0.2</td><td>51.69±0.2</td><td>64.80±0.0</td><td>17.34±0.2</td><td>20.02±0.2</td><td>36.44±0.1</td></tr><tr><td>+ NMS</td><td>44.60±0.1</td><td>51.41±0.1</td><td>64.80±0.1</td><td>17.06±0.1</td><td>19.72±0.1</td><td>36.44±0.2</td></tr><tr><td> + counting module</td><td>49.36±0.1</td><td>57.03±0.0</td><td>65.42±0.1</td><td>23.10±0.2</td><td>26.63±0.2</td><td>37.19±0.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.2.1 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the results on the official VQA v2 leaderboard. The baseline with our component has a significantly higher accuracy on number questions without compromising accuracy on other categories compared to the baseline result. Despite our single-model baseline being substantially worse than the state-of-the-art, by simply adding the counting component we outperform even the 8-model ensemble in Zhou et al. (2017) on the number category. We expect further improvements in number accuracy when incorporating their techniques to improve the quality of attention weights, especially since the current state-of-the-art models suffer from the problems with counting that we mention in section 3. Some qualitative examples of inputs and activations within the counting component are shown in Appendix E. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "We also evaluate our models on the validation set of VQA v2, shown in Table 2. This allows us to consider only the counting questions within number questions, since number questions include questions such as ”what time is it?” as well. We treat any question starting with the words ”how many” as a counting question. As we expect, the benefit of using the counting module on the counting question subset is higher than on number questions in general. Additionally, we try an approach where we simply replace the counting module with NMS, using the average of the attention glimpses as scoring, and one-hot encoding the number of proposals left. The NMS-based approach, using an IoU threshold of 0.5 and no score thresholding based on validation set performance, does not improve on the baseline, which suggests that the piecewise gradient of NMS is a major problem for learning to count in VQA and that conversely, there is a substantial benefit to being able to differentiate through the counting module. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Additionally, we can evaluate the accuracy over balanced pairs as proposed by Teney et al. (2017): the ratio of balanced pairs on which the VQA accuracy for both questions is 1.0. This is a much more difficult metric, since it requires the model to find the subtle details between images instead of being able to rely on question biases in the dataset. First, notice how all balanced pair accuracies are greatly reduced compared to their respective VQA accuracy. More importantly, the absolute accuracy improvement of the counting module is still fully present with the more challenging metric, which is further evidence that the component can properly count rather than simply fitting better to dataset biases. ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "When looking at the activation functions of the trained model, shown in Figure 9, we find that some characteristics of them are shared with high-noise parametrizations of the toy dataset. This suggests that the current attention mechanisms and object proposal network are still very inaccurate, which explains the perhaps small-seeming increase in counting performance. This provides further evidence that the balanced pair accuracy is maybe a more reflective measure of how well current VQA models perform than the overall VQA accuracies of over $70 \\%$ of the current top models. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "After understanding why VQA models struggle to count, we designed a counting component that alleviates this problem through differentiable bounding box deduplication. The component can readily be used alongside any future improvements in VQA models, as long as they still use soft attention as all current top models on VQA v2 do. It has uses outside of VQA as well: for many counting tasks, it can allow an object-proposal-based approach to work without ground-truth objects available as long as there is a – possibly learned – per-proposal scoring (for example using a classification score) and a notion of how dissimilar a pair of proposals are. Since each step in the component has a clear purpose and interpretation, the learned weights of the activation functions are also interpretable. The design of the counting component is an example showing how by encoding inductive biases into a deep learning model, challenging problems such as counting of arbitrary objects can be approached when only relatively little supervisory information is available. ",
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+    },
+    {
+        "type": "text",
+        "text": "For future research, it should be kept in mind that VQA v2 requires a versatile skill set that current models do not have. To make progress on this dataset, we advocate focusing on understanding of what the current shortcomings of models are and finding ways to mitigate them. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A PIECEWISE LINEAR ACTIVATION FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, the interval $[ 0 , 1 ]$ is split into $d$ equal size intervals. Each contains a line segment that is connected to the neighboring line segments at the boundaries of the intervals. These line segments form the shape of the activation function. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For each function $f _ { k }$ , there are $d$ weights $w _ { k 1 } , \\ldots , w _ { k d }$ , where the weight $w _ { k i }$ is the gradient for the interval $[ \\textstyle { \\frac { i - 1 } { d } } , \\textstyle { \\frac { i } { d } } )$ . We arbitrarily fix $d$ to be 16 in this paper, observing no significant difference when changing it to 8 and 32 in preliminary experiments. All $w _ { k i }$ are enforced to be non-negative by always using the absolute value of them, which yields the monotonicity property. Dividing the weights by $\\Sigma _ { m } ^ { d } \\mid w _ { k m } \\mid$ yields the property that $f ( 1 ) = 1$ . The function can be written as ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/658f47b9282626a8b3087bedbdc789dc0ff6db78e6d7364432393fe1366f666f.jpg",
+        "text": "$$\nf _ { k } ( x ) = \\sum _ { i = 1 } ^ { d } \\operatorname* { m a x } ( 0 , 1 - | d x - i | ) \\frac { \\sum _ { j = 1 } ^ { i } | w _ { k j } | } { \\sum _ { m = 1 } ^ { d } | w _ { k m } | }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In essence, the max term selects the two nearest boundary values of an interval, which are normalized cumulative sums over the $w _ { k }$ weights, and linearly interpolates between the two. This approach is similar to the subgradient approach by Jaderberg et al. (2015) to make sampling from indices differentiable. All $w _ { k i }$ are initialized to 1, which makes the functions linear on initialization. When applying $f _ { k } ( { \\bf x } )$ to a vector-valued input $\\mathbf { x }$ , it is assumed to be applied elementwise. By caching the normalized cumulative sum $\\begin{array} { r } { \\sum _ { j } ^ { i } | w _ { k j } | / \\sum _ { m } ^ { d } | w _ { k m } | } \\end{array}$ , this function has linear time complexity with respect to $d$ and is efficiently implementable on GPUs. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Extensions to this are possible through Deep Lattice Networks (You et al., 2017), which preserve monotonicity across several nonlinear neural network layers. They would allow A and $\\mathbf { D }$ to be combined in more sophisticated ways beyond an elementwise product, possibly improving counting performance as long as the property of the range lying within $[ 0 , 1 ]$ is still enforced in some way. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B BASELINE ARCHITECTURE",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This model is based on the work of Kazemi & Elqursh (2017), who outperformed most previous VQA models on the VQA v1 dataset with a simple baseline architecture. We adapt the model to the VQA v2 dataset and make various tweaks that improve validation accuracy slightly. The architecture is illustrated in Figure 6. Details not mentioned here can be assumed to be the same as in their paper. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The most significant change that we make is the use of object proposal features by Anderson et al. (2017) as previously mentioned. The following tweaks were made without considering the performance impact on the counting component; only the validation accuracy of the baseline was optimized. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To fuse vision features $\\mathbf { x }$ and question features y, the baseline concatenates and linearly projects them, followed by a ReLU activation. This is equivalent to ReLU $( \\mathbf { W } _ { x } \\mathbf { x } + \\mathbf { W } _ { y } \\mathbf { y } )$ . We include an additional term that measures how different the projected $\\mathbf { x }$ is from the projected y, changing the fusion mechanism to $\\mathbf { x } \\odot \\mathbf { y } = \\operatorname { R e L U } ( \\mathbf { W } _ { x } \\mathbf { x } + \\mathbf { W } _ { y } \\mathbf { y } ) - ( \\mathbf { W } _ { x } \\mathbf { x } - \\mathbf { W } _ { y } \\mathbf { y } ) ^ { 2 }$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The LSTM (Hochreiter & Schmidhuber, 1997) for question encoding is replaced with a GRU (Cho et al., 2014) with the same hidden size with dynamic per-example unrolling instead of a fixed 14 words per question. We apply batch normalization (Ioffe & Szegedy, 2015) before the last linear projection in the classifier to the 3000 classes. The learning rate is increased from 0.001 to 0.0015 and the batch size is doubled to 256. The model is trained for 100 epochs (1697 iterations per epoch to train on the training set, 2517 iterations per epoch to train on both training and validation sets) instead of 100,000 iterations, roughly in line with the doubling of dataset size when going from VQA v1 to VQA v2. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Note that this single-model baseline is regularized with dropout (Srivastava et al., 2014), while the other current top models skip this and rely on ensembling to reduce overfitting. This explains why our single-model baseline outperforms most single-model results of the state-of-the-art models. We found ensembling of the regularized baseline to provide a much smaller benefit in preliminary experiments compared to the results of ensembling unregularized networks reported in Teney et al. (2017). ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/490c1f6c072e0a062d39ff4d6cc97980d0cf6261d8fcaf1d9d6a6f9cad6001a1.jpg",
+        "image_caption": [
+            "Figure 6: Schematic view of a model using our counting component. The modifications made to the baseline model when including the counting component are marked in red. Blue blocks mark components with trainable parameters, gray blocks mark components without trainable parameters. White $\\textsuperscript { \\textregistered }$ mark linear layers, either linear projections or convolutions with a spatial size of 1 depending on the context. Dropout with drop probability 0.5 is applied before the GRU and every $\\textsuperscript { \\textregistered }$ , except before the $\\textsuperscript { \\textregistered }$ after the counting component. $\\diamond$ stands for the fusion function we define in Appendix B, BN stands for batch normalization, $\\sigma$ stands for a logistic, and Embedding is a word embedding that has been fed through a tanh function. The two glimpses of the attention mechanism are represented with the two lines exiting the $\\textsuperscript { \\textregistered }$ . Note that one of the two glimpses is shared with the counting component. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
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+    {
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+        "img_path": "images/07d2e4d6eedacaf38bf7ab5fb51fab2d11a25b373c83069db69f688b7b40cae0.jpg",
+        "image_caption": [
+            "Figure 7: Shape of activation functions as $l$ is varied for $q = 0 . 5$ on the toy dataset. Each line shows the shape of the activation function when $l$ is set to the value associated to its color. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/e630ceffcd3a5654e3eb6233016450532e537c9e33e24a644459ee0f3beb38ae.jpg",
+        "image_caption": [
+            "Figure 8: Shape of activation functions as $q$ is varied for $l = 0 . 5$ on the toy dataset. Each line shows the shape of the activation function when $q$ is set to the value associated to its color. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 14
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+    {
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+        "img_path": "images/0c0fbd90d5fdb814b3a5f433d4b3e097fa2287a734697d8678539536c723c7d2.jpg",
+        "image_caption": [
+            "Figure 9: Shape of activation functions for a model trained on the train and validation sets of VQA v2 (thick black), compared against the shapes when parametrizing the toy dataset with $q$ around 0.4 (green), 0.7 (orange), or 1.0 (red) with fixed $l = 0 . 2$ . Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/82a95f366fb3f29067d4526bec9b7457d9cdecee732615090ede2cab04ee24a8.jpg",
+        "image_caption": [
+            "Figure 10: Example toy dataset data for varying bounding box side lengths $l$ and noise $q$ . The ground truth column shows bounding boxes of randomly placed true objects (blue) and of irrelevant objects (red). The data column visualizes the samples that are actually used as input (dark blues represent weights close to 1, dark reds represent weights close to 0, lighter colors represent weights closer to 0.5). The weight of the ith bounding box $b _ { i }$ is defined as $a _ { i } = ( 1 - q )$ score $+ q z$ where the score is the maximum overlap of $b _ { i }$ with any true bounding box or 0 if there are no true bounding boxes and $z$ is drawn from $U ( 0 , 1 )$ . Note how this turns red bounding boxes that overlap a lot with a blue bounding box in the ground truth column into a blue bounding box in the data column, which simulates the duplicate proposal that we have to deal with. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            215,
+            142,
+            766,
+            787
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f649731cb7ae201b34fb60357ccbb8c13f0f0e0a6906d1cd4d0372f1be0a82e.jpg",
+        "image_caption": [
+            "Figure 11: Selection of validation images with overlaid bounding boxes, values of the attention matrix A, distance matrix D, and the resulting count matrix C. White entries represent values close to 1, black entries represent values close to 0. The count $c$ is the usual square root of the sum over the elements of C. Notice how particularly in the third example, A clearly contains more rows/columns with high activations than there are actual objects (a sign of overlapping bounding boxes) and the counting module successfully removes intra- and inter-object edges to arrive at the correct prediction regardless. The prediction is not necessarily – though often is – the rounded value of $c$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            127,
+            810,
+            818
+        ],
+        "page_idx": 16
+    }
+]

parse/train/Bkxonh4Ywr/Bkxonh4Ywr.md

@@ -0,0 +1,284 @@
+# LOCALIZING AND AMORTIZING: EFFICIENT INFERENCE FOR GAUSSIAN PROCESSES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The inference of Gaussian Processes concerns the distribution of the underlying function given observed data points. GP inference based on local ranges of data points is able to capture fine-scale correlations and allow fine-grained decomposition of the computation. Following this direction, we propose a new inference model that considers the correlations and observations of the $K$ nearest neighbors for the inference at a data point. Compared with previous works, we also eliminate the data ordering prerequisite to simplify the inference process. Additionally, the inference task is decomposed to small subtasks with several technique innovations, making our model well suits the stochastic optimization. Since the decomposed small subtasks have the same structure, we further speed up the inference procedure with amortized inference. Our model runs efficiently and achieves good performances on several benchmark tasks.
+
+# 1 INTRODUCTION
+
+Gaussian processes (GP) (Rasmussen & Williams, 2006) are flexible non-parametric models with a wide range of applications. GP poses a Gaussian prior over function values f and assumes observations y are generated independently given f. GP inference considers the calculation of the posterior of these function values (Matthews et al., 2016) given observations, namely $p ( \mathbf { f } | \mathbf { y } )$ . Direct computation of the posterior is often intractable on large datasets, motivating people to consider its approximations. Variational inference (Jordan et al., 1999; Blei et al., 2017) for GP (Rasmussen & Williams, 2006) has achieved great successes recently. Variational inference constructs a variational distribution, which is usually a multivariate Gaussian distribution, to approximate the posterior. The approximation is done by minimizing the KL divergence from the posterior to the variational distribution (Blei et al., 2017). The variational distribution is often constructed with some special structures to reduce the number of variational parameters and speed up the computation.
+
+Inducing-point methods (Quinonero-Candela & Rasmussen, 2005; Titsias, 2009; Hensman et al., ˜ 2013; 2015) define variational distributions on a small number $M$ of inducing points and then derive the distribution of non-inducing points conditioned on these inducing points. Inducing points summarize the entire posterior distribution, and their number $M$ balances the computational cost and the quality of the approximation. Inducing-point methods are further improved in several directions, such as generic inference for non-Gaussian likelihoods (Sheth et al., 2015; Dezfouli & Bonilla, 2015; Krauth et al., 2016; Hensman et al., 2015), inter-domain and subspace inducing points (Hensman et al., 2017; Panos et al., 2018), and decoupled approximation with two different sets of inducing points (Cheng & Boots, 2017; Salimbeni et al., 2018). Burt et al. (2019) provide theoretical analysis to show that a relatively small $M$ is sufficient to produce a reliable variational approximation when the input dimension is low.
+
+While inducing-point methods capture global correlations among data points through inducing points, inference methods based on local neighbors focus more on correlation structures at local scales. These methods consider only local-range dependencies to save computation because localrange correlations are often much stronger than distant ones. Nguyen-Tuong et al. (2009); Park & Apley (2018) partition the input space into subregions, fit local models over subregions and then stitch local models into one. Other works examine neighbors of each data point directly. Gramacy & Apley (2015) investigate the properties of GP predictive equation and construct a local predictive approximator. Covariance tapering (Furrer et al., 2006; Kaufman et al., 2008) gains computational efficiency by constructing a sparse correlation matrix with zero correlations between distant data points. Methods based on Vecchia’s approximation (Vecchia, 1988; Datta et al., 2016; Liu & Liu, 2019; Finley et al., 2019) decompose the joint probability of data points into conditionals according to a data ordering and then neglect far data points that are conditioned on.
+
+Recently, Liu & Liu (2019) propose the AIGP method, which extends the idea of local inference to GP models with non-Gaussian likelihoods. They use directed graphical models to approximate both the prior and the posterior. With this construction, the inference task decomposes into local inference subtasks, then they introduce amortized inference and use inference networks to identify solutions to these subtasks (Kingma & Welling, 2013; Dai et al., 2015; Miao et al., 2016). Amortization reduces the number of optimization parameters and greatly speeds up the inference procedure. However, this method has two drawbacks. First, the inference at a data point considers a few of its nearest neighbors but not all of them; therefore, it may lose some important correlations. Second, it depends on a data ordering. A bad ordering often deteriorates the performance, but it is hard to guard against such a bad situation. There are no easy fixes of the two issues, because all these designs in AIGP serve the purpose of decomposition.
+
+In this work, we propose a new GP inference method, Localized and Amortized Inference based on Nearest neighbors (LAIN). LAIN considers $K$ nearest neighbors for the inference at each data point. Particularly, LAIN uses a variational distribution whose covariance is parameterized by a sparse decomposition. The decomposition focuses on the correlations between every data point and its $K$ nearest neighbors 1. LAIN also eliminates the need for a data ordering. These nice properties come after several technical innovations. First, the new distribution does not admit a decomposable entropy calculation. We overcome this difficulty by using a decomposable lower bound of the entropy (Ranganath et al., 2016; Louizos & Welling, 2017). Second, to decompose the logarithm of the prior, AIGP and previous methods use a directed graphical model as an approximation of the prior. We follow this idea, but we consider all possible orderings of data points and collapse them to local combinations, making the computation manageable. With these techniques, LAIN still decomposes the inference task into subtasks, so amortized inference can apply. It is worthing noting that subtasks in LAIN are generated from the same mechanism while those in AIGP are not. We argue that subtasks sharing the same “distribution” are more appropriate for amortization.
+
+Our empirical evaluations show that the LAIN method outperforms baseline methods including AIGP in several learning tasks. Our investigation also indicates that LAIN can achieve decent performance even only a few neighbors are considered.
+
+# 2 BACKGROUND
+
+Gaussian Processes. Suppose we have a dataset containing a feature matrix $\mathbf { X } = ( \mathbf { x } _ { i } ) _ { i = 1 } ^ { N }$ and observations $\mathbf { y } = \mathbf { \Psi } ( y _ { i } ) _ { i = 1 } ^ { N }$ . We assume there is a latent function $f$ that generates $y _ { i }$ from $\mathbf { x } _ { i }$ for each $i$ . Particularly, each $y _ { i }$ is generated by a likelihood model $p ( y _ { i } | f _ { i } )$ with $f _ { i } = f ( \mathbf { x } _ { i } )$ . Denote $\mathbf { f } = ( f _ { i } ) _ { i = 1 } ^ { N }$ , then $\begin{array} { r } { p ( \mathbf { y } | \mathbf { f } ) = \prod _ { i = 1 } ^ { N } p ( y _ { i } | f _ { i } ) } \end{array}$ . The likelihood $p ( y _ { i } | f _ { i } )$ can be very general – here we only assume that $\log p ( y _ { i } | f _ { i } )$ is differentiable with respect to $f _ { i }$ . This mild assumption allows a wide range of data distributions. For example, if $y _ { i }$ is binary, $p ( y _ { i } | f _ { i } )$ is a Bernoulli distribution with $f _ { i }$ as the logit.
+
+We put a GP prior with a mean function $\nu ( \cdot )$ and a kernel function $\kappa ( \cdot , \cdot )$ over the latent function $f$ . The kernel function encodes the prior knowledge of the smoothness of $f$ . One commonly used kernel function is the Radial Basis Function (RBF) kernel, $\kappa ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) = r ^ { 2 } \exp ( - 0 . 5 \| \mathbf { x } _ { i } - \mathbf { x } _ { j } \| _ { 2 } ^ { 2 } / \sigma ^ { 2 } )$ , with $r$ and $\sigma$ as parameters. With this prior, function values in f follow a multivariate Gaussian, f $\sim$ $\mathcal { N } (  { \boldsymbol \nu } ,  { \Sigma } )$ , with the mean ${ \pmb { \nu } } = ( \nu ( x _ { i } ) ) _ { i = 1 } ^ { \mathbf { \hat { N } } }$ and the covariance matrix $\pmb { \Sigma }$ with $\Sigma _ { i , j } = \kappa ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) \ \forall i , j$ .
+
+GP inference concerns the calculation of the posterior $p ( \mathbf { f } | \mathbf { y } )$ (Matthews et al., 2016), from which we can infer the function value $f _ { \star }$ for any new input $\mathbf { x } _ { \star }$ with integral $\begin{array} { r } { \int _ { \mathbf { f } } p ( f _ { \star } | \mathbf { f } ) p ( \mathbf { f } | \mathbf { y } ) \mathrm { d } \mathbf { f } } \end{array}$ . The posterior $p ( \mathbf { f } | \mathbf { y } )$ is generally not tractable, so we appeal to approximate inference.
+
+Variational Inference for GP. Variational inference approximates the posterior $p ( \mathbf { f } | \mathbf { y } )$ with a variational distribution $q ( \mathbf { f } )$ , which is defined as a multivariate Gaussian distribution, $q ( \mathbf { f } ) \sim \mathcal { N } ( \mu , \mathbf { V } )$
+
+![](images/14ddfde5436b3c8fabab96a6fd00f543492efb6a505615591341ebaa4874b1ef.jpg)  
+Figure 1: The structure of the variational distribution. The left box shows the amortization, which fits $\mu _ { i }$ -s and $R _ { i j }$ -s from their related prior kernel and observations. The right part shows the generation process of $f _ { i }$ -s.
+
+The inference is carried out by maximizing the Evidence Lower BOund (ELBO) with respect to $q ( \mathbf { f } )$ (Blei et al., 2017).
+
+$$
+\log p ( \mathbf { y } | \mathbf { X } ) \geq \operatorname* { m a x } _ { q ( \mathbf { f } ) } \underbrace { { \mathbb { E } } _ { q } \left[ \log p ( \mathbf { y } | \mathbf { f } ) \right] } _ { L _ { e l l } } + \underbrace { { \mathbb { E } } _ { q } \left[ \log p ( \mathbf { f } ) \right] } _ { L _ { c r o s s } } \underbrace { - { \mathbb { E } } _ { q } \left[ \log q ( \mathbf { f } ) \right] } _ { L _ { e n t } }
+$$
+
+Here we name the three terms in the ELBO for easy reference later. Typically the ELBO is maximized by gradient-based optimization, preferably stochastic gradient optimization when $N$ is large. Direct optimization of the ELBO is challenging, since the kernel matrix $\pmb { \Sigma }$ and the variational covariance $\mathbf { V }$ are both large and have size $N \times N$ .
+
+Inducing-point methods define $\begin{array} { r } { q ( \mathbf { f } ) ~ = ~ \int _ { \mathbf { f } _ { I } } q ( \mathbf { f } _ { I } ) p ( \mathbf { f } | \mathbf { f } _ { I } ) ~ \mathrm { d } \mathbf { f } _ { I } } \end{array}$ , where $q ( \mathbf { f } _ { I } )$ is the distribution over inducing points $I$ , and $p ( \mathbf { f } | \mathbf { f } _ { I } )$ is derived from the prior. The computation is reduced mainly because only the small distribution $q ( \mathbf { f } _ { I } )$ is optimized, while the conditional $p ( \mathbf { f } | \mathbf { f } _ { I } )$ is fixed when the prior is given.
+
+AIGP parameterizes $\mathbf { V }$ by a Cholesky decomposition, $\mathbf { V } = \mathbf { L L } ^ { \top }$ . Here $\mathbf { L }$ is a sparse lower triangular matrix, and each row of $\mathbf { L }$ has at most $K$ non-zero entries. AIGP uses a triangular $\mathbf { L }$ for easy entropy computation. It also approximates $\log p ( \mathbf { f } )$ with a directed graphical model. Both the lower triangular matrix $\mathbf { L }$ and the directed graph require an ordering of data points.
+
+# 3 METHOD
+
+# 3.1 THE VARIATIONAL DISTRIBUTION
+
+Following previous works, we also define the variational distribution $q ( \mathbf { f } )$ to be a multivariate Gaussian $\mathcal { N } ( \mu , \mathbf { V } )$ . We parameterize $\mathbf { V } = \mathbf { R } \mathbf { R } ^ { \top } + \delta ^ { 2 } \mathbf { I }$ with $\mathbf { R }$ being a sparse matrix and $\delta$ being a small constant. Note that we do not require $\mathbf { R }$ to be triangular. The sparse pattern of $\mathbf { R }$ is decided by the nearest neighbors: $R _ { i j } \neq 0$ only when $j \in n ( i )$ . Here $n ( i )$ is the neighbor set containing data points that have the largest covariance with $i$ in the prior (by definition $n ( i )$ includes $i$ ). In this work, we fix the size of $n ( i )$ to be $K$ , though our derivation works for varied sizes of $n ( i )$ . The row $\mathbf { R } _ { i }$ can be viewed as a representation of $f _ { i }$ in the variational distribution: $\mathbf { R } _ { i }$ informs $f _ { i }$ ’s correlation with other function values, just like a word embedding informs its relation with other words (Mikolov et al., 2013).
+
+Efficient sampling from the marginal is critical for the decomposition of the ELBO later. Owing to the sparse decomposition of the covariance matrix, we can cheaply draw marginal samples for an $f _ { i }$ from $q ( \mathbf { f } )$ with a linear transformation of white noise. The sampling scheme is shown in (2) and pictured in the right part of Figure 1.
+
+$$
+f _ { i } = \mu _ { i } + { \bf R } _ { i } \epsilon + \delta \xi = \mu _ { i } + { \bf R } _ { i , n ( i ) } \epsilon _ { n ( i ) } + \delta \xi , \epsilon \sim \mathcal { N } ( { \bf 0 , I } ) , \xi \sim \mathcal { N } ( { \bf 0 , 1 } ) .
+$$
+
+The constructed distribution $q ( \mathbf { f } )$ well approximates the strong correlations in the prior. From (2), $f _ { i }$ and $f _ { j }$ correlate in $q ( \mathbf { f } )$ by sharing noise entries in $n ( i ) \cap n ( j )$ when the intersection is not empty. In this case, either $f _ { i }$ neighbors $f _ { j }$ , or $f _ { j }$ neighbors $f _ { i }$ , or $f _ { i } , f _ { j }$ share common neighbors. When the neighbor sets are large enough, most strong correlations will be approximated by some non-zero entries in $\mathbf { V }$ .
+
+# 3.2 OPTIMIZATION OF THE ELBO
+
+We optimize the ELBO in (1) to find a good $q ( \mathbf { f } )$ to approximate the GP posterior. To apply stochastic optimization, we will decompose the three terms in the ELBO. We mainly consider the decomposition of $L _ { c r o s s }$ and $L _ { e n t }$ , as the decomposition of $L _ { e l l }$ is easy.
+
+We first decompose the cross entropy $L _ { c r o s s }$ . By convention, the GP prior has a zero mean. Though there is a closed-form calculation of $L _ { c r o s s }$ with both $q ( \mathbf { f } )$ and $p ( \mathbf { f } )$ being multivariate Gaussian, it involves expensive calculations of $\operatorname* { d e t } ( \pmb { \Sigma } )$ and $\Sigma ^ { - 1 }$ . Previous works approximate the prior with Vecchia’s method for easy decomposition and good approximation (Vecchia, 1988; Stein et al., 2004; Datta et al., 2016; Liu & Liu, 2019; Finley et al., 2019). The idea is to build a directed graphical model and approximate $\begin{array} { r } { p ( \mathbf { f } ) \approx \prod _ { i = 1 } ^ { N } p ( f _ { i } | f _ { \alpha ( i ) } ) } \end{array}$ with $\alpha ( i )$ being a small parent set of $i$ . In the original work, Vecchia (1988) first set an order to data points and then choose $\alpha ( i )$ as the $K$ nearest parents of $i$ . But it is not easy to guarantee a good ordering of data points (Banerjee et al., 2014; Guinness, 2018). Here we consider all possible orderings and take the average of approximations to address the data ordering concern.
+
+stimate for eac $L _ { c r o s s }$ as follows. First, we randomly saen we approximate the log-prior by $n ^ { \prime } ( i ) \subset n ( i )$ with , with $i \not \in$ $n ^ { \prime } ( i )$ $i$ $\begin{array} { r } { \log p ( \mathbf { f } ) \approx \sum _ { i = 1 } ^ { N } \log p ( f _ { i } | f _ { n ^ { \prime } ( i ) } ) } \end{array}$ conditional distribution $p ( f _ { i } | f _ { n ^ { \prime } ( i ) } )$ derived from the joint Gaussian $p ( f _ { i } , f _ { n ^ { \prime } ( i ) } )$ in the prior. Then $L _ { c r o s s }$ is estimated by a random batch of terms. The complete calculation is given as
+
+$$
+L _ { c r o s s } \approx \tilde { L } _ { c r o s s } = \frac { N } { | S | } \sum _ { i \in S } \mathbb { E } _ { q ( f _ { i } , f _ { n ^ { \prime } ( i ) } ) } \Big [ \log p ( f _ { i } | f _ { n ^ { \prime } ( i ) } ) \Big ] , \mathrm { ~ r a n d o m ~ s e t ~ } n ^ { \prime } ( i ) \subset n ( i ) .
+$$
+
+Here $S$ is a random batch of data points.
+
+Now we justify that this is an average over all data orderings. Suppose there is a data order $\pi ( \cdot )$ , such that we can define a directed graphical model over $p ( \mathbf { f } )$ by assigning every $i$ a parent set $n _ { \pi } ^ { \prime } ( i ) = \{ j : j \in n ( i ) , \pi ( j ) < \pi ( i ) \}$ . Denote $\Pi$ as all permutations of $N$ data points, with each permutation inducing a graphical model. The average of the log densities of all graphical models can be collapsed to the average computed from local neighborhoods. Denote $\Pi _ { n ( i ) }$ as permutations of indices in the set $n ( i )$ , then we have
+
+$$
+\frac { 1 } { N ! } \sum _ { \pi \in \Pi } \sum _ { i = 1 } ^ { N } \mathbb { E } _ { q ( f _ { i } , f _ { n _ { \pi } ^ { \prime } ( i ) } ) } \Big [ \log p ( f _ { i } | f _ { n _ { \pi } ^ { \prime } ( i ) } ) \Big ] = \sum _ { i = 1 } ^ { N } \frac { 1 } { K ! } \sum _ { \pi \in \Pi _ { n ( i ) } } \mathbb { E } _ { q ( f _ { i } , f _ { n _ { \pi } ^ { \prime } ( i ) } ) } \Big [ \log p ( f _ { i } | f _ { n _ { \pi } ^ { \prime } ( i ) } ) \Big ] .
+$$
+
+Here we only need to consider permutations of data points within $n ( i )$ for each $i$ . Then we obtain (3) by estimating the inner summation by a single random permutation of $n ( i )$ and the outer summation by a random batch $S$ .
+
+We then decompose the entropy $L _ { e n t }$ . The entropy of $q ( \mathbf { f } )$ requires the expensive computation of $\operatorname* { d e t } ( \mathbf { V } )$ . To circumvent this difficulty, we find a decomposable lower bound of the entropy by using an auxiliary distribution (Ranganath et al., 2016; Louizos $\&$ Welling, 2017). Note that we always prefer a lower bound of the objective in this maximization problem. With an arbitrary distribution $r ( \epsilon | \mathbf { f } )$ , a lower bound of $L _ { e n t }$ is
+
+$$
+L _ { e n t } = - \mathbb { E } _ { q } \left[ \log q ( \mathbf { f } ) \right] \geq - \mathbb { E } _ { q ( \mathbf { f } , \epsilon ) } \left[ \log q ( \mathbf { f } | \epsilon ) + \log q ( \epsilon ) - \log r ( \epsilon | \mathbf { f } ) \right] .
+$$
+
+The bound is tight when $r ( \epsilon | \mathbf { f } )$ matches $q ( \epsilon | \mathbf { f } )$ . In this work, we try to let $r ( \epsilon | \mathbf { f } )$ match $q ( \epsilon | \mathbf { f } )$ . Particularly, we set $\begin{array} { r } { r ( \epsilon | \mathbf { f } ) = \prod _ { i } q ( \epsilon _ { i } | \mathbf { f } _ { n ( i ) } ) } \end{array}$ , where the conditional $q \bigl ( \epsilon _ { i } | \mathbf { f } _ { n ( i ) } \bigr )$ is derived from the joint Gaussian distribution $q \bigl ( \epsilon _ { i } , \mathbf { f } _ { n ( i ) } \bigr )$ . Then all terms in the lower bound in (5) are Gaussian loglikelihoods and are decomposable over data points. We can then reach the estimation of the entropy lower bound with a batch of data points.
+
+$$
+L _ { e n t } \geq \tilde { L } _ { e n t } = - \frac { 1 } { 2 } \frac { N } { | S | } \sum _ { i \in S } \log \left( 1 - \mathbf { R } _ { n ( i ) , i } ^ { \top } \left( \mathbf { R } _ { n ( i ) , : } \mathbf { R } _ { n ( i ) , : } ^ { \top } \right) ^ { - 1 } \mathbf { R } _ { n ( i ) , i } \right) + c o n s t .
+$$
+
+We finally decompose the likelihood $L _ { e l l }$ . The likelihood term $\log p ( \mathbf { y } | \mathbf { f } )$ naturally decomposes because $y _ { i }$ -s are conditionally independent given $f _ { i }$ -s.
+
+$$
+L _ { e l l } = \sum _ { i = 1 } ^ { N } \mathbb { E } _ { q ( f _ { i } ) } \left[ \log p ( y _ { i } | f _ { i } ) \right] , \quad \tilde { L } _ { e l l } = \frac { N } { | S | } \sum _ { i \in S } \log p ( y _ { i } | \hat { f } _ { i } ) .
+$$
+
+Here for each term $i$ in the summation, the expectation is estimated by a Monte Carlo sample $\hat { f } _ { i }$ from $q ( f _ { i } )$ . The gradients of variational parameters are propagated through ${ \hat { f } } _ { i }$ via reparameterization (Kingma & Welling, 2013).
+
+Finally, the ELBO has a decomposable approximation $\tilde { L } _ { e l l } + \tilde { L } _ { c r o s s } + \tilde { L } _ { e n t }$ to enable efficient stochastic optimization. From the derivations above, we see the objective can be decomposed by data points. The computation for a data point only involves itself and its $K$ nearest neighbors. Therefore, each stochastic gradient calculation takes time only $O ( K ^ { 3 } )$ . There are $N ( K + \bar { 1 } )$ parameters in $\pmb { \mu }$ and $\mathbf { R }$ to optimize, so the optimization takes at least $O ( N )$ time. We further reduce the number of parameters by amortizing the cost through a shared inference model, taking advantage of the fact that the inference for each data point $i$ only needs its $K$ nearest neighbors.
+
+# 3.3 AMORTIZED INFERENCE
+
+Following AIGP, we also apply amortized inference to GP inference. Particularly, we train an inference network to identify variational parameters ${ \bf \nabla } _ { \mu _ { i } }$ and $\mathbf { R } _ { i , n ( i ) } )$ for each data point $i$ . Since node correlations at a neighborhood can be easily treated as a weighted graph, we use Graph Convolutional Networks (GCNs) (Kipf & Welling, 2017) as our inference network.
+
+A GCN takes an adjacency matrix $\mathbf { A } \in \mathbb { R } ^ { k \times k }$ of graph and the node features $\mathbf { H } ^ { ( 0 ) } \in \mathbb { R } ^ { k \times d _ { 0 } }$ as the input and then makes predictions for all graph nodes. Let $\bar { \mathbf A }$ be the normalized adjacency matrix, $\bar { \mathbf { A } } = \mathbf { D } ^ { - \frac { 1 } { 2 } } \mathbf { A } \mathbf { D } ^ { - \frac { 1 } { 2 } }$ , with $\mathbf { D }$ being the diagonal degree matrix. A GCN layer $\ell$ with the input $\mathbf { H } ^ { ( \ell - 1 ) }$ is defined by $\mathbf { H } ^ { ( \ell ) } = g _ { \ell } ( \mathbf { H } ^ { ( \ell - 1 ) } , \mathbf { A } ) : = \overset { - } { \sigma } \big ( \bar { \mathbf { A } } \mathbf { H } ^ { ( \bar { \ell } - 1 ) } \mathbf { W } ^ { ( \ell ) } \big )$ . Here $\mathbf { W } ^ { ( l ) } \in \mathbb { R } ^ { d _ { \ell - 1 } \times d _ { \ell } }$ is the weight matrix of the layer $\ell$ . $\sigma ( \cdot )$ is the activation function. An $L$ -layer GCN computes its output by $\mathbf { H } = g c n ( \mathbf { H } ^ { 0 } , \mathbf { A } ) : = g _ { L } ( \mathbf { \sigma } _ { \cdot } \dots g _ { 1 } ( \mathbf { H } ^ { 0 } , \mathbf { A } ) \dots , \mathbf { A } )$ . We use two GCNs for the inference task, $g c n _ { 1 }$ for the calculation of $\mu _ { i }$ and $g c n _ { 2 }$ for $\mathbf { R } _ { i , n ( i ) }$ :
+
+$$
+\begin{array} { r } { \mu _ { i } = { \bf a } ^ { \top } g c n _ { 1 } \left( \left[ { \bf y } _ { n \left( i \right) } , { \bf e } _ { i } \right] , { \bf \Sigma } _ { n \left( i \right) , n \left( i \right) } \right) , { \bf R } _ { i , n \left( i \right) } = g c n _ { 2 } \left( \left[ { \bf y } _ { n \left( i \right) } , { \bf e } _ { i } \right] , { \bf \Sigma } _ { { n \left( i \right) } , n \left( i \right) } \right) . } \end{array}
+$$
+
+Here we use $\Sigma _ { n ( i ) , n ( i ) }$ as the adjacency matrix and stack the observation $\mathbf { y } _ { n ( i ) }$ and the one-hot vector $\mathbf { e } _ { i }$ as the input feature. The vector $\mathbf { e } _ { i }$ indicates the element $i$ for which the inference is running for. We choose the activation $\sigma ( \cdot )$ to be ReLU for intermediate layers and identity for the last layer. The last layer of each GCN has size 1 to output a $K \times 1$ vector. a is an averaging vector with all $K$ elements as $\textstyle { \frac { 1 } { K } }$ . The dashed box in Figure 1 shows the amortization.
+
+LAIN defines an inference subtask on a data point and its nearest neighbors, while AIGP defines a subtask on a data point and its parents. Due to this difference, LAIN has two advantages. First, the inference network of LAIN uses the observations from all the $K$ nearest neighbors, while the inference network of AIGP uses observations from parents only but not children. Second, inference subtasks of LAIN are generated with the same mechanism because the nearest-neighbor relationship is homogeneous across all data points. However, the parent-child relationship in AIGP depends on the ordering of data points (e.g. the first one in the order does not have parents). As a learning model, the inference network prefers subtasks from the same “distribution”.
+
+The computational cost of GCN is $O ( K ^ { 2 } )$ by treating the network size as constant. The complexity of one gradient calculation is $O ( K ^ { 3 } )$ . The optimization procedure converges fast since it only optimizes a constant number of variational parameters. In practice, we often observe that the optimization procedure converges in less than one epoch, which is not possible for methods without amortization. Finding nearest neighbors is the only step with running time bounds to the data size, but it only needs one run and is often fast on medium to large data sizes. If the data has a very large size, we can use k-d trees for low-dimensional data and approximate algorithms (Arya et al., 1998; Datar et al., 2004) for high dimensional data.
+
+# 3.4 PREDICTION
+
+For a new data point $\mathbf { x } _ { \star }$ with its $K$ nearest neighbors $n ( \star )$ in the prior, the predictive distribution is
+
+$$
+p ( y _ { \star } | \mathbf { x } _ { \star } , \mathbf { X } , \mathbf { y } ) \approx \int _ { f _ { \star } } p ( y _ { \star } | f _ { \star } ) q ( f _ { \star } | \mathbf { x } _ { \star } , \mathbf { X } _ { n ( \star ) } , \mathbf { y } _ { n ( \star ) } ) \mathrm { d } f _ { \star } \approx \frac { 1 } { | F | } \sum _ { \widehat { f } _ { \star } \in F } p ( y _ { \star } | \widehat { f } _ { \star } ) .
+$$
+
+![](images/d66507f0b5b95720272e2d3439961525cb73c16e79e9c1099008221c027b8159.jpg)  
+Figure 2: The first two plots compare predictive distributions of full GP and LAIN with $K = 1 0$ . The right three plots show how SVGP, AIGP, and LAIN perform with a very small number of inducing points/neighbors. Data points in blue circles are not well fitted.
+
+Here $\begin{array} { r } { q ( f _ { \star } | \mathbf { x } _ { \star } , \mathbf { X } _ { n ( \star ) } , \mathbf { y } _ { n ( \star ) } ) = \int _ { \mathbf { f } _ { n ( \star ) } } p ( f _ { \star } | \mathbf { f } _ { n ( \star ) } ) q ( \mathbf { f } _ { n ( \star ) } ) \mathrm { d } \mathbf { f } _ { n ( \star ) } } \end{array}$ is a Gaussian with parameters,
+
+$$
+\begin{array} { r } { \mu _ { \star } = \mathbf { b } _ { \star } \mu _ { n ( \star ) } , \qquad \sigma _ { \star } ^ { 2 } = \Sigma _ { \star , \star } - \Sigma _ { \star , n ( \star ) } \mathbf { b } _ { \star } ^ { \top } + \mathbf { b } _ { \star } ( \mathbf { R } _ { n ( \star ) } \mathbf { R } _ { n ( \star ) } ^ { T } ) \mathbf { b } _ { \star } ^ { \top } , } \end{array}
+$$
+
+with $\mathbf { b _ { \star } } = \pmb { \Sigma } _ { \star , n ( \star ) } \pmb { \Sigma } _ { n ( \star ) , n ( \star ) } ^ { - 1 }$ . $F$ is a set of Monte Carlo samples from $q ( f _ { \star } | \mathbf { x } _ { \star } , \mathbf { X } _ { n ( \star ) } , \mathbf { y } _ { n ( \star ) } )$ . The
+
+# 4 EXPERIMENT
+
+We compare our method with five state-of-the-art methods: SVGP (Hensman et al., 2015), SAVIGP (Dezfouli & Bonilla, 2015), DGP (Cheng & Boots, 2017), VFF (Hensman et al., 2017), and AIGP (Liu & Liu, 2019). The first three methods are based on inducing points, VFF uses inter-domain inducing points, and AIGP uses local neighbors. Through all experiments, we use RBF as the default kernel, except for VFF we use Ma´tern- $\frac { 3 } { 2 }$ kernel (the code does not provide RBF kernel). We use the implementation of SVGP from GPFlow (Matthews et al., 2017), the implementation of DGP from Faust (2018), and implementations of all other algorithms from their authors.
+
+For SVGP, SAVIGP, and VFF, we vary the number of inducing points, $M \in \{ 2 0 0 , 1 0 0 0 , 2 0 0 0 \}$ , to check their performances. DGP has separate inducing points for mean approximation and those for variance approximation. We use 256 inducing points for variance approximation and vary the number of inducing points for mean approximation from 200 to 2000. We vary the number of neighbors, $K \in \{ 1 0 , 2 0 , 4 0 \}$ , for AIGP and LAIN. GCNs used in these two methods have three hidden layers with dimensions [20, 10, 1]. We randomly split each dataset into training $( 7 5 \% )$ and testing $( 2 5 \% )$ and report both the predictive performance on the test set and the inference running time. To save the space, we report results from two settings for each competing method: one setting is $M = 2 0 0$ or $K = 1 0$ , with which all methods have their fastest speed (marked by $\pmb { \mathscr { z } }$ ), and another setting giving the best predictive performance (marked by $\checkmark$ ).
+
+# 4.1 A TOY EXAMPLE
+
+In this section, we test different methods on a one-dimensional toy example studied in (Snelson & Ghahramani, 2006). The dataset contains 200 data points, shown as black dots in figure 2. We assume Gaussian likelihood in this experiment and run exact inference as the baseline. A smaller GCN (hidden dimensions [10, 5, 1]) is used in this task.
+
+The predictive mean and variance from the exact inference and LAIN with $K = 1 0$ are shown in the first two plots of Figure 2. The result of LAIN is very similar to that of exact inference, except that the mean curve of LAIN is less smooth, which does not really hurt the predictive performance.
+
+We test different methods with very small $M$ and $K$ and observe how they behave. We are likely to face this situation when we work on large datasets in high-dimensional spaces. The last three plots of Figure 2 exhibit predictive distributions of SVGP with $M = 2$ inducing points, AIGP with $K = 2$ parents, and LAIN with $K = 2$ nearest neighbors. When there are not enough inducing points, SVGP over-smooths the prediction and performs poorly for a good fraction of data points. AIGP does not have a good predictive mean either, because under a random ordering the directed graph constructed by AIGP cannot well capture neighboring relations. The predictive mean of LAIN does not deviate far from the ground-truth in the area with training instances, though the curve is rugged due to local variations.
+
+Table 1: Comparison on the eBird dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Config</td><td rowspan=1 colspan=1>Pred NLL</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>SVGP</td><td rowspan=1 colspan=1>M=2004M=1000√</td><td rowspan=1 colspan=1>1.90±.031.88±.02</td><td rowspan=1 colspan=1>107s4.5ks</td></tr><tr><td rowspan=1 colspan=1>SAVIGP</td><td rowspan=1 colspan=1>M=2004M=2000√</td><td rowspan=1 colspan=1>2.04±.031.99±.03</td><td rowspan=1 colspan=1>167s50ks</td></tr><tr><td rowspan=1 colspan=1>VFF</td><td rowspan=1 colspan=1>M=200HM=2000</td><td rowspan=1 colspan=1>1.91±.021.91±.02</td><td rowspan=1 colspan=1>1.3ks13ks</td></tr><tr><td rowspan=1 colspan=1>DGP</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>1.82±.021.80±.02</td><td rowspan=1 colspan=1>96s213s</td></tr><tr><td rowspan=1 colspan=1>AIGP</td><td rowspan=1 colspan=1>K=10  MK=20  √</td><td rowspan=1 colspan=1>1.79±.051.71±.05</td><td rowspan=1 colspan=1>45s125s</td></tr><tr><td rowspan=1 colspan=1>LAIN</td><td rowspan=1 colspan=1>K=10K=20K=40</td><td rowspan=1 colspan=1>1.69±.031.65±.031.60±.02</td><td rowspan=1 colspan=1>55s384s1.3ks</td></tr></table>
+
+Table 2: Comparison on the precipitation dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Config</td><td rowspan=1 colspan=1>Pred NLL</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>SVGP</td><td rowspan=1 colspan=1>M=200HM=2000</td><td rowspan=1 colspan=1>1.57±.031.28±.03</td><td rowspan=1 colspan=1>2.5ks42ks</td></tr><tr><td rowspan=1 colspan=1>SAVIGP</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>1.70±.021.58±.02</td><td rowspan=1 colspan=1>2.8ks50ks</td></tr><tr><td rowspan=1 colspan=1>VFF</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>1.54±.031.53±.03</td><td rowspan=1 colspan=1>9.1ks32ks</td></tr><tr><td rowspan=1 colspan=1>DGP</td><td rowspan=1 colspan=1>M=200HM=2000</td><td rowspan=1 colspan=1>1.07±.051.00±.05</td><td rowspan=1 colspan=1>402s889s</td></tr><tr><td rowspan=1 colspan=1>AIGP</td><td rowspan=1 colspan=1>K=10  4K=10  √</td><td rowspan=1 colspan=1>0.96±.030.96±.03</td><td rowspan=1 colspan=1>155s155s</td></tr><tr><td rowspan=1 colspan=1>LAIN</td><td rowspan=1 colspan=1>K=10K=20K=40</td><td rowspan=1 colspan=1>0.74±.050.72±.050.69±.04</td><td rowspan=1 colspan=1>129s903s2.3ks</td></tr></table>
+
+![](images/4e7a03685eb2a699e7d27c999ce631f8d156786ab9745e1016d46a92fbe32fdc.jpg)  
+Figure 3: ELBO trajectories of LAIN with and without inference networks.
+
+Table 3: Comparison on the MNIST dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Config</td><td rowspan=1 colspan=1>Pred NLL</td><td rowspan=1 colspan=1>Accuracy</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>SVGP</td><td rowspan=1 colspan=1>M=200王M=1000√</td><td rowspan=1 colspan=1>0.053±.0040.051±.004</td><td rowspan=1 colspan=1>98.498.5</td><td rowspan=1 colspan=1>623s23ks</td></tr><tr><td rowspan=1 colspan=1>SAVIGP</td><td rowspan=1 colspan=1>M=200HM=200√</td><td rowspan=1 colspan=1>0.339±.0080.339±.008</td><td rowspan=1 colspan=1>51.751.7</td><td rowspan=1 colspan=1>6.5ks6.5ks</td></tr><tr><td rowspan=1 colspan=1>DGP</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>0.059±.0050.052±.005</td><td rowspan=1 colspan=1>98.198.3</td><td rowspan=1 colspan=1>292s2.1ks</td></tr><tr><td rowspan=1 colspan=1>AIGP</td><td rowspan=1 colspan=1>K=10  4K=40  √</td><td rowspan=1 colspan=1>0.293±.0020.215±.003</td><td rowspan=1 colspan=1>98.098.2</td><td rowspan=1 colspan=1>3.9ks24ks</td></tr><tr><td rowspan=1 colspan=1>LAIN</td><td rowspan=1 colspan=1>K=10K=20K=40</td><td rowspan=1 colspan=1>0.053±.0030.050±.0030.051±.003</td><td rowspan=1 colspan=1>98.999.099.1</td><td rowspan=1 colspan=1>128s632s2.9ks</td></tr><tr><td rowspan=1 colspan=1>KNN</td><td rowspan=1 colspan=1>K=9K=19K=39</td><td rowspan=1 colspan=1>N.A.N.A.N.A.</td><td rowspan=1 colspan=1>98.698.397.6</td><td rowspan=1 colspan=1>24s26s28s</td></tr></table>
+
+# 4.2 BIRD ABUNDANCE ESTIMATION
+
+In this experiment, we estimate the spatial abundance of a bird species (Savannah Sparrow) using eBird dataset (Munson et al., 2015). The dataset has 14,393 observations, each of which is a reported bird count at a GPS location. We model the observed counts with GPS locations as the input. We set the likelihood to be a Poisson distribution, with rate given by $\lambda _ { i } = \exp ( f _ { i } )$ .
+
+We compare different inference methods in terms of Negative predictive Log-Likelihood (NLL, the smaller the better predictive performance). Table 1 shows the results. We can see that LAIN achieves the best predictive performance at $K = 4 0$ . Methods based on inducing points generally perform worse. In this dataset, observations have strong correlations in local areas, but inducing points are not efficient to capture the posterior at such a fine scale. In terms of running speed, LAIN is comparable to AIGP and DGP but faster than other methods. In our experiment, we have also tried to increase inducing points for DGP, but it does not improve its performance.
+
+In this experiment, we also investigate whether inference networks work correctly. We run LAIN without inference networks and optimize $\pmb { \mu }$ and $\mathbf { L }$ for the variational distribution directly. Then we compare LAIN models with and without inference networks by checking their optimization procedure. In this task, we fix hyperparameters, so the two methods solve a pure inference problem. Figure 3 is the trace plot of the negative ELBO versus training epochs. The figure shows the ELBO of the two LAIN models eventually converge to very similar values, though the ELBO without inference networks is slightly better after 50 epochs (likely due to the amortization gap). LAIN with inference networks significantly reduces the number of optimization epochs – the inference networks are well trained after only 0.01 epoch (about 100 iterations). In summary, the result indicates that inference networks can effectively identify the variational parameters using local information.
+
+# 4.3 PRECIPITATION LEVEL ESTIMATION
+
+In this task, we evaluate LAIN on a rainfall dataset. We process the precipitation dataset (Climate Data Online) and obtain the average precipitation level in May at 8,832 stations that are spatially distributed in the US. The GP inputs are GPS locations of these stations, and the observations are the average precipitation levels. We use the log-normal distribution as the likelihood, with its mean as function value $f$ from GP and variance as a hyperparameter learned from the data.
+
+Table 2 summaries the experimental results. LAIN has better predictive performance, and its running speed is comparable to or faster than other methods.
+
+We also analyze the goodness of our prior approximation since we can compute the exact $L _ { c r o s s }$ on this dataset. We compute $L _ { c r o s s }$ with the optimized $q ( \mathbf { f } )$ distribution as well as $\tilde { L } _ { c r o s s }$ . The true value $L _ { c r o s s }$ and the approximation $\tilde { L } _ { c r o s s }$ are: 5,465 versus 5,396 when $K \ : = \ : 1 0$ , 9,905 versus 8,490 when $K = 2 0$ , and 10,086 versus 9,009 when $K = 4 0$ . This result indicates that the approximation $\tilde { L } _ { c r o s s }$ is relatively accurate. Furthermore, $\tilde { L } _ { c r o s s }$ tends to be smaller than the true value and can be considered as a lower bound in such cases.
+
+# 4.4 HAND-WRITTEN DIGIT CLASSIFICATION
+
+In this experiment, we explore a high-dimensional inference problem, GP classification of MNIST digits (LeCun & Cortes, 2010). We consider a binary classification on handwritten images of 5 and 8. To make performance values of different methods more differentiable, we randomly choose a subset of size 7,858 from the original dataset. Pixel values are normalized to [0,1] in the preprocessing step. In the results, we also report the accuracy obtained by different methods.
+
+The results are shown in Table 3. We see that LAIN performs the best in terms of classification accuracy. Its predictive NLL and running speed also overperform competing methods, though not very significant. We also observe that AIGP makes less confident predictions than other methods, which accounts for its worse predictive NLL but high accuracy. We do not report results from VFF due to memory issues.
+
+We also examine KNN in this experiment. From the results, we notice that a small number of neighbors are often sufficient for KNN and LAIN models to perform well. By checking the running time of KNN, we also see that the time of finding nearest neighbors is only a small fraction of the total inference time on this dataset. There are slight differences regarding the test accuracy between KNN and LAIN, presumably due to different weighting schemes: LAIN weights different nearest neighbors according to their correlations, while KNN treats all nearest neighbors uniformly.
+
+# 5 CONCLUSION
+
+In this work, we propose a novel approach for GP inference. We construct a variational distribution that has a sparse decomposition on its covariance matrix. With this distribution, function value at a data point is inferred from its nearest neighbors, encouraging the inference efficiently focuses on approximating strong correlations posed by the prior. The proposed variational distribution is expressive to approximate the GP posterior and also provides a decent structure for efficient ELBO optimization. We further decompose the ELBO into homogeneous subtasks and therefore enable stochastic optimization. Finally, we devise inference networks to perform these subtasks and significantly reduce the number of variational parameters. Our proposed method performs well in terms of predictive performance and running speed on a series of benchmark tasks.
+
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+        "type": "text",
+        "text": "LOCALIZING AND AMORTIZING: EFFICIENT INFERENCE FOR GAUSSIAN PROCESSES ",
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+    {
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+    },
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+        "text": "The inference of Gaussian Processes concerns the distribution of the underlying function given observed data points. GP inference based on local ranges of data points is able to capture fine-scale correlations and allow fine-grained decomposition of the computation. Following this direction, we propose a new inference model that considers the correlations and observations of the $K$ nearest neighbors for the inference at a data point. Compared with previous works, we also eliminate the data ordering prerequisite to simplify the inference process. Additionally, the inference task is decomposed to small subtasks with several technique innovations, making our model well suits the stochastic optimization. Since the decomposed small subtasks have the same structure, we further speed up the inference procedure with amortized inference. Our model runs efficiently and achieves good performances on several benchmark tasks. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Gaussian processes (GP) (Rasmussen & Williams, 2006) are flexible non-parametric models with a wide range of applications. GP poses a Gaussian prior over function values f and assumes observations y are generated independently given f. GP inference considers the calculation of the posterior of these function values (Matthews et al., 2016) given observations, namely $p ( \\mathbf { f } | \\mathbf { y } )$ . Direct computation of the posterior is often intractable on large datasets, motivating people to consider its approximations. Variational inference (Jordan et al., 1999; Blei et al., 2017) for GP (Rasmussen & Williams, 2006) has achieved great successes recently. Variational inference constructs a variational distribution, which is usually a multivariate Gaussian distribution, to approximate the posterior. The approximation is done by minimizing the KL divergence from the posterior to the variational distribution (Blei et al., 2017). The variational distribution is often constructed with some special structures to reduce the number of variational parameters and speed up the computation. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Inducing-point methods (Quinonero-Candela & Rasmussen, 2005; Titsias, 2009; Hensman et al., ˜ 2013; 2015) define variational distributions on a small number $M$ of inducing points and then derive the distribution of non-inducing points conditioned on these inducing points. Inducing points summarize the entire posterior distribution, and their number $M$ balances the computational cost and the quality of the approximation. Inducing-point methods are further improved in several directions, such as generic inference for non-Gaussian likelihoods (Sheth et al., 2015; Dezfouli & Bonilla, 2015; Krauth et al., 2016; Hensman et al., 2015), inter-domain and subspace inducing points (Hensman et al., 2017; Panos et al., 2018), and decoupled approximation with two different sets of inducing points (Cheng & Boots, 2017; Salimbeni et al., 2018). Burt et al. (2019) provide theoretical analysis to show that a relatively small $M$ is sufficient to produce a reliable variational approximation when the input dimension is low. ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "While inducing-point methods capture global correlations among data points through inducing points, inference methods based on local neighbors focus more on correlation structures at local scales. These methods consider only local-range dependencies to save computation because localrange correlations are often much stronger than distant ones. Nguyen-Tuong et al. (2009); Park & Apley (2018) partition the input space into subregions, fit local models over subregions and then stitch local models into one. Other works examine neighbors of each data point directly. Gramacy & Apley (2015) investigate the properties of GP predictive equation and construct a local predictive approximator. Covariance tapering (Furrer et al., 2006; Kaufman et al., 2008) gains computational efficiency by constructing a sparse correlation matrix with zero correlations between distant data points. Methods based on Vecchia’s approximation (Vecchia, 1988; Datta et al., 2016; Liu & Liu, 2019; Finley et al., 2019) decompose the joint probability of data points into conditionals according to a data ordering and then neglect far data points that are conditioned on. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Recently, Liu & Liu (2019) propose the AIGP method, which extends the idea of local inference to GP models with non-Gaussian likelihoods. They use directed graphical models to approximate both the prior and the posterior. With this construction, the inference task decomposes into local inference subtasks, then they introduce amortized inference and use inference networks to identify solutions to these subtasks (Kingma & Welling, 2013; Dai et al., 2015; Miao et al., 2016). Amortization reduces the number of optimization parameters and greatly speeds up the inference procedure. However, this method has two drawbacks. First, the inference at a data point considers a few of its nearest neighbors but not all of them; therefore, it may lose some important correlations. Second, it depends on a data ordering. A bad ordering often deteriorates the performance, but it is hard to guard against such a bad situation. There are no easy fixes of the two issues, because all these designs in AIGP serve the purpose of decomposition. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a new GP inference method, Localized and Amortized Inference based on Nearest neighbors (LAIN). LAIN considers $K$ nearest neighbors for the inference at each data point. Particularly, LAIN uses a variational distribution whose covariance is parameterized by a sparse decomposition. The decomposition focuses on the correlations between every data point and its $K$ nearest neighbors 1. LAIN also eliminates the need for a data ordering. These nice properties come after several technical innovations. First, the new distribution does not admit a decomposable entropy calculation. We overcome this difficulty by using a decomposable lower bound of the entropy (Ranganath et al., 2016; Louizos & Welling, 2017). Second, to decompose the logarithm of the prior, AIGP and previous methods use a directed graphical model as an approximation of the prior. We follow this idea, but we consider all possible orderings of data points and collapse them to local combinations, making the computation manageable. With these techniques, LAIN still decomposes the inference task into subtasks, so amortized inference can apply. It is worthing noting that subtasks in LAIN are generated from the same mechanism while those in AIGP are not. We argue that subtasks sharing the same “distribution” are more appropriate for amortization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our empirical evaluations show that the LAIN method outperforms baseline methods including AIGP in several learning tasks. Our investigation also indicates that LAIN can achieve decent performance even only a few neighbors are considered. ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Gaussian Processes. Suppose we have a dataset containing a feature matrix $\\mathbf { X } = ( \\mathbf { x } _ { i } ) _ { i = 1 } ^ { N }$ and observations $\\mathbf { y } = \\mathbf { \\Psi } ( y _ { i } ) _ { i = 1 } ^ { N }$ . We assume there is a latent function $f$ that generates $y _ { i }$ from $\\mathbf { x } _ { i }$ for each $i$ . Particularly, each $y _ { i }$ is generated by a likelihood model $p ( y _ { i } | f _ { i } )$ with $f _ { i } = f ( \\mathbf { x } _ { i } )$ . Denote $\\mathbf { f } = ( f _ { i } ) _ { i = 1 } ^ { N }$ , then $\\begin{array} { r } { p ( \\mathbf { y } | \\mathbf { f } ) = \\prod _ { i = 1 } ^ { N } p ( y _ { i } | f _ { i } ) } \\end{array}$ . The likelihood $p ( y _ { i } | f _ { i } )$ can be very general – here we only assume that $\\log p ( y _ { i } | f _ { i } )$ is differentiable with respect to $f _ { i }$ . This mild assumption allows a wide range of data distributions. For example, if $y _ { i }$ is binary, $p ( y _ { i } | f _ { i } )$ is a Bernoulli distribution with $f _ { i }$ as the logit. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We put a GP prior with a mean function $\\nu ( \\cdot )$ and a kernel function $\\kappa ( \\cdot , \\cdot )$ over the latent function $f$ . The kernel function encodes the prior knowledge of the smoothness of $f$ . One commonly used kernel function is the Radial Basis Function (RBF) kernel, $\\kappa ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) = r ^ { 2 } \\exp ( - 0 . 5 \\| \\mathbf { x } _ { i } - \\mathbf { x } _ { j } \\| _ { 2 } ^ { 2 } / \\sigma ^ { 2 } )$ , with $r$ and $\\sigma$ as parameters. With this prior, function values in f follow a multivariate Gaussian, f $\\sim$ $\\mathcal { N } (  { \\boldsymbol \\nu } ,  { \\Sigma } )$ , with the mean ${ \\pmb { \\nu } } = ( \\nu ( x _ { i } ) ) _ { i = 1 } ^ { \\mathbf { \\hat { N } } }$ and the covariance matrix $\\pmb { \\Sigma }$ with $\\Sigma _ { i , j } = \\kappa ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) \\ \\forall i , j$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "GP inference concerns the calculation of the posterior $p ( \\mathbf { f } | \\mathbf { y } )$ (Matthews et al., 2016), from which we can infer the function value $f _ { \\star }$ for any new input $\\mathbf { x } _ { \\star }$ with integral $\\begin{array} { r } { \\int _ { \\mathbf { f } } p ( f _ { \\star } | \\mathbf { f } ) p ( \\mathbf { f } | \\mathbf { y } ) \\mathrm { d } \\mathbf { f } } \\end{array}$ . The posterior $p ( \\mathbf { f } | \\mathbf { y } )$ is generally not tractable, so we appeal to approximate inference. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Variational Inference for GP. Variational inference approximates the posterior $p ( \\mathbf { f } | \\mathbf { y } )$ with a variational distribution $q ( \\mathbf { f } )$ , which is defined as a multivariate Gaussian distribution, $q ( \\mathbf { f } ) \\sim \\mathcal { N } ( \\mu , \\mathbf { V } )$ ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/14ddfde5436b3c8fabab96a6fd00f543492efb6a505615591341ebaa4874b1ef.jpg",
+        "image_caption": [
+            "Figure 1: The structure of the variational distribution. The left box shows the amortization, which fits $\\mu _ { i }$ -s and $R _ { i j }$ -s from their related prior kernel and observations. The right part shows the generation process of $f _ { i }$ -s. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The inference is carried out by maximizing the Evidence Lower BOund (ELBO) with respect to $q ( \\mathbf { f } )$ (Blei et al., 2017). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a62a0650906b932165f829056b1e4345d1d06c80d7d71c79450eaa71942f7be1.jpg",
+        "text": "$$\n\\log p ( \\mathbf { y } | \\mathbf { X } ) \\geq \\operatorname* { m a x } _ { q ( \\mathbf { f } ) } \\underbrace { { \\mathbb { E } } _ { q } \\left[ \\log p ( \\mathbf { y } | \\mathbf { f } ) \\right] } _ { L _ { e l l } } + \\underbrace { { \\mathbb { E } } _ { q } \\left[ \\log p ( \\mathbf { f } ) \\right] } _ { L _ { c r o s s } } \\underbrace { - { \\mathbb { E } } _ { q } \\left[ \\log q ( \\mathbf { f } ) \\right] } _ { L _ { e n t } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Here we name the three terms in the ELBO for easy reference later. Typically the ELBO is maximized by gradient-based optimization, preferably stochastic gradient optimization when $N$ is large. Direct optimization of the ELBO is challenging, since the kernel matrix $\\pmb { \\Sigma }$ and the variational covariance $\\mathbf { V }$ are both large and have size $N \\times N$ . ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Inducing-point methods define $\\begin{array} { r } { q ( \\mathbf { f } ) ~ = ~ \\int _ { \\mathbf { f } _ { I } } q ( \\mathbf { f } _ { I } ) p ( \\mathbf { f } | \\mathbf { f } _ { I } ) ~ \\mathrm { d } \\mathbf { f } _ { I } } \\end{array}$ , where $q ( \\mathbf { f } _ { I } )$ is the distribution over inducing points $I$ , and $p ( \\mathbf { f } | \\mathbf { f } _ { I } )$ is derived from the prior. The computation is reduced mainly because only the small distribution $q ( \\mathbf { f } _ { I } )$ is optimized, while the conditional $p ( \\mathbf { f } | \\mathbf { f } _ { I } )$ is fixed when the prior is given. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "AIGP parameterizes $\\mathbf { V }$ by a Cholesky decomposition, $\\mathbf { V } = \\mathbf { L L } ^ { \\top }$ . Here $\\mathbf { L }$ is a sparse lower triangular matrix, and each row of $\\mathbf { L }$ has at most $K$ non-zero entries. AIGP uses a triangular $\\mathbf { L }$ for easy entropy computation. It also approximates $\\log p ( \\mathbf { f } )$ with a directed graphical model. Both the lower triangular matrix $\\mathbf { L }$ and the directed graph require an ordering of data points. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 THE VARIATIONAL DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following previous works, we also define the variational distribution $q ( \\mathbf { f } )$ to be a multivariate Gaussian $\\mathcal { N } ( \\mu , \\mathbf { V } )$ . We parameterize $\\mathbf { V } = \\mathbf { R } \\mathbf { R } ^ { \\top } + \\delta ^ { 2 } \\mathbf { I }$ with $\\mathbf { R }$ being a sparse matrix and $\\delta$ being a small constant. Note that we do not require $\\mathbf { R }$ to be triangular. The sparse pattern of $\\mathbf { R }$ is decided by the nearest neighbors: $R _ { i j } \\neq 0$ only when $j \\in n ( i )$ . Here $n ( i )$ is the neighbor set containing data points that have the largest covariance with $i$ in the prior (by definition $n ( i )$ includes $i$ ). In this work, we fix the size of $n ( i )$ to be $K$ , though our derivation works for varied sizes of $n ( i )$ . The row $\\mathbf { R } _ { i }$ can be viewed as a representation of $f _ { i }$ in the variational distribution: $\\mathbf { R } _ { i }$ informs $f _ { i }$ ’s correlation with other function values, just like a word embedding informs its relation with other words (Mikolov et al., 2013). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Efficient sampling from the marginal is critical for the decomposition of the ELBO later. Owing to the sparse decomposition of the covariance matrix, we can cheaply draw marginal samples for an $f _ { i }$ from $q ( \\mathbf { f } )$ with a linear transformation of white noise. The sampling scheme is shown in (2) and pictured in the right part of Figure 1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/633cce3f1de09ac24bab3053dbff802118e36ae300edc0ea9acb4a469d309b85.jpg",
+        "text": "$$\nf _ { i } = \\mu _ { i } + { \\bf R } _ { i } \\epsilon + \\delta \\xi = \\mu _ { i } + { \\bf R } _ { i , n ( i ) } \\epsilon _ { n ( i ) } + \\delta \\xi , \\epsilon \\sim \\mathcal { N } ( { \\bf 0 , I } ) , \\xi \\sim \\mathcal { N } ( { \\bf 0 , 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The constructed distribution $q ( \\mathbf { f } )$ well approximates the strong correlations in the prior. From (2), $f _ { i }$ and $f _ { j }$ correlate in $q ( \\mathbf { f } )$ by sharing noise entries in $n ( i ) \\cap n ( j )$ when the intersection is not empty. In this case, either $f _ { i }$ neighbors $f _ { j }$ , or $f _ { j }$ neighbors $f _ { i }$ , or $f _ { i } , f _ { j }$ share common neighbors. When the neighbor sets are large enough, most strong correlations will be approximated by some non-zero entries in $\\mathbf { V }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 OPTIMIZATION OF THE ELBO",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We optimize the ELBO in (1) to find a good $q ( \\mathbf { f } )$ to approximate the GP posterior. To apply stochastic optimization, we will decompose the three terms in the ELBO. We mainly consider the decomposition of $L _ { c r o s s }$ and $L _ { e n t }$ , as the decomposition of $L _ { e l l }$ is easy. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We first decompose the cross entropy $L _ { c r o s s }$ . By convention, the GP prior has a zero mean. Though there is a closed-form calculation of $L _ { c r o s s }$ with both $q ( \\mathbf { f } )$ and $p ( \\mathbf { f } )$ being multivariate Gaussian, it involves expensive calculations of $\\operatorname* { d e t } ( \\pmb { \\Sigma } )$ and $\\Sigma ^ { - 1 }$ . Previous works approximate the prior with Vecchia’s method for easy decomposition and good approximation (Vecchia, 1988; Stein et al., 2004; Datta et al., 2016; Liu & Liu, 2019; Finley et al., 2019). The idea is to build a directed graphical model and approximate $\\begin{array} { r } { p ( \\mathbf { f } ) \\approx \\prod _ { i = 1 } ^ { N } p ( f _ { i } | f _ { \\alpha ( i ) } ) } \\end{array}$ with $\\alpha ( i )$ being a small parent set of $i$ . In the original work, Vecchia (1988) first set an order to data points and then choose $\\alpha ( i )$ as the $K$ nearest parents of $i$ . But it is not easy to guarantee a good ordering of data points (Banerjee et al., 2014; Guinness, 2018). Here we consider all possible orderings and take the average of approximations to address the data ordering concern. ",
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+    {
+        "type": "text",
+        "text": "stimate for eac $L _ { c r o s s }$ as follows. First, we randomly saen we approximate the log-prior by $n ^ { \\prime } ( i ) \\subset n ( i )$ with , with $i \\not \\in$ $n ^ { \\prime } ( i )$ $i$ $\\begin{array} { r } { \\log p ( \\mathbf { f } ) \\approx \\sum _ { i = 1 } ^ { N } \\log p ( f _ { i } | f _ { n ^ { \\prime } ( i ) } ) } \\end{array}$ conditional distribution $p ( f _ { i } | f _ { n ^ { \\prime } ( i ) } )$ derived from the joint Gaussian $p ( f _ { i } , f _ { n ^ { \\prime } ( i ) } )$ in the prior. Then $L _ { c r o s s }$ is estimated by a random batch of terms. The complete calculation is given as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b7f718c4176380b8e148c8b138e9d6dcb801bd415b82a2163b1b6dac9764ec5f.jpg",
+        "text": "$$\nL _ { c r o s s } \\approx \\tilde { L } _ { c r o s s } = \\frac { N } { | S | } \\sum _ { i \\in S } \\mathbb { E } _ { q ( f _ { i } , f _ { n ^ { \\prime } ( i ) } ) } \\Big [ \\log p ( f _ { i } | f _ { n ^ { \\prime } ( i ) } ) \\Big ] , \\mathrm { ~ r a n d o m ~ s e t ~ } n ^ { \\prime } ( i ) \\subset n ( i ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here $S$ is a random batch of data points. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now we justify that this is an average over all data orderings. Suppose there is a data order $\\pi ( \\cdot )$ , such that we can define a directed graphical model over $p ( \\mathbf { f } )$ by assigning every $i$ a parent set $n _ { \\pi } ^ { \\prime } ( i ) = \\{ j : j \\in n ( i ) , \\pi ( j ) < \\pi ( i ) \\}$ . Denote $\\Pi$ as all permutations of $N$ data points, with each permutation inducing a graphical model. The average of the log densities of all graphical models can be collapsed to the average computed from local neighborhoods. Denote $\\Pi _ { n ( i ) }$ as permutations of indices in the set $n ( i )$ , then we have ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8810f96867d5fbb5d1d54ade9bc672e4e76d94966f382403f266db0085137ab.jpg",
+        "text": "$$\n\\frac { 1 } { N ! } \\sum _ { \\pi \\in \\Pi } \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { q ( f _ { i } , f _ { n _ { \\pi } ^ { \\prime } ( i ) } ) } \\Big [ \\log p ( f _ { i } | f _ { n _ { \\pi } ^ { \\prime } ( i ) } ) \\Big ] = \\sum _ { i = 1 } ^ { N } \\frac { 1 } { K ! } \\sum _ { \\pi \\in \\Pi _ { n ( i ) } } \\mathbb { E } _ { q ( f _ { i } , f _ { n _ { \\pi } ^ { \\prime } ( i ) } ) } \\Big [ \\log p ( f _ { i } | f _ { n _ { \\pi } ^ { \\prime } ( i ) } ) \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here we only need to consider permutations of data points within $n ( i )$ for each $i$ . Then we obtain (3) by estimating the inner summation by a single random permutation of $n ( i )$ and the outer summation by a random batch $S$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We then decompose the entropy $L _ { e n t }$ . The entropy of $q ( \\mathbf { f } )$ requires the expensive computation of $\\operatorname* { d e t } ( \\mathbf { V } )$ . To circumvent this difficulty, we find a decomposable lower bound of the entropy by using an auxiliary distribution (Ranganath et al., 2016; Louizos $\\&$ Welling, 2017). Note that we always prefer a lower bound of the objective in this maximization problem. With an arbitrary distribution $r ( \\epsilon | \\mathbf { f } )$ , a lower bound of $L _ { e n t }$ is ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df9440cfcab546ee770274c1012946515afcd2f77bf42eb2505de9fe9558cd0a.jpg",
+        "text": "$$\nL _ { e n t } = - \\mathbb { E } _ { q } \\left[ \\log q ( \\mathbf { f } ) \\right] \\geq - \\mathbb { E } _ { q ( \\mathbf { f } , \\epsilon ) } \\left[ \\log q ( \\mathbf { f } | \\epsilon ) + \\log q ( \\epsilon ) - \\log r ( \\epsilon | \\mathbf { f } ) \\right] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The bound is tight when $r ( \\epsilon | \\mathbf { f } )$ matches $q ( \\epsilon | \\mathbf { f } )$ . In this work, we try to let $r ( \\epsilon | \\mathbf { f } )$ match $q ( \\epsilon | \\mathbf { f } )$ . Particularly, we set $\\begin{array} { r } { r ( \\epsilon | \\mathbf { f } ) = \\prod _ { i } q ( \\epsilon _ { i } | \\mathbf { f } _ { n ( i ) } ) } \\end{array}$ , where the conditional $q \\bigl ( \\epsilon _ { i } | \\mathbf { f } _ { n ( i ) } \\bigr )$ is derived from the joint Gaussian distribution $q \\bigl ( \\epsilon _ { i } , \\mathbf { f } _ { n ( i ) } \\bigr )$ . Then all terms in the lower bound in (5) are Gaussian loglikelihoods and are decomposable over data points. We can then reach the estimation of the entropy lower bound with a batch of data points. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57b662d893da90f880d178712ea205f405620a4d2a1428d5387f311f29f4da67.jpg",
+        "text": "$$\nL _ { e n t } \\geq \\tilde { L } _ { e n t } = - \\frac { 1 } { 2 } \\frac { N } { | S | } \\sum _ { i \\in S } \\log \\left( 1 - \\mathbf { R } _ { n ( i ) , i } ^ { \\top } \\left( \\mathbf { R } _ { n ( i ) , : } \\mathbf { R } _ { n ( i ) , : } ^ { \\top } \\right) ^ { - 1 } \\mathbf { R } _ { n ( i ) , i } \\right) + c o n s t .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We finally decompose the likelihood $L _ { e l l }$ . The likelihood term $\\log p ( \\mathbf { y } | \\mathbf { f } )$ naturally decomposes because $y _ { i }$ -s are conditionally independent given $f _ { i }$ -s. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccd5bf474f2a9d4cd415f610e5d28cd69597f1c5e3f03b5ac9231edad846a87e.jpg",
+        "text": "$$\nL _ { e l l } = \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { q ( f _ { i } ) } \\left[ \\log p ( y _ { i } | f _ { i } ) \\right] , \\quad \\tilde { L } _ { e l l } = \\frac { N } { | S | } \\sum _ { i \\in S } \\log p ( y _ { i } | \\hat { f } _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            877,
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+            921
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here for each term $i$ in the summation, the expectation is estimated by a Monte Carlo sample $\\hat { f } _ { i }$ from $q ( f _ { i } )$ . The gradients of variational parameters are propagated through ${ \\hat { f } } _ { i }$ via reparameterization (Kingma & Welling, 2013). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, the ELBO has a decomposable approximation $\\tilde { L } _ { e l l } + \\tilde { L } _ { c r o s s } + \\tilde { L } _ { e n t }$ to enable efficient stochastic optimization. From the derivations above, we see the objective can be decomposed by data points. The computation for a data point only involves itself and its $K$ nearest neighbors. Therefore, each stochastic gradient calculation takes time only $O ( K ^ { 3 } )$ . There are $N ( K + \\bar { 1 } )$ parameters in $\\pmb { \\mu }$ and $\\mathbf { R }$ to optimize, so the optimization takes at least $O ( N )$ time. We further reduce the number of parameters by amortizing the cost through a shared inference model, taking advantage of the fact that the inference for each data point $i$ only needs its $K$ nearest neighbors. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 AMORTIZED INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following AIGP, we also apply amortized inference to GP inference. Particularly, we train an inference network to identify variational parameters ${ \\bf \\nabla } _ { \\mu _ { i } }$ and $\\mathbf { R } _ { i , n ( i ) } )$ for each data point $i$ . Since node correlations at a neighborhood can be easily treated as a weighted graph, we use Graph Convolutional Networks (GCNs) (Kipf & Welling, 2017) as our inference network. ",
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+    },
+    {
+        "type": "text",
+        "text": "A GCN takes an adjacency matrix $\\mathbf { A } \\in \\mathbb { R } ^ { k \\times k }$ of graph and the node features $\\mathbf { H } ^ { ( 0 ) } \\in \\mathbb { R } ^ { k \\times d _ { 0 } }$ as the input and then makes predictions for all graph nodes. Let $\\bar { \\mathbf A }$ be the normalized adjacency matrix, $\\bar { \\mathbf { A } } = \\mathbf { D } ^ { - \\frac { 1 } { 2 } } \\mathbf { A } \\mathbf { D } ^ { - \\frac { 1 } { 2 } }$ , with $\\mathbf { D }$ being the diagonal degree matrix. A GCN layer $\\ell$ with the input $\\mathbf { H } ^ { ( \\ell - 1 ) }$ is defined by $\\mathbf { H } ^ { ( \\ell ) } = g _ { \\ell } ( \\mathbf { H } ^ { ( \\ell - 1 ) } , \\mathbf { A } ) : = \\overset { - } { \\sigma } \\big ( \\bar { \\mathbf { A } } \\mathbf { H } ^ { ( \\bar { \\ell } - 1 ) } \\mathbf { W } ^ { ( \\ell ) } \\big )$ . Here $\\mathbf { W } ^ { ( l ) } \\in \\mathbb { R } ^ { d _ { \\ell - 1 } \\times d _ { \\ell } }$ is the weight matrix of the layer $\\ell$ . $\\sigma ( \\cdot )$ is the activation function. An $L$ -layer GCN computes its output by $\\mathbf { H } = g c n ( \\mathbf { H } ^ { 0 } , \\mathbf { A } ) : = g _ { L } ( \\mathbf { \\sigma } _ { \\cdot } \\dots g _ { 1 } ( \\mathbf { H } ^ { 0 } , \\mathbf { A } ) \\dots , \\mathbf { A } )$ . We use two GCNs for the inference task, $g c n _ { 1 }$ for the calculation of $\\mu _ { i }$ and $g c n _ { 2 }$ for $\\mathbf { R } _ { i , n ( i ) }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3e0d11298448926b7122bd381a5aff55a4751b266150335db43a221501d54bb1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mu _ { i } = { \\bf a } ^ { \\top } g c n _ { 1 } \\left( \\left[ { \\bf y } _ { n \\left( i \\right) } , { \\bf e } _ { i } \\right] , { \\bf \\Sigma } _ { n \\left( i \\right) , n \\left( i \\right) } \\right) , { \\bf R } _ { i , n \\left( i \\right) } = g c n _ { 2 } \\left( \\left[ { \\bf y } _ { n \\left( i \\right) } , { \\bf e } _ { i } \\right] , { \\bf \\Sigma } _ { { n \\left( i \\right) } , n \\left( i \\right) } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            477,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here we use $\\Sigma _ { n ( i ) , n ( i ) }$ as the adjacency matrix and stack the observation $\\mathbf { y } _ { n ( i ) }$ and the one-hot vector $\\mathbf { e } _ { i }$ as the input feature. The vector $\\mathbf { e } _ { i }$ indicates the element $i$ for which the inference is running for. We choose the activation $\\sigma ( \\cdot )$ to be ReLU for intermediate layers and identity for the last layer. The last layer of each GCN has size 1 to output a $K \\times 1$ vector. a is an averaging vector with all $K$ elements as $\\textstyle { \\frac { 1 } { K } }$ . The dashed box in Figure 1 shows the amortization. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "LAIN defines an inference subtask on a data point and its nearest neighbors, while AIGP defines a subtask on a data point and its parents. Due to this difference, LAIN has two advantages. First, the inference network of LAIN uses the observations from all the $K$ nearest neighbors, while the inference network of AIGP uses observations from parents only but not children. Second, inference subtasks of LAIN are generated with the same mechanism because the nearest-neighbor relationship is homogeneous across all data points. However, the parent-child relationship in AIGP depends on the ordering of data points (e.g. the first one in the order does not have parents). As a learning model, the inference network prefers subtasks from the same “distribution”. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The computational cost of GCN is $O ( K ^ { 2 } )$ by treating the network size as constant. The complexity of one gradient calculation is $O ( K ^ { 3 } )$ . The optimization procedure converges fast since it only optimizes a constant number of variational parameters. In practice, we often observe that the optimization procedure converges in less than one epoch, which is not possible for methods without amortization. Finding nearest neighbors is the only step with running time bounds to the data size, but it only needs one run and is often fast on medium to large data sizes. If the data has a very large size, we can use k-d trees for low-dimensional data and approximate algorithms (Arya et al., 1998; Datar et al., 2004) for high dimensional data. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For a new data point $\\mathbf { x } _ { \\star }$ with its $K$ nearest neighbors $n ( \\star )$ in the prior, the predictive distribution is ",
+        "bbox": [
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+            856,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24443d9005e78dcd3e1eb0494355f24f02ed5076e2b997721102527ee67b3d97.jpg",
+        "text": "$$\np ( y _ { \\star } | \\mathbf { x } _ { \\star } , \\mathbf { X } , \\mathbf { y } ) \\approx \\int _ { f _ { \\star } } p ( y _ { \\star } | f _ { \\star } ) q ( f _ { \\star } | \\mathbf { x } _ { \\star } , \\mathbf { X } _ { n ( \\star ) } , \\mathbf { y } _ { n ( \\star ) } ) \\mathrm { d } f _ { \\star } \\approx \\frac { 1 } { | F | } \\sum _ { \\widehat { f } _ { \\star } \\in F } p ( y _ { \\star } | \\widehat { f } _ { \\star } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/d66507f0b5b95720272e2d3439961525cb73c16e79e9c1099008221c027b8159.jpg",
+        "image_caption": [
+            "Figure 2: The first two plots compare predictive distributions of full GP and LAIN with $K = 1 0$ . The right three plots show how SVGP, AIGP, and LAIN perform with a very small number of inducing points/neighbors. Data points in blue circles are not well fitted. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here $\\begin{array} { r } { q ( f _ { \\star } | \\mathbf { x } _ { \\star } , \\mathbf { X } _ { n ( \\star ) } , \\mathbf { y } _ { n ( \\star ) } ) = \\int _ { \\mathbf { f } _ { n ( \\star ) } } p ( f _ { \\star } | \\mathbf { f } _ { n ( \\star ) } ) q ( \\mathbf { f } _ { n ( \\star ) } ) \\mathrm { d } \\mathbf { f } _ { n ( \\star ) } } \\end{array}$ is a Gaussian with parameters, ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/34d6dbae929f2d80e9ddab52afea71fd6a0d6b768e7e8c590021052ac9acbff8.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mu _ { \\star } = \\mathbf { b } _ { \\star } \\mu _ { n ( \\star ) } , \\qquad \\sigma _ { \\star } ^ { 2 } = \\Sigma _ { \\star , \\star } - \\Sigma _ { \\star , n ( \\star ) } \\mathbf { b } _ { \\star } ^ { \\top } + \\mathbf { b } _ { \\star } ( \\mathbf { R } _ { n ( \\star ) } \\mathbf { R } _ { n ( \\star ) } ^ { T } ) \\mathbf { b } _ { \\star } ^ { \\top } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            281,
+            725,
+            304
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "with $\\mathbf { b _ { \\star } } = \\pmb { \\Sigma } _ { \\star , n ( \\star ) } \\pmb { \\Sigma } _ { n ( \\star ) , n ( \\star ) } ^ { - 1 }$ . $F$ is a set of Monte Carlo samples from $q ( f _ { \\star } | \\mathbf { x } _ { \\star } , \\mathbf { X } _ { n ( \\star ) } , \\mathbf { y } _ { n ( \\star ) } )$ . The ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare our method with five state-of-the-art methods: SVGP (Hensman et al., 2015), SAVIGP (Dezfouli & Bonilla, 2015), DGP (Cheng & Boots, 2017), VFF (Hensman et al., 2017), and AIGP (Liu & Liu, 2019). The first three methods are based on inducing points, VFF uses inter-domain inducing points, and AIGP uses local neighbors. Through all experiments, we use RBF as the default kernel, except for VFF we use Ma´tern- $\\frac { 3 } { 2 }$ kernel (the code does not provide RBF kernel). We use the implementation of SVGP from GPFlow (Matthews et al., 2017), the implementation of DGP from Faust (2018), and implementations of all other algorithms from their authors. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For SVGP, SAVIGP, and VFF, we vary the number of inducing points, $M \\in \\{ 2 0 0 , 1 0 0 0 , 2 0 0 0 \\}$ , to check their performances. DGP has separate inducing points for mean approximation and those for variance approximation. We use 256 inducing points for variance approximation and vary the number of inducing points for mean approximation from 200 to 2000. We vary the number of neighbors, $K \\in \\{ 1 0 , 2 0 , 4 0 \\}$ , for AIGP and LAIN. GCNs used in these two methods have three hidden layers with dimensions [20, 10, 1]. We randomly split each dataset into training $( 7 5 \\% )$ and testing $( 2 5 \\% )$ and report both the predictive performance on the test set and the inference running time. To save the space, we report results from two settings for each competing method: one setting is $M = 2 0 0$ or $K = 1 0$ , with which all methods have their fastest speed (marked by $\\pmb { \\mathscr { z } }$ ), and another setting giving the best predictive performance (marked by $\\checkmark$ ). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 A TOY EXAMPLE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            660,
+            331,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we test different methods on a one-dimensional toy example studied in (Snelson & Ghahramani, 2006). The dataset contains 200 data points, shown as black dots in figure 2. We assume Gaussian likelihood in this experiment and run exact inference as the baseline. A smaller GCN (hidden dimensions [10, 5, 1]) is used in this task. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The predictive mean and variance from the exact inference and LAIN with $K = 1 0$ are shown in the first two plots of Figure 2. The result of LAIN is very similar to that of exact inference, except that the mean curve of LAIN is less smooth, which does not really hurt the predictive performance. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We test different methods with very small $M$ and $K$ and observe how they behave. We are likely to face this situation when we work on large datasets in high-dimensional spaces. The last three plots of Figure 2 exhibit predictive distributions of SVGP with $M = 2$ inducing points, AIGP with $K = 2$ parents, and LAIN with $K = 2$ nearest neighbors. When there are not enough inducing points, SVGP over-smooths the prediction and performs poorly for a good fraction of data points. AIGP does not have a good predictive mean either, because under a random ordering the directed graph constructed by AIGP cannot well capture neighboring relations. The predictive mean of LAIN does not deviate far from the ground-truth in the area with training instances, though the curve is rugged due to local variations. ",
+        "bbox": [
+            173,
+            797,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/38fafe4668b852128c05867b2fac2d13b4f5017b77121d0dac6986a6f87605f6.jpg",
+        "table_caption": [
+            "Table 1: Comparison on the eBird dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Config</td><td rowspan=1 colspan=1>Pred NLL</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>SVGP</td><td rowspan=1 colspan=1>M=2004M=1000√</td><td rowspan=1 colspan=1>1.90±.031.88±.02</td><td rowspan=1 colspan=1>107s4.5ks</td></tr><tr><td rowspan=1 colspan=1>SAVIGP</td><td rowspan=1 colspan=1>M=2004M=2000√</td><td rowspan=1 colspan=1>2.04±.031.99±.03</td><td rowspan=1 colspan=1>167s50ks</td></tr><tr><td rowspan=1 colspan=1>VFF</td><td rowspan=1 colspan=1>M=200HM=2000</td><td rowspan=1 colspan=1>1.91±.021.91±.02</td><td rowspan=1 colspan=1>1.3ks13ks</td></tr><tr><td rowspan=1 colspan=1>DGP</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>1.82±.021.80±.02</td><td rowspan=1 colspan=1>96s213s</td></tr><tr><td rowspan=1 colspan=1>AIGP</td><td rowspan=1 colspan=1>K=10  MK=20  √</td><td rowspan=1 colspan=1>1.79±.051.71±.05</td><td rowspan=1 colspan=1>45s125s</td></tr><tr><td rowspan=1 colspan=1>LAIN</td><td rowspan=1 colspan=1>K=10K=20K=40</td><td rowspan=1 colspan=1>1.69±.031.65±.031.60±.02</td><td rowspan=1 colspan=1>55s384s1.3ks</td></tr></table>",
+        "bbox": [
+            187,
+            135,
+            470,
+            315
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/fff338e9618ab7f14f700241ab71af426ba86a27dc8d477ad88b35175291d3cc.jpg",
+        "table_caption": [
+            "Table 2: Comparison on the precipitation dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Config</td><td rowspan=1 colspan=1>Pred NLL</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>SVGP</td><td rowspan=1 colspan=1>M=200HM=2000</td><td rowspan=1 colspan=1>1.57±.031.28±.03</td><td rowspan=1 colspan=1>2.5ks42ks</td></tr><tr><td rowspan=1 colspan=1>SAVIGP</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>1.70±.021.58±.02</td><td rowspan=1 colspan=1>2.8ks50ks</td></tr><tr><td rowspan=1 colspan=1>VFF</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>1.54±.031.53±.03</td><td rowspan=1 colspan=1>9.1ks32ks</td></tr><tr><td rowspan=1 colspan=1>DGP</td><td rowspan=1 colspan=1>M=200HM=2000</td><td rowspan=1 colspan=1>1.07±.051.00±.05</td><td rowspan=1 colspan=1>402s889s</td></tr><tr><td rowspan=1 colspan=1>AIGP</td><td rowspan=1 colspan=1>K=10  4K=10  √</td><td rowspan=1 colspan=1>0.96±.030.96±.03</td><td rowspan=1 colspan=1>155s155s</td></tr><tr><td rowspan=1 colspan=1>LAIN</td><td rowspan=1 colspan=1>K=10K=20K=40</td><td rowspan=1 colspan=1>0.74±.050.72±.050.69±.04</td><td rowspan=1 colspan=1>129s903s2.3ks</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/4e7a03685eb2a699e7d27c999ce631f8d156786ab9745e1016d46a92fbe32fdc.jpg",
+        "image_caption": [
+            "Figure 3: ELBO trajectories of LAIN with and without inference networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            362,
+            429,
+            508
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b2f79cfcd0313c2826534c2146f77dbe4f915b522ad033f4f26a2793f28b8994.jpg",
+        "table_caption": [
+            "Table 3: Comparison on the MNIST dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Config</td><td rowspan=1 colspan=1>Pred NLL</td><td rowspan=1 colspan=1>Accuracy</td><td rowspan=1 colspan=1>Time</td></tr><tr><td rowspan=1 colspan=1>SVGP</td><td rowspan=1 colspan=1>M=200王M=1000√</td><td rowspan=1 colspan=1>0.053±.0040.051±.004</td><td rowspan=1 colspan=1>98.498.5</td><td rowspan=1 colspan=1>623s23ks</td></tr><tr><td rowspan=1 colspan=1>SAVIGP</td><td rowspan=1 colspan=1>M=200HM=200√</td><td rowspan=1 colspan=1>0.339±.0080.339±.008</td><td rowspan=1 colspan=1>51.751.7</td><td rowspan=1 colspan=1>6.5ks6.5ks</td></tr><tr><td rowspan=1 colspan=1>DGP</td><td rowspan=1 colspan=1>M=2004M=2000</td><td rowspan=1 colspan=1>0.059±.0050.052±.005</td><td rowspan=1 colspan=1>98.198.3</td><td rowspan=1 colspan=1>292s2.1ks</td></tr><tr><td rowspan=1 colspan=1>AIGP</td><td rowspan=1 colspan=1>K=10  4K=40  √</td><td rowspan=1 colspan=1>0.293±.0020.215±.003</td><td rowspan=1 colspan=1>98.098.2</td><td rowspan=1 colspan=1>3.9ks24ks</td></tr><tr><td rowspan=1 colspan=1>LAIN</td><td rowspan=1 colspan=1>K=10K=20K=40</td><td rowspan=1 colspan=1>0.053±.0030.050±.0030.051±.003</td><td rowspan=1 colspan=1>98.999.099.1</td><td rowspan=1 colspan=1>128s632s2.9ks</td></tr><tr><td rowspan=1 colspan=1>KNN</td><td rowspan=1 colspan=1>K=9K=19K=39</td><td rowspan=1 colspan=1>N.A.N.A.N.A.</td><td rowspan=1 colspan=1>98.698.397.6</td><td rowspan=1 colspan=1>24s26s28s</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 BIRD ABUNDANCE ESTIMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we estimate the spatial abundance of a bird species (Savannah Sparrow) using eBird dataset (Munson et al., 2015). The dataset has 14,393 observations, each of which is a reported bird count at a GPS location. We model the observed counts with GPS locations as the input. We set the likelihood to be a Poisson distribution, with rate given by $\\lambda _ { i } = \\exp ( f _ { i } )$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare different inference methods in terms of Negative predictive Log-Likelihood (NLL, the smaller the better predictive performance). Table 1 shows the results. We can see that LAIN achieves the best predictive performance at $K = 4 0$ . Methods based on inducing points generally perform worse. In this dataset, observations have strong correlations in local areas, but inducing points are not efficient to capture the posterior at such a fine scale. In terms of running speed, LAIN is comparable to AIGP and DGP but faster than other methods. In our experiment, we have also tried to increase inducing points for DGP, but it does not improve its performance. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we also investigate whether inference networks work correctly. We run LAIN without inference networks and optimize $\\pmb { \\mu }$ and $\\mathbf { L }$ for the variational distribution directly. Then we compare LAIN models with and without inference networks by checking their optimization procedure. In this task, we fix hyperparameters, so the two methods solve a pure inference problem. Figure 3 is the trace plot of the negative ELBO versus training epochs. The figure shows the ELBO of the two LAIN models eventually converge to very similar values, though the ELBO without inference networks is slightly better after 50 epochs (likely due to the amortization gap). LAIN with inference networks significantly reduces the number of optimization epochs – the inference networks are well trained after only 0.01 epoch (about 100 iterations). In summary, the result indicates that inference networks can effectively identify the variational parameters using local information. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 PRECIPITATION LEVEL ESTIMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this task, we evaluate LAIN on a rainfall dataset. We process the precipitation dataset (Climate Data Online) and obtain the average precipitation level in May at 8,832 stations that are spatially distributed in the US. The GP inputs are GPS locations of these stations, and the observations are the average precipitation levels. We use the log-normal distribution as the likelihood, with its mean as function value $f$ from GP and variance as a hyperparameter learned from the data. ",
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 summaries the experimental results. LAIN has better predictive performance, and its running speed is comparable to or faster than other methods. ",
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+    },
+    {
+        "type": "text",
+        "text": "We also analyze the goodness of our prior approximation since we can compute the exact $L _ { c r o s s }$ on this dataset. We compute $L _ { c r o s s }$ with the optimized $q ( \\mathbf { f } )$ distribution as well as $\\tilde { L } _ { c r o s s }$ . The true value $L _ { c r o s s }$ and the approximation $\\tilde { L } _ { c r o s s }$ are: 5,465 versus 5,396 when $K \\ : = \\ : 1 0$ , 9,905 versus 8,490 when $K = 2 0$ , and 10,086 versus 9,009 when $K = 4 0$ . This result indicates that the approximation $\\tilde { L } _ { c r o s s }$ is relatively accurate. Furthermore, $\\tilde { L } _ { c r o s s }$ tends to be smaller than the true value and can be considered as a lower bound in such cases. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 HAND-WRITTEN DIGIT CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we explore a high-dimensional inference problem, GP classification of MNIST digits (LeCun & Cortes, 2010). We consider a binary classification on handwritten images of 5 and 8. To make performance values of different methods more differentiable, we randomly choose a subset of size 7,858 from the original dataset. Pixel values are normalized to [0,1] in the preprocessing step. In the results, we also report the accuracy obtained by different methods. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results are shown in Table 3. We see that LAIN performs the best in terms of classification accuracy. Its predictive NLL and running speed also overperform competing methods, though not very significant. We also observe that AIGP makes less confident predictions than other methods, which accounts for its worse predictive NLL but high accuracy. We do not report results from VFF due to memory issues. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also examine KNN in this experiment. From the results, we notice that a small number of neighbors are often sufficient for KNN and LAIN models to perform well. By checking the running time of KNN, we also see that the time of finding nearest neighbors is only a small fraction of the total inference time on this dataset. There are slight differences regarding the test accuracy between KNN and LAIN, presumably due to different weighting schemes: LAIN weights different nearest neighbors according to their correlations, while KNN treats all nearest neighbors uniformly. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            635,
+            318,
+            651
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a novel approach for GP inference. We construct a variational distribution that has a sparse decomposition on its covariance matrix. With this distribution, function value at a data point is inferred from its nearest neighbors, encouraging the inference efficiently focuses on approximating strong correlations posed by the prior. The proposed variational distribution is expressive to approximate the GP posterior and also provides a decent structure for efficient ELBO optimization. We further decompose the ELBO into homogeneous subtasks and therefore enable stochastic optimization. Finally, we devise inference networks to perform these subtasks and significantly reduce the number of variational parameters. Our proposed method performs well in terms of predictive performance and running speed on a series of benchmark tasks. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
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parse/train/ByqFhGZCW/ByqFhGZCW.md

@@ -0,0 +1,359 @@
+# MACHINE VS MACHINE: MINIMAX-OPTIMAL DEFENSE AGAINST ADVERSARIAL EXAMPLES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recently, researchers have discovered that the state-of-the-art object classifiers can be fooled easily by small perturbations in the input unnoticeable to human eyes. It is known that an attacker can generate strong adversarial examples if she knows the classifier parameters. Conversely, a defender can robustify the classifier by retraining if she has the adversarial examples. The cat-and-mouse game nature of attacks and defenses raises the question of the presence of equilibria in the dynamics. In this paper, we present a neural-network based attack class to approximate a larger but intractable class of attacks, and formulate the attacker-defender interaction as a zero-sum leader-follower game. We present sensitivity-penalized optimization algorithms to find minimax solutions, which are the best worst-case defenses against whitebox attacks. Advantages of the learning-based attacks and defenses compared to gradient-based attacks and defenses are demonstrated with MNIST and CIFAR-10.
+
+# 1 INTRODUCTION
+
+Recently, researchers have made an unsettling discovery that the state-of-the-art object classifiers can be fooled easily by small perturbations in the input unnoticeable to human eyes (Szegedy et al., 2013; Goodfellow et al., 2014b). Following studies tried to explain the cause of the seeming failure of deep learning toward such adversarial examples. The vulnerability was ascribed to linearity (Szegedy et al., 2013), low flexibility (Fawzi et al., 2015), or the flatness/curvedness of decision boundaries (Moosavi-Dezfooli et al., 2017), but a more complete picture is still under research. This is troublesome since such a vulnerability can be exploited in critical situations such as an autonomous car misreading traffic signs or a facial recognition system granting access to an impersonator without being noticed. Several methods of generating adversarial examples were proposed (Goodfellow et al., 2014b; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2017), most of which use the knowledge of the classifier to craft examples. In response, a few defense methods were proposed: retraining target classifiers with adversarial examples called adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b); suppressing gradient by retraining with soft labels called defensive distillation (Papernot et al., 2016); hardening target classifiers by training with an ensemble of adversarial examples (Tramer et al., 2017). \`
+
+In this paper we focus on whitebox attacks, that is, the model and the parameters of the classifier are known to the attacker. This requires a more robust classifier or defense method than simply relying on the secrecy of the parameters as defense. When the classifier parameters are known to an attacker, existing attack methods are very successful at fooling the classifiers. Conversely, when the attack is known to the classifier, e.g., in the form of adversarial examples, one can weaken the attack by retraining the classifier with adversarial examples, called adversarial training. However, if we repeat adversarial sample generation and adversarial training back-to-back, it is observed that the current adversarially-trained classifier is no longer robust to previous attacks (see Sec. 3.1.) To find the classifier robust against the class of gradient-based attacks, we first propose a sensitivitypenalized optimization procedure. Experiments show that the classifier from the procedure is more robust than adversarially-trained classifiers against previous attacks, but it still remains vulnerable to some degrees. This raises the main question of the paper: Can a classifier be robust to all types of attacks? The answer seems to be negative in light of the strong adversarial examples that can be crafted by direct optimization procedures from Huang et al. (2015) or Carlini & Wagner (2017). Note that the class of optimization-based attack is very large, as there is no restriction on the adversarial patterns that can be generated except for certain bounds such as $l _ { p }$ -norm bounds. The vastness of the optimization-based attack class is a hindrance to the study of the problem, as the defender cannot learn efficiently about the attack class from a finite number of samples. To study the problem analytically, we use a class of learning-based attack that can be generated by a class of neural networks. This class of attack can be considered an approximation of the class of optimization -based attacks, in that the search space of optimal perturbation is restricted to the parameter space of a neural network architecture, e.g., all perturbations that can be generated by fully-connected 3- layer ReLU networks. Similar to what we propose, others have recently considered training neural networks to generate adversarial examples (Nguyen & Sinha, 2017; Baluja & Fischer, 2017). While the proposed learning-based attack is weaker than the optimization-based attack, it can generate adversarial examples in test time with only single feedforward passes, which makes real-time attacks possible. We also show that the class of neural-network based attacks is quite different from the the class of gradient-based attacks (see Sec. 4.1.)
+
+Using the learning-based attack class, we introduce a continuous game formulation for analyzing the dynamics of attack-defense. The game is played by an attacker and a defender/classifier 1, where the attacker tries to maximize the risk of the classification task by perturbing input samples under certain constraints such as $l _ { p }$ -norm bounds, and the defender/classifier tries to adjust its parameters to minimize the same risk given the perturbed inputs. It is important to note that for adversarial attack problems, the performance of an attack or a defense cannot be measured in isolation, but only in pairs of (attack, defense). This is because the effectiveness of an attack/defense depends on the defense/attack it is against. As a two-player game, there may not be a dominant defense that is no less robust than all other defenses against all attacks. However, there is a natural notion of the best defense or attack in the worst case. Suppose one player moves first by choosing her parameters and the other player responds with the knowledge of the first player’s move. This is an example of a leader-follower game (Bruckner & Scheffer, 2011) for which there are two well-known ¨ states, the minimax and the maximin solutions if it is a constant-sum game. To find those solutions empirically, we propose a new continuous optimization method using the sensitivity penalization term. We show that the minimax solution from the proposed method is indeed different from the solution from the conventional alternating descent/ascent and is also more robust. We also show that the strength/weakness of the minimax-trained classifier is different from that of adversarially-trained classifiers for gradient-based attacks. The contributions of this paper are summarized as follows.
+
+• We provide a continuous game model to analyze adversarial example attacks and defenses, using the neural network-based attack class as a feasible approximation to a larger but intractable class of optimization-based attacks.   
+We demonstrate the difficulty of defending against multiple attack types and present the minimax defense as the best worst-case defense methods.   
+We propose a sensitivity-penalized optimization method (Alg. 1) to numerically find continuous minimax solutions, which is better than alternating descent/ascent. The proposed optimization method can also be used for other minimax problems beyond the adversarial example problem.
+
+The proposed methods are demonstrated with the MNIST and the CIFAR-10 datasets. For readability, details about experimental settings and the results with CIFAR-10 are presented in the appendix.
+
+# 2 RELATED WORK
+
+Making a classifier robust to test-time adversarial attacks has been studied for linear (kernel) hyperplanes (Lanckriet et al., 2002), naive Bayes (Dalvi et al., 2004) and SVM (Globerson & Roweis, 2006), which also showed the game-theoretic nature of the robust classification problems. Since the recent discovery of adversarial examples for deep neural networks, several methods of generating adversarial samples were proposed (Szegedy et al., 2013; Goodfellow et al., 2014b; Huang et al., 2015; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2017) as well as several methods of defense (Szegedy et al., 2013; Goodfellow et al., 2014b; Papernot et al., 2016; Tramer et al., 2017). These \` papers considered static scenarios, where the attack/defense is constructed against a fixed opponent.
+
+A few researchers have also proposed using a detector to detect and reject adversarial examples (Meng & Chen, 2017; Lu et al., 2017; Metzen et al., 2017). While we do not use detectors in this work, the minimax approach we proposed in the paper can be applied to train the detectors.
+
+The idea of using neural networks to generate adversarial samples has appeared concurrently (Baluja & Fischer, 2017; Nguyen & Sinha, 2017). Similar to our paper, the two papers demonstrates that it is possible to generate strong adversarial samples by a learning approach. Baluja & Fischer (2017) explored different architectures for the “adversarial transformation networks” against several different classifiers. Nguyen & Sinha (2017) proposed “attack learning neural networks” to map clean samples to a region in the feature space where misclassification occurs and “defense learning neural networks” to map them back to the safe region. Instead of prepending the defense layers before the fixed classifier (Nguyen & Sinha, 2017), we retrain the whole classifier as a defense method. However, the key difference of our work to the two papers is that we consider the dynamics of a learning-based defense stacked with a learning-based attack, and the numerical computation of the optimal defense/attack by continuous optimization.
+
+The alternating gradient-descent method for finding an equilibrium of a game has gained renewed interest since the introduction of Generative Adversarial Networks (GAN) (Goodfellow et al., 2014a). However, the instability of the alternating gradient-descent method has been known, and the “unrolling” method (Metz et al., 2016) was proposed to speed up the GAN training. The optimization algorithm proposed in the paper has a similarity with the unrolling method, but it is simpler (corresponding to a single-step unrolling) and involves a gradient-norm regularization which can be interpreted intuitively as sensitivity penalization (Gu & Rigazio, 2014; Lyu et al., 2015). Lastly, the framework of minimax risks was also studied in Hamm (2016) for the purpose of privacy preservation. We propose a different algorithm in this paper, but we also show that the attack on classification and the attack on privacy are the two sides of the same optimization problem with the opposite goals.
+
+# 3 CAT-AND-MOUSE GAME
+
+A classifier whose parameters are known to an attacker is easy to attack. Conversely, an attacker whose sample-generating method is known to a classifier is easy to defend from. In this section, we demonstrate the cat-and-mouse nature of the interaction, using adversarial training (Adv Train) as defense and the fast gradient sign method (FGSM) (Goodfellow et al., 2014b) and the iterative version (IFGSM) (Kurakin et al., 2016a) as attacks. We then show that the equilibrium, if it exists, can be found more efficiently by directly solving a sensitivity-penalized optimization problem.
+
+# 3.1 A NAIVE APPROACH
+
+Suppose $g$ is a classifier $g : \mathcal { X }  \mathcal { Y }$ and $l ( g ( x ) , y )$ is a loss function. The FGSM attack generates a perturbed example $z ( x )$ given the clean sample $x$ as follows:
+
+$$
+z ( x ) = x + \eta \mathrm { s i g n } ( \nabla _ { x } l ( g ( x ) , y ) ) .
+$$
+
+The clean input images we use here are $l _ { \infty }$ -normalized, that is, all pixel values are in the range $[ - 1 , 1 ]$ . It was argued that the use of true label $y$ results in “label leaking” (Kurakin et al., 2016b), but we use will true labels in the paper for simplicity. For another attack example, the IFGSM attack iteratively refines an adversarial example by the following update
+
+$$
+\begin{array} { r } { z _ { i + 1 } = \mathrm { c l i p } _ { x , \eta } ( z _ { i } + \eta \mathrm { s i g n } ( \nabla _ { z } l ( g ( z _ { i } ) , y ) ) ) , } \end{array}
+$$
+
+where the clipping used in this paper is $\begin{array} { r } { \mathrm { c l i p } _ { x , \eta } ( x ^ { \prime } ) \triangleq \operatorname* { m i n } \{ 1 , \ x + \eta , \ \operatorname* { m a x } \{ - 1 , \ x - \eta , \ x ^ { \prime } \} \} . } \end{array}$
+
+Existing attack methods such as FGSM and IFGSM are very effective at fooling the classifier. Table 1 shows that the two methods are able to perfectly fool a convolutional neural network trained with clean images from MNIST. (Details of the classifier architecture and the settings are in the appendix.)
+
+On the other hand, these attacks, if known to the classifier, can be weakened by retraining the classifier with the original dataset augmented by adversarial examples with ground-truth labels, known as adversarial training. In this paper we use the 1:1 mixture of the clean and the adversarial samples for adversarial training. Table 2 shows the result of adversarial training for different attacks.
+
+<table><tr><td rowspan="2">Defense\Attack</td><td rowspan="2">No attack</td><td colspan="4">FGSM</td><td colspan="4">IFGSM</td></tr><tr><td>n=0.3</td><td>m=0.4</td><td>m=0.5</td><td>n=0.6</td><td>m=0.3</td><td>m=0.4</td><td>n=0.5</td><td>n=0.6</td></tr><tr><td>No defense</td><td>0.006</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr></table>
+
+Table 1: Test error rates of FGSM and IFGSM attacks on an undefended convolutional neural network for MNIST. These attacks can cause perfect misclassification for the given range of $\eta$ .
+
+The test error rates for adversarial test examples after training become below $1 \%$ indicating nearperfect avoidance. This is in stark contrast with the perfect misclassification of the undefended classifier in Table 1.
+
+Table 2: Error rates of FGSM and IFGSM attacks on adversarially-trained classifiers for MNIST. This defense can avert the attacks and achieve the error rates of the no-attack case.   
+
+<table><tr><td rowspan="2">Defense\Attack</td><td rowspan="2">No attack</td><td colspan="4">FGSM</td><td colspan="4">IFGSM</td></tr><tr><td>n=0.3</td><td>n=0.4</td><td>m=0.5</td><td>n=0.6</td><td>n=0.3</td><td>n=0.4</td><td>m=0.5</td><td>n=0.6</td></tr><tr><td>Adv train</td><td>n/a</td><td>0.004</td><td>0.003</td><td>0.003</td><td>0.005</td><td>0.003</td><td>0.003</td><td>0.004</td><td>0.010</td></tr></table>
+
+A question arises as to what would happen if the procedure of 1) adversarial sample generation using the current classifier, and 2) retraining classifier using the current adversarial examples is repeated for many rounds. The answer to this cat-and-mouse game is easy to experiment although time-consuming. Let’s denote the attack on the original classifier as FGSM1, and the corresponding retrained classifier as Adv FGSM1. Repeating the procedure above generates the sequence of models $\mathrm { F G S M 1 } \to \mathrm { A d v } \ \mathrm { F G S M 1 } \to \mathrm { F G S M 2 } \to \mathrm { A d v } \ \mathrm { F C }$ GSM2, etc. Fig. 1 shows one such trial with $8 0 +$ 80 rounds of the procedure. Initially, the attacker achieves near-perfect attacks (i.e., error rate $\simeq 1$ ), and the defender achieves near-perfect defense (i.e., error rate $\simeq 0$ ). As the iteration increases, the attacker becomes weaker with error rate $\simeq 0 . 5$ , but the defense is still very successful, and the rate seems to oscillate persistently. While we can run more iterations to see if it converges, this is not a very principled nor efficient approach to find an equilibrium, if it exists.
+
+![](images/da60f7db9bd6b022adc7b3bbf142be95b5b59cdb173c6d20c6bf19032f4ee2e7.jpg)  
+Figure 1: A cat-and-mouse game of FGSM attacks and adversarial training for MNIST. The upper red points are the error rates after adversarial training, and the lower green points are the error rates after FGSM attack $\eta = 0 . 3 )$ . After 160 iterations, the error rate is still oscillating between 0 and 0.5.
+
+# 3.2 GRADIENT-BASED ATTACKS AND SENSITIVITY PENALTY
+
+We can perform the cat-and-mouse simulation more efficiently by an optimization approach. Instead of training the classifier fully with adversarial examples and then regenerating adversarial examples, suppose we only update the classifier with a single gradient-descent step then regenerate adversarial examples. To emphasize the parameters $u$ of the classifier/defender $g ( x ; u )$ , let’s rewrite the empirical risk of classifying the perturbed data as
+
+$$
+f ( u , Z ) \triangleq \frac { 1 } { N } \sum _ { i = 1 } ^ { N } l ( g ( z ( x _ { i } ) ; u ) , y _ { i } ) ,
+$$
+
+where $z ( x )$ denote an FGSM-like attack based on the loss gradient
+
+$$
+\begin{array} { r } { z ( { \boldsymbol x } ) \gets { \boldsymbol x } + \eta \nabla _ { z } l ( g ( z ( { \boldsymbol x } ) ; { \boldsymbol u } ) , y ) , } \end{array}
+$$
+
+and $Z = ( z _ { 1 } , . . . , z _ { N } ) \triangleq ( z ( x _ { 1 } ) , . . . , z ( x _ { N } ) )$ is the sequence of perturbed examples. In expectation of the attack, the defender should choose $u$ to minimize $f ( u , Z ( u ) )$ where the dependence of the attack on the classifier $u$ is expressed explicitly. If we minimize $f$ using gradient descent
+
+$$
+u  u - \lambda \frac { d f ( u , Z ) } { d u } ,
+$$
+
+then from the chain rule, the total derivative $\textstyle { \frac { d f } { d u } }$ is
+
+$$
+{ \frac { d f } { d u } } = { \frac { \partial f } { \partial u } } + { \frac { \partial Z } { \partial u } } { \frac { \partial f } { \partial Z } } = { \frac { \partial f } { \partial u } } + \sum _ { i } { \frac { \partial z _ { i } } { \partial u } } { \frac { \partial f } { \partial z _ { i } } } = { \frac { \partial f } { \partial u } } + { \frac { \eta } { N } } \sum _ { i } { \frac { \partial ^ { 2 } l } { \partial z _ { i } \partial u } } { \frac { \partial l } { \partial z _ { i } } }
+$$
+
+from (3) and (4).
+
+Interestingly, this total derivative (6) at the current state coincides with the gradient $\nabla _ { u }$ of the following cost
+
+$$
+f _ { \mathrm { s e n s } } ( u ) \triangleq f ( u , Z ) + \frac { \gamma } { 2 } \left\| \frac { \partial f ( u , Z ) } { \partial Z } \right\| ^ { 2 } = f ( u , Z ) + \frac { \eta } { 2 N } \sum _ { i = 1 } ^ { N } \left\| \frac { \partial l ( g ( z _ { i } ; u ) , y _ { i } ) } { \partial z _ { i } } \right\| ^ { 2 }
+$$
+
+where $\gamma = \eta N$ . There are two implications. Interpretation-wise, this cost function is the sum of the original risk $f$ and the ‘sensitivity’ term $\| \partial f / \partial Z \| ^ { 2 }$ which penalizes abrupt changes of the risk w.r.t. the input. Therefore, $u$ is chosen at each iteration to not only decrease the risk but also to make the classifier insensitive to input perturbation so that the attacker cannot take advantage of large gradients. The idea of minimizing the sensitivity to input is a familiar approach in robustifying classifiers (Gu & Rigazio, 2014; Lyu et al., 2015). Secondly, the new formulation can be implemented easily. The gradient descent update using the seemingly complicated gradient (6) can be replaced by the gradient descent update of (7). The capability of automatic differentiation (Rall, 1981) in modern machine learning libraries can be used to compute the gradient of (7) efficiently. Using this direct approach, we can find the defense parameters $u$ which will be robust to gradient-based attacks. Fig. 2 shows the decrease of test error during training using the this gradient descent approach for MNIST. It only takes a very small fraction of time to reach the final states of the Fig. 2 compared to that of Fig. 1.
+
+![](images/7d60a9f66246467edbdc5b42d1f2c5b63e1328b9fe50e235eff32588b2fd4e1a.jpg)  
+Figure 2: Convergence of test error rates for sensitivity-penalized optimization (7) with MNIST.
+
+There is also an important difference between the solution of the cat-and-mouse game and the minimizer of (7). Table 3 shows that the adversarially trained classifier (Adv FGSM1) is robust to both clean data and FGSM1 attack, but is susceptible to FGSM2 attack, displaying the cat-and-mouse nature. The same holds for Adv FGSM2, Adv FGSM3, etc. After 80 rounds of the cat-and-mouse procedure, the classifier Adv FGSM80 becomes robust to FGSM80 as well as moderately robust to other attacks including FGSM81 $\circleddash$ FGSM-curr). However, the classifier Sens FGSM from direct minimization of (7) is even more robust toward FGSM-curr than Adv FGSM80 and is overall the best. To see the advantage of the sensitivity term in (7), we also performed the minimization of (7) without the sensitivity term under the same conditions as Sens FGSM. This optimization method is similar to the method proposed in Huang et al. (2015), referred to as Learning with Adversaries (LWA FGSM). In the table, one can see that Sens FGSM is also better than LWA FGSM overall, although the difference is small.
+
+Note that Sens FGSM is better than other adversarially-trained classifiers, it too is still vulnerable to attacks such as FGSM80. This vulnerability raises the question if it is possible to make a classifier robust to any type of attacks, or more practically, robust to at least a large class of attacks. We discuss this issue in the next section.
+
+Table 3: Error rates of different attacks on various adversarially-trained classifiers for MNIST. FGSM-curr means the FGSM attack on the specific classifier on the left. Adv FGSM is the classifier adversarially trained with FGSM attacks. Sens FGSM is the result of minimizing (7) by gradient descent (5). LWA FGSM is the result of minimizing (7) without the gradient-norm term.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Defense\Attack</td><td rowspan="2">No attack</td><td colspan="3">FGSM</td><td rowspan="2">FGSM-curr</td></tr><tr><td>FGSM1</td><td>FGSM2 ：</td><td>FGSM80</td></tr><tr><td rowspan="5">n=0.3</td><td>No defense Adv FGSM1</td><td>0.026 0.012</td><td>1.000 0.004</td><td>0.881 0.995</td><td>：</td><td>0.355 1.000</td></tr><tr><td></td><td></td><td>0.999</td><td>：</td><td>0.499</td><td>0.995</td></tr><tr><td>Adv FGSM2</td><td>0.012</td><td></td><td>0.002 ：</td><td>0.505</td><td>0.995</td></tr><tr><td>AdvFGSM80</td><td>0.009</td><td>0.335</td><td>0.273 ：</td><td>0.009</td><td>0.442</td></tr><tr><td>LWAFGSM Sens FGSM</td><td>0.008</td><td>0.121</td><td>0.188 ：</td><td>0.210 0.194</td><td>0.048 0.048</td></tr><tr><td rowspan="6">m=0.4</td><td>No defense</td><td>0.009 0.026</td><td>0.104 1.000</td><td>0.176 0.944</td><td>： ：</td><td>0.528 1.000</td></tr><tr><td>AdvFGSM1</td><td>0.013</td><td>0.003</td><td>0.984</td><td>0.589 ：</td><td>0.984</td></tr><tr><td>AdvFGSM2</td><td>0.017</td><td>0.999</td><td>0.005 ：</td><td>0.549</td><td>0.999</td></tr><tr><td>AdvFGSM80</td><td>0.009</td><td>0.509</td><td>0.525</td><td>： 0.024</td><td>0.131</td></tr><tr><td>LWAFGSM</td><td>0.009</td><td>0.204</td><td>0.284</td><td>： 0.336</td><td>0.043</td></tr><tr><td>Sens FGSM</td><td>0.009</td><td>0.128</td><td>0.234</td><td>· 0.296</td><td>0.038</td></tr><tr><td rowspan="6">m=0.5</td><td>No defense</td><td>0.026</td><td>1.000</td><td>0.931</td><td>：</td><td>0.662</td><td>1.000</td></tr><tr><td>AdvFGSM1</td><td>0.010</td><td>0.002</td><td>0.970</td><td>：</td><td>0.724</td><td>0.970</td></tr><tr><td>Adv FGSM2</td><td>0.010</td><td>0.866</td><td>0.006</td><td>：</td><td>0.604</td><td>0.871</td></tr><tr><td>AdvFGSM80</td><td>0.008</td><td>0.653</td><td>0.559</td><td>：</td><td>0.023</td><td>0.089</td></tr><tr><td>LWAFGSM</td><td>0.009</td><td>0.248</td><td>0.260</td><td>：</td><td>0.432</td><td>0.035</td></tr><tr><td>Sens FGSM</td><td>0.009</td><td>0.266</td><td>0.285</td><td>：</td><td>0.365</td><td>0.039</td></tr><tr><td rowspan="6">n=0.6</td><td>No defense</td><td>0.026</td><td>1.000</td><td>0.963</td><td>：</td><td>0.803</td><td>1.000</td></tr><tr><td>AdvFGSM1</td><td>0.012</td><td>0.003</td><td>0.889</td><td>：</td><td>0.790</td><td>0.889</td></tr><tr><td>Adv FGSM2</td><td>0.008</td><td>0.649</td><td>0.007</td><td>：</td><td>0.687</td><td>0.767</td></tr><tr><td>AdvFGSM80</td><td>0.009</td><td>0.439</td><td>0.426</td><td>：</td><td>0.020</td><td>0.021</td></tr><tr><td>LWAFGSM</td><td>0.011</td><td>0.317</td><td>0.315</td><td>：</td><td>0.488</td><td>0.034</td></tr><tr><td>Sens FGSM</td><td>0.010</td><td>0.264</td><td>0.244</td><td>：</td><td>0.465</td><td>0.033</td></tr></table>
+
+# 4 GAME FORMULATION
+
+In this section, we consider the class of optimization-based attack and the class of neural-network based attacks as an approximation of the former. Using the neural-network based attack class, we formulate the attacker-defender dynamics as a game and discuss two types of equilibria – the minimax and the maximin solutions. We present algorithms that generalize the approach presented in the previous section.
+
+# 4.1 LEARNING-BASED ATTACK
+
+An attacker $z ( x ) : \mathcal { X }  \mathcal { X }$ can be more general than a specific class of attacks such as FGSM. Again, let $g : \mathcal { X }  \mathcal { Y }$ is a classifier parameterized by $u$ and $l ( g ( x ; u ) , y )$ is a loss function. If time complexity is not an issue, the following optimization-based attack (Huang et al., 2015)
+
+$$
+\operatorname* { m a x } _ { Z = ( z _ { 1 } , \ldots , z _ { N } ) } \left[ f ( u , Z ) \triangleq \frac { 1 } { N } \sum _ { i } l ( g ( z _ { i } ; u ) , y _ { i } ) \right] = \frac { 1 } { N } \sum _ { i } \operatorname* { m a x } _ { z _ { i } } \ l ( g ( z _ { i } ; u ) , y _ { i } ) ,
+$$
+
+which is also related to the CW attack (Carlini & Wagner, 2017), can generate strong adversarial examples, where adversarial patterns $Z = ( z _ { 1 } , . . . , z _ { N } )$ are unrestricted except for the bounds such as $\| z _ { i } - x _ { i } \| _ { p } \leq \eta$ . The corresponding class of adversarial patterns $Z$ is very large, which results in strong but non-generalizable adversarial examples. Non-generalizable means the perturbation $z ( x )$ has to be recomputed for every new test sample $x$ . While the class of optimization-based attacks is powerful, its large size makes it difficult to analytically study the optimal defense methods. To make the problem learnable, we restrict the class of patterns $Z$ to that which can be generated by a flexible but manageable class of perturbation $\{ z ( \cdot ; v ) \mid \forall v \in V \}$ , e.g., an autoencoder of a fixed architecture where the parameter $v$ is the network weights. This class is a clearly an approximation to the class of full optimization-based attacks, but is generalizable, i.e., no time-consuming optimization is required in the test phase but only single feedforward passes. The attack network (AttNet), as we call it, can be of any class of appropriate neural networks. Here we use a three-layer fully-connected network with 300 hiddens units per layer in this paper. Different from Nguyen & Sinha (2017) or Baluja & Fischer (2017), we feed the label $y$ into the input of the network along with the features $x$ . This is analogous to using the true label $y$ in the original FGSM. While this label input is optional but it can make the training of the attacker network easier. As with other attacks, we impose the $l _ { \infty }$ -norm constraint on $z$ , i.e., $\| z ( x ) - x \| _ { \infty } \leq \eta$ .
+
+Suppose now $f ( u , v )$ is the empirical risk of a classifier-attacker pair where the input $x$ is first transformed by attack network $z ( x ; v )$ and then fed to the classifier $g ( z ( x ; v ) ; u )$ . The attack network can be trained by gradient descent as well. Given a classifier $u$ , we can use gradient descent
+
+$$
+v  v + \sigma { \frac { \partial f ( u , v ) } { \partial v } }
+$$
+
+to find an optimal attacker $v$ that maximizes the risk $f$ assuming the classifier $u$ is fixed. Table 4 compares the error rates of the FGSM attacks and the attack network (AttNet). The table shows that AttNet is better than or comparable to FGSM in all cases. In particular, we already observed that the FGSM attack is no more effective against the classifier hardened against gradient-based attacks (Adv FGSM80 or Sens FGSM), but the AttNet can incur significant error $( > \sim 0 . 9 )$ for those hardened defenders. This indicates that the class of learning-based attacks is indeed different from the class of gradient-based attacks.
+
+<table><tr><td rowspan="2">Defense\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan="2">n=0.3</td><td colspan="2">n=0.4</td></tr><tr><td>No defense</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>AdvFGSM1</td><td>0.996</td><td>1.000</td><td>0.984</td><td>1.000</td></tr><tr><td>AdvFGSM80</td><td>0.473</td><td>0.899</td><td>0.131</td><td>0.903</td></tr><tr><td>Sens FGSM</td><td>0.048</td><td>0.965</td><td>0.038</td><td>0.902</td></tr><tr><td rowspan="4">No defense Adv FGSM1 AdvFGSM80</td><td colspan="2">m=0.5</td><td colspan="2">m=0.6</td></tr><tr><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>0.985</td><td>1.000</td><td>0.966</td><td>1.000</td></tr><tr><td>0.089</td><td>0.897 1.000</td><td>0.021</td><td>0.897</td></tr><tr><td rowspan="2">Sens FGSM</td><td colspan="2">0.039</td><td colspan="2">0.033 0.903</td></tr></table>
+
+Table 4: Error rates of FGSM vs learning-based attack network (AttNet) on various adversariallytrained classifiers for MNIST. FGSM-curr/AttNet-curr means they are computed/trained for the specific classifier on the leftmost column. Note that FGSM fails to attack hardened networks (Adv FGSM80 and Sens FGSM), whereas AttNet can still attack them successfully.
+
+# 4.2 MINIMAX GAME FOR LEARNING-BASED ATTACKS
+
+Finally, we consider the dynamics of the pair of classifier-attacker when each player can change its parameters. Given the current classifier $u$ , an optimal whitebox attacker parameter $v$ is the maximizer of the risk $f ( u , v )$
+
+$$
+v ^ { * } ( u ) \triangleq \arg \operatorname* { m a x } _ { v } f ( u , v ) .
+$$
+
+Consequently, the defender should choose the classifier parameters $u$ such that the maximum risk is minimized
+
+$$
+u ^ { * } \triangleq \arg \operatorname* { m i n } _ { u } \operatorname* { m a x } _ { v } f ( u , v ) = \arg \operatorname* { m i n } _ { u } f ( u , v ^ { * } ( u ) ) .
+$$
+
+This solution to the continuous minimax problem has a natural interpretation as the best worst-case solution. Assuming the attacker is optimal, i.e., it chooses the best attack from (10) given $u$ , no other defense can achieve a lower risk than the minimax defense $u ^ { * }$ in (11). The minimax defense is also a conservative defense. If the attacker is not optimal, and/or if the attack does not know the defense $u$ exactly (as in blackbox attacks), the actual risk can be lower than what the minimax solution $f ( u ^ { * } , v ^ { * } ( u ^ { * } ) )$ predicts. Before proceeding further, we point out that the claims above apply to the global minimizer $u ^ { * }$ and the maximizer function $v ^ { \ast } ( \cdot )$ , but in practice we can only find local solutions for complex risk functions of deep classifiers and attackers.
+
+To solve (11), we analyze the problem similarly to (5)-(7) from the previous section. At each iteration, the defender should choose $u$ in expectation of the attack and minimize $f ( u , v ^ { * } ( u ) )$ . We use
+
+gradient descent
+
+$$
+u  u - \lambda \frac { d f ( u , v ^ { * } ( u ) ) } { d u } ,
+$$
+
+where the total derivative $\textstyle { \frac { d f } { d u } }$ is
+
+$$
+\frac { d f } { d u } = \frac { \partial f ( u , v ^ { * } ( u ) ) } { \partial u } + \frac { \partial v ^ { * } ( u ) } { \partial u } \frac { \partial f ( u , v ) } { \partial v } .
+$$
+
+Since the exact maximizer $v ^ { * } ( u )$ is difficult to find, we only update $v$ incrementally by one (or more) steps of gradient-ascent update
+
+$$
+v  v + \sigma \frac { \partial f ( u , v ) } { \partial v } .
+$$
+
+The resulting formulation is closely related to the unrolled optimization (Metz et al., 2016) proposed for training GANs, although the latter has a very different cost function $f$ . Using the single update (14), the total derivative is
+
+$$
+\frac { d f } { d u } = \frac { \partial f ( u , v ^ { * } ( u ) ) } { \partial u } + \sigma \frac { \partial ^ { 2 } f ( u , v ) } { \partial u \partial v } \frac { \partial f ( u , v ) } { \partial v } .
+$$
+
+Similar to hardening a classifier against gradient-based attacks by minimizing (7) at each iteration, the gradient update of $u$ for $f ( u , v )$ can be done using the gradient of the following sensitivitypenalized function
+
+$$
+f _ { \mathrm { s e n s } } ( u ) \triangleq f ( u , v ) + { \frac { \sigma } { 2 } } \left\| { \frac { \partial f ( u , v ) } { \partial v } } \right\| ^ { 2 } .
+$$
+
+In other words, $u$ is chosen not only to minimize the risk but also to prevent the attacker from exploiting the sensitivity of $f$ to $v$ . The algorithm is summarized in Alg. 1.
+
+# Algorithm 1 Minimax Optimization by Sensitivity Penalization
+
+<table><tr><td>Input: risk f(u,v),#of iterations T,learning rates (oi),(入i),(Yi)</td></tr><tr><td>Output: (u*,u*(u*))</td></tr><tr><td>Initialize uo,Uo Begin</td></tr><tr><td>for i=1, ...,T do</td></tr><tr><td>Max step:Ui=Ui-1+Oi af(ui-1,Ui-1) du</td></tr><tr><td>a Yi-1 of(ui-1,Ui-1 Min step: Ui = Ui-1- Xi f(ui-1,Ui-1)+</td></tr><tr><td>du 2 du</td></tr><tr><td>end for Return (UT,UT).</td></tr></table>
+
+Note that this algorithm is actually independent of the adversarial example problem, and can be used for other minimax problems as well.
+
+# 4.3 MINIMAX VS MAXIMIN PROBLEMS
+
+In analogy with the minimax problem, we can also consider the maximin solution defined by
+
+$$
+v ^ { * } \triangleq \arg \operatorname* { m a x } _ { v } \operatorname* { m i n } _ { u } f ( u , v ) = \arg \operatorname* { m a x } _ { v } f ( u ^ { * } ( v ) , v ) .
+$$
+
+where
+
+$$
+u ^ { * } ( v ) \triangleq \arg \operatorname* { m i n } _ { u } f ( u , v )
+$$
+
+is the minimizer function. Here we are abusing the notations for the minimax solution $u ^ { * }$ , the maximin solution $v ^ { * }$ , the minimizer $u ^ { * } ( \cdot )$ , and the maximizer $v ^ { \ast } ( \cdot )$ . Similar to the minimax solution, the maximin solution has an intuitive meaning – it is the best worst-case solution for the attacker. Assuming the defender is optimal, i.e., it chooses the best defense from (18) that minimizes the risk $f ( u , v )$ given the attack $v$ , no other attack can inflict a higher risk than the maximin attack $v ^ { * }$ . It is also a conservative attack. If the defender is not optimal, and/or if the defender does not know the attack $v$ exactly, the actual risk can be higher than what the solution $f ( u ^ { * } ( v ^ { * } ) , v ^ { * } )$ predicts. Note that the maximin scenario where the defender knows the attack method is not very realistic but is the opposite of the minimax scenario and provides the lower bound.
+
+To summarize, minimax and maximin defenses and attacks have the following inherent properties.
+
+Lemma 1. Let $u ^ { * } , v ^ { * } ( u ) , v ^ { * } , u ^ { * } ( v )$ be the solutions of $( I I ) , ( I O ) , ( I 7 ) , ( I 8 ) .$ .
+
+1. $f ( u , v ^ { * } ( u ) ) \geq f ( u , v )$ : For any given defense $u$ , the max attack $v ^ { * } ( u )$ is the most effective attack.   
+2. $f ( u ^ { * } , v ^ { * } ( u ^ { * } ) ) \leq f ( u , v ^ { * } ( u ) )$ : Against the optimal attack $v ^ { * } ( u )$ , the minimax defense $u ^ { * }$ is the most effective defense.   
+3. $f ( u ^ { * } ( v ) , v ) \leq f ( u , v )$ : For any given attack $v$ , the min defense $u ^ { * } ( v )$ is the most effective defense.   
+4. $f ( u ^ { * } ( v ) , v ^ { * } ) \geq f ( u ^ { * } ( v ) , v )$ : Against the optimal defense $u ^ { * } ( v )$ , the maximin attack $v ^ { * }$ is the most effective attack.   
+5. $\begin{array} { r } { \operatorname* { m a x } _ { v } \operatorname* { m i n } _ { u } f ( u , v ) \leq \operatorname* { m i n } _ { u } \operatorname* { m a x } _ { v } f ( u , v ) . } \end{array}$ : The risk of the best worst-case attack is lower than that of the best worst-case defense.
+
+These properties follow directly from the definitions. The lemma helps us to better understand the dependence of defense and attack, and gives us the range of the possible risk values which can be measured empirically. To find maximin solutions, we use the same algorithm (Alg. 1) except that the variables $u$ and $v$ are switched and the sign of $f$ is flipped before the algorithm is called.
+
+# 4.4 EXPERIMENTS
+
+In addition to minimax and maximin optimization, we also consider as a reference algorithm the alternating descent/ascent method used in GAN training Goodfellow et al. (2014a)
+
+$$
+u  u - \lambda \frac { \partial f } { \partial u } , \quad v  v + \sigma \frac { \partial f } { \partial v } .
+$$
+
+Note that alternating descent/ascent finds local saddle points which are not necessarily minimax or maximin solutions, and therefore its solution will in general be different from the solution from Alg. 1. The difference of the solutions from three optimizations – Minimax, Maximin, and Alternating descent/ascent (Alt) – applied to a common problem, is demonstrated in Fig. 3. The figure shows the test error over the course of optimization starting from random initializations. One can see that Minimax (top blue curves) and Alt (middle green curves) converge to different values suggesting the learned classifiers will also be different.
+
+![](images/d6e579cfdb700ba5473f7cc533f56179bcabe9de8be7ad689d37ca03e59cf7fd.jpg)  
+Figure 3: Convergence of the test error rates for Minimax optimization (blue), Alternating ascent/descent (green), and Maximin optimization (red) for MNIST.
+
+Table 5 compares the robustness of the classifiers trained by Minimax and Alt against the AttNet attack (1st/2nd rows and 2nd column for each $\eta$ .) Minimax defense is more robust than Alt defense at $\eta = 0 . 3$ (0.020 vs 0.104) and at $\eta = 0 . 4$ (0.552 vs 0.873). For larger $\eta$ ’s, both are unusably vulnerable. Different performance of the two classifiers implies that the minimax solution found by Alg. 1 is different from the local saddle point found by alternating descent/ascent. In addition, against FGSM attacks, Minimax is moderately robust $( 0 . 2 1 8 - 0 . 3 4 2 )$ despite that the classifiers are not specifically trained against gradient-based attacks. In contrast, Sens FGSM is very vulnerable (0.902 – 1.000) against AttNet which we have already observed. This result suggests that the class of AttNet attacks and the class of gradient-based attacks are indeed different, and the former class is larger than the latter.
+
+Table 5: Error rates of Minimax-, Alt-, and adversarially-trained (Sens FGSM) classifiers for MNIST. Minimax is overall better than Alt against AttNet-curr, and is also moderately robust against the out-of-class attack (FGSM-curr).   
+
+<table><tr><td rowspan="2">Defense\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan="2">m=0.3</td><td colspan="2">n=0.4</td></tr><tr><td>Minimax</td><td>0.218</td><td>0.020</td><td>0.238</td><td>0.552</td></tr><tr><td>Alt</td><td>0.244</td><td>0.104</td><td>0.503</td><td>0.873</td></tr><tr><td>Sens FGSM</td><td>0.048</td><td>0.965</td><td>0.038</td><td>0.902</td></tr><tr><td rowspan="3">Minimax Alt</td><td>m=0.5</td><td></td><td>m=0.6</td><td></td></tr><tr><td>0.342</td><td>1.000</td><td>0.299</td><td>1.000</td></tr><tr><td>0.289</td><td>0.902</td><td>0.157</td><td>0.899</td></tr><tr><td>Sens FGSM</td><td>0.039</td><td>1.000</td><td>0.033</td><td>0.903</td></tr></table>
+
+Lastly, the adversarial examples generated by various attacks in the paper have diverse patterns and are shown in Fig. 4 of the appendix.
+
+# 5 DISCUSSION
+
+# 5.1 ROBUSTNESS AGAINST MULTIPLE ATTACK TYPES
+
+We discuss some limitations of the framework and also propose an extension. Ideally, a defender should find a robust classifier against the worst attack from a very large class of attacks such as optimization-based attacks. However, it is difficult to train classifiers against attacks from a large class. On the other hand, if the class is too small, then the worst attack from that class is not representative of all possible worst attacks, and therefore the minimax defense found will not be robust to out-of-class attacks. The trade-off seems inevitable.
+
+It is, however, possible to build a defense against multiple specific types of attacks. Suppose $z _ { 1 } ( u ) , . . . , z _ { m } ( u )$ are $m$ different types of attacks, e.g., $z _ { \mathrm { 1 } } \mathrm { = F G S M }$ , $z _ { \mathrm { 2 } } { = } \mathrm { I F G S M }$ , etc. The minimax defense for the combined attack is the solution to the mixed continuous-discrete problem
+
+$$
+\operatorname* { m i n } _ { u } \operatorname* { m a x } \{ f ( u , z _ { 1 } ( u ) ) , . . . , f ( u , z _ { m } ( u ) ) \} .
+$$
+
+Additionally, suppose $z _ { m + 1 } ( u , v ) , . . . , z _ { m + n } ( u , v )$ are $n$ different types of learning-based attacks, e.g., $z _ { m + 1 } = 2$ -layer dense net, $z _ { m + 2 } = 5$ -layer convolutional nets, etc. The minimax defense against the mixture of multiple fixed-type and learning-based attacks can be found by solving
+
+$$
+\operatorname* { m i n } _ { u } \operatorname* { m a x } \{ f ( u , z _ { 1 } ( u ) ) , \dots , f ( u , z _ { m } ( u ) ) , \operatorname* { m a x } _ { v } f ( u , z _ { m + 1 } ( u , v ) ) , \dots , \operatorname* { m a x } _ { v } f ( u , z _ { m + n } ( u , v ) ) \} .
+$$
+
+Due to the huge computational demand to solve (21), we leave it as a future work.
+
+# 5.2 ADVERSARIAL EXAMPLES AND PRIVACY ATTACKS
+
+Lastly, we discuss a bigger picture of the game between adversarial players. The minimax optimization arises in the leader-follower game (Bruckner & Scheffer, 2011) with the constant sum constraint. ¨ The leader-follower setting makes sense because the defense $=$ classifier parameters) is often public knowledge and the attacker exploits the knowledge. Interestingly, the problem of the attack on privacy (Hamm, 2016) has a very similar formulation as the adversarial attack problem, different only in that the classifier is an attacker and the data perturbator is a defender. In the problem of privacy preservation against inference, the defender is a data transformer $z ( x )$ (parameterized by $u$ ) which perturbs the raw data, and the attacker is a classifier (parameterized by $v$ ) who tries to extract sensitive information such as identity from the perturbed data such as online activity of a person. The transformer is the leader, such as when the privacy mechanism is public knowledge, and the classifier is the follower as it attacks the given perturbed data. The risk for the defender is therefore the accuracy of the inference of sensitive information measured by $- E [ l ( \boldsymbol { z } ( \boldsymbol { x } ; \boldsymbol { u } ) , \boldsymbol { y } ; \boldsymbol { v } ) ]$ . Solving the minimax risk problem $\begin{array} { r l } {  { ( \operatorname* { m i n } _ { u } \operatorname* { m a x } _ { v } - E [ l ( z ( x ; u ) , y ; v ) ] ) } \quad } & { { } } \end{array}$ gives us the best worst-case defense when the classifier/attacker knows the transformer/defender parameters, which therefore gives us a robust data transformer to preserve the privacy against the best inference attack (among the given class of attacks.) On the other hand, solving the maximin risk problem $\begin{array} { r l } { } & { { } ( \operatorname* { m a x } _ { v } \operatorname* { m i n } _ { u } - \bar { E } [ l ( \bar { z ( x ; u ) } , y ; v ) ] ) } \end{array}$ gives us the best worst-case classifier/attacker when its parameters are known to the transformer. As one can see, the problems of adversarial attack and privacy attack are two sides of the same coin which can be addressed by similar frameworks and optimization algorithms.
+
+# 6 CONCLUSION
+
+In this paper, we present a continuous game formulation of adversarial attacks and defenses using a learning-based attack class implemented by neural networks. We show that this class of attacks is quite different from the gradient-based attacks. While a classifier robust to all types of attack may yet be an elusive goal, the minimax defense against the neural network-based attack class is well-defined and practically achievable. We show that the proposed optimization method can find minimax defenses which are more robust than adversarially-trained classifiers and the classifiers from simple alternating descent/ascent. We demonstrate these with MNIST and CIFAR-10.
+
+# REFERENCES
+
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+
+# A RESULTS WITH MNIST
+
+The architecture of the MNIST classifier is similar to the Tensorflow model 2, and is trained with the following hyperparameters: $\{ B a t c h s i z e = I 2 8 \}$ , optimizer $=$ AdamOptimizer with $\lambda = 1 0 ^ { - 4 }$ , total # of iteration $\scriptstyle : = 5 0 , 0 0 0 . \}$
+
+The attack network has three hidden fully-connected layers of 300 units, trained with the following hyperparameters:   
+{Batch $s i z e ~ = ~ I 2 8$ , dropout rate $= ~ 0 . 5$ , optimizer $=$ AdamOptimizer with $1 0 ^ { - 3 }$ , total $\#$ of iteration $\scriptstyle : = 3 0 , 0 0 0 . \}$
+
+For minimax, alt, and maximin optimization, the total number of iteration was 100,000. The sensitivity-penalty coefficient of $\gamma = 1$ was used in Alg. 1.
+
+![](images/46763f54f338dd040d61c182c0c65ee517429ea079f203508ac7da6af10f2095.jpg)  
+Figure 4: Adversarial samples generated from different attacks at $\eta = 0 . 2$ . (a) Original data (b) FGSM1 (c) FGSM80 (d) IFGSM1 (e) Minimax (f) Alt (g) Maximin. Note the diversity of patterns.
+
+# B RESULTS WITH CIFAR-10
+
+We preprocess the CIFAR-10 dataset by removing the mean and normalizing the pixel values with the standard deviation of all pixels in the image. It is followed by clipping the values to $\pm 2$ standard deviations and rescaling to $[ - 1 , 1 ]$ . The architecture of the CIFAR classifier is similar to the Tensorflow model 3 but is simplified further by removing the local response normalization layers. With the simple structure, we attained $\sim 7 8 \%$ accuracy with the test data. The classifier is trained with the following hyperparameters:
+
+$\{ B a t c h s i z e = I 2 8 ,$ , optimizer $=$ AdamOptimizer with $\lambda = 1 0 ^ { - 4 }$ , total # of iteration $\scriptstyle : = I O O , O O O .  \}$
+
+The attack network has three hidden fully-connected layers of 300 units, trained with the following hyperparameters:   
+$\{ B a t c h \ s i z e \ = \ I 2 8 ,$ , dropout rate $= 0 . 5$ , optimizer $=$ AdamOptimizer with $\sigma = 1 0 ^ { - 3 }$ , total # of iteration $\scriptstyle : = 3 0 , 0 0 0 . \}$
+
+For minimax, alt, and maximin optimization, the total number of iteration was 100,000. The sensitivity-penalty coefficient of $\gamma = 1$ was used in Alg. 1.
+
+In the rest of the appendix, we repeat all the experiments with the MNIST dataset using the CIFAR10 dataset.
+
+<table><tr><td rowspan="2">Defense\Attack</td><td rowspan="2">No attack</td><td colspan="4">FGSM</td><td colspan="4">IFGSM</td></tr><tr><td>n=0.1</td><td>n=0.2</td><td>n=0.3</td><td>m=0.4</td><td>n=0.1</td><td>m=0.2</td><td>n=0.3</td><td>m=0.4</td></tr><tr><td>No defense</td><td>0.222</td><td>0.976</td><td>0.825</td><td>0.869</td><td>0.884</td><td>0.668</td><td>0.907</td><td>0.959</td><td>0.971</td></tr></table>
+
+Table 6: Error rates of FGSM and IFGSM attacks on the original classifier for cifar10. These attacks can cause large misclassification for the given range of $\eta$ .
+
+<table><tr><td rowspan="2">Defense\Attack</td><td rowspan="2">No attack</td><td colspan="4">FGSM</td><td colspan="4">IFGSM</td></tr><tr><td>n=0.1</td><td>n=0.2</td><td>n=0.3</td><td>n=0.4</td><td>n=0.1</td><td>n=0.2</td><td>m=0.3</td><td>n=0.4</td></tr><tr><td>Adv train</td><td>n/a</td><td>0.196</td><td>0.642</td><td>0.668</td><td>0.702</td><td>0.373</td><td>0.658</td><td>0.741</td><td>0.750</td></tr></table>
+
+Table 7: Error rates of FGSM and IFGSM attacks on the adversarially-trained classifiers for CIFAR10. This defense can significantly lower the errors from the attacks, although not as low as the MNIST problem.
+
+![](images/25c2a7b8f76e7f3af510741ee0bca0ba313bdf0750f34346abc872740838414e.jpg)  
+Figure 5: Cat and mouse game of FGSM attacks and adversarial training for CIFAR-10. The upper green points are the error rates after adversarial training, and the lower orange points are the error rates after FGSM attack. After 160 iterations $\eta = 0 . 3 )$ , the error rate is still oscillating.
+
+![](images/ab800da7f6508864830e13eb093463215686949fbfe0fc6bc9669047e0667041.jpg)  
+Figure 6: Convergence of test error rates for sensitivity-penalized optimization with MNIST.
+
+![](images/b047abf440dc0ad31a90a50bcd45e3e6e00c2c669bbe9a856a56ac7d52ef84d7.jpg)  
+Figure 7: Convergence of the test error rates for Minimax optimization (blue), Alternating ascent/descent (green), and Maximin optimization (red) for CIFAR-10.
+
+Table 8: Error rates of different attacks on various adversarially-trained classifiers for CIFAR-10. FGSM-curr means the FGSM attack on the specific classifier on the leftmost column. Adv FGSM is the classifier adversally trained with FGSM attacks. Sens FGSM is the result of minimizing the sensitivity penalty (7). LWA FGSM is the result of minimizing (7) without the gradient-norm term.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Defense\Attack</td><td rowspan="2">No attack</td><td colspan="4">FGSM</td><td rowspan="2">FGSM-curr</td></tr><tr><td>FGSM-1</td><td>FGSM-2</td><td>：</td><td>FGSM-80</td></tr><tr><td rowspan="5">n=0.1</td><td>No defense AdvFGSM1</td><td>0.222 0.220</td><td>0.976 0.196</td><td>0.671 0.680</td><td>： ：</td><td>0.595 0.616</td><td>0.976 0.245</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Adv FGSM2</td><td>0.258</td><td>0.640</td><td>0.484</td><td>：</td><td>0.612</td><td>0.708</td></tr><tr><td>AdvFGSM80</td><td>0.228</td><td>0.644</td><td>0.529</td><td>：</td><td>0.087</td><td>0.086</td></tr><tr><td>LWAFGSM Sens FGSM</td><td>0.223 0.223</td><td>0.283 0.342</td><td>0.692 0.701</td><td>：</td><td>0.652 0.663</td><td>0.125 0.106</td></tr><tr><td rowspan="6">n=0.2</td><td>No defense</td><td>0.222</td><td>0.825</td><td>0.692</td><td>： ：</td><td>0.819</td><td>0.969</td></tr><tr><td>AdvFGSM1</td><td>0.216</td><td>0.642</td><td>0.630</td><td>：</td><td>0.609</td><td>0.264</td></tr><tr><td>Adv FGSM2</td><td>0.305</td><td>0.579</td><td>0.290</td><td>：</td><td>0.599</td><td>0.556</td></tr><tr><td>AdvFGSM80</td><td>0.218</td><td>0.445</td><td>0.502</td><td>·</td><td>0.078</td><td>0.078</td></tr><tr><td>LWAFGSM</td><td>0.209</td><td>0.689</td><td>0.666</td><td>··</td><td>0.615</td><td>0.105</td></tr><tr><td>Sens FGSM</td><td>0.209</td><td>0.713</td><td>0.672</td><td>：</td><td>0.637</td><td>0.073</td></tr><tr><td rowspan="6">n=0.3</td><td>No defense AdvFGSM1</td><td>0.222</td><td>0.869</td><td>0.891</td><td>：</td><td>0.877</td><td>0.955</td></tr><tr><td></td><td>0.214</td><td>0.668</td><td>0.628</td><td>：</td><td>0.642</td><td>0.424</td></tr><tr><td>Adv FGSM2</td><td>0.205</td><td>0.499</td><td>0.407</td><td>：</td><td>0.514</td><td>0.389</td></tr><tr><td>AdvFGSM80</td><td>0.223</td><td>0.471</td><td>0.324</td><td>：</td><td>0.081</td><td>0.084</td></tr><tr><td>LWAFGSM</td><td>0.215</td><td>0.686</td><td>0.634</td><td>：</td><td>0.640</td><td>0.215</td></tr><tr><td>Sens FGSM</td><td>0.213</td><td>0.715</td><td>0.628</td><td>·</td><td>0.652</td><td>0.089</td></tr><tr><td rowspan="6">n=0.4</td><td>No defense AdvFGSM1</td><td>0.222</td><td>0.884</td><td>0.899</td><td>：</td><td>0.892</td><td>0.941</td></tr><tr><td></td><td>0.208</td><td>0.702</td><td>0.687</td><td>：</td><td>0.697</td><td>0.536</td></tr><tr><td>Adv FGSM2</td><td>0.206</td><td>0.592</td><td>0.546</td><td>：</td><td>0.618</td><td>0.545</td></tr><tr><td>AdvFGSM80</td><td>0.225</td><td>0.497</td><td>0.385</td><td>：</td><td>0.121</td><td>0.124</td></tr><tr><td>LWAFGSM</td><td>0.210</td><td>0.693</td><td>0.639</td><td>：</td><td>0.626</td><td>0.173</td></tr><tr><td>Sens FGSM</td><td>0.214</td><td>0.714</td><td>0.635</td><td>·</td><td>0.640</td><td>0.109</td></tr></table>
+
+Table 9: Error rates of FGSM vs learning-based attack network (AttNet) on various adversariallytrained classifiers for CIFAR-10. FGSM-curr/AttNet-curr means they are computed/trained for the specific classifier on the leftmost column. Note that FGSM fails to attack against the ‘hardened’ networks (Adv FGSM80 and Sens FGSM), but AttNet can still attack them successfully.   
+
+<table><tr><td rowspan="2">Defense\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan="2">n=0.1</td><td colspan="2">=0.2</td></tr><tr><td>No defense</td><td>0.976</td><td>0.740</td><td>0.969</td><td>0.905</td></tr><tr><td>Adv FGSM1</td><td>0.245</td><td>0.999</td><td>0.264</td><td>1.000</td></tr><tr><td>Adv FGSM80</td><td>0.086</td><td>1.000</td><td>0.078</td><td>1.000</td></tr><tr><td>Sens FGSM</td><td>0.106</td><td>0.898</td><td>0.073</td><td>0.979</td></tr><tr><td rowspan="4">No defense Adv FGSM1 AdvFGSM80</td><td colspan="2">m=0.3</td><td colspan="2">m=0.4</td></tr><tr><td>0.955</td><td>0.888</td><td>0.941</td><td>0.999</td></tr><tr><td>0.424</td><td>1.000</td><td>0.536</td><td>1.000</td></tr><tr><td>0.084</td><td>1.000</td><td>0.124</td><td>0.900</td></tr><tr><td rowspan="2">Sens FGSM</td><td rowspan="2">0.089</td><td rowspan="2">1.000</td><td rowspan="2">0.109</td></tr><tr><td>1.000</td></tr></table>
+
+<table><tr><td rowspan="2">Defense\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan="2">n=0.1</td><td colspan="2">m=0.2</td></tr><tr><td>Minimax</td><td>0.967</td><td>0.276</td><td>0.980</td><td>0.418</td></tr><tr><td>Alt</td><td>0.994</td><td>0.264</td><td>0.996</td><td>0.857</td></tr><tr><td>Sens FGSM</td><td>0.106</td><td>0.898</td><td>0.073</td><td>0.979</td></tr><tr><td rowspan="3">Minimax Alt</td><td>n=0.3</td><td></td><td>m=0.4</td><td></td></tr><tr><td>0.967</td><td>0.875</td><td>0.931</td><td>0.994</td></tr><tr><td>0.987</td><td>0.896</td><td>0.958</td><td>1.000</td></tr><tr><td>Sens FGSM</td><td>0.089</td><td>1.000</td><td>0.109</td><td>1.000</td></tr></table>
+
+Table 10: Error rates of Minimax-, Alt-, and adversarially-trained (Sens FGSM) classifiers for MNIST. While Minimax and Alt are both vulnerable to AttNet attacks, Minimax is much less vulnerable than Alt at $\eta = 0 . 2$ .

parse/train/ByqFhGZCW/ByqFhGZCW_content_list.json

@@ -0,0 +1,1675 @@
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+    {
+        "type": "text",
+        "text": "MACHINE VS MACHINE: MINIMAX-OPTIMAL DEFENSE AGAINST ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
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+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, researchers have discovered that the state-of-the-art object classifiers can be fooled easily by small perturbations in the input unnoticeable to human eyes. It is known that an attacker can generate strong adversarial examples if she knows the classifier parameters. Conversely, a defender can robustify the classifier by retraining if she has the adversarial examples. The cat-and-mouse game nature of attacks and defenses raises the question of the presence of equilibria in the dynamics. In this paper, we present a neural-network based attack class to approximate a larger but intractable class of attacks, and formulate the attacker-defender interaction as a zero-sum leader-follower game. We present sensitivity-penalized optimization algorithms to find minimax solutions, which are the best worst-case defenses against whitebox attacks. Advantages of the learning-based attacks and defenses compared to gradient-based attacks and defenses are demonstrated with MNIST and CIFAR-10. ",
+        "bbox": [
+            233,
+            265,
+            764,
+            444
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            469,
+            336,
+            486
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, researchers have made an unsettling discovery that the state-of-the-art object classifiers can be fooled easily by small perturbations in the input unnoticeable to human eyes (Szegedy et al., 2013; Goodfellow et al., 2014b). Following studies tried to explain the cause of the seeming failure of deep learning toward such adversarial examples. The vulnerability was ascribed to linearity (Szegedy et al., 2013), low flexibility (Fawzi et al., 2015), or the flatness/curvedness of decision boundaries (Moosavi-Dezfooli et al., 2017), but a more complete picture is still under research. This is troublesome since such a vulnerability can be exploited in critical situations such as an autonomous car misreading traffic signs or a facial recognition system granting access to an impersonator without being noticed. Several methods of generating adversarial examples were proposed (Goodfellow et al., 2014b; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2017), most of which use the knowledge of the classifier to craft examples. In response, a few defense methods were proposed: retraining target classifiers with adversarial examples called adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b); suppressing gradient by retraining with soft labels called defensive distillation (Papernot et al., 2016); hardening target classifiers by training with an ensemble of adversarial examples (Tramer et al., 2017). \\` ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we focus on whitebox attacks, that is, the model and the parameters of the classifier are known to the attacker. This requires a more robust classifier or defense method than simply relying on the secrecy of the parameters as defense. When the classifier parameters are known to an attacker, existing attack methods are very successful at fooling the classifiers. Conversely, when the attack is known to the classifier, e.g., in the form of adversarial examples, one can weaken the attack by retraining the classifier with adversarial examples, called adversarial training. However, if we repeat adversarial sample generation and adversarial training back-to-back, it is observed that the current adversarially-trained classifier is no longer robust to previous attacks (see Sec. 3.1.) To find the classifier robust against the class of gradient-based attacks, we first propose a sensitivitypenalized optimization procedure. Experiments show that the classifier from the procedure is more robust than adversarially-trained classifiers against previous attacks, but it still remains vulnerable to some degrees. This raises the main question of the paper: Can a classifier be robust to all types of attacks? The answer seems to be negative in light of the strong adversarial examples that can be crafted by direct optimization procedures from Huang et al. (2015) or Carlini & Wagner (2017). Note that the class of optimization-based attack is very large, as there is no restriction on the adversarial patterns that can be generated except for certain bounds such as $l _ { p }$ -norm bounds. The vastness of the optimization-based attack class is a hindrance to the study of the problem, as the defender cannot learn efficiently about the attack class from a finite number of samples. To study the problem analytically, we use a class of learning-based attack that can be generated by a class of neural networks. This class of attack can be considered an approximation of the class of optimization -based attacks, in that the search space of optimal perturbation is restricted to the parameter space of a neural network architecture, e.g., all perturbations that can be generated by fully-connected 3- layer ReLU networks. Similar to what we propose, others have recently considered training neural networks to generate adversarial examples (Nguyen & Sinha, 2017; Baluja & Fischer, 2017). While the proposed learning-based attack is weaker than the optimization-based attack, it can generate adversarial examples in test time with only single feedforward passes, which makes real-time attacks possible. We also show that the class of neural-network based attacks is quite different from the the class of gradient-based attacks (see Sec. 4.1.) ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Using the learning-based attack class, we introduce a continuous game formulation for analyzing the dynamics of attack-defense. The game is played by an attacker and a defender/classifier 1, where the attacker tries to maximize the risk of the classification task by perturbing input samples under certain constraints such as $l _ { p }$ -norm bounds, and the defender/classifier tries to adjust its parameters to minimize the same risk given the perturbed inputs. It is important to note that for adversarial attack problems, the performance of an attack or a defense cannot be measured in isolation, but only in pairs of (attack, defense). This is because the effectiveness of an attack/defense depends on the defense/attack it is against. As a two-player game, there may not be a dominant defense that is no less robust than all other defenses against all attacks. However, there is a natural notion of the best defense or attack in the worst case. Suppose one player moves first by choosing her parameters and the other player responds with the knowledge of the first player’s move. This is an example of a leader-follower game (Bruckner & Scheffer, 2011) for which there are two well-known ¨ states, the minimax and the maximin solutions if it is a constant-sum game. To find those solutions empirically, we propose a new continuous optimization method using the sensitivity penalization term. We show that the minimax solution from the proposed method is indeed different from the solution from the conventional alternating descent/ascent and is also more robust. We also show that the strength/weakness of the minimax-trained classifier is different from that of adversarially-trained classifiers for gradient-based attacks. The contributions of this paper are summarized as follows. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We provide a continuous game model to analyze adversarial example attacks and defenses, using the neural network-based attack class as a feasible approximation to a larger but intractable class of optimization-based attacks.   \nWe demonstrate the difficulty of defending against multiple attack types and present the minimax defense as the best worst-case defense methods.   \nWe propose a sensitivity-penalized optimization method (Alg. 1) to numerically find continuous minimax solutions, which is better than alternating descent/ascent. The proposed optimization method can also be used for other minimax problems beyond the adversarial example problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed methods are demonstrated with the MNIST and the CIFAR-10 datasets. For readability, details about experimental settings and the results with CIFAR-10 are presented in the appendix. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            753,
+            341,
+            770
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Making a classifier robust to test-time adversarial attacks has been studied for linear (kernel) hyperplanes (Lanckriet et al., 2002), naive Bayes (Dalvi et al., 2004) and SVM (Globerson & Roweis, 2006), which also showed the game-theoretic nature of the robust classification problems. Since the recent discovery of adversarial examples for deep neural networks, several methods of generating adversarial samples were proposed (Szegedy et al., 2013; Goodfellow et al., 2014b; Huang et al., 2015; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2017) as well as several methods of defense (Szegedy et al., 2013; Goodfellow et al., 2014b; Papernot et al., 2016; Tramer et al., 2017). These \\` papers considered static scenarios, where the attack/defense is constructed against a fixed opponent. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A few researchers have also proposed using a detector to detect and reject adversarial examples (Meng & Chen, 2017; Lu et al., 2017; Metzen et al., 2017). While we do not use detectors in this work, the minimax approach we proposed in the paper can be applied to train the detectors. ",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The idea of using neural networks to generate adversarial samples has appeared concurrently (Baluja & Fischer, 2017; Nguyen & Sinha, 2017). Similar to our paper, the two papers demonstrates that it is possible to generate strong adversarial samples by a learning approach. Baluja & Fischer (2017) explored different architectures for the “adversarial transformation networks” against several different classifiers. Nguyen & Sinha (2017) proposed “attack learning neural networks” to map clean samples to a region in the feature space where misclassification occurs and “defense learning neural networks” to map them back to the safe region. Instead of prepending the defense layers before the fixed classifier (Nguyen & Sinha, 2017), we retrain the whole classifier as a defense method. However, the key difference of our work to the two papers is that we consider the dynamics of a learning-based defense stacked with a learning-based attack, and the numerical computation of the optimal defense/attack by continuous optimization. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The alternating gradient-descent method for finding an equilibrium of a game has gained renewed interest since the introduction of Generative Adversarial Networks (GAN) (Goodfellow et al., 2014a). However, the instability of the alternating gradient-descent method has been known, and the “unrolling” method (Metz et al., 2016) was proposed to speed up the GAN training. The optimization algorithm proposed in the paper has a similarity with the unrolling method, but it is simpler (corresponding to a single-step unrolling) and involves a gradient-norm regularization which can be interpreted intuitively as sensitivity penalization (Gu & Rigazio, 2014; Lyu et al., 2015). Lastly, the framework of minimax risks was also studied in Hamm (2016) for the purpose of privacy preservation. We propose a different algorithm in this paper, but we also show that the attack on classification and the attack on privacy are the two sides of the same optimization problem with the opposite goals. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 CAT-AND-MOUSE GAME ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A classifier whose parameters are known to an attacker is easy to attack. Conversely, an attacker whose sample-generating method is known to a classifier is easy to defend from. In this section, we demonstrate the cat-and-mouse nature of the interaction, using adversarial training (Adv Train) as defense and the fast gradient sign method (FGSM) (Goodfellow et al., 2014b) and the iterative version (IFGSM) (Kurakin et al., 2016a) as attacks. We then show that the equilibrium, if it exists, can be found more efficiently by directly solving a sensitivity-penalized optimization problem. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 A NAIVE APPROACH",
+        "text_level": 1,
+        "bbox": [
+            176,
+            606,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Suppose $g$ is a classifier $g : \\mathcal { X }  \\mathcal { Y }$ and $l ( g ( x ) , y )$ is a loss function. The FGSM attack generates a perturbed example $z ( x )$ given the clean sample $x$ as follows: ",
+        "bbox": [
+            171,
+            631,
+            823,
+            660
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/238fd593deb329b28b9b60c0e96fac886517ea8b06c5b7529e6a20974fbd7992.jpg",
+        "text": "$$\nz ( x ) = x + \\eta \\mathrm { s i g n } ( \\nabla _ { x } l ( g ( x ) , y ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            667,
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+            685
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The clean input images we use here are $l _ { \\infty }$ -normalized, that is, all pixel values are in the range $[ - 1 , 1 ]$ . It was argued that the use of true label $y$ results in “label leaking” (Kurakin et al., 2016b), but we use will true labels in the paper for simplicity. For another attack example, the IFGSM attack iteratively refines an adversarial example by the following update ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/286478ac7be38308cddc4a0b93c786028c334db75c126a469d46706c52b0fc23.jpg",
+        "text": "$$\n\\begin{array} { r } { z _ { i + 1 } = \\mathrm { c l i p } _ { x , \\eta } ( z _ { i } + \\eta \\mathrm { s i g n } ( \\nabla _ { z } l ( g ( z _ { i } ) , y ) ) ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            755,
+            642,
+            773
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the clipping used in this paper is $\\begin{array} { r } { \\mathrm { c l i p } _ { x , \\eta } ( x ^ { \\prime } ) \\triangleq \\operatorname* { m i n } \\{ 1 , \\ x + \\eta , \\ \\operatorname* { m a x } \\{ - 1 , \\ x - \\eta , \\ x ^ { \\prime } \\} \\} . } \\end{array}$ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Existing attack methods such as FGSM and IFGSM are very effective at fooling the classifier. Table 1 shows that the two methods are able to perfectly fool a convolutional neural network trained with clean images from MNIST. (Details of the classifier architecture and the settings are in the appendix.) ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, these attacks, if known to the classifier, can be weakened by retraining the classifier with the original dataset augmented by adversarial examples with ground-truth labels, known as adversarial training. In this paper we use the 1:1 mixture of the clean and the adversarial samples for adversarial training. Table 2 shows the result of adversarial training for different attacks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/61a0d151715028422e0827d8618b1d1ad4f0a97fbe024990b187fd995a56068c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td rowspan=\"2\">No attack</td><td colspan=\"4\">FGSM</td><td colspan=\"4\">IFGSM</td></tr><tr><td>n=0.3</td><td>m=0.4</td><td>m=0.5</td><td>n=0.6</td><td>m=0.3</td><td>m=0.4</td><td>n=0.5</td><td>n=0.6</td></tr><tr><td>No defense</td><td>0.006</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr></table>",
+        "bbox": [
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+            101,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Test error rates of FGSM and IFGSM attacks on an undefended convolutional neural network for MNIST. These attacks can cause perfect misclassification for the given range of $\\eta$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The test error rates for adversarial test examples after training become below $1 \\%$ indicating nearperfect avoidance. This is in stark contrast with the perfect misclassification of the undefended classifier in Table 1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/d52068f43f9a24c47fea6ab25846ffe8092ce9a3072481bfd529abd4f21d60a3.jpg",
+        "table_caption": [
+            "Table 2: Error rates of FGSM and IFGSM attacks on adversarially-trained classifiers for MNIST. This defense can avert the attacks and achieve the error rates of the no-attack case. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td rowspan=\"2\">No attack</td><td colspan=\"4\">FGSM</td><td colspan=\"4\">IFGSM</td></tr><tr><td>n=0.3</td><td>n=0.4</td><td>m=0.5</td><td>n=0.6</td><td>n=0.3</td><td>n=0.4</td><td>m=0.5</td><td>n=0.6</td></tr><tr><td>Adv train</td><td>n/a</td><td>0.004</td><td>0.003</td><td>0.003</td><td>0.005</td><td>0.003</td><td>0.003</td><td>0.004</td><td>0.010</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A question arises as to what would happen if the procedure of 1) adversarial sample generation using the current classifier, and 2) retraining classifier using the current adversarial examples is repeated for many rounds. The answer to this cat-and-mouse game is easy to experiment although time-consuming. Let’s denote the attack on the original classifier as FGSM1, and the corresponding retrained classifier as Adv FGSM1. Repeating the procedure above generates the sequence of models $\\mathrm { F G S M 1 } \\to \\mathrm { A d v } \\ \\mathrm { F G S M 1 } \\to \\mathrm { F G S M 2 } \\to \\mathrm { A d v } \\ \\mathrm { F C }$ GSM2, etc. Fig. 1 shows one such trial with $8 0 +$ 80 rounds of the procedure. Initially, the attacker achieves near-perfect attacks (i.e., error rate $\\simeq 1$ ), and the defender achieves near-perfect defense (i.e., error rate $\\simeq 0$ ). As the iteration increases, the attacker becomes weaker with error rate $\\simeq 0 . 5$ , but the defense is still very successful, and the rate seems to oscillate persistently. While we can run more iterations to see if it converges, this is not a very principled nor efficient approach to find an equilibrium, if it exists. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/da60f7db9bd6b022adc7b3bbf142be95b5b59cdb173c6d20c6bf19032f4ee2e7.jpg",
+        "image_caption": [
+            "Figure 1: A cat-and-mouse game of FGSM attacks and adversarial training for MNIST. The upper red points are the error rates after adversarial training, and the lower green points are the error rates after FGSM attack $\\eta = 0 . 3 )$ . After 160 iterations, the error rate is still oscillating between 0 and 0.5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 GRADIENT-BASED ATTACKS AND SENSITIVITY PENALTY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We can perform the cat-and-mouse simulation more efficiently by an optimization approach. Instead of training the classifier fully with adversarial examples and then regenerating adversarial examples, suppose we only update the classifier with a single gradient-descent step then regenerate adversarial examples. To emphasize the parameters $u$ of the classifier/defender $g ( x ; u )$ , let’s rewrite the empirical risk of classifying the perturbed data as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f618e9f5c6322863f784cdc06c2f8152b296bc9f2da59be15873608ac238e8a7.jpg",
+        "text": "$$\nf ( u , Z ) \\triangleq \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } l ( g ( z ( x _ { i } ) ; u ) , y _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $z ( x )$ denote an FGSM-like attack based on the loss gradient ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aaaf0edcea2839f56c3855401a3bbbc4af2795d4fd743a43f5d1a9aa0b6dbcbc.jpg",
+        "text": "$$\n\\begin{array} { r } { z ( { \\boldsymbol x } ) \\gets { \\boldsymbol x } + \\eta \\nabla _ { z } l ( g ( z ( { \\boldsymbol x } ) ; { \\boldsymbol u } ) , y ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and $Z = ( z _ { 1 } , . . . , z _ { N } ) \\triangleq ( z ( x _ { 1 } ) , . . . , z ( x _ { N } ) )$ is the sequence of perturbed examples. In expectation of the attack, the defender should choose $u$ to minimize $f ( u , Z ( u ) )$ where the dependence of the attack on the classifier $u$ is expressed explicitly. If we minimize $f$ using gradient descent ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c428e338c043f4f1f54a55ee5dc2ef2835960d54a2ab31c78e626eabd21481a.jpg",
+        "text": "$$\nu  u - \\lambda \\frac { d f ( u , Z ) } { d u } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            152,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "then from the chain rule, the total derivative $\\textstyle { \\frac { d f } { d u } }$ is ",
+        "bbox": [
+            176,
+            190,
+            501,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d39790e7acc812c937f96fafaa891caee2dda2377bb2f1e46f5045a591b44ba3.jpg",
+        "text": "$$\n{ \\frac { d f } { d u } } = { \\frac { \\partial f } { \\partial u } } + { \\frac { \\partial Z } { \\partial u } } { \\frac { \\partial f } { \\partial Z } } = { \\frac { \\partial f } { \\partial u } } + \\sum _ { i } { \\frac { \\partial z _ { i } } { \\partial u } } { \\frac { \\partial f } { \\partial z _ { i } } } = { \\frac { \\partial f } { \\partial u } } + { \\frac { \\eta } { N } } \\sum _ { i } { \\frac { \\partial ^ { 2 } l } { \\partial z _ { i } \\partial u } } { \\frac { \\partial l } { \\partial z _ { i } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            263,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "from (3) and (4). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, this total derivative (6) at the current state coincides with the gradient $\\nabla _ { u }$ of the following cost ",
+        "bbox": [
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+            281,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17d8ee8c730a731fc5643461eaaeb2f34ade1c9acfc0f9aa38145562ed2494b3.jpg",
+        "text": "$$\nf _ { \\mathrm { s e n s } } ( u ) \\triangleq f ( u , Z ) + \\frac { \\gamma } { 2 } \\left\\| \\frac { \\partial f ( u , Z ) } { \\partial Z } \\right\\| ^ { 2 } = f ( u , Z ) + \\frac { \\eta } { 2 N } \\sum _ { i = 1 } ^ { N } \\left\\| \\frac { \\partial l ( g ( z _ { i } ; u ) , y _ { i } ) } { \\partial z _ { i } } \\right\\| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            315,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma = \\eta N$ . There are two implications. Interpretation-wise, this cost function is the sum of the original risk $f$ and the ‘sensitivity’ term $\\| \\partial f / \\partial Z \\| ^ { 2 }$ which penalizes abrupt changes of the risk w.r.t. the input. Therefore, $u$ is chosen at each iteration to not only decrease the risk but also to make the classifier insensitive to input perturbation so that the attacker cannot take advantage of large gradients. The idea of minimizing the sensitivity to input is a familiar approach in robustifying classifiers (Gu & Rigazio, 2014; Lyu et al., 2015). Secondly, the new formulation can be implemented easily. The gradient descent update using the seemingly complicated gradient (6) can be replaced by the gradient descent update of (7). The capability of automatic differentiation (Rall, 1981) in modern machine learning libraries can be used to compute the gradient of (7) efficiently. Using this direct approach, we can find the defense parameters $u$ which will be robust to gradient-based attacks. Fig. 2 shows the decrease of test error during training using the this gradient descent approach for MNIST. It only takes a very small fraction of time to reach the final states of the Fig. 2 compared to that of Fig. 1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d60a9f66246467edbdc5b42d1f2c5b63e1328b9fe50e235eff32588b2fd4e1a.jpg",
+        "image_caption": [
+            "Figure 2: Convergence of test error rates for sensitivity-penalized optimization (7) with MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            560,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "There is also an important difference between the solution of the cat-and-mouse game and the minimizer of (7). Table 3 shows that the adversarially trained classifier (Adv FGSM1) is robust to both clean data and FGSM1 attack, but is susceptible to FGSM2 attack, displaying the cat-and-mouse nature. The same holds for Adv FGSM2, Adv FGSM3, etc. After 80 rounds of the cat-and-mouse procedure, the classifier Adv FGSM80 becomes robust to FGSM80 as well as moderately robust to other attacks including FGSM81 $\\circleddash$ FGSM-curr). However, the classifier Sens FGSM from direct minimization of (7) is even more robust toward FGSM-curr than Adv FGSM80 and is overall the best. To see the advantage of the sensitivity term in (7), we also performed the minimization of (7) without the sensitivity term under the same conditions as Sens FGSM. This optimization method is similar to the method proposed in Huang et al. (2015), referred to as Learning with Adversaries (LWA FGSM). In the table, one can see that Sens FGSM is also better than LWA FGSM overall, although the difference is small. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that Sens FGSM is better than other adversarially-trained classifiers, it too is still vulnerable to attacks such as FGSM80. This vulnerability raises the question if it is possible to make a classifier robust to any type of attacks, or more practically, robust to at least a large class of attacks. We discuss this issue in the next section. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/4b21994081895c9731eb34cc38b7dd7573c4aecfee8cfbfa9ecd93d6296def02.jpg",
+        "table_caption": [
+            "Table 3: Error rates of different attacks on various adversarially-trained classifiers for MNIST. FGSM-curr means the FGSM attack on the specific classifier on the left. Adv FGSM is the classifier adversarially trained with FGSM attacks. Sens FGSM is the result of minimizing (7) by gradient descent (5). LWA FGSM is the result of minimizing (7) without the gradient-norm term. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Defense\\Attack</td><td rowspan=\"2\">No attack</td><td colspan=\"3\">FGSM</td><td rowspan=\"2\">FGSM-curr</td></tr><tr><td>FGSM1</td><td>FGSM2 ：</td><td>FGSM80</td></tr><tr><td rowspan=\"5\">n=0.3</td><td>No defense Adv FGSM1</td><td>0.026 0.012</td><td>1.000 0.004</td><td>0.881 0.995</td><td>：</td><td>0.355 1.000</td></tr><tr><td></td><td></td><td>0.999</td><td>：</td><td>0.499</td><td>0.995</td></tr><tr><td>Adv FGSM2</td><td>0.012</td><td></td><td>0.002 ：</td><td>0.505</td><td>0.995</td></tr><tr><td>AdvFGSM80</td><td>0.009</td><td>0.335</td><td>0.273 ：</td><td>0.009</td><td>0.442</td></tr><tr><td>LWAFGSM Sens FGSM</td><td>0.008</td><td>0.121</td><td>0.188 ：</td><td>0.210 0.194</td><td>0.048 0.048</td></tr><tr><td rowspan=\"6\">m=0.4</td><td>No defense</td><td>0.009 0.026</td><td>0.104 1.000</td><td>0.176 0.944</td><td>： ：</td><td>0.528 1.000</td></tr><tr><td>AdvFGSM1</td><td>0.013</td><td>0.003</td><td>0.984</td><td>0.589 ：</td><td>0.984</td></tr><tr><td>AdvFGSM2</td><td>0.017</td><td>0.999</td><td>0.005 ：</td><td>0.549</td><td>0.999</td></tr><tr><td>AdvFGSM80</td><td>0.009</td><td>0.509</td><td>0.525</td><td>： 0.024</td><td>0.131</td></tr><tr><td>LWAFGSM</td><td>0.009</td><td>0.204</td><td>0.284</td><td>： 0.336</td><td>0.043</td></tr><tr><td>Sens FGSM</td><td>0.009</td><td>0.128</td><td>0.234</td><td>· 0.296</td><td>0.038</td></tr><tr><td rowspan=\"6\">m=0.5</td><td>No defense</td><td>0.026</td><td>1.000</td><td>0.931</td><td>：</td><td>0.662</td><td>1.000</td></tr><tr><td>AdvFGSM1</td><td>0.010</td><td>0.002</td><td>0.970</td><td>：</td><td>0.724</td><td>0.970</td></tr><tr><td>Adv FGSM2</td><td>0.010</td><td>0.866</td><td>0.006</td><td>：</td><td>0.604</td><td>0.871</td></tr><tr><td>AdvFGSM80</td><td>0.008</td><td>0.653</td><td>0.559</td><td>：</td><td>0.023</td><td>0.089</td></tr><tr><td>LWAFGSM</td><td>0.009</td><td>0.248</td><td>0.260</td><td>：</td><td>0.432</td><td>0.035</td></tr><tr><td>Sens FGSM</td><td>0.009</td><td>0.266</td><td>0.285</td><td>：</td><td>0.365</td><td>0.039</td></tr><tr><td rowspan=\"6\">n=0.6</td><td>No defense</td><td>0.026</td><td>1.000</td><td>0.963</td><td>：</td><td>0.803</td><td>1.000</td></tr><tr><td>AdvFGSM1</td><td>0.012</td><td>0.003</td><td>0.889</td><td>：</td><td>0.790</td><td>0.889</td></tr><tr><td>Adv FGSM2</td><td>0.008</td><td>0.649</td><td>0.007</td><td>：</td><td>0.687</td><td>0.767</td></tr><tr><td>AdvFGSM80</td><td>0.009</td><td>0.439</td><td>0.426</td><td>：</td><td>0.020</td><td>0.021</td></tr><tr><td>LWAFGSM</td><td>0.011</td><td>0.317</td><td>0.315</td><td>：</td><td>0.488</td><td>0.034</td></tr><tr><td>Sens FGSM</td><td>0.010</td><td>0.264</td><td>0.244</td><td>：</td><td>0.465</td><td>0.033</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 GAME FORMULATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider the class of optimization-based attack and the class of neural-network based attacks as an approximation of the former. Using the neural-network based attack class, we formulate the attacker-defender dynamics as a game and discuss two types of equilibria – the minimax and the maximin solutions. We present algorithms that generalize the approach presented in the previous section. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 LEARNING-BASED ATTACK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "An attacker $z ( x ) : \\mathcal { X }  \\mathcal { X }$ can be more general than a specific class of attacks such as FGSM. Again, let $g : \\mathcal { X }  \\mathcal { Y }$ is a classifier parameterized by $u$ and $l ( g ( x ; u ) , y )$ is a loss function. If time complexity is not an issue, the following optimization-based attack (Huang et al., 2015) ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c61b4b1425c5d1b499893c1ee84a0ca194ab2ed211e7ced7083e8ba394786c1d.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { Z = ( z _ { 1 } , \\ldots , z _ { N } ) } \\left[ f ( u , Z ) \\triangleq \\frac { 1 } { N } \\sum _ { i } l ( g ( z _ { i } ; u ) , y _ { i } ) \\right] = \\frac { 1 } { N } \\sum _ { i } \\operatorname* { m a x } _ { z _ { i } } \\ l ( g ( z _ { i } ; u ) , y _ { i } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which is also related to the CW attack (Carlini & Wagner, 2017), can generate strong adversarial examples, where adversarial patterns $Z = ( z _ { 1 } , . . . , z _ { N } )$ are unrestricted except for the bounds such as $\\| z _ { i } - x _ { i } \\| _ { p } \\leq \\eta$ . The corresponding class of adversarial patterns $Z$ is very large, which results in strong but non-generalizable adversarial examples. Non-generalizable means the perturbation $z ( x )$ has to be recomputed for every new test sample $x$ . While the class of optimization-based attacks is powerful, its large size makes it difficult to analytically study the optimal defense methods. To make the problem learnable, we restrict the class of patterns $Z$ to that which can be generated by a flexible but manageable class of perturbation $\\{ z ( \\cdot ; v ) \\mid \\forall v \\in V \\}$ , e.g., an autoencoder of a fixed architecture where the parameter $v$ is the network weights. This class is a clearly an approximation to the class of full optimization-based attacks, but is generalizable, i.e., no time-consuming optimization is required in the test phase but only single feedforward passes. The attack network (AttNet), as we call it, can be of any class of appropriate neural networks. Here we use a three-layer fully-connected network with 300 hiddens units per layer in this paper. Different from Nguyen & Sinha (2017) or Baluja & Fischer (2017), we feed the label $y$ into the input of the network along with the features $x$ . This is analogous to using the true label $y$ in the original FGSM. While this label input is optional but it can make the training of the attacker network easier. As with other attacks, we impose the $l _ { \\infty }$ -norm constraint on $z$ , i.e., $\\| z ( x ) - x \\| _ { \\infty } \\leq \\eta$ . ",
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+        "type": "text",
+        "text": "Suppose now $f ( u , v )$ is the empirical risk of a classifier-attacker pair where the input $x$ is first transformed by attack network $z ( x ; v )$ and then fed to the classifier $g ( z ( x ; v ) ; u )$ . The attack network can be trained by gradient descent as well. Given a classifier $u$ , we can use gradient descent ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/568704f18005db088c80e92dca454ff83663c8fad4433282fa2700015e9acea5.jpg",
+        "text": "$$\nv  v + \\sigma { \\frac { \\partial f ( u , v ) } { \\partial v } }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "to find an optimal attacker $v$ that maximizes the risk $f$ assuming the classifier $u$ is fixed. Table 4 compares the error rates of the FGSM attacks and the attack network (AttNet). The table shows that AttNet is better than or comparable to FGSM in all cases. In particular, we already observed that the FGSM attack is no more effective against the classifier hardened against gradient-based attacks (Adv FGSM80 or Sens FGSM), but the AttNet can incur significant error $( > \\sim 0 . 9 )$ for those hardened defenders. This indicates that the class of learning-based attacks is indeed different from the class of gradient-based attacks. ",
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+    {
+        "type": "table",
+        "img_path": "images/4e362c1d948f8b5cbafeb904b26919a7028880a023240bdff465e4c53968ac8b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan=\"2\">n=0.3</td><td colspan=\"2\">n=0.4</td></tr><tr><td>No defense</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>AdvFGSM1</td><td>0.996</td><td>1.000</td><td>0.984</td><td>1.000</td></tr><tr><td>AdvFGSM80</td><td>0.473</td><td>0.899</td><td>0.131</td><td>0.903</td></tr><tr><td>Sens FGSM</td><td>0.048</td><td>0.965</td><td>0.038</td><td>0.902</td></tr><tr><td rowspan=\"4\">No defense Adv FGSM1 AdvFGSM80</td><td colspan=\"2\">m=0.5</td><td colspan=\"2\">m=0.6</td></tr><tr><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>0.985</td><td>1.000</td><td>0.966</td><td>1.000</td></tr><tr><td>0.089</td><td>0.897 1.000</td><td>0.021</td><td>0.897</td></tr><tr><td rowspan=\"2\">Sens FGSM</td><td colspan=\"2\">0.039</td><td colspan=\"2\">0.033 0.903</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 4: Error rates of FGSM vs learning-based attack network (AttNet) on various adversariallytrained classifiers for MNIST. FGSM-curr/AttNet-curr means they are computed/trained for the specific classifier on the leftmost column. Note that FGSM fails to attack hardened networks (Adv FGSM80 and Sens FGSM), whereas AttNet can still attack them successfully. ",
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+    {
+        "type": "text",
+        "text": "4.2 MINIMAX GAME FOR LEARNING-BASED ATTACKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Finally, we consider the dynamics of the pair of classifier-attacker when each player can change its parameters. Given the current classifier $u$ , an optimal whitebox attacker parameter $v$ is the maximizer of the risk $f ( u , v )$ ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/abda00d2f2aa78c8ec1257d5f2630101d6a8543b6ff5d006066fcdd42d27be7a.jpg",
+        "text": "$$\nv ^ { * } ( u ) \\triangleq \\arg \\operatorname* { m a x } _ { v } f ( u , v ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Consequently, the defender should choose the classifier parameters $u$ such that the maximum risk is minimized ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3591a0c32e527f1326da459016c3149fb64cb1d322df9501f62495e2ce7d864f.jpg",
+        "text": "$$\nu ^ { * } \\triangleq \\arg \\operatorname* { m i n } _ { u } \\operatorname* { m a x } _ { v } f ( u , v ) = \\arg \\operatorname* { m i n } _ { u } f ( u , v ^ { * } ( u ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This solution to the continuous minimax problem has a natural interpretation as the best worst-case solution. Assuming the attacker is optimal, i.e., it chooses the best attack from (10) given $u$ , no other defense can achieve a lower risk than the minimax defense $u ^ { * }$ in (11). The minimax defense is also a conservative defense. If the attacker is not optimal, and/or if the attack does not know the defense $u$ exactly (as in blackbox attacks), the actual risk can be lower than what the minimax solution $f ( u ^ { * } , v ^ { * } ( u ^ { * } ) )$ predicts. Before proceeding further, we point out that the claims above apply to the global minimizer $u ^ { * }$ and the maximizer function $v ^ { \\ast } ( \\cdot )$ , but in practice we can only find local solutions for complex risk functions of deep classifiers and attackers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To solve (11), we analyze the problem similarly to (5)-(7) from the previous section. At each iteration, the defender should choose $u$ in expectation of the attack and minimize $f ( u , v ^ { * } ( u ) )$ . We use ",
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+    {
+        "type": "text",
+        "text": "gradient descent ",
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+    {
+        "type": "equation",
+        "img_path": "images/8408588fb63d97e2a23750d31b06dc979d87743e54ff87543aaf27503f46f6ae.jpg",
+        "text": "$$\nu  u - \\lambda \\frac { d f ( u , v ^ { * } ( u ) ) } { d u } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where the total derivative $\\textstyle { \\frac { d f } { d u } }$ is ",
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+    {
+        "type": "equation",
+        "img_path": "images/0c6d8f778d027fa1e57203f07a6ef88c61e1ecb9b90c361e8d80d96e84dbf2d4.jpg",
+        "text": "$$\n\\frac { d f } { d u } = \\frac { \\partial f ( u , v ^ { * } ( u ) ) } { \\partial u } + \\frac { \\partial v ^ { * } ( u ) } { \\partial u } \\frac { \\partial f ( u , v ) } { \\partial v } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Since the exact maximizer $v ^ { * } ( u )$ is difficult to find, we only update $v$ incrementally by one (or more) steps of gradient-ascent update ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/04abe8a1d6f7d7ae70cdcd35615888684727ba2974bc65eb980effb4a1618dc1.jpg",
+        "text": "$$\nv  v + \\sigma \\frac { \\partial f ( u , v ) } { \\partial v } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The resulting formulation is closely related to the unrolled optimization (Metz et al., 2016) proposed for training GANs, although the latter has a very different cost function $f$ . Using the single update (14), the total derivative is ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4b1b9e328efcfe6296a09fceced6bb261ba8d1e08812b1205a01db60d5af8fb5.jpg",
+        "text": "$$\n\\frac { d f } { d u } = \\frac { \\partial f ( u , v ^ { * } ( u ) ) } { \\partial u } + \\sigma \\frac { \\partial ^ { 2 } f ( u , v ) } { \\partial u \\partial v } \\frac { \\partial f ( u , v ) } { \\partial v } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Similar to hardening a classifier against gradient-based attacks by minimizing (7) at each iteration, the gradient update of $u$ for $f ( u , v )$ can be done using the gradient of the following sensitivitypenalized function ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/02b9234e9b7fc236058a2c414896dc8163be554267595dd8a22727352f48daa8.jpg",
+        "text": "$$\nf _ { \\mathrm { s e n s } } ( u ) \\triangleq f ( u , v ) + { \\frac { \\sigma } { 2 } } \\left\\| { \\frac { \\partial f ( u , v ) } { \\partial v } } \\right\\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In other words, $u$ is chosen not only to minimize the risk but also to prevent the attacker from exploiting the sensitivity of $f$ to $v$ . The algorithm is summarized in Alg. 1. ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Minimax Optimization by Sensitivity Penalization ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/a19d8d885972b682b106fde6c2846722aa774b01498d1c23cd261e1e270f20b8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input: risk f(u,v),#of iterations T,learning rates (oi),(入i),(Yi)</td></tr><tr><td>Output: (u*,u*(u*))</td></tr><tr><td>Initialize uo,Uo Begin</td></tr><tr><td>for i=1, ...,T do</td></tr><tr><td>Max step:Ui=Ui-1+Oi af(ui-1,Ui-1) du</td></tr><tr><td>a Yi-1 of(ui-1,Ui-1 Min step: Ui = Ui-1- Xi f(ui-1,Ui-1)+</td></tr><tr><td>du 2 du</td></tr><tr><td>end for Return (UT,UT).</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that this algorithm is actually independent of the adversarial example problem, and can be used for other minimax problems as well. ",
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+    {
+        "type": "text",
+        "text": "4.3 MINIMAX VS MAXIMIN PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In analogy with the minimax problem, we can also consider the maximin solution defined by ",
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+    {
+        "type": "equation",
+        "img_path": "images/3cf2780c689d2a1fcd0fadbee49705076488fc818b583a716fac7cad77a2b68b.jpg",
+        "text": "$$\nv ^ { * } \\triangleq \\arg \\operatorname* { m a x } _ { v } \\operatorname* { m i n } _ { u } f ( u , v ) = \\arg \\operatorname* { m a x } _ { v } f ( u ^ { * } ( v ) , v ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d0f4652784378d25da69a61ab3eaba14011656e43cc34940f4a25e1b569eda9.jpg",
+        "text": "$$\nu ^ { * } ( v ) \\triangleq \\arg \\operatorname* { m i n } _ { u } f ( u , v )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "is the minimizer function. Here we are abusing the notations for the minimax solution $u ^ { * }$ , the maximin solution $v ^ { * }$ , the minimizer $u ^ { * } ( \\cdot )$ , and the maximizer $v ^ { \\ast } ( \\cdot )$ . Similar to the minimax solution, the maximin solution has an intuitive meaning – it is the best worst-case solution for the attacker. Assuming the defender is optimal, i.e., it chooses the best defense from (18) that minimizes the risk $f ( u , v )$ given the attack $v$ , no other attack can inflict a higher risk than the maximin attack $v ^ { * }$ . It is also a conservative attack. If the defender is not optimal, and/or if the defender does not know the attack $v$ exactly, the actual risk can be higher than what the solution $f ( u ^ { * } ( v ^ { * } ) , v ^ { * } )$ predicts. Note that the maximin scenario where the defender knows the attack method is not very realistic but is the opposite of the minimax scenario and provides the lower bound. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "To summarize, minimax and maximin defenses and attacks have the following inherent properties. ",
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+    {
+        "type": "text",
+        "text": "Lemma 1. Let $u ^ { * } , v ^ { * } ( u ) , v ^ { * } , u ^ { * } ( v )$ be the solutions of $( I I ) , ( I O ) , ( I 7 ) , ( I 8 ) .$ . ",
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+    {
+        "type": "text",
+        "text": "1. $f ( u , v ^ { * } ( u ) ) \\geq f ( u , v )$ : For any given defense $u$ , the max attack $v ^ { * } ( u )$ is the most effective attack.   \n2. $f ( u ^ { * } , v ^ { * } ( u ^ { * } ) ) \\leq f ( u , v ^ { * } ( u ) )$ : Against the optimal attack $v ^ { * } ( u )$ , the minimax defense $u ^ { * }$ is the most effective defense.   \n3. $f ( u ^ { * } ( v ) , v ) \\leq f ( u , v )$ : For any given attack $v$ , the min defense $u ^ { * } ( v )$ is the most effective defense.   \n4. $f ( u ^ { * } ( v ) , v ^ { * } ) \\geq f ( u ^ { * } ( v ) , v )$ : Against the optimal defense $u ^ { * } ( v )$ , the maximin attack $v ^ { * }$ is the most effective attack.   \n5. $\\begin{array} { r } { \\operatorname* { m a x } _ { v } \\operatorname* { m i n } _ { u } f ( u , v ) \\leq \\operatorname* { m i n } _ { u } \\operatorname* { m a x } _ { v } f ( u , v ) . } \\end{array}$ : The risk of the best worst-case attack is lower than that of the best worst-case defense. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These properties follow directly from the definitions. The lemma helps us to better understand the dependence of defense and attack, and gives us the range of the possible risk values which can be measured empirically. To find maximin solutions, we use the same algorithm (Alg. 1) except that the variables $u$ and $v$ are switched and the sign of $f$ is flipped before the algorithm is called. ",
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+    {
+        "type": "text",
+        "text": "4.4 EXPERIMENTS ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In addition to minimax and maximin optimization, we also consider as a reference algorithm the alternating descent/ascent method used in GAN training Goodfellow et al. (2014a) ",
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/147e2ee2190ccf320aae1d447badbf4e4464176c5ea733e0e9d1bbbe1aeca6f5.jpg",
+        "text": "$$\nu  u - \\lambda \\frac { \\partial f } { \\partial u } , \\quad v  v + \\sigma \\frac { \\partial f } { \\partial v } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that alternating descent/ascent finds local saddle points which are not necessarily minimax or maximin solutions, and therefore its solution will in general be different from the solution from Alg. 1. The difference of the solutions from three optimizations – Minimax, Maximin, and Alternating descent/ascent (Alt) – applied to a common problem, is demonstrated in Fig. 3. The figure shows the test error over the course of optimization starting from random initializations. One can see that Minimax (top blue curves) and Alt (middle green curves) converge to different values suggesting the learned classifiers will also be different. ",
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+    {
+        "type": "image",
+        "img_path": "images/d6e579cfdb700ba5473f7cc533f56179bcabe9de8be7ad689d37ca03e59cf7fd.jpg",
+        "image_caption": [
+            "Figure 3: Convergence of the test error rates for Minimax optimization (blue), Alternating ascent/descent (green), and Maximin optimization (red) for MNIST. "
+        ],
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+    {
+        "type": "text",
+        "text": "Table 5 compares the robustness of the classifiers trained by Minimax and Alt against the AttNet attack (1st/2nd rows and 2nd column for each $\\eta$ .) Minimax defense is more robust than Alt defense at $\\eta = 0 . 3$ (0.020 vs 0.104) and at $\\eta = 0 . 4$ (0.552 vs 0.873). For larger $\\eta$ ’s, both are unusably vulnerable. Different performance of the two classifiers implies that the minimax solution found by Alg. 1 is different from the local saddle point found by alternating descent/ascent. In addition, against FGSM attacks, Minimax is moderately robust $( 0 . 2 1 8 - 0 . 3 4 2 )$ despite that the classifiers are not specifically trained against gradient-based attacks. In contrast, Sens FGSM is very vulnerable (0.902 – 1.000) against AttNet which we have already observed. This result suggests that the class of AttNet attacks and the class of gradient-based attacks are indeed different, and the former class is larger than the latter. ",
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+        "table_caption": [
+            "Table 5: Error rates of Minimax-, Alt-, and adversarially-trained (Sens FGSM) classifiers for MNIST. Minimax is overall better than Alt against AttNet-curr, and is also moderately robust against the out-of-class attack (FGSM-curr). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan=\"2\">m=0.3</td><td colspan=\"2\">n=0.4</td></tr><tr><td>Minimax</td><td>0.218</td><td>0.020</td><td>0.238</td><td>0.552</td></tr><tr><td>Alt</td><td>0.244</td><td>0.104</td><td>0.503</td><td>0.873</td></tr><tr><td>Sens FGSM</td><td>0.048</td><td>0.965</td><td>0.038</td><td>0.902</td></tr><tr><td rowspan=\"3\">Minimax Alt</td><td>m=0.5</td><td></td><td>m=0.6</td><td></td></tr><tr><td>0.342</td><td>1.000</td><td>0.299</td><td>1.000</td></tr><tr><td>0.289</td><td>0.902</td><td>0.157</td><td>0.899</td></tr><tr><td>Sens FGSM</td><td>0.039</td><td>1.000</td><td>0.033</td><td>0.903</td></tr></table>",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lastly, the adversarial examples generated by various attacks in the paper have diverse patterns and are shown in Fig. 4 of the appendix. ",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 ROBUSTNESS AGAINST MULTIPLE ATTACK TYPES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We discuss some limitations of the framework and also propose an extension. Ideally, a defender should find a robust classifier against the worst attack from a very large class of attacks such as optimization-based attacks. However, it is difficult to train classifiers against attacks from a large class. On the other hand, if the class is too small, then the worst attack from that class is not representative of all possible worst attacks, and therefore the minimax defense found will not be robust to out-of-class attacks. The trade-off seems inevitable. ",
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+    },
+    {
+        "type": "text",
+        "text": "It is, however, possible to build a defense against multiple specific types of attacks. Suppose $z _ { 1 } ( u ) , . . . , z _ { m } ( u )$ are $m$ different types of attacks, e.g., $z _ { \\mathrm { 1 } } \\mathrm { = F G S M }$ , $z _ { \\mathrm { 2 } } { = } \\mathrm { I F G S M }$ , etc. The minimax defense for the combined attack is the solution to the mixed continuous-discrete problem ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a810ae2dd4c734c8b67a09d862522c09079497b8185dcd881090e50216b734e0.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { u } \\operatorname* { m a x } \\{ f ( u , z _ { 1 } ( u ) ) , . . . , f ( u , z _ { m } ( u ) ) \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Additionally, suppose $z _ { m + 1 } ( u , v ) , . . . , z _ { m + n } ( u , v )$ are $n$ different types of learning-based attacks, e.g., $z _ { m + 1 } = 2$ -layer dense net, $z _ { m + 2 } = 5$ -layer convolutional nets, etc. The minimax defense against the mixture of multiple fixed-type and learning-based attacks can be found by solving ",
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/507b5a5e542f60ddeb4b9887d8ac22c243f0d2cc509730d509e7828835464155.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { u } \\operatorname* { m a x } \\{ f ( u , z _ { 1 } ( u ) ) , \\dots , f ( u , z _ { m } ( u ) ) , \\operatorname* { m a x } _ { v } f ( u , z _ { m + 1 } ( u , v ) ) , \\dots , \\operatorname* { m a x } _ { v } f ( u , z _ { m + n } ( u , v ) ) \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Due to the huge computational demand to solve (21), we leave it as a future work. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 ADVERSARIAL EXAMPLES AND PRIVACY ATTACKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Lastly, we discuss a bigger picture of the game between adversarial players. The minimax optimization arises in the leader-follower game (Bruckner & Scheffer, 2011) with the constant sum constraint. ¨ The leader-follower setting makes sense because the defense $=$ classifier parameters) is often public knowledge and the attacker exploits the knowledge. Interestingly, the problem of the attack on privacy (Hamm, 2016) has a very similar formulation as the adversarial attack problem, different only in that the classifier is an attacker and the data perturbator is a defender. In the problem of privacy preservation against inference, the defender is a data transformer $z ( x )$ (parameterized by $u$ ) which perturbs the raw data, and the attacker is a classifier (parameterized by $v$ ) who tries to extract sensitive information such as identity from the perturbed data such as online activity of a person. The transformer is the leader, such as when the privacy mechanism is public knowledge, and the classifier is the follower as it attacks the given perturbed data. The risk for the defender is therefore the accuracy of the inference of sensitive information measured by $- E [ l ( \\boldsymbol { z } ( \\boldsymbol { x } ; \\boldsymbol { u } ) , \\boldsymbol { y } ; \\boldsymbol { v } ) ]$ . Solving the minimax risk problem $\\begin{array} { r l } {  { ( \\operatorname* { m i n } _ { u } \\operatorname* { m a x } _ { v } - E [ l ( z ( x ; u ) , y ; v ) ] ) } \\quad } & { { } } \\end{array}$ gives us the best worst-case defense when the classifier/attacker knows the transformer/defender parameters, which therefore gives us a robust data transformer to preserve the privacy against the best inference attack (among the given class of attacks.) On the other hand, solving the maximin risk problem $\\begin{array} { r l } { } & { { } ( \\operatorname* { m a x } _ { v } \\operatorname* { m i n } _ { u } - \\bar { E } [ l ( \\bar { z ( x ; u ) } , y ; v ) ] ) } \\end{array}$ gives us the best worst-case classifier/attacker when its parameters are known to the transformer. As one can see, the problems of adversarial attack and privacy attack are two sides of the same coin which can be addressed by similar frameworks and optimization algorithms. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            304,
+            318,
+            320
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a continuous game formulation of adversarial attacks and defenses using a learning-based attack class implemented by neural networks. We show that this class of attacks is quite different from the gradient-based attacks. While a classifier robust to all types of attack may yet be an elusive goal, the minimax defense against the neural network-based attack class is well-defined and practically achievable. We show that the proposed optimization method can find minimax defenses which are more robust than adversarially-trained classifiers and the classifiers from simple alternating descent/ascent. We demonstrate these with MNIST and CIFAR-10. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            454,
+            285,
+            468
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Shumeet Baluja and Ian Fischer. Adversarial transformation networks: Learning to generate adversarial examples. arXiv preprint arXiv:1703.09387, 2017.   \nMichael Bruckner and Tobias Scheffer. Stackelberg games for adversarial prediction problems. In ¨ Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 547–555. ACM, 2011.   \nNicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In Security and Privacy (SP), 2017 IEEE Symposium on, pp. 39–57. IEEE, 2017.   \nNilesh Dalvi, Pedro Domingos, Sumit Sanghai, Deepak Verma, et al. Adversarial classification. In Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 99–108. ACM, 2004.   \nAlhussein Fawzi, Omar Fawzi, and Pascal Frossard. Analysis of classifiers’ robustness to adversarial perturbations. arXiv preprint arXiv:1502.02590, 2015.   \nAmir Globerson and Sam Roweis. Nightmare at test time: robust learning by feature deletion. In Proceedings of the 23rd international conference on Machine learning, pp. 353–360. ACM, 2006.   \nIan Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems, pp. 2672–2680, 2014a.   \nIan J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014b.   \nShixiang Gu and Luca Rigazio. Towards deep neural network architectures robust to adversarial examples. arXiv preprint arXiv:1412.5068, 2014.   \nJihun Hamm. Minimax filter: Learning to preserve privacy from inference attacks. arXiv preprint arXiv:1610.03577, 2016.   \nRuitong Huang, Bing Xu, Dale Schuurmans, and Csaba Szepesvari. Learning with a strong adversary. ´ arXiv preprint arXiv:1511.03034, 2015.   \nAlexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial examples in the physical world. arXiv preprint arXiv:1607.02533, 2016a.   \nAlexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial machine learning at scale. arXiv preprint arXiv:1611.01236, 2016b.   \nGert RG Lanckriet, Laurent El Ghaoui, Chiranjib Bhattacharyya, and Michael I Jordan. A robust minimax approach to classification. Journal of Machine Learning Research, 3(Dec):555–582, 2002.   \nJiajun Lu, Theerasit Issaranon, and David Forsyth. Safetynet: Detecting and rejecting adversarial examples robustly. arXiv preprint arXiv:1704.00103, 2017.   \nChunchuan Lyu, Kaizhu Huang, and Hai-Ning Liang. A unified gradient regularization family for adversarial examples. In Data Mining (ICDM), 2015 IEEE International Conference on, pp. 301–309. IEEE, 2015.   \nDongyu Meng and Hao Chen. Magnet: a two-pronged defense against adversarial examples. arXiv preprint arXiv:1705.09064, 2017.   \nLuke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163, 2016.   \nJan Hendrik Metzen, Tim Genewein, Volker Fischer, and Bastian Bischoff. On detecting adversarial perturbations. arXiv preprint arXiv:1702.04267, 2017.   \nSeyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, Omar Fawzi, and Pascal Frossard. Universal adversarial perturbations. arXiv preprint arXiv:1610.08401, 2016.   \nSeyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, Omar Fawzi, Pascal Frossard, and Stefano Soatto. Analysis of universal adversarial perturbations. arXiv preprint arXiv:1705.09554, 2017.   \nLinh Nguyen and Arunesh Sinha. A learning approach to secure learning. arXiv preprint arXiv:1709.04447, 2017.   \nNicolas Papernot, Patrick McDaniel, Xi Wu, Somesh Jha, and Ananthram Swami. Distillation as a defense to adversarial perturbations against deep neural networks. In Security and Privacy (SP), 2016 IEEE Symposium on, pp. 582–597. IEEE, 2016.   \nLouis B Rall. Automatic differentiation: Techniques and applications. 1981.   \nChristian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.   \nFlorian Tramer, Alexey Kurakin, Nicolas Papernot, Dan Boneh, and Patrick McDaniel. Ensemble adversarial \\` training: Attacks and defenses. arXiv preprint arXiv:1705.07204, 2017. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A RESULTS WITH MNIST ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The architecture of the MNIST classifier is similar to the Tensorflow model 2, and is trained with the following hyperparameters: $\\{ B a t c h s i z e = I 2 8 \\}$ , optimizer $=$ AdamOptimizer with $\\lambda = 1 0 ^ { - 4 }$ , total # of iteration $\\scriptstyle : = 5 0 , 0 0 0 . \\}$ ",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The attack network has three hidden fully-connected layers of 300 units, trained with the following hyperparameters:   \n{Batch $s i z e ~ = ~ I 2 8$ , dropout rate $= ~ 0 . 5$ , optimizer $=$ AdamOptimizer with $1 0 ^ { - 3 }$ , total $\\#$ of iteration $\\scriptstyle : = 3 0 , 0 0 0 . \\}$ ",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For minimax, alt, and maximin optimization, the total number of iteration was 100,000. The sensitivity-penalty coefficient of $\\gamma = 1$ was used in Alg. 1. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/46763f54f338dd040d61c182c0c65ee517429ea079f203508ac7da6af10f2095.jpg",
+        "image_caption": [
+            "Figure 4: Adversarial samples generated from different attacks at $\\eta = 0 . 2$ . (a) Original data (b) FGSM1 (c) FGSM80 (d) IFGSM1 (e) Minimax (f) Alt (g) Maximin. Note the diversity of patterns. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "B RESULTS WITH CIFAR-10 ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We preprocess the CIFAR-10 dataset by removing the mean and normalizing the pixel values with the standard deviation of all pixels in the image. It is followed by clipping the values to $\\pm 2$ standard deviations and rescaling to $[ - 1 , 1 ]$ . The architecture of the CIFAR classifier is similar to the Tensorflow model 3 but is simplified further by removing the local response normalization layers. With the simple structure, we attained $\\sim 7 8 \\%$ accuracy with the test data. The classifier is trained with the following hyperparameters: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\{ B a t c h s i z e = I 2 8 ,$ , optimizer $=$ AdamOptimizer with $\\lambda = 1 0 ^ { - 4 }$ , total # of iteration $\\scriptstyle : = I O O , O O O .  \\}$ ",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The attack network has three hidden fully-connected layers of 300 units, trained with the following hyperparameters:   \n$\\{ B a t c h \\ s i z e \\ = \\ I 2 8 ,$ , dropout rate $= 0 . 5$ , optimizer $=$ AdamOptimizer with $\\sigma = 1 0 ^ { - 3 }$ , total # of iteration $\\scriptstyle : = 3 0 , 0 0 0 . \\}$ ",
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+    {
+        "type": "text",
+        "text": "For minimax, alt, and maximin optimization, the total number of iteration was 100,000. The sensitivity-penalty coefficient of $\\gamma = 1$ was used in Alg. 1. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the rest of the appendix, we repeat all the experiments with the MNIST dataset using the CIFAR10 dataset. ",
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+        "type": "table",
+        "img_path": "images/86653185356d02e5af803f0548c8400c789ba0baacade8ea1ba9cf01c2f4ac2b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td rowspan=\"2\">No attack</td><td colspan=\"4\">FGSM</td><td colspan=\"4\">IFGSM</td></tr><tr><td>n=0.1</td><td>n=0.2</td><td>n=0.3</td><td>m=0.4</td><td>n=0.1</td><td>m=0.2</td><td>n=0.3</td><td>m=0.4</td></tr><tr><td>No defense</td><td>0.222</td><td>0.976</td><td>0.825</td><td>0.869</td><td>0.884</td><td>0.668</td><td>0.907</td><td>0.959</td><td>0.971</td></tr></table>",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Error rates of FGSM and IFGSM attacks on the original classifier for cifar10. These attacks can cause large misclassification for the given range of $\\eta$ . ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/3600f6849ae75ac79029f60ed2e1db6fbdd495786a5561c401f2ac5ae72649d8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td rowspan=\"2\">No attack</td><td colspan=\"4\">FGSM</td><td colspan=\"4\">IFGSM</td></tr><tr><td>n=0.1</td><td>n=0.2</td><td>n=0.3</td><td>n=0.4</td><td>n=0.1</td><td>n=0.2</td><td>m=0.3</td><td>n=0.4</td></tr><tr><td>Adv train</td><td>n/a</td><td>0.196</td><td>0.642</td><td>0.668</td><td>0.702</td><td>0.373</td><td>0.658</td><td>0.741</td><td>0.750</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 7: Error rates of FGSM and IFGSM attacks on the adversarially-trained classifiers for CIFAR10. This defense can significantly lower the errors from the attacks, although not as low as the MNIST problem. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/25c2a7b8f76e7f3af510741ee0bca0ba313bdf0750f34346abc872740838414e.jpg",
+        "image_caption": [
+            "Figure 5: Cat and mouse game of FGSM attacks and adversarial training for CIFAR-10. The upper green points are the error rates after adversarial training, and the lower orange points are the error rates after FGSM attack. After 160 iterations $\\eta = 0 . 3 )$ , the error rate is still oscillating. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/ab800da7f6508864830e13eb093463215686949fbfe0fc6bc9669047e0667041.jpg",
+        "image_caption": [
+            "Figure 6: Convergence of test error rates for sensitivity-penalized optimization with MNIST. "
+        ],
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+        "img_path": "images/b047abf440dc0ad31a90a50bcd45e3e6e00c2c669bbe9a856a56ac7d52ef84d7.jpg",
+        "image_caption": [
+            "Figure 7: Convergence of the test error rates for Minimax optimization (blue), Alternating ascent/descent (green), and Maximin optimization (red) for CIFAR-10. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/83db0c85dcb0bfe41f5e5735bc31939ca3d1326f9891ec84cbefbb7507678aef.jpg",
+        "table_caption": [
+            "Table 8: Error rates of different attacks on various adversarially-trained classifiers for CIFAR-10. FGSM-curr means the FGSM attack on the specific classifier on the leftmost column. Adv FGSM is the classifier adversally trained with FGSM attacks. Sens FGSM is the result of minimizing the sensitivity penalty (7). LWA FGSM is the result of minimizing (7) without the gradient-norm term. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Defense\\Attack</td><td rowspan=\"2\">No attack</td><td colspan=\"4\">FGSM</td><td rowspan=\"2\">FGSM-curr</td></tr><tr><td>FGSM-1</td><td>FGSM-2</td><td>：</td><td>FGSM-80</td></tr><tr><td rowspan=\"5\">n=0.1</td><td>No defense AdvFGSM1</td><td>0.222 0.220</td><td>0.976 0.196</td><td>0.671 0.680</td><td>： ：</td><td>0.595 0.616</td><td>0.976 0.245</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Adv FGSM2</td><td>0.258</td><td>0.640</td><td>0.484</td><td>：</td><td>0.612</td><td>0.708</td></tr><tr><td>AdvFGSM80</td><td>0.228</td><td>0.644</td><td>0.529</td><td>：</td><td>0.087</td><td>0.086</td></tr><tr><td>LWAFGSM Sens FGSM</td><td>0.223 0.223</td><td>0.283 0.342</td><td>0.692 0.701</td><td>：</td><td>0.652 0.663</td><td>0.125 0.106</td></tr><tr><td rowspan=\"6\">n=0.2</td><td>No defense</td><td>0.222</td><td>0.825</td><td>0.692</td><td>： ：</td><td>0.819</td><td>0.969</td></tr><tr><td>AdvFGSM1</td><td>0.216</td><td>0.642</td><td>0.630</td><td>：</td><td>0.609</td><td>0.264</td></tr><tr><td>Adv FGSM2</td><td>0.305</td><td>0.579</td><td>0.290</td><td>：</td><td>0.599</td><td>0.556</td></tr><tr><td>AdvFGSM80</td><td>0.218</td><td>0.445</td><td>0.502</td><td>·</td><td>0.078</td><td>0.078</td></tr><tr><td>LWAFGSM</td><td>0.209</td><td>0.689</td><td>0.666</td><td>··</td><td>0.615</td><td>0.105</td></tr><tr><td>Sens FGSM</td><td>0.209</td><td>0.713</td><td>0.672</td><td>：</td><td>0.637</td><td>0.073</td></tr><tr><td rowspan=\"6\">n=0.3</td><td>No defense AdvFGSM1</td><td>0.222</td><td>0.869</td><td>0.891</td><td>：</td><td>0.877</td><td>0.955</td></tr><tr><td></td><td>0.214</td><td>0.668</td><td>0.628</td><td>：</td><td>0.642</td><td>0.424</td></tr><tr><td>Adv FGSM2</td><td>0.205</td><td>0.499</td><td>0.407</td><td>：</td><td>0.514</td><td>0.389</td></tr><tr><td>AdvFGSM80</td><td>0.223</td><td>0.471</td><td>0.324</td><td>：</td><td>0.081</td><td>0.084</td></tr><tr><td>LWAFGSM</td><td>0.215</td><td>0.686</td><td>0.634</td><td>：</td><td>0.640</td><td>0.215</td></tr><tr><td>Sens FGSM</td><td>0.213</td><td>0.715</td><td>0.628</td><td>·</td><td>0.652</td><td>0.089</td></tr><tr><td rowspan=\"6\">n=0.4</td><td>No defense AdvFGSM1</td><td>0.222</td><td>0.884</td><td>0.899</td><td>：</td><td>0.892</td><td>0.941</td></tr><tr><td></td><td>0.208</td><td>0.702</td><td>0.687</td><td>：</td><td>0.697</td><td>0.536</td></tr><tr><td>Adv FGSM2</td><td>0.206</td><td>0.592</td><td>0.546</td><td>：</td><td>0.618</td><td>0.545</td></tr><tr><td>AdvFGSM80</td><td>0.225</td><td>0.497</td><td>0.385</td><td>：</td><td>0.121</td><td>0.124</td></tr><tr><td>LWAFGSM</td><td>0.210</td><td>0.693</td><td>0.639</td><td>：</td><td>0.626</td><td>0.173</td></tr><tr><td>Sens FGSM</td><td>0.214</td><td>0.714</td><td>0.635</td><td>·</td><td>0.640</td><td>0.109</td></tr></table>",
+        "bbox": [
+            199,
+            102,
+            799,
+            433
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/f237f1b06ef6077dedd701f06efc3c8a3f4f90c8e38bab5c098897dc35cba28c.jpg",
+        "table_caption": [
+            "Table 9: Error rates of FGSM vs learning-based attack network (AttNet) on various adversariallytrained classifiers for CIFAR-10. FGSM-curr/AttNet-curr means they are computed/trained for the specific classifier on the leftmost column. Note that FGSM fails to attack against the ‘hardened’ networks (Adv FGSM80 and Sens FGSM), but AttNet can still attack them successfully. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan=\"2\">n=0.1</td><td colspan=\"2\">=0.2</td></tr><tr><td>No defense</td><td>0.976</td><td>0.740</td><td>0.969</td><td>0.905</td></tr><tr><td>Adv FGSM1</td><td>0.245</td><td>0.999</td><td>0.264</td><td>1.000</td></tr><tr><td>Adv FGSM80</td><td>0.086</td><td>1.000</td><td>0.078</td><td>1.000</td></tr><tr><td>Sens FGSM</td><td>0.106</td><td>0.898</td><td>0.073</td><td>0.979</td></tr><tr><td rowspan=\"4\">No defense Adv FGSM1 AdvFGSM80</td><td colspan=\"2\">m=0.3</td><td colspan=\"2\">m=0.4</td></tr><tr><td>0.955</td><td>0.888</td><td>0.941</td><td>0.999</td></tr><tr><td>0.424</td><td>1.000</td><td>0.536</td><td>1.000</td></tr><tr><td>0.084</td><td>1.000</td><td>0.124</td><td>0.900</td></tr><tr><td rowspan=\"2\">Sens FGSM</td><td rowspan=\"2\">0.089</td><td rowspan=\"2\">1.000</td><td rowspan=\"2\">0.109</td></tr><tr><td>1.000</td></tr></table>",
+        "bbox": [
+            261,
+            515,
+            735,
+            660
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/c7207949c3acf48b7f88a9707e607bd12c7c2190fdb4058f86e39e275bbf5bc2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Defense\\Attack</td><td>FGSM-curr</td><td>AttNet-curr</td><td>FGSM-curr</td><td>AttNet-curr</td></tr><tr><td colspan=\"2\">n=0.1</td><td colspan=\"2\">m=0.2</td></tr><tr><td>Minimax</td><td>0.967</td><td>0.276</td><td>0.980</td><td>0.418</td></tr><tr><td>Alt</td><td>0.994</td><td>0.264</td><td>0.996</td><td>0.857</td></tr><tr><td>Sens FGSM</td><td>0.106</td><td>0.898</td><td>0.073</td><td>0.979</td></tr><tr><td rowspan=\"3\">Minimax Alt</td><td>n=0.3</td><td></td><td>m=0.4</td><td></td></tr><tr><td>0.967</td><td>0.875</td><td>0.931</td><td>0.994</td></tr><tr><td>0.987</td><td>0.896</td><td>0.958</td><td>1.000</td></tr><tr><td>Sens FGSM</td><td>0.089</td><td>1.000</td><td>0.109</td><td>1.000</td></tr></table>",
+        "bbox": [
+            263,
+            741,
+            733,
+            862
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 10: Error rates of Minimax-, Alt-, and adversarially-trained (Sens FGSM) classifiers for MNIST. While Minimax and Alt are both vulnerable to AttNet attacks, Minimax is much less vulnerable than Alt at $\\eta = 0 . 2$ . ",
+        "bbox": [
+            178,
+            872,
+            823,
+            915
+        ],
+        "page_idx": 14
+    }
+]

parse/train/H1lmhaVtvr/H1lmhaVtvr.md

@@ -0,0 +1,323 @@
+# DYNAMICAL DISTANCE LEARNING FOR SEMI-SUPERVISED AND UNSUPERVISED SKILL DISCOVERY
+
+Kristian Hartikainen∗ University of California, Berkeley University of Oxford
+
+Xinyang Geng University of California, Berkeley
+
+Tuomas Haarnoja†   
+University of California, Berkeley   
+Google DeepMind
+
+Sergey Levine† University of California, Berkeley
+
+# ABSTRACT
+
+Reinforcement learning requires manual specification of a reward function to learn a task. While in principle this reward function only needs to specify the task goal, in practice reinforcement learning can be very time-consuming or even infeasible unless the reward function is shaped so as to provide a smooth gradient towards a successful outcome. This shaping is difficult to specify by hand, particularly when the task is learned from raw observations, such as images. In this paper, we study how we can automatically learn dynamical distances: a measure of the expected number of time steps to reach a given goal state from any other state. These dynamical distances can be used to provide well-shaped reward functions for reaching new goals, making it possible to learn complex tasks efficiently. We show that dynamical distances can be used in a semi-supervised regime, where unsupervised interaction with the environment is used to learn the dynamical distances, while a small amount of preference supervision is used to determine the task goal, without any manually engineered reward function or goal examples. We evaluate our method both on a real-world robot and in simulation. We show that our method can learn to turn a valve with a real-world 9-DoF hand, using raw image observations and just ten preference labels, without any other supervision. Videos of the learned skills can be found on the project website: https://sites.google.com/view/dynamical-distance-learning.
+
+# 1 INTRODUCTION
+
+The manual design of reward functions represents a major barrier to the adoption of reinforcement learning (RL), particularly in robotics, where vision-based policies can be learned end-toend (Levine et al., 2016; Haarnoja et al., 2018c), but still require reward functions that themselves might need visual detectors to be designed by hand (Singh et al., 2019). While in principle the reward only needs to specify the goal of the task, in practice RL can be exceptionally time-consuming or even infeasible unless the reward function is shaped so as to provide a smooth gradient towards a successful outcome. Prior work tackles such situations with dedicated exploration methods (Houthooft et al., 2016; Osband et al., 2016; Andrychowicz et al., 2017), or by using large amounts of random exploration (Mnih et al., 2015), which is feasible in simulation but infeasible for real-world robotic learning. It is also common to employ heuristic shaping, such as the Cartesian distance to a goal for an object relocation task (Mahmood et al., 2018; Haarnoja et al., 2018a). However, this kind of shaping is brittle and requires manual insight, and is often impossible when ground truth state observations are unavailable, such as when learning from image observations.
+
+![](images/47f0104fb94672688f8075e4e1408411ced19042e07825379df049b062940adf.jpg)  
+Figure 1: We present a dynamical distance learning (DDL) method that can learn a 9-DoF real-world dexterous manipulation task directly from raw image observations. DDL does not assume access to the true reward function and solves the 180 degree valve-rotation task in 8 hours by relying only on 10 human-provided preference labels.
+
+In this paper, we aim to address these challenges by introducing dynamical distance learning (DDL), a general method for learning distance functions that can provide effective shaping for goal-reaching tasks without manual engineering. Instead of imposing heuristic metrics that have no relationship to the system dynamics, we quantify the distance between two states in terms of the number of time steps needed to transition between them. This is a natural choice for dynamical systems, and prior works have explored learning such distances in simple and low-dimensional domains (Kaelbling, 1993). While such distances can be learned using standard model-free reinforcement learning algorithms, such as Q-learning, we show that such methods generally struggle to acquire meaningful distances for more complex systems, particularly with high-dimensional observations such as images. We present a simple method that employs supervised regression to fit dynamical distances, and then uses these distances to provide reward shaping, guide exploration, and discover distinct skills.
+
+The most direct use of DDL is to provide reward shaping for a standard deep RL algorithm, to optimize a policy to reach a given goal state. We can also formulate a semi-supervised skill learning method, where a user expresses preferences over goals, and the agent autonomously collects experience to learn dynamical distances in a self-supervised way. Finally, we can use DDL in a fully unsupervised method, where the most distant states are selected for exploration, resulting in an unsupervised reinforcement learning procedure that discovers difficult skills that reach dynamically distant states from a given start state. All of these applications avoid the need for manually designed reward functions, demonstrations, or user-provided examples, and involve minimal modification to existing deep RL algorithms.
+
+DDL is a simple and scalable approach to learning dynamical distances that can readily accommodate raw image inputs and, as shown in our experiments, substantially outperforms prior methods that learn goal-conditioned policies or distances using approximate dynamic programming techniques, such as Q-learning. We show that using dynamical distances as a reward function in standard reinforcement learning methods results in policies that take the shortest path to a given goal, despite the additional shaping. Empirically, we compare the semi-supervised variant of our method to prior techniques for learning from preferences. We also compare our method to prior methods for unsupervised skill discovery on tasks ranging from 2D navigation to quadrupedal locomotion. Our experimental evaluation demonstrates that DDL can learn complex locomotion skills without any supervision at all, and that the preferences-based version of DDL can learn to turn a valve with a real-world 9-DoF hand, using raw image observations and 10 human-provided preference labels, without any other supervision.
+
+# 2 RELATED WORK
+
+Dynamical distance learning is most closely related to methods that learn goal-conditioned policies or value functions (Schaul et al., 2015; Sutton et al., 2011). Many of these works learn goal-reaching directly via model-free RL, often by using temporal difference updates to learn the distance function as a value function (Kaelbling et al., 1996; Schaul et al., 2015; Andrychowicz et al., 2017; Pong et al., 2018; Nair et al., 2018; Florensa et al., 2019). For example, Kaelbling (1993) learns a goal conditioned Q-function to represent the shortest path between any two states, and Andrychowicz et al. (2017) learns a value function that resembles a distance to goals, under a user-specified lowdimensional goal representation. Unlike these methods, DDL learns policy-conditioned distances with an explicit supervised learning procedure, and then employs these distances to recover a reward function for RL. We experimentally compare to RL-based distance learning methods, and show that
+
+DDL attains substantially better results, especially with complex observations. Another line of prior work uses a learned distance to build a search graph over a set of visited states (Savinov et al., 2018; Eysenbach et al., 2019), which can then be used to plan to reach new states via the shortest path. Our method also learns a distance function separately from the policy, but instead of using it to build a graph, we use it to obtain a reward function for a separate model-free RL algorithm.
+
+The semi-supervised variant of DDL is guided by a small number of preference queries. Prior work has explored several ways to elicit goals from users, such as using outcome examples and a small number of label queries (Singh et al., 2019), or using a large number of relatively cheap preferences (Christiano et al., 2017). The preference queries that our semi-supervised method uses are easy to obtain and, in contrast to prior work (Christiano et al., 2017), we only need a small number of these queries to learn a policy that reliably achieves the user’s desired goal. Our method is also well suited for fully unsupervised learning, in which case DDL uses the distance function to propose goals for unsupervised skill discovery. Prior work on unsupervised reinforcement learning has proposed choosing goals based on a variety of unsupervised criteria, typically with the aim of attaining broad state coverage (Nair et al., 2018; Florensa et al., 2018; Eysenbach et al., 2018; Warde-Farley et al., 2018; Pong et al., 2019). Our method instead repeatedly chooses the most distant state as the goal, which produces rapid exploration and quickly discovers relatively complex skills. We provide a comparative evaluation in our experiments.
+
+# 3 PRELIMINARIES
+
+In this work, we study control of systems defined by fully observed Markovian dynamics $p ( \mathbf { s } ^ { \prime } | \mathbf { s } , \mathbf { a } ) :$ $s \times s \times { \mathcal { A } } \to { \mathbb { R } } _ { > 0 }$ , where $s$ and $\mathcal { A }$ are continuous state and action spaces. We aim to learn a stochastic policy $\pi ( \mathbf { \bar { a } } | \mathbf { s } ) : \mathcal { A } \times \mathcal { S } \to \mathbb { R } _ { \geq 0 }$ , to reach a goal state $\mathbf { g } \in { \mathcal { S } }$ . We will denote a trajectory with $\boldsymbol { \tau } \triangleq ( \mathbf { s } _ { 0 } , \mathbf { a } _ { 0 } , . . . , \mathbf { s } _ { T } ) \sim \rho _ { \pi }$ , where $\rho _ { \pi }$ is a the trajectory distribution induced by the policy $\pi$ , and $\mathbf { s } _ { 0 }$ is sampled from an initial state distribution $\rho ( \mathbf { s } _ { 0 } )$ . The policy can be optimized using any reinforcement learning algorithm by maximizing
+
+$$
+\mathcal { L } ( \pi ) = \mathbb { E } _ { \tau \sim \rho _ { \pi } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { \mathbf { g } } ( \mathbf { s } _ { t } , \mathbf { a } _ { t } ) \right] ,
+$$
+
+where $r _ { \mathbf { g } } : \mathcal { S } \times \mathcal { A }  [ - R _ { \operatorname* { m i n } } , R _ { \operatorname* { m a x } } ]$ is a bounded reward function and $\gamma \in [ 0 , 1 )$ is a discount factor.1 However, we do not assume that we have access to a shaped reward function. In principle, we could set the reward to $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } ) = 0$ if $\mathbf { s } = \mathbf { g }$ and $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } ) = - 1$ otherwise to learn a policy to reach the goal in as few time steps as possible. Unfortunately, such a sparse reward signal is extremely hard to optimize, as it does not provide any gradient towards the optimal solution until the goal is actually reached. Instead, in Section 4, we will show that we can efficiently learn to reach goals by making use of a learned dynamical distance function.
+
+# 4 DYNAMICAL DISTANCE LEARNING
+
+The aim of our method is to learn policies that reach goal states. These goal states can be selected either in an unsupervised fashion, to discover complex skills, or selected manually by the user. The learning process alternates between two steps: in the distance evaluation step, we learn a policyspecific dynamical distance, which is defined in the following subsection. In the policy improvement step, the policy is optimized to reach the desired goal by using the distance function as the negative reward. This process will lead to a sequence of policies and dynamical distance functions that converge to an effective goal-reaching policy. Under certain assumptions, we can prove that this process converges to a policy that minimizes the distance from any state to any goal, as discussed in Appendix B. In this section, we define dynamical distances and describe our dynamical distance learning (DDL) procedure. In Section 5, we will describe the different ways that the goals can be chosen to instantiate our method as a semi-supervised or unsupervised skill learning procedure.
+
+# 4.1 DYNAMICAL DISTANCE FUNCTIONS
+
+The dynamical distance associated with a policy $\pi$ , which we write as $d ^ { \pi } ( \mathbf { s } _ { i } , \mathbf { s } _ { j } )$ , is defined as the expected number of time steps it took for $\pi$ to reach a state ${ \bf s } _ { j }$ from a state $\mathbf { s } _ { i }$ , given that the two were visited in the same episode.2 Mathematically, the distance is defined as:
+
+$$
+d ^ { \pi } ( \mathbf { s } , \mathbf { s } ^ { \prime } ) \triangleq \mathbb { E } _ { \tau \sim \pi | \mathbf { s } _ { i } = \mathbf { s } , \mathbf { s } _ { j } = \mathbf { s } ^ { \prime } , \ j \geq i } \left[ \sum _ { { t = i } } ^ { j - 1 } \gamma ^ { t - i } c ( \mathbf { s } _ { t } , \mathbf { s } _ { { t + 1 } } ) \right] ,
+$$
+
+where $\tau$ is sampled from the conditional distribution of trajectories that passes through first s and then $\mathbf { s } ^ { \prime }$ , and where $c$ is some local cost of moving from $\mathbf { s } _ { i }$ to $\mathbf { s } _ { i + 1 }$ . For example, in a typical case in the absence of supervision, we can set $c ( \mathbf { s } _ { t } , \mathbf { s } _ { t + 1 } ) \equiv 1$ analogously to the binary reward function in Equation 1, in which case the sum reduces to $j - i$ , and we recover the expected number of time steps to reach $\mathbf { s } ^ { \prime }$ . In principle, we could also trivially incorporate more complex local costs $c$ , for example to include action costs. This modification would be straightforward, though we focus on the simple $c ( \mathbf { s } _ { t } , \mathbf { s } _ { t + 1 } ) \equiv 1$ in our derivation and experiments. We include the discount factor to extend the definition to infinitely long trajectories, but in practice we set $\gamma = 1$ .
+
+# 4.2 DISTANCE EVALUATION
+
+In the distance evaluation step, we learn a distance function $d _ { \psi } ^ { \pi } ( { \bf s } , { \bf s } ^ { \prime } )$ , parameterized by $\psi$ , to estimate the dynamical distance between pairs of states visited by a given policy $\pi _ { \phi }$ , parameterized by $\phi$ . We first roll out the policy multiple times to sample trajectories $\tau _ { k }$ of length $T$ . The empirical distance between states $\mathbf { s } _ { i } , \mathbf { s } _ { j } \in \tau _ { k }$ , where $0 \leq i \leq j \leq T$ , is given by $j - i$ . Because the trajectories have a finite length, we are effectively ignoring the cases where reaching ${ \bf s } _ { j }$ from $\mathbf { s } _ { i }$ would take more than $T - i$ steps, biasing this estimate toward zero, but since the bias becomes smaller for shorter distances, we did not find this to be a major limitation in practice. We can now learn the distance function via supervised regression by minimizing
+
+$$
+\mathcal { L } _ { d } ( \psi ) = \frac { 1 } { 2 } \mathbb { E } _ { \stackrel { \tau \sim \rho _ { \pi } } { i \sim \left[ 0 , T \right] } } \left[ \left( d _ { \psi } ^ { \pi } ( \mathbf { s } _ { i } , \mathbf { s } _ { j } ) - ( j - i ) ) \right) ^ { 2 } \right] .
+$$
+
+As we will show in our experimental evaluation, this supervised regression approach makes it feasible to learn dynamical distances for complex tasks with raw image observations, something that has proven exceptionally challenging for methods that learn distances via goal-conditioned policies or value functions and rely on temporal difference-style methods. In direct comparisons, we find that such methods generally struggle to learn on the more complex tasks with image observations. On the other hand, a disadvantage of supervised regression is that it requires on-policy experience, potentially leading to poor sample efficiency. However, because we use the distance as an intermediate representation that guides off-policy policy learning, as we will discuss in Section 4.3, we did not find the on-policy updates for the distance to slow down learning. Indeed, our experiments in Section 6.1 show that we can learn a manipulation task on a real robot with roughly the same amount of experience as is necessary when using a well-shaped and hand-tuned reward function.
+
+# 4.3 POLICY IMPROVEMENT
+
+In the policy improvement step, we use $d _ { \psi } ^ { \pi }$ to optimize a policy $\pi _ { \phi }$ , parameterized by $\phi$ , to reach a goal g. In principle, we could optimize the policy by choosing actions that greedily minimize the distance to the goal, which essentially treats negative distances as the values of a value function, and would be equivalent to the policy improvement step in standard policy iteration. However, acting greedily with respect to the dynamical distance defined in Equation 2 would result in a policy that is optimistic with respect to the dynamics.
+
+This is because the dynamical distance is defined as the expected number of time steps conditioned on the policy successfully reaching the second state from the first state, and therefore does not account for the case where the second state is not reached successfully. In some cases, this results in pathologically bad value functions. For example, consider the MDP shown on the right, where the agent can reach the goal g using one of two paths. The first path has one intermediate state that leads to the target state with probability $p$ , and an absorbing terminal state $\mathbf { s _ { T } }$ with probability $1 - p$ . The other path has two intermediate states, but allows the agent to reach the target every time. The optimal dynamical distance will be 2, regardless of the value of $p$ , causing the policy to always choose the risky path and potentially miss the target completely.
+
+![](images/e9a6ed0cece8d8511f10716fc928dbf9c1345baf3e098e8fd917c3352889a752.jpg)
+
+The definition of dynamical distances in Equation 2 follows directly from how we learn the distance function, by choosing both $\mathbf { s } _ { i }$ and ${ \bf s } _ { j }$ from the same trajectory. Conditioning on both $\mathbf { s } _ { i }$ and ${ \bf s } _ { j }$ is needed when the state space is continuous or large, since visiting two states by chance has zero or near-zero probability. We instead propose to use the distance as a negative reward, and apply reinforcement learning to minimize the cumulative distance on the path to the goal:
+
+$$
+\mathcal { L } _ { \pi } ( \phi ) = \mathbb { E } _ { \tau \sim \rho _ { \pi } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } d _ { \psi } ^ { \pi } ( \mathbf { s } _ { t } , \mathbf { g } ) \right] .
+$$
+
+This amounts to minimizing the cumulative distance over visited states, and thus taking a risky action becomes unfavourable if it takes the agent to a state that is far from the target at a later time. We further show that, under certain assumption, the policy that optimizes Equation 4 will indeed acquire the correct behavior, as discussed in Appendix A, and will converge to a policy that takes the shortest path to the goal, as we show in Appendix B.
+
+We note that our simulated experiments below are run in deterministic environments and we do not fully understand why cumulative distances work better than greedily minimizing the distances even in those cases. A comparison between these two cases is shown in Section 6.2.
+
+# 4.4 ALGORITHM SUMMARY
+
+The dynamical distance learning (DDL) algorithm is described in Figure 1. Our implementation uses soft actor-critic (SAC) (Haarnoja et al., 2018c) as the policy optimizer, but one could also use any other off-the-shelf algorithm. In each iteration, DDL first samples a trajectory using the current policy, and saves it in a replay pool $\mathcal { D }$ . In the second step, DDL updates the distance function by minimizing the loss in Equation 3. The distance function is optimized for a fixed number of $N _ { d }$ stochastic gradient steps. Note that this method requires that we use recent experience from $\mathcal { D }$ , so
+
+# Algorithm 1 Dynamical Distance Learning
+
+1: Input: φ, ψ . Initial policy and distance parameters   
+2: Input: D . Empty replay pool repeat $\tau \sim \rho _ { \pi }$ , $\mathcal { D }  \mathcal { D } \cup \tau$ . Sample a new trajectory for $i = 0$ to $N _ { d }$ do $\psi  \psi - \lambda _ { d } \hat { \nabla } \mathcal { L } _ { d } ( \psi ; \pi )$ . Minimize distance loss end for $\mathbf { g } $ choose goal $( \mathcal { D } )$ . Choose goal state for $i = 0$ to $N _ { \pi }$ do   
+10: $\phi  \phi - \dot { \lambda } _ { \pi } \hat { \nabla } \mathcal { L } _ { \pi } ( \phi ; d , \mathbf { g } ) \circ$ . Minimize policy loss   
+11: end for   
+12: until converged
+
+as to learn the distance corresponding to the current policy. In the third step, DDL chooses a goal state from the recent experience buffer. We will describe two methods to choose these goal states in Section 5. In the fourth step, DDL updates the policy by taking $N _ { \pi }$ gradient steps to minimize the loss in Equation 4. The implementation of this step depends on the RL algorithm of choice. These steps are then repeated until convergence.
+
+# 5 GOAL PROPOSALS
+
+In the previous section, we discussed how we can utilize a learned distance function to efficiently optimize a goal-reaching policy. However, a learned distance function is only meaningful if evaluated at states from the distribution it has been trained on, suggesting that the goal states should be chosen from the replay pool. Choosing a goal that the policy can already reach might at first appear strange, but it turns out to yield efficient directed exploration, as explained next.
+
+Simple random exploration, such as $\epsilon$ -greedy exploration or other strategies that add noise to the actions, can effectively cover states that are close to the starting state, in terms of dynamical distance. However, when high-reward states or goal states are far away from the start state, such na¨ıve strategies are unlikely to reach them. From this observation, we can devise a simple and effective exploration strategy that leverages the learned dynamical distances: we first use the policy to reach a known goal as quickly as possible and then explore the vicinity of that goal. This way more time is
+
+![](images/ad3a2c2957db89c38776b94caa2984eebe492828532eba334b8d4af505ff1cd3.jpg)  
+Figure 2: We evaluate our method both in simulation and on a real-world robot. We show that our method can learn to turn a valve with a real-world 9-DoF hand (a), and run ablations in the simulated version of the same task (b). We also demonstrate that our method can learn pole balancing (c) and locomotion (d, e, f) skills in simulation.
+
+left to randomly explore states far from the initial state and this way likely discovering useful states.   
+We propose two different strategies for choosing the goals below.
+
+# 5.1 SEMI-SUPERVISED LEARNING FROM PREFERENCES
+
+DDL can be used to learn to reach specific goals elicited from a user. The simplest way to do this is for a user to provide the goal state directly, either by specifying the full state, or selecting the state manually from the replay pool. However, we can also provide a more convenient way to elicit the desired state with preference queries. In this setting, the user is repeatedly presented with a small slate of candidate states from the replay pool, and asked to select the one that they prefer most. In practice, we present the user with a visualization of the final state in several of the most recent episodes, and the user selects the one that they consider closest to their desired goal.
+
+For example, if the user wishes to train a legged robot to walk forward, they might pick the state where the robot has progressed the largest distance in the desired direction. The required user effort in selecting these states is minimal, and most of the agent’s experience is still unsupervised, simply using the latest user-chosen state as the goal. In our experiments, we show that this semi-supervised learning procedure, which we call dynamical distance learning from preferences (DDLfP) can learn to rotate a valve with real-world hand from just ten queries, and can learn simulated locomotion tasks using 100 simulated queries.
+
+# 5.2 UNSUPERVISED EXPLORATION AND SKILL ACQUISITION
+
+We can also use DDL to efficiently acquire complex behaviors, such as locomotion skills, in a completely unsupervised fashion. From the observation that many high-reward states are far away from the start state, we can devise a simple and effective exploration strategy that leverages our learned dynamical distances: we can simply select goals that are far from the initial state according to their estimated dynamical distance. We call this variant of our method “dynamical distance learning - unsupervised” (DDLUS).
+
+Intuitively, this method causes the agent to explore the “frontier” of hard-to-reach states, either discovering shorter paths for reaching them and thus making them no longer be on the frontier, or else finding new states further on the fringe through additive random exploration. In practice, we find that this allows the agent to quickly explore distant states in a directed fashion. In Section 6, we show that, by setting choose go $\begin{array} { r } { \mathrm { a l } ( \mathscr { D } ) \equiv \mathrm { \bar { a r g } m a x } _ { \mathbf { g } \in \mathscr { D } } d _ { \psi } ^ { \pi } ( \mathbf { s } _ { 0 } , \mathbf { g } ) } \end{array}$ , where $\mathbf { s } _ { 0 }$ is the initial state, we can acquire effective running gaits and pole balancing skills in a variety of simulated settings. While this approach is not guaranteed to discover interesting and useful skills in general, we find that, on a variety of commonly used benchmark tasks, this approach to unsupervised goal selection actually discovers behaviors that perform better with respect to the (unknown) task reward than previously proposed unsupervised reinforcement learning objectives.
+
+# 6 EXPERIMENTS
+
+Our experimental evaluation aims to study the following empirical questions: (1) Does supervised regression provide a good estimator of the true dynamical distance? (2) Is DDL applicable to realworld, vision-based robotic control tasks? (3) Does DDL provide an efficient method of learning skills a) from user-provided preferences, and b) completely unsupervised?
+
+We evaluate our method both in the real world and in simulation on a set of state- and visionbased continuous control tasks. We consider a 9-DoF real-world dexterous manipulation task and 4 standard OpenAI Gym tasks (Hopper-v3, HalfCheetah-v3, Ant-v3, and InvertedDoublePendulumv2). For all of the tasks, we parameterize our distance function as a neural network, and use soft actor-critic (SAC) (Haarnoja et al., 2018b) with the default hyperparameters to learn the policy. For state-based tasks, we use feed-forward neural networks and for the vision-based tasks we add a convolutional preprocessing network before these fully connected layers. The image observation for all the vision-based tasks are 3072 dimensional $3 2 \mathrm { x } 3 2$ RGB images). Further details are presented in Appendix E.
+
+We study question (1) using a simple didactic example involving navigation through a twodimensional S-shaped maze, which we present in Appendix C. The other two research questions are studied in the following sections.
+
+# 6.1 VISION-BASED REAL-WORLD MANIPULATION FROM HUMAN PREFERENCES
+
+To study the question (2), we apply DDLfP to a real-world vision-based robotic manipulation task. The domain consists of a 9-DoF “DClaw” hand introduced by Ahn et al. (2019), and the manipulation task requires the hand to rotate a valve 180 degrees, as shown in Figure 1. The human operator is queried for a preference every 10K environment steps. Both the visionand state-based experiments with the real robot use 10 queries during the first 4 hours of an 8- hour training period. Note that, for this and all the subsequent experiments, DDLfP does not have access to the true reward, and must learn entirely from preference queries, which in this case are provided by a human operator.
+
+Figure 3 presents the performance over the course of training. DDLfP uses 10 preference queries to learn the task and its performance is comparable to that of SAC trained with a ground truth shaped reward function. We also
+
+![](images/0e8638fd707890840c834f07574f1f91dee0ea9f7124ec973dd7253b6efddd6f.jpg)  
+Figure 3: (Left) learning curves for the valve rotation task learned from state. (Right) Same task from vision. The curves correspond to the final distance (measured in radians) of the valve from the target angle during a rollout. Our method (DDLfP, orange) solves the task in 8 hours. Its performance is comparable to that of SAC with true rewards, and VICE with example outcome images. DDLfP only requires 10 preference queries, and learns without true rewards or outcome images. We compare our method in the simulated version of this task in Figure 5.
+
+show a comparison to variational inverse control with events (VICE) (Singh et al., 2019), a recent classifier-based reward specification framework. Instead of preference queries, VICE requires the user to provide examples of the desired goal state at the beginning of training (20 images in this case). For vision-based tasks, VICE involves directly showing images of the desired outcome to the user, which requires physically arranging a scene and taking a picture of it. Preferences, on the other hand, require a user to simply select one state out of a small set, which can be done with a button press and done e.g. remotely, thus often making it substantially less labor-intensive than VICE. As we can see in the experiments, DDLfP achieves similar performance with substantially less operator effort, using only a small number of preference queries. The series of goal preferences queried from the human operator are shown in Appendix D.
+
+# 6.2 ABLATIONS, COMPARISONS, AND ANALYSIS
+
+Next, we analyze design decisions in our method and compare it to prior methods in simulation. First, we replace the cumulative objective in Equation 4 with objective that greedily minimizes the distance function trained with supervised loss. This objective is unable to learn the task from either state or vision observations. Next, we replace the supervised loss in Equation 3 of our DDL method with a temporal difference (TD) Q-learning style update rule that learns dynamical distances with approximate dynamic programming. The results in Figure 5 show that, all else being equal, the TD-based method fails to learn successfully from both low-dimensional state and vision observations. Figure 5 further shows a comparison between using the dynamical distance as the reward in comparison to a reward of -1 for each step until the goal is reached, which corresponds to hindsight experience replay (HER) with goal sampling replaced with preference goals (Andrychowicz et al., 2017). We see that dynamical distances allow the policy to reach the goal when learning both from state and from images, while HER is only successful when learning from low-dimensional states.
+
+![](images/8cc14e198c178ab985ae1fbee23f574bd912e277d52d63c9b8c36da42d31afb6.jpg)  
+Figure 4: Learning curves for MuJoCo tasks with DDLfP. The y-axis presents the true return of the task. We compare DDLfP to SAC trained directly from the true reward function, which provides an oracle upper bound baseline, and the prior method proposed by Christiano et al. (2017). The prior method uses an on-policy RL algorithm which typically requires more samples than off-policy algorithms, and thus we also plot its final performance after 20M training steps with red star. At the time of the submission, the Ant-v3 run is still in progress and the complete learning curve will be included in the final.
+
+These results are corroborated by prior results in the literature that have found that temporal difference learning struggles to capture the true value accurately (Lillicrap et al., 2015; Fujimoto et al., 2018). Note that prior work work does not use the full state as the goal, but rather manually selects a low-dimensional subspace, such as the location of an object, forcing the distance to focus on task-relevant objects (Andrychowicz et al., 2017). Our method learns distances between full image states (3072-dimensional) while HER uses 3- dimensional goals, a difference of two orders of magnitude in dimensionality. This difficulty of learning complex image-based goals is further corroborated in prior work (Pong et al., 2018; Nair et al., 2018; Pong et al., 2019; WardeFarley et al., 2018).
+
+![](images/0b3a0827d435fb404120c1d115f3062e035d2886233bfc9d096b8d61e9c3e894.jpg)  
+Figure 5: We compare DDL against alternative methods for learning distances on the simulated valve turning task, when learning from the underlying low-dimensional state (left) and from images (right). Dynamical distances used greedily (orange) or learned with TD (green) generally perform poorly. HER (red) can learn from lowdimensional states, but fails to learn from images. Our method, DDLfP (blue) successfully learns the task from either states or images.
+
+Figure 4 presents results for learning from preferences via DDLfP (in green) on a set of continuous control tasks to further study the question (3,a). The plots show the true reward for each method on each task. DDLfP receives only sparse preferences as task-specific supervision, and the preferences in this case are provided synthetically, choosing the state that has progressed the largest distance from the initial state in the desired direction, i.e. the state with largest x-coordinate value. However, this still provides substantially less supervision signal than access to the true reward for all samples. We compare to (Christiano et al., 2017), which also uses preferences for learning skills, but without the use of dynamical distances. The prior method is provided with 750 preference queries over the course of training, while our method uses 100 for all locomotion tasks, and only a single query for the InvertedDoublePendulum-v2, as the initial state and the goal states coincides.3 Note that Christiano et al. (2017) utilizes an on-policy RL algorithms, which is less efficient than SAC. However, DDLfP outperforms this prior method in terms of both final performance and learning speed on all tasks, except for the Hopper-v3 task.
+
+Locomotion tasks like the ones considered here do not fit into DDL framework directly. In this particular case of locomotion tasks, we can fix the issue by considering a case where the ultimate task is to reach a specific goal, i.e. the operator would always choose the goal to be the state closest to the ”ultimate task goal”. In that case, we can see the locomotion task to be the limit case where the ultimate goal is as far as possibly reachable within the maximum episode length.
+
+![](images/802d9fd3e2568aae649bb0f14a735f9102ce64347008b9a463a07431197fd1e3.jpg)  
+Figure 6: (Top) Learning curves for DDLUS. The y-axis plots the environment return (not accessible during the training) for InvertedDoublePendulum-v3, and the L2-distance travelled from the origin for Hopper-v3, HalfCheetah-v3, and Ant-v3. (Bottom) Frequency histograms of skills learned with DDLUS (blue) and DIAYN (orange) (Eysenbach et al., 2018) across different training runs, evaluated according to the travelled L2-distance from the origin.
+
+# 6.3 ACQUIRING UNSUPERVISED SKILLS
+
+Finally, we study question (3,b) in order to understand how well DDLUS can acquire skills without any supervision. We structure these experiments analogously to the unsupervised skill learning experiments proposed by Eysenbach et al. (2018), and compare to the DIAYN algorithm, another unsupervised skill discovery method, proposed in their prior work. While our method maximizes the complexity of the learned skills by attempting to reach the furthest possible goal, DIAYN maximizes the diversity of learned skills. This of course produces different biases in the skills produced by the two methods. Figure 6 shows both learning curves and histograms of the skills learned in the locomotion tasks with the two methods, evaluated according to how far the simulated robot in each domain travels from the initial state. Our DDLUS method learns skills that travel further than DIAYN, while still providing a variety of different behaviors (e.g., travel in different directions). This experiment aims to provide a direct comparison to the DIAYN algorithm (Eysenbach et al., 2018), though a reasonable criticism is that maximizing dynamical distance is particularly wellsuited for the criteria proposed by Eysenbach et al. (2018). We also evaluated DDLUS on the InvertedDoublePendulum-v2 domain, where the task is to balance a pole on a cart. As can be seen from Figure 6, DDLUS can efficiently solve the task without the true reward, as reaching dynamically far states amounts to avoiding failure as far as possible.
+
+# 7 CONCLUSION
+
+We presented dynamical distance learning (DDL), an algorithm for learning dynamical distances that can be used to specify reward functions for goal reaching policies, and support both unsupervised and semi-supervised exploration and skill discovery. Our algorithm uses a simple and stable supervised learning procedure to learn dynamical distances, which are then used to provide a reward function for a standard reinforcement learning method. This makes DDL straightforward to apply even with complex and high-dimensional observations, such as images. By removing the need for manual reward function design and manual reward shaping, our method makes it substantially more practical to employ deep reinforcement learning to acquire skills even with real-world robotic systems. We demonstrate this by learning a valve-turning task with a real-world robotic hand, using 10 preference queries from a human, without any manual reward design or other examples or supervision. One of the main limitations of our current approach is that, although it can be used with an off-policy reinforcement learning algorithm, it requires on-policy data collection for learning the dynamical distances. While the resulting method is still efficient enough to learn directly in the real world, the efficiency of our approach can likely be improved in future work by lifting this limitation. This would not only make learning faster but would also make it possible to pre-train dynamical distances using previously collected experience, potentially making it feasible to scale our method to a multi-task learning setting, where the same dynamical distance function can be used to learn multiple distinct skills.
+
+# ACKNOWLEDGMENTS
+
+We thank Vikash Kumar for the DClaw robot design, Nicolas Heess for helpful discussion, and Henry Zhu and Justin Yu for their help on setting up and running the hardware experiments. This research was supported by the Office of Naval Research, the National Science Foundation through IIS-1651843 and IIS-1700696, and Berkeley DeepDrive.
+
+# REFERENCES
+
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+
+Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. In Advances in Neural Information Processing Systems (NeurIPS), pp. 5048–5058, 2017.
+
+Paul F Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei. Deep reinforcement learning from human preferences. In Advances in Neural Information Processing Systems (NeurIPS), pp. 4299–4307, 2017.
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+Benjamin Eysenbach, Ruslan Salakhutdinov, and Sergey Levine. Search on the replay buffer: Bridging planning and reinforcement learning. arXiv preprint arXiv:1906.05253, 2019.
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+
+# Appendices
+
+# A CORRECT BEHAVIOR IN THE PATHOLOGICAL MDP
+
+In this appendix we show that the policy that maximizes the objective in Equation 1, with the reward $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } ) { \bar { \mathbf { \eta } } } = - d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ , where $d ^ { \pi }$ is given by Equation 2, prefers safe actions over risky actions.
+
+Assume that $c ( \mathbf { s } _ { t } , \mathbf { s } _ { t + 1 } ) = \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { t } \right]$ is an indicator function that is 0 if $\mathbf { s } _ { t }$ is a goal state or terminal state and 1 for all the other states. We can now write the definition of $d ^ { \pi }$ as an infinite sum and substitute $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } )  - d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ in Equation 1:
+
+$$
+\mathcal { L } ( \pi ) = - \mathbb { E } _ { \tau \sim \pi } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { E } _ { \tau ^ { \prime } \sim \pi } \left[ \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { k } \right] \middle | \mathbf { s } _ { 0 } ^ { \prime } = \mathbf { s } _ { t } , \mathbf { a } _ { 0 } ^ { \prime } = \mathbf { a } _ { t } \right] \right] .
+$$
+
+The first term $k = 0 ,$ ) in the inner sum depends only on $\mathbf { s } _ { 0 } ^ { \prime }$ , which is given, and the term can thus be moved outside the inner expectation:
+
+$$
+\mathcal { L } ( \boldsymbol { \pi } ) = - \mathbb { E } _ { \boldsymbol { \tau } \sim \boldsymbol { \pi } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { t } \right] + \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { E } _ { \boldsymbol { \tau } ^ { \prime } \sim \boldsymbol { \pi } } \left[ \sum _ { k = 1 } ^ { \infty } \gamma ^ { k } \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { k } ^ { \prime } \right] \bigg \vert \mathbf { s } _ { 0 } ^ { \prime } = \mathbf { s } _ { t } , \mathbf { a } _ { 0 } ^ { \prime } = \mathbf { a } _ { t } \right] \right] .
+$$
+
+Next, note that the statistics of the inner expectation over $( \mathbf { s } _ { 1 } ^ { \prime } , \mathbf { a } _ { 1 } ^ { \prime } )$ are the same as the outer expectation over $( \mathbf { s } _ { 1 } , \mathbf { a } _ { 1 } )$ , as they are both conditioned on the same $\left( \mathbf { s } _ { t } , \mathbf { a } _ { t } \right)$ . Thus, we can condition the second expectation directly on $( \mathbf { s } _ { 1 } ^ { \prime } , \mathbf { a } _ { 1 } ^ { \prime } ) = ( \mathbf { s } _ { t + 1 } , \mathbf { a } _ { t + 1 } )$ :
+
+$$
+\mathcal { L } ( \boldsymbol { \pi } ) = - \mathbb { E } _ { \boldsymbol { \tau } \sim \boldsymbol { \pi } } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { 1 } _ { \mathbf { g } } [ \mathbf { s } _ { t } ] + \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { E } _ { \boldsymbol { \tau } ^ { \prime } \sim \boldsymbol { \pi } } [ \sum _ { k = 1 } ^ { \infty } \gamma ^ { k } \mathbb { 1 } _ { \mathbf { g } } [ \mathbf { s } _ { k } ^ { \prime } ] | \mathbf { s } _ { 1 } ^ { \prime } = \mathbf { s } _ { t + 1 } , \mathbf { a } _ { 1 } ^ { \prime } = \mathbf { a } _ { t + 1 } ] ] .
+$$
+
+We can now apply the same argument as before and move $\mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { 1 } ^ { \prime } \right]$ outside the inner expectation. Repeating these steps multiple times yields
+
+$$
+\begin{array} { l } { { \displaystyle { \mathcal { L } } ( \boldsymbol { \pi } ) = - \mathbb { E } _ { \boldsymbol { \tau } \sim \boldsymbol { \pi } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { t } \right] + \sum _ { t = 0 } ^ { \infty } \gamma ^ { t + 1 } \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { t + 1 } \right] + \sum _ { t = 0 } ^ { \infty } \gamma ^ { t + 2 } \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { t + 2 } \right] + . . . \right] } } \\ { { \displaystyle ~ = - \mathbb { E } _ { \boldsymbol { \tau } \sim \boldsymbol { \pi } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } ( t + 1 ) \mathbb { 1 } _ { \mathbf { g } } \left[ \mathbf { s } _ { t } \right] \right] . } } \end{array}
+$$
+
+Assuming that the agent always reaches the goal relatively quickly compared to the discount factor, such that $\gamma ^ { t } \approx 1$ , the trajectories that take longer dominate the loss due to the $( t + 1 )$ factor. Therefore, an optimal agent prefers actions that reduce the risk of long, highly suboptimal trajectories, avoiding the pathological behavior discussed in Section 4.3.
+
+# B POLICY IMPROVEMENT WHEN USING DISTANCE AS REWARD
+
+In this appendix we show that, when we use the negative dynamical distance $- d ^ { \pi }$ as the reward function in RL, we can learn an optimal policy with respect to the true dynamical distance, leading to policies that optimize the actual number of time steps needed to reach the goal. This result is nontrivial, since the reward function does not at first glance directly optimize for shortest paths. Our proof relies on the assumption that the MDP has deterministic dynamics. However, this assumption holds in all of our experiments, since the MuJoCo benchmark tasks are governed by deterministic dynamics. Under this assumption, DDL will learn policies that take the shortest path to the goal at convergence, despite using the negative dynamical distance as the reward.
+
+Let $d ^ { * } ( { \bf s } , { \bf g } ) = \operatorname* { m i n } _ { \pi } d ^ { \pi } ( { \bf s } , { \bf g } )$ be the optimal distance from state s to goal state $\mathbf { g }$ . Let $\pi ^ { \prime }$ be the optimal policy for the reinforcement learning problem with reward $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } ) = - d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ . DDL can be viewed as alternating between fitting $d ^ { \pi }$ to the current policy $\pi$ , and learning a new policy $\pi ^ { \prime }$ that is optimal with respect to the reward function given by $- d ^ { \pi }$ .4 We can now state our main theorem as follows:
+
+Theorem 1. Under deterministic dynamics, for any state s and g, we have:
+
+1. $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \leq d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) .$ .   
+2. If $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) = d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ , then $d ^ { \pi ^ { \prime } } ( { \bf s } , { \bf g } ) = d ^ { \ast } ( { \bf s } , { \bf g } ) .$
+
+This implies that, when the policy converges, such that $\pi ^ { \prime } = \pi$ , the policy $\pi ^ { \prime }$ achieves the optimal distance to any goal, and therefore is the optimal policy for the shortest path reward function (e.g., the reward function that assigns a reward of $- 1$ for any step that does not reach the goal).
+
+Proof.
+
+Part 1 Without loss of generality, we assume that our policy is deterministic, since the set of optimal policies in an MDP always includes at least one deterministic policy. We also assume that g is a terminal state and thus $d ( \mathbf { g } , \mathbf { g } ) = 0$ . Let us denote the action of policy $\pi$ on state s as $\pi ( \mathbf { s } )$ . We start by showing that $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \leq d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ . We fix a particular goal $\mathbf { g }$ . Let $S _ { k } = \left\{ \mathbf { s } ; d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) = k \right\}$ be the set of states that takes $k$ steps under $\pi$ to reach the goal. We show that $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \leq d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) =$ $k$ for all $\mathbf { s } \in S _ { k }$ for each $k$ by contradiction.
+
+For $k = 0$ , $S _ { 0 } = \{ \mathbf { g } \}$ is just the single goal state and ${ d ^ { \pi } } ^ { \prime } ( { \bf g } , { \bf g } ) = { d ^ { \pi } } ( { \bf g } , { \bf g } ) = 0$ by definition. For $k = 1$ , for all $\mathbf { s } \in S _ { 1 }$ , there is an action a that reaches the goal state as the direct next state. Therefore, the optimized policy $\pi ^ { \prime }$ would still take the same action a on these states and $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) = 1$ .
+
+Now assume that the opposite is true, that $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) > d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ for some states. Then, there must be a smallest number $K > 1$ and a state ${ \bf s } _ { 0 } \in { \cal S } _ { K }$ such that $d ^ { \pi ^ { \prime } } ( \mathbf { s } _ { 0 } , \mathbf { g } ) = T > d ^ { \pi } ( \mathbf { s } _ { 0 } , \mathbf { g } ) = K$ . Now let us denote the trajectory of states taken by $\pi$ starting from ${ \bf s } _ { 0 }$ as $\{ \mathbf { s } _ { 0 } , \mathbf { s } _ { 1 } , . . . , \mathbf { s } _ { K } = g \}$ , and the trajectory taken by $\pi ^ { \prime }$ as $\{ \mathbf { s } _ { 0 } ^ { \prime } = \mathbf { s } _ { 0 } , \mathbf { s } _ { 1 } ^ { \prime } , . . . , \mathbf { s } _ { T } ^ { \prime } = g \}$ . Let $\mathcal { L } _ { \pi } ( \cdot )$ denote the accumulated discounted sum of distance as defined in Equation 4. By our assumption $T > K$ , and since $\pi ^ { \prime }$ is optimal with respect to the reward $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } ) = - d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ , we have
+
+$$
+\mathcal { L } _ { \pi } ( \pi ^ { \prime } ) = \sum _ { i = 0 } ^ { T - 1 } \gamma ^ { i } d ^ { \pi } ( \mathbf { s } _ { i } ^ { \prime } , \mathbf { g } ) \leq \mathcal { L } _ { \pi } ( \pi ) = \sum _ { i = 0 } ^ { K - 1 } \gamma ^ { i } d ^ { \pi } ( \mathbf { s } _ { i } , \mathbf { g } ) = \sum _ { i = 0 } ^ { K - 1 } \gamma ^ { i } ( K - 1 - i )
+$$
+
+Then there must be a time $\hat { t } < K$ such that $d ^ { \pi } ( \mathbf { s } _ { \hat { t } } ^ { \prime } , \mathbf { g } ) < d ^ { \pi } ( \mathbf { s } _ { \hat { t } } , \mathbf { g } ) = K - 1 - \hat { t }$ . Therefore $\mathbf { s } _ { \hat { t } } ^ { \prime } \in S _ { k }$ for some $k < K - 1 - \hat { t }$ . However, starting from $\mathbf { s } _ { \hat { t } } ^ { \prime }$ , we have $d ^ { \pi ^ { \prime } } ( \mathbf { s } _ { \hat { t } } ^ { \prime } , \mathbf { g } ) = T - 1 - \hat { t } > K - 1 - \hat { t } =$ $d ^ { \pi } ( \mathbf { s } _ { \hat { t } } ^ { \prime } , \mathbf { g } )$ . Therefore, we reached a contradiction with our assumption that $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \leq d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ for all s, $k < K$ such that $\mathbf { s } \in S _ { k }$ . Therefore, $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \leq d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ holds for all states.
+
+Part 2 Now we show the second part: if $d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) = d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } )$ , then $d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) = d ^ { * } ( \mathbf { s } , \mathbf { g } )$ . We prove this with a similar argument, grouping states by distance. Let $S _ { k } ^ { * } = \{ \mathbf { s } ; d ^ { * } ( \mathbf { s } , \mathbf { g } ) = k \}$ be the set of states that takes $k$ steps under the optimal policy to reach the goal. Note that, for any arbitrary policy $\pi$ , we have $d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) \geq d ^ { * } ( \mathbf { s } , \mathbf { g } )$ by definition, since $d ^ { * }$ is the optimal distance.
+
+Suppose that $d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) > d ^ { * } ( \mathbf { s } , \mathbf { g } )$ for some state s. Then there must be a smallest integer $K \geq 0$ such that there exists a state ${ \bf s } _ { 0 } \in { \cal S } _ { K } ^ { * }$ where $d ^ { \pi } ( { \bf s } _ { 0 } , { \bf g } ) > d ^ { * } ( { \bf s } _ { 0 } , { \bf g } )$ . For all $k \ < \ K$ , we have $d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) = d ^ { * } ( \mathbf { s } , \mathbf { g } )$ for all $\mathbf { s } \in S _ { k } ^ { * }$ . Now starting from that state ${ \bf s } _ { 0 }$ , let the trajectory of states taken by $\pi$ be $\{ \mathbf { s } _ { 0 } , \mathbf { s } _ { 1 } , . . . , \mathbf { s } _ { T } = g \}$ . Note that since $d ^ { \pi } ( { \bf s } _ { 0 } , { \bf g } ) > d ^ { * } ( { \bf s } _ { 0 } , { \bf g } ) = { \cal K } , $ $T > K$ . Let $\hat { \pi }$ be the policy such that it agrees with $\pi ^ { * }$ on $\mathbf { s } _ { 0 }$ and agrees with $\pi$ everywhere else. At the first step, $\hat { \pi }$ lands on state $\mathbf { s } _ { 1 } ^ { \prime }$ . Since ${ \bf s } _ { 0 }$ is $K$ steps away from $\mathbf { g }$ under $d ^ { * }$ , $\mathbf { s } _ { 1 } ^ { \prime }$ must be $K - 1$ steps away under $d ^ { * }$ and ${ \bf s } _ { 1 } ^ { \prime } \in { \cal S } _ { K - 1 } ^ { * }$ . Therefore, since $\pi$ and $\pi ^ { * }$ agrees on all states that are less than $K$ steps away from goal g, $\hat { \pi }$ would take the same action as $\pi ^ { * }$ and hence take another $K - 1$ steps to goal g. Now let us denote the trajectory taken by $\hat { \pi }$ as $\{ \mathbf { s } _ { 0 } ^ { \prime } = \mathbf { s } _ { 0 } , \mathbf { s } _ { 1 } ^ { \prime } , . . . , \mathbf { s } _ { K } ^ { \prime } = g \}$ . We compare the discounted sum of rewards of $\pi$ and $\hat { \pi }$ under the reward function $r _ { \mathbf { g } } ( \mathbf { s } , \mathbf { a } ) = - d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ .
+
+$$
+\begin{array} { l } { { \displaystyle { \mathcal { L } } _ { \pi } ( \pi ) = \sum _ { i = 0 } ^ { T - 1 } \gamma ^ { i } d ^ { \pi } ( { \bf s } _ { i } , { \bf g } ) = d ^ { \pi } ( { \bf s } _ { 0 } , { \bf g } ) + \sum _ { i = 1 } ^ { T - 1 } \gamma ^ { i } d ^ { \pi } ( { \bf s } _ { i } , { \bf g } ) } \ ~ } \\ { { \displaystyle ~ = d ^ { \pi } ( { \bf s } _ { 0 } , { \bf g } ) + \sum _ { i = 1 } ^ { T - 1 } \gamma ^ { i } ( T - i ) \geq d ^ { \pi } ( { \bf s } _ { 0 } , { \bf g } ) + \sum _ { i = 1 } ^ { K - 1 } \gamma ^ { i } ( K - i ) } \ ~ } \\ { { \displaystyle ~ = d ^ { \pi } ( { \bf s } _ { 0 } , { \bf g } ) + \sum _ { i = 1 } ^ { K - 1 } \gamma ^ { i } d ^ { \pi } ( { \bf s } _ { i } ^ { \prime } , { \bf g } ) = \sum _ { i = 0 } ^ { K - 1 } \gamma ^ { i } d ^ { \pi } ( { \bf s } _ { i } ^ { \prime } , { \bf g } ) = { \mathcal L } _ { \pi } ( \hat { \pi } ) } \ ~ } \end{array}
+$$
+
+Therefore, we can see that $\hat { \pi }$ is a better policy than $\pi$ . Then the optimal policy $\pi ^ { \prime }$ under this reward must be different from $\pi$ on at least one state. Hence $d ^ { \pi } ( \mathbf { s } , \mathbf { g } ) \neq d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } )$ .
+
+We’ve now reached the conclusion that if $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \neq d ^ { * } ( \mathbf { s } , \mathbf { g } )$ , then $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) \neq d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ . Hence, by contraposition, if $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) = d ^ { \pi } ( \mathbf { s } , \mathbf { g } )$ , then it must be that $d ^ { \pi ^ { \prime } } ( \mathbf { s } , \mathbf { g } ) = d ^ { \ast } ( \mathbf { s } , \mathbf { g } )$ . Our proof is thus complete.
+
+C DIDACTIC EXAMPLE
+
+Our didactic example involves a simple 2D point robot navigating an S-shaped maze. The state space is two-dimensional, and the action is a two-dimensional velocity vector. This experiment is visualized in Figure 7. The black rectangles correspond to walls, and the goal is depicted with a blue star. The learned distance from all points in the maze to the goal is illustrated with a heat map, in which lighter colors correspond to closer states and darker colors to distant states. During the training, the initial state is chosen uniformly at random, and the policy is trained to reach the goal state. From the visualization, it is apparent that DDL learns an accurate estimate of the true dynamical distances in this domain. Note that, in contrast to na¨ıve metrics, such as Euclidean distance, the dynamical distances conform to the walls and provide an accurate estimate of reachability, making them ideally suited for reward shaping.
+
+![](images/0b1a512d4116173ff2294981ff2c4240e5fd3f1a154be461354f90236d4173dd.jpg)  
+Figure 7: Evaluation of the learned distance in a 2D point environment. The state is the xycoordinates of the point, and action corresponds to 2D velocities. The black bars denote walls, blue star is a goal state, and the heat map denotes the estimated distance to the goal. (a) Our method learns an accurate estimate of the shape of the distance function. (b) Ground-truth distance.
+
+# D PREFERENCE QUERIES FOR REAL-WORLD DCLAW EXPERIMENT
+
+![](images/1f5b8406ba52b303a00f5db5e76a224ae4a6bbbbe39040a18a95ab863a5b6bfb.jpg)  
+Figure 8: Human preference queries for the vision-based DClaw experiment presented in Section 6.1. Each image row presents the set of images shown to the human operator on a single query round. On each row, the first 10 images correspond to the last states of the most recent rollouts and the right-most image corresponds to the last goal. For each query, the human operator picks a new goal by inputting its index (between 0-10) into a text-based interface. The goals selected by human are highlighted with white borders.
+
+# E TECHNICAL DETAILS
+
+All our experiments use Soft Actor-Critic as the policy optimizer, trained the default parameters by provided by the authors in (Haarnoja et al., 2018c).
+
+For all of the tasks, we parameterize our distance function as a neural network. For state-based tasks, we use feed-forward neural networks with two 256-unit hidden layers. For the vision-based tasks we add a convolutional preprocessing network before these fully-connected layers, consisting of four convolutional layers, each with $6 4 3 \mathrm { x } 3 $ filters. Both cases use Adam optimizer with learning rate 3e4 and TensorFlow‘s default momentum parameters. The image observation for all the vision-based tasks are 3072 dimensional (32x32 RGB images).
+
+Most important hyperparameters that we swept over in the final experiments, namely the size of the on-policy pool for training the distance function and the number of gradient steps per environment samples, are presented in Table 1 below:
+
+Table 1: Distance estimator hyperparameters.   
+
+<table><tr><td>Environment</td><td>gradient steps per environment steps</td><td>on-policy pool size</td></tr><tr><td>InvertedDoublePendulum-v2</td><td>1/64</td><td>100k</td></tr><tr><td>Hopper-v3</td><td>1/64</td><td>16k</td></tr><tr><td>HalfCheetah-v3</td><td>1/16</td><td>16k</td></tr><tr><td>Ant-v3</td><td>1/64</td><td>10k</td></tr><tr><td>DClaw (both state and vision)</td><td>1/16</td><td>100k</td></tr></table>
+
+For the DDLUS goal proposals, we consider all the samples in the distance on-policy pool as the goal candidates. For DDLfP, we present the operator the last states $( s _ { T - 1 } )$ of the last $N$ episodes, where $N = 5$ for all the simulated experiments, and $N = 1 0$ for the hardware DClaw.
+
+As discussed in Section 5, for both DDLUS and DDLfP, the agent needs to explore in the vicinity of the goal state. In practice, we implement this by switching to a random uniform policy after $0 . 9 \mathrm { T }$ timesteps of each episode, where $\mathrm { T }$ is the maximum episode length (1000 for all the mujoco tasks and 200 for the DClaw task).

parse/train/H1lmhaVtvr/H1lmhaVtvr_content_list.json

@@ -0,0 +1,1671 @@
+[
+    {
+        "type": "text",
+        "text": "DYNAMICAL DISTANCE LEARNING FOR SEMI-SUPERVISED AND UNSUPERVISED SKILL DISCOVERY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            653,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kristian Hartikainen∗ University of California, Berkeley University of Oxford ",
+        "bbox": [
+            184,
+            194,
+            408,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Xinyang Geng University of California, Berkeley ",
+        "bbox": [
+            504,
+            195,
+            730,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tuomas Haarnoja†   \nUniversity of California, Berkeley   \nGoogle DeepMind ",
+        "bbox": [
+            184,
+            257,
+            406,
+            300
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sergey Levine† University of California, Berkeley ",
+        "bbox": [
+            506,
+            257,
+            732,
+            286
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            337,
+            544,
+            352
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning requires manual specification of a reward function to learn a task. While in principle this reward function only needs to specify the task goal, in practice reinforcement learning can be very time-consuming or even infeasible unless the reward function is shaped so as to provide a smooth gradient towards a successful outcome. This shaping is difficult to specify by hand, particularly when the task is learned from raw observations, such as images. In this paper, we study how we can automatically learn dynamical distances: a measure of the expected number of time steps to reach a given goal state from any other state. These dynamical distances can be used to provide well-shaped reward functions for reaching new goals, making it possible to learn complex tasks efficiently. We show that dynamical distances can be used in a semi-supervised regime, where unsupervised interaction with the environment is used to learn the dynamical distances, while a small amount of preference supervision is used to determine the task goal, without any manually engineered reward function or goal examples. We evaluate our method both on a real-world robot and in simulation. We show that our method can learn to turn a valve with a real-world 9-DoF hand, using raw image observations and just ten preference labels, without any other supervision. Videos of the learned skills can be found on the project website: https://sites.google.com/view/dynamical-distance-learning. ",
+        "bbox": [
+            233,
+            371,
+            764,
+            633
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            666,
+            336,
+            683
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The manual design of reward functions represents a major barrier to the adoption of reinforcement learning (RL), particularly in robotics, where vision-based policies can be learned end-toend (Levine et al., 2016; Haarnoja et al., 2018c), but still require reward functions that themselves might need visual detectors to be designed by hand (Singh et al., 2019). While in principle the reward only needs to specify the goal of the task, in practice RL can be exceptionally time-consuming or even infeasible unless the reward function is shaped so as to provide a smooth gradient towards a successful outcome. Prior work tackles such situations with dedicated exploration methods (Houthooft et al., 2016; Osband et al., 2016; Andrychowicz et al., 2017), or by using large amounts of random exploration (Mnih et al., 2015), which is feasible in simulation but infeasible for real-world robotic learning. It is also common to employ heuristic shaping, such as the Cartesian distance to a goal for an object relocation task (Mahmood et al., 2018; Haarnoja et al., 2018a). However, this kind of shaping is brittle and requires manual insight, and is often impossible when ground truth state observations are unavailable, such as when learning from image observations. ",
+        "bbox": [
+            173,
+            700,
+            825,
+            881
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/47f0104fb94672688f8075e4e1408411ced19042e07825379df049b062940adf.jpg",
+        "image_caption": [
+            "Figure 1: We present a dynamical distance learning (DDL) method that can learn a 9-DoF real-world dexterous manipulation task directly from raw image observations. DDL does not assume access to the true reward function and solves the 180 degree valve-rotation task in 8 hours by relying only on 10 human-provided preference labels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            103,
+            820,
+            166
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we aim to address these challenges by introducing dynamical distance learning (DDL), a general method for learning distance functions that can provide effective shaping for goal-reaching tasks without manual engineering. Instead of imposing heuristic metrics that have no relationship to the system dynamics, we quantify the distance between two states in terms of the number of time steps needed to transition between them. This is a natural choice for dynamical systems, and prior works have explored learning such distances in simple and low-dimensional domains (Kaelbling, 1993). While such distances can be learned using standard model-free reinforcement learning algorithms, such as Q-learning, we show that such methods generally struggle to acquire meaningful distances for more complex systems, particularly with high-dimensional observations such as images. We present a simple method that employs supervised regression to fit dynamical distances, and then uses these distances to provide reward shaping, guide exploration, and discover distinct skills. ",
+        "bbox": [
+            174,
+            261,
+            825,
+            426
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The most direct use of DDL is to provide reward shaping for a standard deep RL algorithm, to optimize a policy to reach a given goal state. We can also formulate a semi-supervised skill learning method, where a user expresses preferences over goals, and the agent autonomously collects experience to learn dynamical distances in a self-supervised way. Finally, we can use DDL in a fully unsupervised method, where the most distant states are selected for exploration, resulting in an unsupervised reinforcement learning procedure that discovers difficult skills that reach dynamically distant states from a given start state. All of these applications avoid the need for manually designed reward functions, demonstrations, or user-provided examples, and involve minimal modification to existing deep RL algorithms. ",
+        "bbox": [
+            174,
+            434,
+            825,
+            559
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "DDL is a simple and scalable approach to learning dynamical distances that can readily accommodate raw image inputs and, as shown in our experiments, substantially outperforms prior methods that learn goal-conditioned policies or distances using approximate dynamic programming techniques, such as Q-learning. We show that using dynamical distances as a reward function in standard reinforcement learning methods results in policies that take the shortest path to a given goal, despite the additional shaping. Empirically, we compare the semi-supervised variant of our method to prior techniques for learning from preferences. We also compare our method to prior methods for unsupervised skill discovery on tasks ranging from 2D navigation to quadrupedal locomotion. Our experimental evaluation demonstrates that DDL can learn complex locomotion skills without any supervision at all, and that the preferences-based version of DDL can learn to turn a valve with a real-world 9-DoF hand, using raw image observations and 10 human-provided preference labels, without any other supervision. ",
+        "bbox": [
+            173,
+            566,
+            825,
+            733
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            752,
+            344,
+            768
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Dynamical distance learning is most closely related to methods that learn goal-conditioned policies or value functions (Schaul et al., 2015; Sutton et al., 2011). Many of these works learn goal-reaching directly via model-free RL, often by using temporal difference updates to learn the distance function as a value function (Kaelbling et al., 1996; Schaul et al., 2015; Andrychowicz et al., 2017; Pong et al., 2018; Nair et al., 2018; Florensa et al., 2019). For example, Kaelbling (1993) learns a goal conditioned Q-function to represent the shortest path between any two states, and Andrychowicz et al. (2017) learns a value function that resembles a distance to goals, under a user-specified lowdimensional goal representation. Unlike these methods, DDL learns policy-conditioned distances with an explicit supervised learning procedure, and then employs these distances to recover a reward function for RL. We experimentally compare to RL-based distance learning methods, and show that ",
+        "bbox": [
+            174,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "DDL attains substantially better results, especially with complex observations. Another line of prior work uses a learned distance to build a search graph over a set of visited states (Savinov et al., 2018; Eysenbach et al., 2019), which can then be used to plan to reach new states via the shortest path. Our method also learns a distance function separately from the policy, but instead of using it to build a graph, we use it to obtain a reward function for a separate model-free RL algorithm. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            172
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The semi-supervised variant of DDL is guided by a small number of preference queries. Prior work has explored several ways to elicit goals from users, such as using outcome examples and a small number of label queries (Singh et al., 2019), or using a large number of relatively cheap preferences (Christiano et al., 2017). The preference queries that our semi-supervised method uses are easy to obtain and, in contrast to prior work (Christiano et al., 2017), we only need a small number of these queries to learn a policy that reliably achieves the user’s desired goal. Our method is also well suited for fully unsupervised learning, in which case DDL uses the distance function to propose goals for unsupervised skill discovery. Prior work on unsupervised reinforcement learning has proposed choosing goals based on a variety of unsupervised criteria, typically with the aim of attaining broad state coverage (Nair et al., 2018; Florensa et al., 2018; Eysenbach et al., 2018; Warde-Farley et al., 2018; Pong et al., 2019). Our method instead repeatedly chooses the most distant state as the goal, which produces rapid exploration and quickly discovers relatively complex skills. We provide a comparative evaluation in our experiments. ",
+        "bbox": [
+            174,
+            180,
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+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we study control of systems defined by fully observed Markovian dynamics $p ( \\mathbf { s } ^ { \\prime } | \\mathbf { s } , \\mathbf { a } ) :$ $s \\times s \\times { \\mathcal { A } } \\to { \\mathbb { R } } _ { > 0 }$ , where $s$ and $\\mathcal { A }$ are continuous state and action spaces. We aim to learn a stochastic policy $\\pi ( \\mathbf { \\bar { a } } | \\mathbf { s } ) : \\mathcal { A } \\times \\mathcal { S } \\to \\mathbb { R } _ { \\geq 0 }$ , to reach a goal state $\\mathbf { g } \\in { \\mathcal { S } }$ . We will denote a trajectory with $\\boldsymbol { \\tau } \\triangleq ( \\mathbf { s } _ { 0 } , \\mathbf { a } _ { 0 } , . . . , \\mathbf { s } _ { T } ) \\sim \\rho _ { \\pi }$ , where $\\rho _ { \\pi }$ is a the trajectory distribution induced by the policy $\\pi$ , and $\\mathbf { s } _ { 0 }$ is sampled from an initial state distribution $\\rho ( \\mathbf { s } _ { 0 } )$ . The policy can be optimized using any reinforcement learning algorithm by maximizing ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b63fc3dfeb86a9581ff8309a23e78dc7fed3834b7f7d7331c677c96705f43d7.jpg",
+        "text": "$$\n\\mathcal { L } ( \\pi ) = \\mathbb { E } _ { \\tau \\sim \\rho _ { \\pi } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { \\mathbf { g } } ( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            515,
+            617,
+            559
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $r _ { \\mathbf { g } } : \\mathcal { S } \\times \\mathcal { A }  [ - R _ { \\operatorname* { m i n } } , R _ { \\operatorname* { m a x } } ]$ is a bounded reward function and $\\gamma \\in [ 0 , 1 )$ is a discount factor.1 However, we do not assume that we have access to a shaped reward function. In principle, we could set the reward to $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } ) = 0$ if $\\mathbf { s } = \\mathbf { g }$ and $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } ) = - 1$ otherwise to learn a policy to reach the goal in as few time steps as possible. Unfortunately, such a sparse reward signal is extremely hard to optimize, as it does not provide any gradient towards the optimal solution until the goal is actually reached. Instead, in Section 4, we will show that we can efficiently learn to reach goals by making use of a learned dynamical distance function. ",
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+    {
+        "type": "text",
+        "text": "4 DYNAMICAL DISTANCE LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The aim of our method is to learn policies that reach goal states. These goal states can be selected either in an unsupervised fashion, to discover complex skills, or selected manually by the user. The learning process alternates between two steps: in the distance evaluation step, we learn a policyspecific dynamical distance, which is defined in the following subsection. In the policy improvement step, the policy is optimized to reach the desired goal by using the distance function as the negative reward. This process will lead to a sequence of policies and dynamical distance functions that converge to an effective goal-reaching policy. Under certain assumptions, we can prove that this process converges to a policy that minimizes the distance from any state to any goal, as discussed in Appendix B. In this section, we define dynamical distances and describe our dynamical distance learning (DDL) procedure. In Section 5, we will describe the different ways that the goals can be chosen to instantiate our method as a semi-supervised or unsupervised skill learning procedure. ",
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+    {
+        "type": "text",
+        "text": "4.1 DYNAMICAL DISTANCE FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The dynamical distance associated with a policy $\\pi$ , which we write as $d ^ { \\pi } ( \\mathbf { s } _ { i } , \\mathbf { s } _ { j } )$ , is defined as the expected number of time steps it took for $\\pi$ to reach a state ${ \\bf s } _ { j }$ from a state $\\mathbf { s } _ { i }$ , given that the two were visited in the same episode.2 Mathematically, the distance is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/c10623499119277042473119512cec966f7ec7ee1f2c7ac75048a119c5c677b0.jpg",
+        "text": "$$\nd ^ { \\pi } ( \\mathbf { s } , \\mathbf { s } ^ { \\prime } ) \\triangleq \\mathbb { E } _ { \\tau \\sim \\pi | \\mathbf { s } _ { i } = \\mathbf { s } , \\mathbf { s } _ { j } = \\mathbf { s } ^ { \\prime } , \\ j \\geq i } \\left[ \\sum _ { { t = i } } ^ { j - 1 } \\gamma ^ { t - i } c ( \\mathbf { s } _ { t } , \\mathbf { s } _ { { t + 1 } } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is sampled from the conditional distribution of trajectories that passes through first s and then $\\mathbf { s } ^ { \\prime }$ , and where $c$ is some local cost of moving from $\\mathbf { s } _ { i }$ to $\\mathbf { s } _ { i + 1 }$ . For example, in a typical case in the absence of supervision, we can set $c ( \\mathbf { s } _ { t } , \\mathbf { s } _ { t + 1 } ) \\equiv 1$ analogously to the binary reward function in Equation 1, in which case the sum reduces to $j - i$ , and we recover the expected number of time steps to reach $\\mathbf { s } ^ { \\prime }$ . In principle, we could also trivially incorporate more complex local costs $c$ , for example to include action costs. This modification would be straightforward, though we focus on the simple $c ( \\mathbf { s } _ { t } , \\mathbf { s } _ { t + 1 } ) \\equiv 1$ in our derivation and experiments. We include the discount factor to extend the definition to infinitely long trajectories, but in practice we set $\\gamma = 1$ . ",
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+    {
+        "type": "text",
+        "text": "4.2 DISTANCE EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the distance evaluation step, we learn a distance function $d _ { \\psi } ^ { \\pi } ( { \\bf s } , { \\bf s } ^ { \\prime } )$ , parameterized by $\\psi$ , to estimate the dynamical distance between pairs of states visited by a given policy $\\pi _ { \\phi }$ , parameterized by $\\phi$ . We first roll out the policy multiple times to sample trajectories $\\tau _ { k }$ of length $T$ . The empirical distance between states $\\mathbf { s } _ { i } , \\mathbf { s } _ { j } \\in \\tau _ { k }$ , where $0 \\leq i \\leq j \\leq T$ , is given by $j - i$ . Because the trajectories have a finite length, we are effectively ignoring the cases where reaching ${ \\bf s } _ { j }$ from $\\mathbf { s } _ { i }$ would take more than $T - i$ steps, biasing this estimate toward zero, but since the bias becomes smaller for shorter distances, we did not find this to be a major limitation in practice. We can now learn the distance function via supervised regression by minimizing ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e0d466bd80a9f43ec48179f8ce199e4b7024d998263a53c025b66e431fc13218.jpg",
+        "text": "$$\n\\mathcal { L } _ { d } ( \\psi ) = \\frac { 1 } { 2 } \\mathbb { E } _ { \\stackrel { \\tau \\sim \\rho _ { \\pi } } { i \\sim \\left[ 0 , T \\right] } } \\left[ \\left( d _ { \\psi } ^ { \\pi } ( \\mathbf { s } _ { i } , \\mathbf { s } _ { j } ) - ( j - i ) ) \\right) ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As we will show in our experimental evaluation, this supervised regression approach makes it feasible to learn dynamical distances for complex tasks with raw image observations, something that has proven exceptionally challenging for methods that learn distances via goal-conditioned policies or value functions and rely on temporal difference-style methods. In direct comparisons, we find that such methods generally struggle to learn on the more complex tasks with image observations. On the other hand, a disadvantage of supervised regression is that it requires on-policy experience, potentially leading to poor sample efficiency. However, because we use the distance as an intermediate representation that guides off-policy policy learning, as we will discuss in Section 4.3, we did not find the on-policy updates for the distance to slow down learning. Indeed, our experiments in Section 6.1 show that we can learn a manipulation task on a real robot with roughly the same amount of experience as is necessary when using a well-shaped and hand-tuned reward function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 POLICY IMPROVEMENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the policy improvement step, we use $d _ { \\psi } ^ { \\pi }$ to optimize a policy $\\pi _ { \\phi }$ , parameterized by $\\phi$ , to reach a goal g. In principle, we could optimize the policy by choosing actions that greedily minimize the distance to the goal, which essentially treats negative distances as the values of a value function, and would be equivalent to the policy improvement step in standard policy iteration. However, acting greedily with respect to the dynamical distance defined in Equation 2 would result in a policy that is optimistic with respect to the dynamics. ",
+        "bbox": [
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+        "type": "text",
+        "text": "This is because the dynamical distance is defined as the expected number of time steps conditioned on the policy successfully reaching the second state from the first state, and therefore does not account for the case where the second state is not reached successfully. In some cases, this results in pathologically bad value functions. For example, consider the MDP shown on the right, where the agent can reach the goal g using one of two paths. The first path has one intermediate state that leads to the target state with probability $p$ , and an absorbing terminal state $\\mathbf { s _ { T } }$ with probability $1 - p$ . The other path has two intermediate states, but allows the agent to reach the target every time. The optimal dynamical distance will be 2, regardless of the value of $p$ , causing the policy to always choose the risky path and potentially miss the target completely. ",
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+    {
+        "type": "image",
+        "img_path": "images/e9a6ed0cece8d8511f10716fc928dbf9c1345baf3e098e8fd917c3352889a752.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The definition of dynamical distances in Equation 2 follows directly from how we learn the distance function, by choosing both $\\mathbf { s } _ { i }$ and ${ \\bf s } _ { j }$ from the same trajectory. Conditioning on both $\\mathbf { s } _ { i }$ and ${ \\bf s } _ { j }$ is needed when the state space is continuous or large, since visiting two states by chance has zero or near-zero probability. We instead propose to use the distance as a negative reward, and apply reinforcement learning to minimize the cumulative distance on the path to the goal: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05a89532e07f87b8fa9e1c1161122aa32df557ece271e0eddd048518d2e884a3.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\pi } ( \\phi ) = \\mathbb { E } _ { \\tau \\sim \\rho _ { \\pi } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } d _ { \\psi } ^ { \\pi } ( \\mathbf { s } _ { t } , \\mathbf { g } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This amounts to minimizing the cumulative distance over visited states, and thus taking a risky action becomes unfavourable if it takes the agent to a state that is far from the target at a later time. We further show that, under certain assumption, the policy that optimizes Equation 4 will indeed acquire the correct behavior, as discussed in Appendix A, and will converge to a policy that takes the shortest path to the goal, as we show in Appendix B. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We note that our simulated experiments below are run in deterministic environments and we do not fully understand why cumulative distances work better than greedily minimizing the distances even in those cases. A comparison between these two cases is shown in Section 6.2. ",
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+    {
+        "type": "text",
+        "text": "4.4 ALGORITHM SUMMARY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The dynamical distance learning (DDL) algorithm is described in Figure 1. Our implementation uses soft actor-critic (SAC) (Haarnoja et al., 2018c) as the policy optimizer, but one could also use any other off-the-shelf algorithm. In each iteration, DDL first samples a trajectory using the current policy, and saves it in a replay pool $\\mathcal { D }$ . In the second step, DDL updates the distance function by minimizing the loss in Equation 3. The distance function is optimized for a fixed number of $N _ { d }$ stochastic gradient steps. Note that this method requires that we use recent experience from $\\mathcal { D }$ , so ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Dynamical Distance Learning ",
+        "text_level": 1,
+        "bbox": [
+            519,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: Input: φ, ψ . Initial policy and distance parameters   \n2: Input: D . Empty replay pool repeat $\\tau \\sim \\rho _ { \\pi }$ , $\\mathcal { D }  \\mathcal { D } \\cup \\tau$ . Sample a new trajectory for $i = 0$ to $N _ { d }$ do $\\psi  \\psi - \\lambda _ { d } \\hat { \\nabla } \\mathcal { L } _ { d } ( \\psi ; \\pi )$ . Minimize distance loss end for $\\mathbf { g } $ choose goal $( \\mathcal { D } )$ . Choose goal state for $i = 0$ to $N _ { \\pi }$ do   \n10: $\\phi  \\phi - \\dot { \\lambda } _ { \\pi } \\hat { \\nabla } \\mathcal { L } _ { \\pi } ( \\phi ; d , \\mathbf { g } ) \\circ$ . Minimize policy loss   \n11: end for   \n12: until converged ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "as to learn the distance corresponding to the current policy. In the third step, DDL chooses a goal state from the recent experience buffer. We will describe two methods to choose these goal states in Section 5. In the fourth step, DDL updates the policy by taking $N _ { \\pi }$ gradient steps to minimize the loss in Equation 4. The implementation of this step depends on the RL algorithm of choice. These steps are then repeated until convergence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 GOAL PROPOSALS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the previous section, we discussed how we can utilize a learned distance function to efficiently optimize a goal-reaching policy. However, a learned distance function is only meaningful if evaluated at states from the distribution it has been trained on, suggesting that the goal states should be chosen from the replay pool. Choosing a goal that the policy can already reach might at first appear strange, but it turns out to yield efficient directed exploration, as explained next. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Simple random exploration, such as $\\epsilon$ -greedy exploration or other strategies that add noise to the actions, can effectively cover states that are close to the starting state, in terms of dynamical distance. However, when high-reward states or goal states are far away from the start state, such na¨ıve strategies are unlikely to reach them. From this observation, we can devise a simple and effective exploration strategy that leverages the learned dynamical distances: we first use the policy to reach a known goal as quickly as possible and then explore the vicinity of that goal. This way more time is ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/ad3a2c2957db89c38776b94caa2984eebe492828532eba334b8d4af505ff1cd3.jpg",
+        "image_caption": [
+            "Figure 2: We evaluate our method both in simulation and on a real-world robot. We show that our method can learn to turn a valve with a real-world 9-DoF hand (a), and run ablations in the simulated version of the same task (b). We also demonstrate that our method can learn pole balancing (c) and locomotion (d, e, f) skills in simulation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "left to randomly explore states far from the initial state and this way likely discovering useful states.   \nWe propose two different strategies for choosing the goals below. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 SEMI-SUPERVISED LEARNING FROM PREFERENCES",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DDL can be used to learn to reach specific goals elicited from a user. The simplest way to do this is for a user to provide the goal state directly, either by specifying the full state, or selecting the state manually from the replay pool. However, we can also provide a more convenient way to elicit the desired state with preference queries. In this setting, the user is repeatedly presented with a small slate of candidate states from the replay pool, and asked to select the one that they prefer most. In practice, we present the user with a visualization of the final state in several of the most recent episodes, and the user selects the one that they consider closest to their desired goal. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For example, if the user wishes to train a legged robot to walk forward, they might pick the state where the robot has progressed the largest distance in the desired direction. The required user effort in selecting these states is minimal, and most of the agent’s experience is still unsupervised, simply using the latest user-chosen state as the goal. In our experiments, we show that this semi-supervised learning procedure, which we call dynamical distance learning from preferences (DDLfP) can learn to rotate a valve with real-world hand from just ten queries, and can learn simulated locomotion tasks using 100 simulated queries. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 UNSUPERVISED EXPLORATION AND SKILL ACQUISITION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+    },
+    {
+        "type": "text",
+        "text": "We can also use DDL to efficiently acquire complex behaviors, such as locomotion skills, in a completely unsupervised fashion. From the observation that many high-reward states are far away from the start state, we can devise a simple and effective exploration strategy that leverages our learned dynamical distances: we can simply select goals that are far from the initial state according to their estimated dynamical distance. We call this variant of our method “dynamical distance learning - unsupervised” (DDLUS). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, this method causes the agent to explore the “frontier” of hard-to-reach states, either discovering shorter paths for reaching them and thus making them no longer be on the frontier, or else finding new states further on the fringe through additive random exploration. In practice, we find that this allows the agent to quickly explore distant states in a directed fashion. In Section 6, we show that, by setting choose go $\\begin{array} { r } { \\mathrm { a l } ( \\mathscr { D } ) \\equiv \\mathrm { \\bar { a r g } m a x } _ { \\mathbf { g } \\in \\mathscr { D } } d _ { \\psi } ^ { \\pi } ( \\mathbf { s } _ { 0 } , \\mathbf { g } ) } \\end{array}$ , where $\\mathbf { s } _ { 0 }$ is the initial state, we can acquire effective running gaits and pole balancing skills in a variety of simulated settings. While this approach is not guaranteed to discover interesting and useful skills in general, we find that, on a variety of commonly used benchmark tasks, this approach to unsupervised goal selection actually discovers behaviors that perform better with respect to the (unknown) task reward than previously proposed unsupervised reinforcement learning objectives. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our experimental evaluation aims to study the following empirical questions: (1) Does supervised regression provide a good estimator of the true dynamical distance? (2) Is DDL applicable to realworld, vision-based robotic control tasks? (3) Does DDL provide an efficient method of learning skills a) from user-provided preferences, and b) completely unsupervised? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate our method both in the real world and in simulation on a set of state- and visionbased continuous control tasks. We consider a 9-DoF real-world dexterous manipulation task and 4 standard OpenAI Gym tasks (Hopper-v3, HalfCheetah-v3, Ant-v3, and InvertedDoublePendulumv2). For all of the tasks, we parameterize our distance function as a neural network, and use soft actor-critic (SAC) (Haarnoja et al., 2018b) with the default hyperparameters to learn the policy. For state-based tasks, we use feed-forward neural networks and for the vision-based tasks we add a convolutional preprocessing network before these fully connected layers. The image observation for all the vision-based tasks are 3072 dimensional $3 2 \\mathrm { x } 3 2$ RGB images). Further details are presented in Appendix E. ",
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+        "type": "text",
+        "text": "We study question (1) using a simple didactic example involving navigation through a twodimensional S-shaped maze, which we present in Appendix C. The other two research questions are studied in the following sections. ",
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+        "type": "text",
+        "text": "6.1 VISION-BASED REAL-WORLD MANIPULATION FROM HUMAN PREFERENCES ",
+        "text_level": 1,
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+        "text": "To study the question (2), we apply DDLfP to a real-world vision-based robotic manipulation task. The domain consists of a 9-DoF “DClaw” hand introduced by Ahn et al. (2019), and the manipulation task requires the hand to rotate a valve 180 degrees, as shown in Figure 1. The human operator is queried for a preference every 10K environment steps. Both the visionand state-based experiments with the real robot use 10 queries during the first 4 hours of an 8- hour training period. Note that, for this and all the subsequent experiments, DDLfP does not have access to the true reward, and must learn entirely from preference queries, which in this case are provided by a human operator. ",
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+        "text": "Figure 3 presents the performance over the course of training. DDLfP uses 10 preference queries to learn the task and its performance is comparable to that of SAC trained with a ground truth shaped reward function. We also ",
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+    {
+        "type": "image",
+        "img_path": "images/0e8638fd707890840c834f07574f1f91dee0ea9f7124ec973dd7253b6efddd6f.jpg",
+        "image_caption": [
+            "Figure 3: (Left) learning curves for the valve rotation task learned from state. (Right) Same task from vision. The curves correspond to the final distance (measured in radians) of the valve from the target angle during a rollout. Our method (DDLfP, orange) solves the task in 8 hours. Its performance is comparable to that of SAC with true rewards, and VICE with example outcome images. DDLfP only requires 10 preference queries, and learns without true rewards or outcome images. We compare our method in the simulated version of this task in Figure 5. "
+        ],
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+    {
+        "type": "text",
+        "text": "show a comparison to variational inverse control with events (VICE) (Singh et al., 2019), a recent classifier-based reward specification framework. Instead of preference queries, VICE requires the user to provide examples of the desired goal state at the beginning of training (20 images in this case). For vision-based tasks, VICE involves directly showing images of the desired outcome to the user, which requires physically arranging a scene and taking a picture of it. Preferences, on the other hand, require a user to simply select one state out of a small set, which can be done with a button press and done e.g. remotely, thus often making it substantially less labor-intensive than VICE. As we can see in the experiments, DDLfP achieves similar performance with substantially less operator effort, using only a small number of preference queries. The series of goal preferences queried from the human operator are shown in Appendix D. ",
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+        "type": "text",
+        "text": "6.2 ABLATIONS, COMPARISONS, AND ANALYSIS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Next, we analyze design decisions in our method and compare it to prior methods in simulation. First, we replace the cumulative objective in Equation 4 with objective that greedily minimizes the distance function trained with supervised loss. This objective is unable to learn the task from either state or vision observations. Next, we replace the supervised loss in Equation 3 of our DDL method with a temporal difference (TD) Q-learning style update rule that learns dynamical distances with approximate dynamic programming. The results in Figure 5 show that, all else being equal, the TD-based method fails to learn successfully from both low-dimensional state and vision observations. Figure 5 further shows a comparison between using the dynamical distance as the reward in comparison to a reward of -1 for each step until the goal is reached, which corresponds to hindsight experience replay (HER) with goal sampling replaced with preference goals (Andrychowicz et al., 2017). We see that dynamical distances allow the policy to reach the goal when learning both from state and from images, while HER is only successful when learning from low-dimensional states. ",
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+        "img_path": "images/8cc14e198c178ab985ae1fbee23f574bd912e277d52d63c9b8c36da42d31afb6.jpg",
+        "image_caption": [
+            "Figure 4: Learning curves for MuJoCo tasks with DDLfP. The y-axis presents the true return of the task. We compare DDLfP to SAC trained directly from the true reward function, which provides an oracle upper bound baseline, and the prior method proposed by Christiano et al. (2017). The prior method uses an on-policy RL algorithm which typically requires more samples than off-policy algorithms, and thus we also plot its final performance after 20M training steps with red star. At the time of the submission, the Ant-v3 run is still in progress and the complete learning curve will be included in the final. "
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+        "type": "text",
+        "text": "These results are corroborated by prior results in the literature that have found that temporal difference learning struggles to capture the true value accurately (Lillicrap et al., 2015; Fujimoto et al., 2018). Note that prior work work does not use the full state as the goal, but rather manually selects a low-dimensional subspace, such as the location of an object, forcing the distance to focus on task-relevant objects (Andrychowicz et al., 2017). Our method learns distances between full image states (3072-dimensional) while HER uses 3- dimensional goals, a difference of two orders of magnitude in dimensionality. This difficulty of learning complex image-based goals is further corroborated in prior work (Pong et al., 2018; Nair et al., 2018; Pong et al., 2019; WardeFarley et al., 2018). ",
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+        "img_path": "images/0b3a0827d435fb404120c1d115f3062e035d2886233bfc9d096b8d61e9c3e894.jpg",
+        "image_caption": [
+            "Figure 5: We compare DDL against alternative methods for learning distances on the simulated valve turning task, when learning from the underlying low-dimensional state (left) and from images (right). Dynamical distances used greedily (orange) or learned with TD (green) generally perform poorly. HER (red) can learn from lowdimensional states, but fails to learn from images. Our method, DDLfP (blue) successfully learns the task from either states or images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 4 presents results for learning from preferences via DDLfP (in green) on a set of continuous control tasks to further study the question (3,a). The plots show the true reward for each method on each task. DDLfP receives only sparse preferences as task-specific supervision, and the preferences in this case are provided synthetically, choosing the state that has progressed the largest distance from the initial state in the desired direction, i.e. the state with largest x-coordinate value. However, this still provides substantially less supervision signal than access to the true reward for all samples. We compare to (Christiano et al., 2017), which also uses preferences for learning skills, but without the use of dynamical distances. The prior method is provided with 750 preference queries over the course of training, while our method uses 100 for all locomotion tasks, and only a single query for the InvertedDoublePendulum-v2, as the initial state and the goal states coincides.3 Note that Christiano et al. (2017) utilizes an on-policy RL algorithms, which is less efficient than SAC. However, DDLfP outperforms this prior method in terms of both final performance and learning speed on all tasks, except for the Hopper-v3 task. ",
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+    {
+        "type": "text",
+        "text": "Locomotion tasks like the ones considered here do not fit into DDL framework directly. In this particular case of locomotion tasks, we can fix the issue by considering a case where the ultimate task is to reach a specific goal, i.e. the operator would always choose the goal to be the state closest to the ”ultimate task goal”. In that case, we can see the locomotion task to be the limit case where the ultimate goal is as far as possibly reachable within the maximum episode length. ",
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+    {
+        "type": "image",
+        "img_path": "images/802d9fd3e2568aae649bb0f14a735f9102ce64347008b9a463a07431197fd1e3.jpg",
+        "image_caption": [
+            "Figure 6: (Top) Learning curves for DDLUS. The y-axis plots the environment return (not accessible during the training) for InvertedDoublePendulum-v3, and the L2-distance travelled from the origin for Hopper-v3, HalfCheetah-v3, and Ant-v3. (Bottom) Frequency histograms of skills learned with DDLUS (blue) and DIAYN (orange) (Eysenbach et al., 2018) across different training runs, evaluated according to the travelled L2-distance from the origin. "
+        ],
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+    {
+        "type": "text",
+        "text": "6.3 ACQUIRING UNSUPERVISED SKILLS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, we study question (3,b) in order to understand how well DDLUS can acquire skills without any supervision. We structure these experiments analogously to the unsupervised skill learning experiments proposed by Eysenbach et al. (2018), and compare to the DIAYN algorithm, another unsupervised skill discovery method, proposed in their prior work. While our method maximizes the complexity of the learned skills by attempting to reach the furthest possible goal, DIAYN maximizes the diversity of learned skills. This of course produces different biases in the skills produced by the two methods. Figure 6 shows both learning curves and histograms of the skills learned in the locomotion tasks with the two methods, evaluated according to how far the simulated robot in each domain travels from the initial state. Our DDLUS method learns skills that travel further than DIAYN, while still providing a variety of different behaviors (e.g., travel in different directions). This experiment aims to provide a direct comparison to the DIAYN algorithm (Eysenbach et al., 2018), though a reasonable criticism is that maximizing dynamical distance is particularly wellsuited for the criteria proposed by Eysenbach et al. (2018). We also evaluated DDLUS on the InvertedDoublePendulum-v2 domain, where the task is to balance a pole on a cart. As can be seen from Figure 6, DDLUS can efficiently solve the task without the true reward, as reaching dynamically far states amounts to avoiding failure as far as possible. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We presented dynamical distance learning (DDL), an algorithm for learning dynamical distances that can be used to specify reward functions for goal reaching policies, and support both unsupervised and semi-supervised exploration and skill discovery. Our algorithm uses a simple and stable supervised learning procedure to learn dynamical distances, which are then used to provide a reward function for a standard reinforcement learning method. This makes DDL straightforward to apply even with complex and high-dimensional observations, such as images. By removing the need for manual reward function design and manual reward shaping, our method makes it substantially more practical to employ deep reinforcement learning to acquire skills even with real-world robotic systems. We demonstrate this by learning a valve-turning task with a real-world robotic hand, using 10 preference queries from a human, without any manual reward design or other examples or supervision. One of the main limitations of our current approach is that, although it can be used with an off-policy reinforcement learning algorithm, it requires on-policy data collection for learning the dynamical distances. While the resulting method is still efficient enough to learn directly in the real world, the efficiency of our approach can likely be improved in future work by lifting this limitation. This would not only make learning faster but would also make it possible to pre-train dynamical distances using previously collected experience, potentially making it feasible to scale our method to a multi-task learning setting, where the same dynamical distance function can be used to learn multiple distinct skills. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We thank Vikash Kumar for the DClaw robot design, Nicolas Heess for helpful discussion, and Henry Zhu and Justin Yu for their help on setting up and running the hardware experiments. This research was supported by the Office of Naval Research, the National Science Foundation through IIS-1651843 and IIS-1700696, and Berkeley DeepDrive. ",
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+    {
+        "type": "text",
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+        "text": "preprint arXiv:1811.11359, 2018. ",
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+    {
+        "type": "text",
+        "text": "Appendices ",
+        "text_level": 1,
+        "bbox": [
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+            98,
+            343,
+            127
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A CORRECT BEHAVIOR IN THE PATHOLOGICAL MDP ",
+        "text_level": 1,
+        "bbox": [
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+            148,
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+            166
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this appendix we show that the policy that maximizes the objective in Equation 1, with the reward $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } ) { \\bar { \\mathbf { \\eta } } } = - d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ , where $d ^ { \\pi }$ is given by Equation 2, prefers safe actions over risky actions. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Assume that $c ( \\mathbf { s } _ { t } , \\mathbf { s } _ { t + 1 } ) = \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { t } \\right]$ is an indicator function that is 0 if $\\mathbf { s } _ { t }$ is a goal state or terminal state and 1 for all the other states. We can now write the definition of $d ^ { \\pi }$ as an infinite sum and substitute $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } )  - d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ in Equation 1: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0ab71c8e3428853bde44c8d0648a4dc52a70ff93948daa5af23d6ad387e422b.jpg",
+        "text": "$$\n\\mathcal { L } ( \\pi ) = - \\mathbb { E } _ { \\tau \\sim \\pi } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { E } _ { \\tau ^ { \\prime } \\sim \\pi } \\left[ \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { k } \\right] \\middle | \\mathbf { s } _ { 0 } ^ { \\prime } = \\mathbf { s } _ { t } , \\mathbf { a } _ { 0 } ^ { \\prime } = \\mathbf { a } _ { t } \\right] \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            270,
+            725,
+            314
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The first term $k = 0 ,$ ) in the inner sum depends only on $\\mathbf { s } _ { 0 } ^ { \\prime }$ , which is given, and the term can thus be moved outside the inner expectation: ",
+        "bbox": [
+            174,
+            323,
+            826,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbdfd8bf6daf077af2fdd7085f6066cf4177190ee888f41e22247f3f759d4b1d.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\pi } ) = - \\mathbb { E } _ { \\boldsymbol { \\tau } \\sim \\boldsymbol { \\pi } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { t } \\right] + \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { E } _ { \\boldsymbol { \\tau } ^ { \\prime } \\sim \\boldsymbol { \\pi } } \\left[ \\sum _ { k = 1 } ^ { \\infty } \\gamma ^ { k } \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { k } ^ { \\prime } \\right] \\bigg \\vert \\mathbf { s } _ { 0 } ^ { \\prime } = \\mathbf { s } _ { t } , \\mathbf { a } _ { 0 } ^ { \\prime } = \\mathbf { a } _ { t } \\right] \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            359,
+            777,
+            404
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Next, note that the statistics of the inner expectation over $( \\mathbf { s } _ { 1 } ^ { \\prime } , \\mathbf { a } _ { 1 } ^ { \\prime } )$ are the same as the outer expectation over $( \\mathbf { s } _ { 1 } , \\mathbf { a } _ { 1 } )$ , as they are both conditioned on the same $\\left( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } \\right)$ . Thus, we can condition the second expectation directly on $( \\mathbf { s } _ { 1 } ^ { \\prime } , \\mathbf { a } _ { 1 } ^ { \\prime } ) = ( \\mathbf { s } _ { t + 1 } , \\mathbf { a } _ { t + 1 } )$ : ",
+        "bbox": [
+            178,
+            412,
+            823,
+            457
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b719aee7587e39fa6c4e41d1ff49a41a705b9239adfd3e0acc0d3632ddb22e4c.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\pi } ) = - \\mathbb { E } _ { \\boldsymbol { \\tau } \\sim \\boldsymbol { \\pi } } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } _ { \\mathbf { g } } [ \\mathbf { s } _ { t } ] + \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { E } _ { \\boldsymbol { \\tau } ^ { \\prime } \\sim \\boldsymbol { \\pi } } [ \\sum _ { k = 1 } ^ { \\infty } \\gamma ^ { k } \\mathbb { 1 } _ { \\mathbf { g } } [ \\mathbf { s } _ { k } ^ { \\prime } ] | \\mathbf { s } _ { 1 } ^ { \\prime } = \\mathbf { s } _ { t + 1 } , \\mathbf { a } _ { 1 } ^ { \\prime } = \\mathbf { a } _ { t + 1 } ] ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            464,
+            784,
+            508
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We can now apply the same argument as before and move $\\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { 1 } ^ { \\prime } \\right]$ outside the inner expectation. Repeating these steps multiple times yields ",
+        "bbox": [
+            173,
+            517,
+            823,
+            547
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b07c1a2f5291c2ef21970a18edb3da7616b3ee59a1dba6716f422715f2ab864.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathcal { L } } ( \\boldsymbol { \\pi } ) = - \\mathbb { E } _ { \\boldsymbol { \\tau } \\sim \\boldsymbol { \\pi } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { t } \\right] + \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t + 1 } \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { t + 1 } \\right] + \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t + 2 } \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { t + 2 } \\right] + . . . \\right] } } \\\\ { { \\displaystyle ~ = - \\mathbb { E } _ { \\boldsymbol { \\tau } \\sim \\boldsymbol { \\pi } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } ( t + 1 ) \\mathbb { 1 } _ { \\mathbf { g } } \\left[ \\mathbf { s } _ { t } \\right] \\right] . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            555,
+            763,
+            643
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Assuming that the agent always reaches the goal relatively quickly compared to the discount factor, such that $\\gamma ^ { t } \\approx 1$ , the trajectories that take longer dominate the loss due to the $( t + 1 )$ factor. Therefore, an optimal agent prefers actions that reduce the risk of long, highly suboptimal trajectories, avoiding the pathological behavior discussed in Section 4.3. ",
+        "bbox": [
+            174,
+            650,
+            825,
+            707
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B POLICY IMPROVEMENT WHEN USING DISTANCE AS REWARD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            729,
+            717,
+            746
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this appendix we show that, when we use the negative dynamical distance $- d ^ { \\pi }$ as the reward function in RL, we can learn an optimal policy with respect to the true dynamical distance, leading to policies that optimize the actual number of time steps needed to reach the goal. This result is nontrivial, since the reward function does not at first glance directly optimize for shortest paths. Our proof relies on the assumption that the MDP has deterministic dynamics. However, this assumption holds in all of our experiments, since the MuJoCo benchmark tasks are governed by deterministic dynamics. Under this assumption, DDL will learn policies that take the shortest path to the goal at convergence, despite using the negative dynamical distance as the reward. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            876
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Let $d ^ { * } ( { \\bf s } , { \\bf g } ) = \\operatorname* { m i n } _ { \\pi } d ^ { \\pi } ( { \\bf s } , { \\bf g } )$ be the optimal distance from state s to goal state $\\mathbf { g }$ . Let $\\pi ^ { \\prime }$ be the optimal policy for the reinforcement learning problem with reward $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } ) = - d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ . DDL can be viewed as alternating between fitting $d ^ { \\pi }$ to the current policy $\\pi$ , and learning a new policy $\\pi ^ { \\prime }$ that is optimal with respect to the reward function given by $- d ^ { \\pi }$ .4 We can now state our main theorem as follows: ",
+        "bbox": [
+            174,
+            881,
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+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Under deterministic dynamics, for any state s and g, we have: ",
+        "bbox": [
+            173,
+            136,
+            673,
+            151
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\leq d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) .$ .   \n2. If $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) = d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ , then $d ^ { \\pi ^ { \\prime } } ( { \\bf s } , { \\bf g } ) = d ^ { \\ast } ( { \\bf s } , { \\bf g } ) .$ ",
+        "bbox": [
+            210,
+            162,
+            560,
+            208
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This implies that, when the policy converges, such that $\\pi ^ { \\prime } = \\pi$ , the policy $\\pi ^ { \\prime }$ achieves the optimal distance to any goal, and therefore is the optimal policy for the shortest path reward function (e.g., the reward function that assigns a reward of $- 1$ for any step that does not reach the goal). ",
+        "bbox": [
+            174,
+            218,
+            825,
+            261
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. ",
+        "bbox": [
+            174,
+            276,
+            215,
+            291
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Part 1 Without loss of generality, we assume that our policy is deterministic, since the set of optimal policies in an MDP always includes at least one deterministic policy. We also assume that g is a terminal state and thus $d ( \\mathbf { g } , \\mathbf { g } ) = 0$ . Let us denote the action of policy $\\pi$ on state s as $\\pi ( \\mathbf { s } )$ . We start by showing that $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\leq d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ . We fix a particular goal $\\mathbf { g }$ . Let $S _ { k } = \\left\\{ \\mathbf { s } ; d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) = k \\right\\}$ be the set of states that takes $k$ steps under $\\pi$ to reach the goal. We show that $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\leq d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) =$ $k$ for all $\\mathbf { s } \\in S _ { k }$ for each $k$ by contradiction. ",
+        "bbox": [
+            173,
+            306,
+            825,
+            396
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For $k = 0$ , $S _ { 0 } = \\{ \\mathbf { g } \\}$ is just the single goal state and ${ d ^ { \\pi } } ^ { \\prime } ( { \\bf g } , { \\bf g } ) = { d ^ { \\pi } } ( { \\bf g } , { \\bf g } ) = 0$ by definition. For $k = 1$ , for all $\\mathbf { s } \\in S _ { 1 }$ , there is an action a that reaches the goal state as the direct next state. Therefore, the optimized policy $\\pi ^ { \\prime }$ would still take the same action a on these states and $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) = 1$ . ",
+        "bbox": [
+            173,
+            404,
+            825,
+            449
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Now assume that the opposite is true, that $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) > d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ for some states. Then, there must be a smallest number $K > 1$ and a state ${ \\bf s } _ { 0 } \\in { \\cal S } _ { K }$ such that $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } _ { 0 } , \\mathbf { g } ) = T > d ^ { \\pi } ( \\mathbf { s } _ { 0 } , \\mathbf { g } ) = K$ . Now let us denote the trajectory of states taken by $\\pi$ starting from ${ \\bf s } _ { 0 }$ as $\\{ \\mathbf { s } _ { 0 } , \\mathbf { s } _ { 1 } , . . . , \\mathbf { s } _ { K } = g \\}$ , and the trajectory taken by $\\pi ^ { \\prime }$ as $\\{ \\mathbf { s } _ { 0 } ^ { \\prime } = \\mathbf { s } _ { 0 } , \\mathbf { s } _ { 1 } ^ { \\prime } , . . . , \\mathbf { s } _ { T } ^ { \\prime } = g \\}$ . Let $\\mathcal { L } _ { \\pi } ( \\cdot )$ denote the accumulated discounted sum of distance as defined in Equation 4. By our assumption $T > K$ , and since $\\pi ^ { \\prime }$ is optimal with respect to the reward $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } ) = - d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ , we have ",
+        "bbox": [
+            173,
+            455,
+            825,
+            545
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97b74743a0aca3dde0a8d6d78155fc7cd4fae4a675b2fc313334717087b449c8.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\pi } ( \\pi ^ { \\prime } ) = \\sum _ { i = 0 } ^ { T - 1 } \\gamma ^ { i } d ^ { \\pi } ( \\mathbf { s } _ { i } ^ { \\prime } , \\mathbf { g } ) \\leq \\mathcal { L } _ { \\pi } ( \\pi ) = \\sum _ { i = 0 } ^ { K - 1 } \\gamma ^ { i } d ^ { \\pi } ( \\mathbf { s } _ { i } , \\mathbf { g } ) = \\sum _ { i = 0 } ^ { K - 1 } \\gamma ^ { i } ( K - 1 - i )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            551,
+            758,
+            595
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Then there must be a time $\\hat { t } < K$ such that $d ^ { \\pi } ( \\mathbf { s } _ { \\hat { t } } ^ { \\prime } , \\mathbf { g } ) < d ^ { \\pi } ( \\mathbf { s } _ { \\hat { t } } , \\mathbf { g } ) = K - 1 - \\hat { t }$ . Therefore $\\mathbf { s } _ { \\hat { t } } ^ { \\prime } \\in S _ { k }$ for some $k < K - 1 - \\hat { t }$ . However, starting from $\\mathbf { s } _ { \\hat { t } } ^ { \\prime }$ , we have $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } _ { \\hat { t } } ^ { \\prime } , \\mathbf { g } ) = T - 1 - \\hat { t } > K - 1 - \\hat { t } =$ $d ^ { \\pi } ( \\mathbf { s } _ { \\hat { t } } ^ { \\prime } , \\mathbf { g } )$ . Therefore, we reached a contradiction with our assumption that $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\leq d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ for all s, $k < K$ such that $\\mathbf { s } \\in S _ { k }$ . Therefore, $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\leq d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ holds for all states. ",
+        "bbox": [
+            173,
+            609,
+            825,
+            683
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Part 2 Now we show the second part: if $d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) = d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } )$ , then $d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) = d ^ { * } ( \\mathbf { s } , \\mathbf { g } )$ . We prove this with a similar argument, grouping states by distance. Let $S _ { k } ^ { * } = \\{ \\mathbf { s } ; d ^ { * } ( \\mathbf { s } , \\mathbf { g } ) = k \\}$ be the set of states that takes $k$ steps under the optimal policy to reach the goal. Note that, for any arbitrary policy $\\pi$ , we have $d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) \\geq d ^ { * } ( \\mathbf { s } , \\mathbf { g } )$ by definition, since $d ^ { * }$ is the optimal distance. ",
+        "bbox": [
+            173,
+            696,
+            825,
+            756
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Suppose that $d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) > d ^ { * } ( \\mathbf { s } , \\mathbf { g } )$ for some state s. Then there must be a smallest integer $K \\geq 0$ such that there exists a state ${ \\bf s } _ { 0 } \\in { \\cal S } _ { K } ^ { * }$ where $d ^ { \\pi } ( { \\bf s } _ { 0 } , { \\bf g } ) > d ^ { * } ( { \\bf s } _ { 0 } , { \\bf g } )$ . For all $k \\ < \\ K$ , we have $d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) = d ^ { * } ( \\mathbf { s } , \\mathbf { g } )$ for all $\\mathbf { s } \\in S _ { k } ^ { * }$ . Now starting from that state ${ \\bf s } _ { 0 }$ , let the trajectory of states taken by $\\pi$ be $\\{ \\mathbf { s } _ { 0 } , \\mathbf { s } _ { 1 } , . . . , \\mathbf { s } _ { T } = g \\}$ . Note that since $d ^ { \\pi } ( { \\bf s } _ { 0 } , { \\bf g } ) > d ^ { * } ( { \\bf s } _ { 0 } , { \\bf g } ) = { \\cal K } , $ $T > K$ . Let $\\hat { \\pi }$ be the policy such that it agrees with $\\pi ^ { * }$ on $\\mathbf { s } _ { 0 }$ and agrees with $\\pi$ everywhere else. At the first step, $\\hat { \\pi }$ lands on state $\\mathbf { s } _ { 1 } ^ { \\prime }$ . Since ${ \\bf s } _ { 0 }$ is $K$ steps away from $\\mathbf { g }$ under $d ^ { * }$ , $\\mathbf { s } _ { 1 } ^ { \\prime }$ must be $K - 1$ steps away under $d ^ { * }$ and ${ \\bf s } _ { 1 } ^ { \\prime } \\in { \\cal S } _ { K - 1 } ^ { * }$ . Therefore, since $\\pi$ and $\\pi ^ { * }$ agrees on all states that are less than $K$ steps away from goal g, $\\hat { \\pi }$ would take the same action as $\\pi ^ { * }$ and hence take another $K - 1$ steps to goal g. Now let us denote the trajectory taken by $\\hat { \\pi }$ as $\\{ \\mathbf { s } _ { 0 } ^ { \\prime } = \\mathbf { s } _ { 0 } , \\mathbf { s } _ { 1 } ^ { \\prime } , . . . , \\mathbf { s } _ { K } ^ { \\prime } = g \\}$ . We compare the discounted sum of rewards of $\\pi$ and $\\hat { \\pi }$ under the reward function $r _ { \\mathbf { g } } ( \\mathbf { s } , \\mathbf { a } ) = - d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ . ",
+        "bbox": [
+            173,
+            761,
+            825,
+            861
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            147
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/03743315f7fc485563c81cff4f4bc782263da69a704e69b919dc35a31977d560.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathcal { L } } _ { \\pi } ( \\pi ) = \\sum _ { i = 0 } ^ { T - 1 } \\gamma ^ { i } d ^ { \\pi } ( { \\bf s } _ { i } , { \\bf g } ) = d ^ { \\pi } ( { \\bf s } _ { 0 } , { \\bf g } ) + \\sum _ { i = 1 } ^ { T - 1 } \\gamma ^ { i } d ^ { \\pi } ( { \\bf s } _ { i } , { \\bf g } ) } \\ ~ } \\\\ { { \\displaystyle ~ = d ^ { \\pi } ( { \\bf s } _ { 0 } , { \\bf g } ) + \\sum _ { i = 1 } ^ { T - 1 } \\gamma ^ { i } ( T - i ) \\geq d ^ { \\pi } ( { \\bf s } _ { 0 } , { \\bf g } ) + \\sum _ { i = 1 } ^ { K - 1 } \\gamma ^ { i } ( K - i ) } \\ ~ } \\\\ { { \\displaystyle ~ = d ^ { \\pi } ( { \\bf s } _ { 0 } , { \\bf g } ) + \\sum _ { i = 1 } ^ { K - 1 } \\gamma ^ { i } d ^ { \\pi } ( { \\bf s } _ { i } ^ { \\prime } , { \\bf g } ) = \\sum _ { i = 0 } ^ { K - 1 } \\gamma ^ { i } d ^ { \\pi } ( { \\bf s } _ { i } ^ { \\prime } , { \\bf g } ) = { \\mathcal L } _ { \\pi } ( \\hat { \\pi } ) } \\ ~ } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            253,
+            722,
+            386
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we can see that $\\hat { \\pi }$ is a better policy than $\\pi$ . Then the optimal policy $\\pi ^ { \\prime }$ under this reward must be different from $\\pi$ on at least one state. Hence $d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } ) \\neq d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } )$ . ",
+        "bbox": [
+            173,
+            479,
+            825,
+            512
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We’ve now reached the conclusion that if $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\neq d ^ { * } ( \\mathbf { s } , \\mathbf { g } )$ , then $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) \\neq d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ . Hence, by contraposition, if $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) = d ^ { \\pi } ( \\mathbf { s } , \\mathbf { g } )$ , then it must be that $d ^ { \\pi ^ { \\prime } } ( \\mathbf { s } , \\mathbf { g } ) = d ^ { \\ast } ( \\mathbf { s } , \\mathbf { g } )$ . Our proof is thus complete. ",
+        "bbox": [
+            174,
+            518,
+            825,
+            565
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C DIDACTIC EXAMPLE ",
+        "bbox": [
+            174,
+            698,
+            382,
+            714
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our didactic example involves a simple 2D point robot navigating an S-shaped maze. The state space is two-dimensional, and the action is a two-dimensional velocity vector. This experiment is visualized in Figure 7. The black rectangles correspond to walls, and the goal is depicted with a blue star. The learned distance from all points in the maze to the goal is illustrated with a heat map, in which lighter colors correspond to closer states and darker colors to distant states. During the training, the initial state is chosen uniformly at random, and the policy is trained to reach the goal state. From the visualization, it is apparent that DDL learns an accurate estimate of the true dynamical distances in this domain. Note that, in contrast to na¨ıve metrics, such as Euclidean distance, the dynamical distances conform to the walls and provide an accurate estimate of reachability, making them ideally suited for reward shaping. ",
+        "bbox": [
+            173,
+            784,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/0b1a512d4116173ff2294981ff2c4240e5fd3f1a154be461354f90236d4173dd.jpg",
+        "image_caption": [
+            "Figure 7: Evaluation of the learned distance in a 2D point environment. The state is the xycoordinates of the point, and action corresponds to 2D velocities. The black bars denote walls, blue star is a goal state, and the heat map denotes the estimated distance to the goal. (a) Our method learns an accurate estimate of the shape of the distance function. (b) Ground-truth distance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
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+            342
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D PREFERENCE QUERIES FOR REAL-WORLD DCLAW EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            751,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/1f5b8406ba52b303a00f5db5e76a224ae4a6bbbbe39040a18a95ab863a5b6bfb.jpg",
+        "image_caption": [
+            "Figure 8: Human preference queries for the vision-based DClaw experiment presented in Section 6.1. Each image row presents the set of images shown to the human operator on a single query round. On each row, the first 10 images correspond to the last states of the most recent rollouts and the right-most image corresponds to the last goal. For each query, the human operator picks a new goal by inputting its index (between 0-10) into a text-based interface. The goals selected by human are highlighted with white borders. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            32,
+            125,
+            965,
+            848
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E TECHNICAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            387,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "All our experiments use Soft Actor-Critic as the policy optimizer, trained the default parameters by provided by the authors in (Haarnoja et al., 2018c). ",
+        "bbox": [
+            174,
+            133,
+            823,
+            161
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "For all of the tasks, we parameterize our distance function as a neural network. For state-based tasks, we use feed-forward neural networks with two 256-unit hidden layers. For the vision-based tasks we add a convolutional preprocessing network before these fully-connected layers, consisting of four convolutional layers, each with $6 4 3 \\mathrm { x } 3 $ filters. Both cases use Adam optimizer with learning rate 3e4 and TensorFlow‘s default momentum parameters. The image observation for all the vision-based tasks are 3072 dimensional (32x32 RGB images). ",
+        "bbox": [
+            173,
+            167,
+            825,
+            252
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Most important hyperparameters that we swept over in the final experiments, namely the size of the on-policy pool for training the distance function and the number of gradient steps per environment samples, are presented in Table 1 below: ",
+        "bbox": [
+            174,
+            260,
+            823,
+            301
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/f72d091c25aa26bc117c478dffbe33a7d9f3c835bfed6250b83bfab708991bab.jpg",
+        "table_caption": [
+            "Table 1: Distance estimator hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Environment</td><td>gradient steps per environment steps</td><td>on-policy pool size</td></tr><tr><td>InvertedDoublePendulum-v2</td><td>1/64</td><td>100k</td></tr><tr><td>Hopper-v3</td><td>1/64</td><td>16k</td></tr><tr><td>HalfCheetah-v3</td><td>1/16</td><td>16k</td></tr><tr><td>Ant-v3</td><td>1/64</td><td>10k</td></tr><tr><td>DClaw (both state and vision)</td><td>1/16</td><td>100k</td></tr></table>",
+        "bbox": [
+            189,
+            314,
+            808,
+            401
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "For the DDLUS goal proposals, we consider all the samples in the distance on-policy pool as the goal candidates. For DDLfP, we present the operator the last states $( s _ { T - 1 } )$ of the last $N$ episodes, where $N = 5$ for all the simulated experiments, and $N = 1 0$ for the hardware DClaw. ",
+        "bbox": [
+            174,
+            440,
+            825,
+            483
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 5, for both DDLUS and DDLfP, the agent needs to explore in the vicinity of the goal state. In practice, we implement this by switching to a random uniform policy after $0 . 9 \\mathrm { T }$ timesteps of each episode, where $\\mathrm { T }$ is the maximum episode length (1000 for all the mujoco tasks and 200 for the DClaw task). ",
+        "bbox": [
+            174,
+            489,
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+            545
+        ],
+        "page_idx": 16
+    }
+]

parse/train/HkxAAvcxx/HkxAAvcxx.md

@@ -0,0 +1,179 @@
+# TRANSFORMATION-BASED MODELS OF VIDEO SEQUENCES
+
+Joost van Amersfoort ∗, Anitha Kannan, Marc’Aurelio Ranzato, Arthur Szlam, Du Tran & Soumith Chintala
+
+Facebook AI Research joost@joo.st, {akannan, ranzato, aszlam, trandu, soumith}@fb.com
+
+# ABSTRACT
+
+In this work we propose a simple unsupervised approach for next frame prediction in video. Instead of directly predicting the pixels in a frame given past frames, we predict the transformations needed for generating the next frame in a sequence, given the transformations of the past frames. This leads to sharper results, while using a smaller prediction model.
+
+In order to enable a fair comparison between different video frame prediction models, we also propose a new evaluation protocol. We use generated frames as input to a classifier trained with ground truth sequences. This criterion guarantees that models scoring high are those producing sequences which preserve discriminative features, as opposed to merely penalizing any deviation, plausible or not, from the ground truth. Our proposed approach compares favourably against more sophisticated ones on the UCF-101 data set, while also being more efficient in terms of the number of parameters and computational cost.
+
+# 1 INTRODUCTION
+
+There has been an increased interest in unsupervised learning of representations from video sequences (Mathieu et al., 2016; Srivastava et al., 2015; Vondrick et al., 2016). A popular formulation of the task is to learn to predict a small number of future frames given the previous K frames; the motivation being that predicting future frames requires understanding how objects interact and what plausible sequences of motion are. These methods directly aim to predict pixel values, with either MSE loss or adversarial loss.
+
+In this paper, we take a different approach to the problem of next frame prediction. In particular, our model operates in the space of transformations between frames, directly modeling the source of variability. We exploit the assumption that the transformations of objects from frame to frame should be smooth, even when the pixel values are not. Instead of predicting pixel values, we directly predict how objects transform. The key insight is that while there are many possible outputs, predicting one such transformation will yield motion that may not correspond to ground truth, yet will be realistic; see fig. 1. We therefore propose a transformation-based model that operates in the space of affine transforms. Given the affine transforms of a few previous frames, the model learns to predict the local affine transforms that can be deterministically applied on the image patches of the previous frame to generate the next frame. The intuition is that estimation errors will lead to a slightly different yet plausible motion. Note that this allows us to keep using the MSE criterion, which is easy to optimize, as long as it is in transformation space. No blur in the pixel space will be introduced since the output of the transformation model is directly applied to the pixels, keeping sharp edges intact. Refer to fig. 5 and our online material 1 for examples.
+
+The other contribution of this work is the evaluation protocol. Typically, generative models of video sequences are evaluated in terms of MSE in pixel space (Srivastava et al., 2015), which is not a good choice since this metric favors blurry predictions over other more realistic looking options that just happen to differ from the ground truth. Instead, we propose to feed the generated frames to a video classifier trained on ground truth sequences. The idea is that the less the classifier’s performance is affected by the generates frames the more the model has preserved distinctive features and the more the generated sequences are plausible. Regardless of whether they resemble the actual ground truth or not. This protocol treats the classifier as a black box to measure how well the generated sequences can serve as surrogate for the truth sequence for the classification task. In this paper we will validate our assumption that motion can be modelled by local affine transforms, after which we will compare our method with networks trained using adversarial training and simple regression on the output frame, using both this new evaluation protocol and by providing samples for qualitative inspection.
+
+![](images/12ee9523c8c7a6b995c4a5f346b3260214950d38e8e359bf96b473cbaa612058.jpg)  
+Figure 1: Motivating toy example. From left to right: the first digit shows what the model is conditioned upon, the second digit shows the frame we would like to predict at the next time step, the third digit shows the blurry prediction if we were to minimize MSE in pixel space, the last digit shows the prediction when minimizing MSE in the space of transformations. While the two models may have the same MSE in pixel space, the transformation-based model generates much sharper outputs. Although the motion is different than the ground truth (second digit), it is still a plausible next frame to the conditioned frame. In practice, the input is a sequence of consecutive frames.
+
+Our experiments show that our simple and efficient model outperforms other baselines, including much more sophisticated models, on benchmarks on the UCF-101 data set (Soomro et al., 2012). We also provide qualitative comparisons to the moving MNIST digit data set (Srivastava et al., 2015).
+
+# 1.1 RELATED WORK
+
+Early work on video modeling focused on predicting small patches (Michalski et al., 2014; Srivastava et al., 2015); unfortunately, these models have not shown to scale to the complexity of highresolution videos. Also these models require a significant amount of parameters and computational power for even relatively simple data.
+
+In Ranzato et al. (2014), the authors circumvented this problem by quantizing the space of image patches. While they were able to predict a few high-resolution frames in the future, it seems dissatisfying to impose such a drastic assumption to simplify the prediction task.
+
+Mathieu et al. (2016) recently proposed to replace MSE in pixel space with a MSE on image gradients, leveraging prior domain knowledge, and further improved using a multi-scale architecture with adversarial training (Goodfellow et al., 2014). While producing better results than earlier methods, the models used require a very large amount of computational power. We make an explicit comparison to this paper in the experiments section 3.
+
+In Oh et al. (2015), frames of a video game are predicted given an action (transformation) taken by the player. While the paper shows great results, the movement in a natural video cannot be described by a simple action and is therefore not widely applicable. Finally, our work is also related to optical flow estimation (Brox et al., 2004). Instead of estimating the flow of pixels, here we estimate the flow of patches and separately predict how these patches transform in future frames.
+
+Prior work relating to the evaluation protocol can be found in Yan et al. (2015). The authors generate images using a set of predefined attributes and later show that they can recover these using a pretrained neural network. Our proposal extends this to videos, which is more complicated since both appearance and motion are needed for correct classification.
+
+![](images/df89c340471c0d73caddfcea4acad4c37effaff1ca0dd8d6869df3386e93421a.jpg)  
+Figure 2: Outline of the transformation-based model. The model is a CNN that takes as input a sequence of consecutive affine transforms between pairs of adjacent video frames. It predicts the affine transform between the last input frame and the next one in the sequence. We compute affine transforms (6 parameters per patch) for overlapping patches of size $8 \times 8$ in each video frame. Learning operates in the space of transformations as shown inside the dashed box. The front-end on the left is a module that estimates the affine transforms between pairs of consecutive input frames. The post-processor on the right reconstructs a frame from the predicted set of affine transforms and it is only used at test time.
+
+# 2 MODEL
+
+The model we propose is based on three key assumptions: 1) just estimating object motion yields sequences that are plausible and relatively sharp, 2) global motion can be estimated by tiling highresolution video frames into patches and estimating motion “convolutionally” at the patch level, and 3) patches at the same spatial location over two consecutive time steps undergo a deformation which can be well described by an affine transformation.
+
+The first assumption is at the core of the proposed method: by considering uncertainty in the space of transformations we produce sequences that may still look plausible. The other two assumptions state that a video sequence can be composed by patches undergoing affine transformations. We agree that these are simplistic assumptions, which ignore how object identity affects motion and do not account for out of plane rotations and more general forms of deformation. However, our qualitative and quantitative evaluation shows the efficacy of these assumptions to real video sequence as can be seen in section 3 and from visualizations in the supplementary material2.
+
+Our approach consists of three steps. First, we estimate affine transforms of every video sequence to build a training set for our model. Second, we train a model that takes the past $N$ affine transforms and predicts the next $M$ affine transforms. Finally, at test time, the model uses the predicted affine transforms to reconstruct pixel values of the generated sequence. We describe the details of each phase in the following sections.
+
+# 2.1 AFFINE TRANSFORM EXTRACTOR
+
+Given a frame $x$ and the subsequent frame $y$ , the goal of the affine transform extractor is to learn mappings that can warp $x$ into $y$ . Since different parts of the scene may undergo different transforms, we tile $x$ into overlapping patches and infer a transformation for each patch. The estimation process couples the transformations at different spatial locations because we minimize the reconstruction error of the entire frame $y$ , as opposed to treating each patch independently.
+
+![](images/81a308020c438f39ad5439f467fe0f3828cc385ad43508393d31c92ed57c46bb.jpg)  
+Figure 3: Outline of the system predicting 4 frames ahead in time. Only affine transforms $A _ { 1 }$ , $A _ { 2 }$ and $A _ { 3 }$ are provided, and the model predicts ${ \tilde { A } } _ { 4 }$ , ${ \tilde { A } } _ { 5 }$ , ${ \tilde { A } } _ { 6 }$ and ${ \tilde { A } } _ { 7 }$ , which are used to reconstruct the next 4 frames. Since affine parameters are continuous values and the whole chain of CNNs is differentiable, the whole unrolled system can be trained by back-propagation of the error. Note that CNNs all share the same parameters
+
+Let $x$ and $y$ have size $D _ { r } \times D _ { c }$ . Let image $x$ be decomposed into a set of overlapping patches, each containing pixels from patches of size $d _ { r } \times d _ { c }$ with $d _ { r } \leq D _ { r }$ and $d _ { c } \leq D _ { c }$ . These patches are laid out on a regular grid with stride $s _ { r }$ and $s _ { c }$ pixels over rows and columns, respectively. Therefore, every pixel participates in $\frac { d _ { r } } { s _ { r } } \frac { d _ { c } } { s _ { c } }$ overlapping patches, not taking into account for the sake of simplicity border effects and non-integer divisions. We denote the whole set of overlapping patches by $\{ X _ { k } \}$ , where index $k$ runs over the whole set of patches. Similarly and using the same coordinate system, we denote by $\left\{ Y _ { k } \right\}$ the set of overlapping patches of $y$ .
+
+We assume that there is an affine mapping $A _ { k }$ that maps $X _ { k }$ to $Y _ { k }$ , for all values of $k$ . $A _ { k }$ is a $2 \times 3$ matrix of free parameters representing a generic affine transform (translation, rotation and scaling) between the coordinates of output and input frame. Let $\tilde { Y } _ { k }$ be the transformed patches obtained when $A _ { k }$ is applied to $X _ { k }$ . Since coordinates overlap between patches, we reconstruct $y$ by averaging all predictions at the same location, yielding the estimate $\tilde { y }$ . The joint set of $A _ { k }$ is then jointly determined by minimizing the mean squared reconstruction error between $y$ and $\tilde { y }$ .
+
+Notice that our approach and aim differs from spatial transformer networks (Jaderberg et al., 2015) since we perform this estimation off-line only for the input frames, computing one transform per patch.
+
+In our experiments, we extracted $1 6 \times 1 6$ pixel patches from the input and we used stride 4 over rows and columns. The input patches are then matched at the output against smaller patches of size $8 \times 8$ pixels, to account for objects moving in and out of the patch region.
+
+# 2.2 AFFINE TRANSFORM PREDICTOR
+
+The affine transform predictor is used to predict the affine transforms between the last input frame and the next frame in the sequence. A schematic illustration of the system is shown in fig. 2. It receives as input the affine transforms between pairs of adjacent frames, as produced by the affine transform extractor described in the previous section. Each transform is arranged in a grid of size $6 \times n \times n$ , where $n$ is the number of patches in a row/column and 6 is the number of parameters of each affine transform. Therefore, if four frames are used to initialize the model, the actual input consists of 18 maps of size $n \times n$ , which are the concatenation of $A _ { t - 2 } , A _ { t - 1 } , A _ { t }$ , where $A _ { t }$ is the collection of patch affine transforms between frame at time $t - 1$ and $t$ .
+
+The model consists of a multi-layer convolutional network without any pooling. The network is the composition of convolutional layers with ReLU non-linearity, computing a component-wise thresholding as in $v = \operatorname* { m a x } ( 0 , u )$ . We learn the parameters in the filters of the convolutional layers by minimizing the mean squared error between the output of the network and the target transforms.
+
+Notice that we do not add any regularization to the model. In particular, we rely on the convolutional structure of the model to smooth out predictions at nearby spatial locations.
+
+# 2.3 MULTI-STEP PREDICTION
+
+In the previous section, we described how to predict the set of affine transforms at the next time step.   
+In practice, we would like to predict several time steps in the future.
+
+A greedy approach would: a) train as described above to minimize the prediction error for the affine transforms at the next time step, and b) at test time, predict one step ahead and then re-circulate the model prediction back to the input to predict the affine transform two steps ahead, etc. Unfortunately, errors may accumulate throughout this process because the model was never exposed to its own predictions at training time.
+
+The approach we propose replicates the model over time, also during training as shown in fig. 3. If we wish to predict $M$ steps in the future, we replicate the CNN $M$ times and pass the output of the CNN at time step $t$ as input to the same CNN at time step $t + 1$ , as we do at test time. Since predictions live in a continuous space, the whole system is differentiable and amenable to standard back-propagation of the error. Since parameters of the CNN are shared across time, the overall system is equivalent to a peculiar recurrent neural network, where affine transforms play the role of recurrent states. The experiments in section 3 demonstrate that this method is more accurate and robust than the greedy approach.
+
+# 2.4 TESTING
+
+At test time, we wish to predict $M$ frames in the future given the past $N$ frames. After extracting the $N - 1$ affine transforms from the frames we condition upon, we replicate the model $M$ times and feed its own prediction back to the input, as explained in the previous section.
+
+Once the affine transforms are predicted, we can reconstruct the actual pixel values. We use the last frame of the sequence and apply the first set of affine transforms to each patch in that frame. Each pixel in the output frame is predicted multiple times, depending on the stride used. We average these predictions and reconstruct the whole frame. As required, we can repeat this process for as many frames as necessary, using the last reconstructed frame and the next affine transform.
+
+In order to evaluate the generation, we propose to feed the generated frames to a trained classifier for a task of interest. For instance, we can condition the generation using frames taken from video clips which have been labeled with the corresponding action. The classifier has been trained on ground truth data but it is evaluated using frames fantasized by the generative model. The performance of the classifier on ground truth data is an upper bound on the performance of any generative model. This evaluation protocol does not penalize any generation that deviates from the ground truth, as standard MSE would. It instead check that discriminative features and the overall semantics of the generated sequence is correct, which is ultimately what we are interested in.
+
+# 3 EXPERIMENTS
+
+In this section, we validate the key assumptions made by our model and compare against state-ofthe-art generative models on two data sets. We strongly encourage the reader to watch the short video clips in the Supplementary Material to better understand the quality of our generations.
+
+In section 2, we discussed the three key assumptions at the foundations of our model: 1) errors in the transformation space look still plausible, 2) a frame can be decomposed into patches, and 3) each patch motion is well modeled by an affine transform. The results in the Supplementary Material 3 validate assumption 2 and 3 qualitatively. Every row shows a sequence from the UCF101 dataset (Soomro et al., 2012). The column on the left shows the original video frames and the one on the right the reconstructions from the estimated affine transforms, as described in section 2.1. As you can see there is barely any noticeable difference between these video sequences, suggesting that video sequences can be very well represented as tiled affine transforms. For a quantitative comparison and for an assessment of how well the first assumption holds, please refer to section 3.2.
+
+![](images/7443870679cdf0ab5ac54a3777ad2932377d7e6b13ddb24a80c1973a6c1827d3.jpg)  
+Figure 4: Predictions of 4 sequences from the moving MNIST dataset. The top row of each pair shows the ground truth frames; the first four frames are used as input to the model. The bottom row shows the predictions of the model.
+
+In the next section, we will first report some results using the toy data set of “moving MNIST digits” (Srivastava et al., 2015). We then discuss generations of natural high-resolution videos using the UCF-101 dataset and compare to current state-of-the-art methods.
+
+# 3.1 MOVING MNIST
+
+For our first experiment, we used the dataset of moving MNIST digits (Srivastava et al., 2015) and perform qualitative analysis4. It consists of one or two MNIST digits, placed at random locations and moving at constant speed inside a $6 4 \times 6 4$ frame. When a digit hits a boundary, it bounces, meaning that velocity in that direction is reversed. Digits can occlude each other and bounce off walls, making the data set challenging.
+
+Using scripts provided by Srivastava et al. (2015), we generated a fixed dataset of 128,000 sequences and used $80 \%$ for training, $10 \%$ for validation and $10 \%$ for testing. Next, we estimated the affine transforms between every pair of adjacent frames to a total of 4 frames, and trained a small CNN in the space of affine transforms. The CNN has 3 convolutional layers and the following number of feature maps: 18, 32, 32, 6. All filters have size $3 \times 3$ .
+
+Fig. 4 shows some representative test sequences and the model outputs. Each subfigure corresponds to a sequence from the test set; the top row corresponds to the ground truth sequence while the bottom row shows the generations. The input to the CNN are three sets of affine transforms corresponding to the first four consecutive frames. The network predicts the next six sets of affine transforms from which we reconstruct the corresponding frames. These results should be compared to fig. 5 in Srivastava et al. (2015). The generations in fig. 4 show that the model has potential to represent and generate video sequences, it learns to move digits in the right direction, to bounce them, and it handles multiple digits well except when occluion makes inputs too ambiguous. The model’s performance is analyzed quantitatively in the next section using high resolution natural videos.
+
+# 3.2 UCF 101 DATA SET
+
+The UCF-101 dataset (Soomro et al., 2012) is a collection of 13320 videos of 101 action categories. Frames have size $2 4 0 \times 3 2 0$ pixels. We train a CNN on patches of size $6 4 \times 6 4$ pixels; the CNN has 6 convolutional layers and the following number of feature maps: 18, 128, 128, 128, 64, 32, 16, 6. All filters have size $3 \times 3$ . The optimal number of filters has been found using cross-validation in order to minimize the estimation error of the affine transform parameters. Unless otherwise stated, we condition generation on 4 ground truth frames and we predict the following 8 frames.
+
+We evaluate several models5: a) a baseline which merely copies the last frame used for conditioning, b) a baseline method which estimates optical flow (Brox et al., 2004) from two consecutive frames and extrapolates flow in subsequent frames under the assumption of constant flow speed, c) an adversarially trained multi-scale CNN (Mathieu et al., 2016) and several variants of our proposed approach.
+
+![](images/9bb8e2779df3278f29e753245446f1e87e7f9f6c450239af241f4b76732a3d70.jpg)  
+Figure 5: Example of predictions produced by different models. Each row shows an example. The first two columns show the ground truth. The two frames are 4 time steps apart. The next two columns show predictions from a baseline model employing optical flow. Next, we show the prediction produced by the adversarially trained CNN proposed by Mathieu et al. (2016). The last two column show the prediction produced by our affine-transformation based approach. All pairs in the same column group are four time steps apart. All methods were conditioned on the same set of 4 input frames (not shown in the figure)
+
+Table 1: Classification accuracy on UCF-101 dataset. The classifier is trained on the actual training video sequences, but it is tested using frames generated by various generative models. Each column shows the accuracy on the test set when taking a different number of input frames as input. Our approach maps $1 6 \times 1 6$ patches into $8 \times 8$ with stride 4, and it takes 4 frames at the input.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1> 4 frames</td><td rowspan=1 colspan=1>8 frames</td></tr><tr><td rowspan=1 colspan=1>Ground truth frames</td><td rowspan=1 colspan=1>72.46</td><td rowspan=1 colspan=1>72.29</td></tr><tr><td rowspan=1 colspan=1>Using ground truth affine transforms</td><td rowspan=1 colspan=1>71.7</td><td rowspan=1 colspan=1>71.28</td></tr><tr><td rowspan=1 colspan=1>Copy last frame</td><td rowspan=1 colspan=1>60.76</td><td rowspan=1 colspan=1>54.27</td></tr><tr><td rowspan=1 colspan=1>Optical Flow</td><td rowspan=1 colspan=1>57.29</td><td rowspan=1 colspan=1>49.37</td></tr><tr><td rowspan=1 colspan=1>Mathieu et al. (2016)</td><td rowspan=1 colspan=1>57.98</td><td rowspan=1 colspan=1>47.01</td></tr><tr><td rowspan=1 colspan=1>ours - one step prediction (not unrolled)</td><td rowspan=1 colspan=1>64.13</td><td rowspan=1 colspan=1>57.63</td></tr><tr><td rowspan=1 colspan=1>ours - four step prediction (unrolled 4 times)</td><td rowspan=1 colspan=1>64.54</td><td rowspan=1 colspan=1>57.88</td></tr></table>
+
+Qualitative comparisons can be seen in the fig. 5 and in the supplementary material6. The first column on the page shows the input, the second the ground truth, followed by results from our model, Mathieu et al. (2016) and optical flow (Brox et al., 2004). Note especially the severe deformations in the last two columns, while our model keeps the frame recognizable. It produces fairly sharp reconstructions validating our first hypothesis that errors in the space of transformations still yield plausible reconstructions (see section 2). However it is also apparent that our approach underestimates movement, which follows directly from using the MSE criterion. As discussed before, MSE in pixel space leads to blurry results, however using MSE in transformation space also has some drawbacks. In practice, the model will predict the average of several likely transformations, which could lead to an understimation of the true movement.
+
+In order to quantify the generation quality we use the metric described in section 2.4. We use C3D network (Tran et al., 2015) as the video action classifier: C3D uses both appearance and temporal information jointly, and is pre-trained with Sports1M (Karpathy et al., 2014) and fine tuned on UCF 101. Due to the model constraints, we trained only two models, that takes 4 and 8 frames as input, respectively.
+
+We evaluate the quality of generation using 4 (the first four predicted frames) and the whole set of 8 predicted frames, for the task of action classification. At test time, we generate frames from each model under consideration, and then use them as input to the corresponding C3D network.
+
+Table 1 shows the accuracy of our approach and several baselines. The best performance is achieved by using ground truth frames, a result comparable to methods recently appeared in the literature (Karpathy et al., 2014; Tran et al., 2015). We see that for ground truth frames, the number of frames (4 or 8) doesn’t make a difference. There is not much additional temporal or spatial signal provided by having greater than four frames. Next, we evaluate how much we lose by representing frames as tiled affine transforms. As the second row shows there is negligible if any loss of accuracy when using frames reconstructed from the estimated affine transforms (using the method described in section 2.1), validating our assumptions at the beginning of section 2 on how video sequences can be represented. The next question is then whether these affine transforms are predictable at all. The last two rows of Table 1 show that this is indeed the case, to some extent. The longer the sequence of generated frames the poorer the performance, since the generation task gets more and more difficult.
+
+Compared to other methods, our approach performs better than optical flow and even the more sophisticated multi-scale CNN proposed in Mathieu et al. (2016) while being computationally cheaper. For instance, our method has less than half a million parameters and requires about 2G floating point operations to generate a frame at test time, while the multi-scale CNN of Mathieu et al. (2016) has 25 times more parameters (not counting the discriminator used at training time) and it requires more than 100 times more floating point operations to generate a single frame.
+
+Finally, we investigate the robustness of the system to its hyper-parameters: a) choice of patch size, b) number of input frames, and c) number of predicted frames. The results reported in Table 2 demonstrate that the model is overall pretty robust to these choices. Using patch sizes that are too big makes reconstructions blocky but within each block motion is coherent. Smaller patch sizes give more flexibility but make the prediction task harder as well. Mapping into patches of size smaller than $1 6 \times 1 6$ seems a good choice. Using only 2 input frames does not seem to provide enough context to the predictor, but anything above 3 works equally well. Training for prediction of the next frame works well, but better results can be achieved by training to predict several frames in the future, overall when evaluating longer sequences.
+
+Table 2: Analysis of the robustness to the choice of hyper-parameters, shows classification scores compared to reference model. The reference model takes 4 frames as input, predicts one frame, and maps $1 2 \times 1 2$ patches onto $8 \times 8$ patches with stride 4.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>4 frames</td><td rowspan=1 colspan=1>8 frames</td></tr><tr><td rowspan=1 colspan=1>reference</td><td rowspan=1 colspan=1>63.57</td><td rowspan=1 colspan=1>57.32</td></tr><tr><td rowspan=1 colspan=1>Varying patch sizefrom 32 × 32 to 16× 16from 16 × 16 to 8 × 8</td><td rowspan=1 colspan=1>61.7363.75</td><td rowspan=1 colspan=1>53.8557.18</td></tr><tr><td rowspan=1 colspan=1>Number of input frames23</td><td rowspan=1 colspan=1>63.663.8</td><td rowspan=1 colspan=1>57.1157.4</td></tr><tr><td rowspan=1 colspan=1>Number of predicted frames24</td><td rowspan=1 colspan=1>64.164.54</td><td rowspan=1 colspan=1>57.557.88</td></tr></table>
+
+# 4 CONCLUSIONS
+
+In this work, we proposed a new approach to generative modeling of video sequences. This model does not make any assumption about the spatio-temporal resolution of video sequences nor about object categories. The key insight of our approach is to model in the space of transformations as opposed to raw pixel space. A priori we lack a good metric to measure how well a frame is reconstructed under uncertainty due to objects motion in natural scenes. Uncertainty about object motion and occlusions causes blurry generations when using MSE in pixel space. Instead, by operating in the space of transformations we aim at predicting how objects move, and estimation errors only yield a different, and possibly still plausible, motion. With this motivation we proposed a simple CNN operating in the space of affine transforms and we showed that it can generate sensible sequences up to about 4 frames. This model produces sequences that are both visually and quantitatively better than previously proposed approaches.
+
+The second contribution of this work is the metric to compare generative models of video sequences. A good metric should not penalize a generative model for producing a sequence which is plausible but different from the ground truth. With this goal in mind and assuming we have at our disposal labeled sequences, we can first train a classifier using ground truth sequences. Next, the classifier is fed with sequences produced by our generative model for evaluation. A good generative model should produce sequences that still retain discriminative features. In other words, plausibility of generation is assessed in terms of how well inherent information is preserved during generation as opposed to necessarily and merely reproducing the ground truth sequences.
+
+The proposed model is relatively simple; straightforward extensions that could improve its prediction accuracy are the use of a multi-scale architecture and the addition of recurrent units. These would enable a better modeling of objects of different sizes moving at varying speeds and to better capture complex temporal dynamics (e.g., cyclical movements like walking). A larger extension would be the addition of an appearance model, which together with our explicit transformation model could lead to learning better feature representations for classification.
+
+In our view, the proposed approach should be considered as a stronger baseline for future research into next frame prediction. Even though our analysis shows improved performance and better looking generations, there are also obvious limitations. The first such limitation is the underestimation of transformations due to usage of the MSE as a criterion. We consider two main avenues worth pursuing in this space. First, we consider modelling a distribution of transformations and sampling one from it. The challenge of this approach is to sample a consistent trajectory. One could model the distribution of an entire trajectory, but that is a complex optimization problem. A second option is to use adversarial training to force the model to pick a plausible action. This option does not guarantee that underestimation of movement will be avoided. This will depend on the discriminator model accepting this as a plausible option.
+
+Another limitation is that the current model does not factor out the “what” from the “where”, appearance from motion. The representation of two distinct objects subject to the same motion, as well as the representation of the same object subject to two different motion patterns are intrinsically different. Instead, it would be more powerful to learn models that can discover such factorization and leverage it to produce more efficient and compact representations.
+
+# ACKNOWLEDGMENTS
+
+Authors thank Camille Couprie and Michael Mathieu for discussions and helping with evaluation of their models.
+
+# REFERENCES
+
+Thomas Brox, Andres Bruhn, Nils Papenberg, and Joachim Weickert. High accuracy optical flow ´ estimation based on a theory for warping. In Computer Vision-ECCV 2004, pp. 25–36. Springer, 2004.
+
+I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In NIPS, 2014.
+
+Max Jaderberg, Karen Simonyan, Andrew Zisserman, and Koray Kavukcuoglu. Spatial transformer networks. NIPS, 2015.
+
+Andrej Karpathy, George Toderici, Sachin Shetty, Tommy Leung, Rahul Sukthankar, and Li FeiFei. Large-scale video classification with convolutional neural networks. In Computer Vision and Pattern Recognition (CVPR), 2014 IEEE Conference on, pp. 1725–1732. IEEE, 2014.
+
+Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. In ICLR, 2016.
+
+Vincent Michalski, Roland Memisevic, and Kishore Konda. Modeling deep temporal dependencies with recurrent grammar cells. In NIPS, 2014.
+
+Junhyuk Oh, Xiaoxiao Guo, Honglak Lee, Richard Lewis, and Satinder Singh. Action-conditional video prediction using deep networks in atari games. NIPS, 2015.
+
+MarcAurelio Ranzato, Arthur Szlam, Joan Bruna, Michael Mathieu, Ronan Collobert, and Sumit Chopra. Video (language) modeling: a baseline for generative models of natural videos. arXiv preprint arXiv:1412.6604, 2014.
+
+Khurram Soomro, Amir Roshan Zamir, and Mubarak Shah. Ucf101: A dataset of 101 human actions classes from videos in the wild. CRCV-TR-12-01, 2012.
+
+Nitish Srivastava, Elman Mansimov, and Ruslan Salakhutdinov. Unsupervised learning of video representations using lstms. CoRR, abs/1502.04681, 2, 2015.
+
+Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal features with 3d convolutional networks. In Proceedings of the IEEE International Conference on Computer Vision, pp. 4489–4497, 2015.
+
+Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Generating videos with scene dynamics. arXiv preprint arXiv:1609.02612, 2016.
+
+Xinchen Yan, Jimei Yang, Kihyuk Sohn, and Honglak Lee. Attribute2image: Conditional image generation from visual attributes. arXiv preprint arXiv:1512.00570, 2015.

parse/train/HkxAAvcxx/HkxAAvcxx_content_list.json

@@ -0,0 +1,1004 @@
+[
+    {
+        "type": "text",
+        "text": "TRANSFORMATION-BASED MODELS OF VIDEO SEQUENCES ",
+        "text_level": 1,
+        "bbox": [
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+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Joost van Amersfoort ∗, Anitha Kannan, Marc’Aurelio Ranzato, Arthur Szlam, Du Tran & Soumith Chintala ",
+        "bbox": [
+            183,
+            172,
+            624,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Facebook AI Research joost@joo.st, {akannan, ranzato, aszlam, trandu, soumith}@fb.com ",
+        "bbox": [
+            186,
+            203,
+            808,
+            229
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            266,
+            544,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work we propose a simple unsupervised approach for next frame prediction in video. Instead of directly predicting the pixels in a frame given past frames, we predict the transformations needed for generating the next frame in a sequence, given the transformations of the past frames. This leads to sharper results, while using a smaller prediction model. ",
+        "bbox": [
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+            297,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In order to enable a fair comparison between different video frame prediction models, we also propose a new evaluation protocol. We use generated frames as input to a classifier trained with ground truth sequences. This criterion guarantees that models scoring high are those producing sequences which preserve discriminative features, as opposed to merely penalizing any deviation, plausible or not, from the ground truth. Our proposed approach compares favourably against more sophisticated ones on the UCF-101 data set, while also being more efficient in terms of the number of parameters and computational cost. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            507,
+            336,
+            523
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There has been an increased interest in unsupervised learning of representations from video sequences (Mathieu et al., 2016; Srivastava et al., 2015; Vondrick et al., 2016). A popular formulation of the task is to learn to predict a small number of future frames given the previous K frames; the motivation being that predicting future frames requires understanding how objects interact and what plausible sequences of motion are. These methods directly aim to predict pixel values, with either MSE loss or adversarial loss. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we take a different approach to the problem of next frame prediction. In particular, our model operates in the space of transformations between frames, directly modeling the source of variability. We exploit the assumption that the transformations of objects from frame to frame should be smooth, even when the pixel values are not. Instead of predicting pixel values, we directly predict how objects transform. The key insight is that while there are many possible outputs, predicting one such transformation will yield motion that may not correspond to ground truth, yet will be realistic; see fig. 1. We therefore propose a transformation-based model that operates in the space of affine transforms. Given the affine transforms of a few previous frames, the model learns to predict the local affine transforms that can be deterministically applied on the image patches of the previous frame to generate the next frame. The intuition is that estimation errors will lead to a slightly different yet plausible motion. Note that this allows us to keep using the MSE criterion, which is easy to optimize, as long as it is in transformation space. No blur in the pixel space will be introduced since the output of the transformation model is directly applied to the pixels, keeping sharp edges intact. Refer to fig. 5 and our online material 1 for examples. ",
+        "bbox": [
+            174,
+            631,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The other contribution of this work is the evaluation protocol. Typically, generative models of video sequences are evaluated in terms of MSE in pixel space (Srivastava et al., 2015), which is not a good choice since this metric favors blurry predictions over other more realistic looking options that just happen to differ from the ground truth. Instead, we propose to feed the generated frames to a video classifier trained on ground truth sequences. The idea is that the less the classifier’s performance is affected by the generates frames the more the model has preserved distinctive features and the more the generated sequences are plausible. Regardless of whether they resemble the actual ground truth or not. This protocol treats the classifier as a black box to measure how well the generated sequences can serve as surrogate for the truth sequence for the classification task. In this paper we will validate our assumption that motion can be modelled by local affine transforms, after which we will compare our method with networks trained using adversarial training and simple regression on the output frame, using both this new evaluation protocol and by providing samples for qualitative inspection. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/12ee9523c8c7a6b995c4a5f346b3260214950d38e8e359bf96b473cbaa612058.jpg",
+        "image_caption": [
+            "Figure 1: Motivating toy example. From left to right: the first digit shows what the model is conditioned upon, the second digit shows the frame we would like to predict at the next time step, the third digit shows the blurry prediction if we were to minimize MSE in pixel space, the last digit shows the prediction when minimizing MSE in the space of transformations. While the two models may have the same MSE in pixel space, the transformation-based model generates much sharper outputs. Although the motion is different than the ground truth (second digit), it is still a plausible next frame to the conditioned frame. In practice, the input is a sequence of consecutive frames. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            161
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our experiments show that our simple and efficient model outperforms other baselines, including much more sophisticated models, on benchmarks on the UCF-101 data set (Soomro et al., 2012). We also provide qualitative comparisons to the moving MNIST digit data set (Srivastava et al., 2015). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            564,
+            330,
+            579
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Early work on video modeling focused on predicting small patches (Michalski et al., 2014; Srivastava et al., 2015); unfortunately, these models have not shown to scale to the complexity of highresolution videos. Also these models require a significant amount of parameters and computational power for even relatively simple data. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In Ranzato et al. (2014), the authors circumvented this problem by quantizing the space of image patches. While they were able to predict a few high-resolution frames in the future, it seems dissatisfying to impose such a drastic assumption to simplify the prediction task. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Mathieu et al. (2016) recently proposed to replace MSE in pixel space with a MSE on image gradients, leveraging prior domain knowledge, and further improved using a multi-scale architecture with adversarial training (Goodfellow et al., 2014). While producing better results than earlier methods, the models used require a very large amount of computational power. We make an explicit comparison to this paper in the experiments section 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In Oh et al. (2015), frames of a video game are predicted given an action (transformation) taken by the player. While the paper shows great results, the movement in a natural video cannot be described by a simple action and is therefore not widely applicable. Finally, our work is also related to optical flow estimation (Brox et al., 2004). Instead of estimating the flow of pixels, here we estimate the flow of patches and separately predict how these patches transform in future frames. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Prior work relating to the evaluation protocol can be found in Yan et al. (2015). The authors generate images using a set of predefined attributes and later show that they can recover these using a pretrained neural network. Our proposal extends this to videos, which is more complicated since both appearance and motion are needed for correct classification. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "image",
+        "img_path": "images/df89c340471c0d73caddfcea4acad4c37effaff1ca0dd8d6869df3386e93421a.jpg",
+        "image_caption": [
+            "Figure 2: Outline of the transformation-based model. The model is a CNN that takes as input a sequence of consecutive affine transforms between pairs of adjacent video frames. It predicts the affine transform between the last input frame and the next one in the sequence. We compute affine transforms (6 parameters per patch) for overlapping patches of size $8 \\times 8$ in each video frame. Learning operates in the space of transformations as shown inside the dashed box. The front-end on the left is a module that estimates the affine transforms between pairs of consecutive input frames. The post-processor on the right reconstructs a frame from the predicted set of affine transforms and it is only used at test time. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2 MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The model we propose is based on three key assumptions: 1) just estimating object motion yields sequences that are plausible and relatively sharp, 2) global motion can be estimated by tiling highresolution video frames into patches and estimating motion “convolutionally” at the patch level, and 3) patches at the same spatial location over two consecutive time steps undergo a deformation which can be well described by an affine transformation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The first assumption is at the core of the proposed method: by considering uncertainty in the space of transformations we produce sequences that may still look plausible. The other two assumptions state that a video sequence can be composed by patches undergoing affine transformations. We agree that these are simplistic assumptions, which ignore how object identity affects motion and do not account for out of plane rotations and more general forms of deformation. However, our qualitative and quantitative evaluation shows the efficacy of these assumptions to real video sequence as can be seen in section 3 and from visualizations in the supplementary material2. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our approach consists of three steps. First, we estimate affine transforms of every video sequence to build a training set for our model. Second, we train a model that takes the past $N$ affine transforms and predicts the next $M$ affine transforms. Finally, at test time, the model uses the predicted affine transforms to reconstruct pixel values of the generated sequence. We describe the details of each phase in the following sections. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 AFFINE TRANSFORM EXTRACTOR ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            784,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a frame $x$ and the subsequent frame $y$ , the goal of the affine transform extractor is to learn mappings that can warp $x$ into $y$ . Since different parts of the scene may undergo different transforms, we tile $x$ into overlapping patches and infer a transformation for each patch. The estimation process couples the transformations at different spatial locations because we minimize the reconstruction error of the entire frame $y$ , as opposed to treating each patch independently. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "image",
+        "img_path": "images/81a308020c438f39ad5439f467fe0f3828cc385ad43508393d31c92ed57c46bb.jpg",
+        "image_caption": [
+            "Figure 3: Outline of the system predicting 4 frames ahead in time. Only affine transforms $A _ { 1 }$ , $A _ { 2 }$ and $A _ { 3 }$ are provided, and the model predicts ${ \\tilde { A } } _ { 4 }$ , ${ \\tilde { A } } _ { 5 }$ , ${ \\tilde { A } } _ { 6 }$ and ${ \\tilde { A } } _ { 7 }$ , which are used to reconstruct the next 4 frames. Since affine parameters are continuous values and the whole chain of CNNs is differentiable, the whole unrolled system can be trained by back-propagation of the error. Note that CNNs all share the same parameters "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            101,
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+            180
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $x$ and $y$ have size $D _ { r } \\times D _ { c }$ . Let image $x$ be decomposed into a set of overlapping patches, each containing pixels from patches of size $d _ { r } \\times d _ { c }$ with $d _ { r } \\leq D _ { r }$ and $d _ { c } \\leq D _ { c }$ . These patches are laid out on a regular grid with stride $s _ { r }$ and $s _ { c }$ pixels over rows and columns, respectively. Therefore, every pixel participates in $\\frac { d _ { r } } { s _ { r } } \\frac { d _ { c } } { s _ { c } }$ overlapping patches, not taking into account for the sake of simplicity border effects and non-integer divisions. We denote the whole set of overlapping patches by $\\{ X _ { k } \\}$ , where index $k$ runs over the whole set of patches. Similarly and using the same coordinate system, we denote by $\\left\\{ Y _ { k } \\right\\}$ the set of overlapping patches of $y$ . ",
+        "bbox": [
+            173,
+            337,
+            825,
+            439
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We assume that there is an affine mapping $A _ { k }$ that maps $X _ { k }$ to $Y _ { k }$ , for all values of $k$ . $A _ { k }$ is a $2 \\times 3$ matrix of free parameters representing a generic affine transform (translation, rotation and scaling) between the coordinates of output and input frame. Let $\\tilde { Y } _ { k }$ be the transformed patches obtained when $A _ { k }$ is applied to $X _ { k }$ . Since coordinates overlap between patches, we reconstruct $y$ by averaging all predictions at the same location, yielding the estimate $\\tilde { y }$ . The joint set of $A _ { k }$ is then jointly determined by minimizing the mean squared reconstruction error between $y$ and $\\tilde { y }$ . ",
+        "bbox": [
+            174,
+            445,
+            825,
+            531
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Notice that our approach and aim differs from spatial transformer networks (Jaderberg et al., 2015) since we perform this estimation off-line only for the input frames, computing one transform per patch. ",
+        "bbox": [
+            174,
+            537,
+            825,
+            580
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we extracted $1 6 \\times 1 6$ pixel patches from the input and we used stride 4 over rows and columns. The input patches are then matched at the output against smaller patches of size $8 \\times 8$ pixels, to account for objects moving in and out of the patch region. ",
+        "bbox": [
+            174,
+            587,
+            825,
+            630
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 AFFINE TRANSFORM PREDICTOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            676,
+            442,
+            690
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The affine transform predictor is used to predict the affine transforms between the last input frame and the next frame in the sequence. A schematic illustration of the system is shown in fig. 2. It receives as input the affine transforms between pairs of adjacent frames, as produced by the affine transform extractor described in the previous section. Each transform is arranged in a grid of size $6 \\times n \\times n$ , where $n$ is the number of patches in a row/column and 6 is the number of parameters of each affine transform. Therefore, if four frames are used to initialize the model, the actual input consists of 18 maps of size $n \\times n$ , which are the concatenation of $A _ { t - 2 } , A _ { t - 1 } , A _ { t }$ , where $A _ { t }$ is the collection of patch affine transforms between frame at time $t - 1$ and $t$ . ",
+        "bbox": [
+            173,
+            713,
+            825,
+            825
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The model consists of a multi-layer convolutional network without any pooling. The network is the composition of convolutional layers with ReLU non-linearity, computing a component-wise thresholding as in $v = \\operatorname* { m a x } ( 0 , u )$ . We learn the parameters in the filters of the convolutional layers by minimizing the mean squared error between the output of the network and the target transforms. ",
+        "bbox": [
+            174,
+            833,
+            825,
+            888
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Notice that we do not add any regularization to the model. In particular, we rely on the convolutional structure of the model to smooth out predictions at nearby spatial locations. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 MULTI-STEP PREDICTION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            395,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the previous section, we described how to predict the set of affine transforms at the next time step.   \nIn practice, we would like to predict several time steps in the future. ",
+        "bbox": [
+            174,
+            132,
+            821,
+            160
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A greedy approach would: a) train as described above to minimize the prediction error for the affine transforms at the next time step, and b) at test time, predict one step ahead and then re-circulate the model prediction back to the input to predict the affine transform two steps ahead, etc. Unfortunately, errors may accumulate throughout this process because the model was never exposed to its own predictions at training time. ",
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+    {
+        "type": "text",
+        "text": "The approach we propose replicates the model over time, also during training as shown in fig. 3. If we wish to predict $M$ steps in the future, we replicate the CNN $M$ times and pass the output of the CNN at time step $t$ as input to the same CNN at time step $t + 1$ , as we do at test time. Since predictions live in a continuous space, the whole system is differentiable and amenable to standard back-propagation of the error. Since parameters of the CNN are shared across time, the overall system is equivalent to a peculiar recurrent neural network, where affine transforms play the role of recurrent states. The experiments in section 3 demonstrate that this method is more accurate and robust than the greedy approach. ",
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+        "text": "2.4 TESTING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "At test time, we wish to predict $M$ frames in the future given the past $N$ frames. After extracting the $N - 1$ affine transforms from the frames we condition upon, we replicate the model $M$ times and feed its own prediction back to the input, as explained in the previous section. ",
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+    {
+        "type": "text",
+        "text": "Once the affine transforms are predicted, we can reconstruct the actual pixel values. We use the last frame of the sequence and apply the first set of affine transforms to each patch in that frame. Each pixel in the output frame is predicted multiple times, depending on the stride used. We average these predictions and reconstruct the whole frame. As required, we can repeat this process for as many frames as necessary, using the last reconstructed frame and the next affine transform. ",
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+        "type": "text",
+        "text": "In order to evaluate the generation, we propose to feed the generated frames to a trained classifier for a task of interest. For instance, we can condition the generation using frames taken from video clips which have been labeled with the corresponding action. The classifier has been trained on ground truth data but it is evaluated using frames fantasized by the generative model. The performance of the classifier on ground truth data is an upper bound on the performance of any generative model. This evaluation protocol does not penalize any generation that deviates from the ground truth, as standard MSE would. It instead check that discriminative features and the overall semantics of the generated sequence is correct, which is ultimately what we are interested in. ",
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+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "In this section, we validate the key assumptions made by our model and compare against state-ofthe-art generative models on two data sets. We strongly encourage the reader to watch the short video clips in the Supplementary Material to better understand the quality of our generations. ",
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+        "type": "text",
+        "text": "In section 2, we discussed the three key assumptions at the foundations of our model: 1) errors in the transformation space look still plausible, 2) a frame can be decomposed into patches, and 3) each patch motion is well modeled by an affine transform. The results in the Supplementary Material 3 validate assumption 2 and 3 qualitatively. Every row shows a sequence from the UCF101 dataset (Soomro et al., 2012). The column on the left shows the original video frames and the one on the right the reconstructions from the estimated affine transforms, as described in section 2.1. As you can see there is barely any noticeable difference between these video sequences, suggesting that video sequences can be very well represented as tiled affine transforms. For a quantitative comparison and for an assessment of how well the first assumption holds, please refer to section 3.2. ",
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+        "img_path": "images/7443870679cdf0ab5ac54a3777ad2932377d7e6b13ddb24a80c1973a6c1827d3.jpg",
+        "image_caption": [
+            "Figure 4: Predictions of 4 sequences from the moving MNIST dataset. The top row of each pair shows the ground truth frames; the first four frames are used as input to the model. The bottom row shows the predictions of the model. "
+        ],
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+        "text": "In the next section, we will first report some results using the toy data set of “moving MNIST digits” (Srivastava et al., 2015). We then discuss generations of natural high-resolution videos using the UCF-101 dataset and compare to current state-of-the-art methods. ",
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+        "text": "3.1 MOVING MNIST ",
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+        "text": "For our first experiment, we used the dataset of moving MNIST digits (Srivastava et al., 2015) and perform qualitative analysis4. It consists of one or two MNIST digits, placed at random locations and moving at constant speed inside a $6 4 \\times 6 4$ frame. When a digit hits a boundary, it bounces, meaning that velocity in that direction is reversed. Digits can occlude each other and bounce off walls, making the data set challenging. ",
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+        "text": "Using scripts provided by Srivastava et al. (2015), we generated a fixed dataset of 128,000 sequences and used $80 \\%$ for training, $10 \\%$ for validation and $10 \\%$ for testing. Next, we estimated the affine transforms between every pair of adjacent frames to a total of 4 frames, and trained a small CNN in the space of affine transforms. The CNN has 3 convolutional layers and the following number of feature maps: 18, 32, 32, 6. All filters have size $3 \\times 3$ . ",
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+    {
+        "type": "text",
+        "text": "Fig. 4 shows some representative test sequences and the model outputs. Each subfigure corresponds to a sequence from the test set; the top row corresponds to the ground truth sequence while the bottom row shows the generations. The input to the CNN are three sets of affine transforms corresponding to the first four consecutive frames. The network predicts the next six sets of affine transforms from which we reconstruct the corresponding frames. These results should be compared to fig. 5 in Srivastava et al. (2015). The generations in fig. 4 show that the model has potential to represent and generate video sequences, it learns to move digits in the right direction, to bounce them, and it handles multiple digits well except when occluion makes inputs too ambiguous. The model’s performance is analyzed quantitatively in the next section using high resolution natural videos. ",
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+        "type": "text",
+        "text": "3.2 UCF 101 DATA SET ",
+        "text_level": 1,
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+        "text": "The UCF-101 dataset (Soomro et al., 2012) is a collection of 13320 videos of 101 action categories. Frames have size $2 4 0 \\times 3 2 0$ pixels. We train a CNN on patches of size $6 4 \\times 6 4$ pixels; the CNN has 6 convolutional layers and the following number of feature maps: 18, 128, 128, 128, 64, 32, 16, 6. All filters have size $3 \\times 3$ . The optimal number of filters has been found using cross-validation in order to minimize the estimation error of the affine transform parameters. Unless otherwise stated, we condition generation on 4 ground truth frames and we predict the following 8 frames. ",
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+        "type": "text",
+        "text": "We evaluate several models5: a) a baseline which merely copies the last frame used for conditioning, b) a baseline method which estimates optical flow (Brox et al., 2004) from two consecutive frames and extrapolates flow in subsequent frames under the assumption of constant flow speed, c) an adversarially trained multi-scale CNN (Mathieu et al., 2016) and several variants of our proposed approach. ",
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+    {
+        "type": "image",
+        "img_path": "images/9bb8e2779df3278f29e753245446f1e87e7f9f6c450239af241f4b76732a3d70.jpg",
+        "image_caption": [
+            "Figure 5: Example of predictions produced by different models. Each row shows an example. The first two columns show the ground truth. The two frames are 4 time steps apart. The next two columns show predictions from a baseline model employing optical flow. Next, we show the prediction produced by the adversarially trained CNN proposed by Mathieu et al. (2016). The last two column show the prediction produced by our affine-transformation based approach. All pairs in the same column group are four time steps apart. All methods were conditioned on the same set of 4 input frames (not shown in the figure) "
+        ],
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+    {
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+        "img_path": "images/88cc11b34a1e13cd49b1472b4421c3a6b31ce9c02df28e2bb51463aac4414f28.jpg",
+        "table_caption": [
+            "Table 1: Classification accuracy on UCF-101 dataset. The classifier is trained on the actual training video sequences, but it is tested using frames generated by various generative models. Each column shows the accuracy on the test set when taking a different number of input frames as input. Our approach maps $1 6 \\times 1 6$ patches into $8 \\times 8$ with stride 4, and it takes 4 frames at the input. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1> 4 frames</td><td rowspan=1 colspan=1>8 frames</td></tr><tr><td rowspan=1 colspan=1>Ground truth frames</td><td rowspan=1 colspan=1>72.46</td><td rowspan=1 colspan=1>72.29</td></tr><tr><td rowspan=1 colspan=1>Using ground truth affine transforms</td><td rowspan=1 colspan=1>71.7</td><td rowspan=1 colspan=1>71.28</td></tr><tr><td rowspan=1 colspan=1>Copy last frame</td><td rowspan=1 colspan=1>60.76</td><td rowspan=1 colspan=1>54.27</td></tr><tr><td rowspan=1 colspan=1>Optical Flow</td><td rowspan=1 colspan=1>57.29</td><td rowspan=1 colspan=1>49.37</td></tr><tr><td rowspan=1 colspan=1>Mathieu et al. (2016)</td><td rowspan=1 colspan=1>57.98</td><td rowspan=1 colspan=1>47.01</td></tr><tr><td rowspan=1 colspan=1>ours - one step prediction (not unrolled)</td><td rowspan=1 colspan=1>64.13</td><td rowspan=1 colspan=1>57.63</td></tr><tr><td rowspan=1 colspan=1>ours - four step prediction (unrolled 4 times)</td><td rowspan=1 colspan=1>64.54</td><td rowspan=1 colspan=1>57.88</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Qualitative comparisons can be seen in the fig. 5 and in the supplementary material6. The first column on the page shows the input, the second the ground truth, followed by results from our model, Mathieu et al. (2016) and optical flow (Brox et al., 2004). Note especially the severe deformations in the last two columns, while our model keeps the frame recognizable. It produces fairly sharp reconstructions validating our first hypothesis that errors in the space of transformations still yield plausible reconstructions (see section 2). However it is also apparent that our approach underestimates movement, which follows directly from using the MSE criterion. As discussed before, MSE in pixel space leads to blurry results, however using MSE in transformation space also has some drawbacks. In practice, the model will predict the average of several likely transformations, which could lead to an understimation of the true movement. ",
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+    {
+        "type": "text",
+        "text": "In order to quantify the generation quality we use the metric described in section 2.4. We use C3D network (Tran et al., 2015) as the video action classifier: C3D uses both appearance and temporal information jointly, and is pre-trained with Sports1M (Karpathy et al., 2014) and fine tuned on UCF 101. Due to the model constraints, we trained only two models, that takes 4 and 8 frames as input, respectively. ",
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+    {
+        "type": "text",
+        "text": "We evaluate the quality of generation using 4 (the first four predicted frames) and the whole set of 8 predicted frames, for the task of action classification. At test time, we generate frames from each model under consideration, and then use them as input to the corresponding C3D network. ",
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+    {
+        "type": "text",
+        "text": "Table 1 shows the accuracy of our approach and several baselines. The best performance is achieved by using ground truth frames, a result comparable to methods recently appeared in the literature (Karpathy et al., 2014; Tran et al., 2015). We see that for ground truth frames, the number of frames (4 or 8) doesn’t make a difference. There is not much additional temporal or spatial signal provided by having greater than four frames. Next, we evaluate how much we lose by representing frames as tiled affine transforms. As the second row shows there is negligible if any loss of accuracy when using frames reconstructed from the estimated affine transforms (using the method described in section 2.1), validating our assumptions at the beginning of section 2 on how video sequences can be represented. The next question is then whether these affine transforms are predictable at all. The last two rows of Table 1 show that this is indeed the case, to some extent. The longer the sequence of generated frames the poorer the performance, since the generation task gets more and more difficult. ",
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+    {
+        "type": "text",
+        "text": "Compared to other methods, our approach performs better than optical flow and even the more sophisticated multi-scale CNN proposed in Mathieu et al. (2016) while being computationally cheaper. For instance, our method has less than half a million parameters and requires about 2G floating point operations to generate a frame at test time, while the multi-scale CNN of Mathieu et al. (2016) has 25 times more parameters (not counting the discriminator used at training time) and it requires more than 100 times more floating point operations to generate a single frame. ",
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+    {
+        "type": "text",
+        "text": "Finally, we investigate the robustness of the system to its hyper-parameters: a) choice of patch size, b) number of input frames, and c) number of predicted frames. The results reported in Table 2 demonstrate that the model is overall pretty robust to these choices. Using patch sizes that are too big makes reconstructions blocky but within each block motion is coherent. Smaller patch sizes give more flexibility but make the prediction task harder as well. Mapping into patches of size smaller than $1 6 \\times 1 6$ seems a good choice. Using only 2 input frames does not seem to provide enough context to the predictor, but anything above 3 works equally well. Training for prediction of the next frame works well, but better results can be achieved by training to predict several frames in the future, overall when evaluating longer sequences. ",
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+        "table_caption": [
+            "Table 2: Analysis of the robustness to the choice of hyper-parameters, shows classification scores compared to reference model. The reference model takes 4 frames as input, predicts one frame, and maps $1 2 \\times 1 2$ patches onto $8 \\times 8$ patches with stride 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>4 frames</td><td rowspan=1 colspan=1>8 frames</td></tr><tr><td rowspan=1 colspan=1>reference</td><td rowspan=1 colspan=1>63.57</td><td rowspan=1 colspan=1>57.32</td></tr><tr><td rowspan=1 colspan=1>Varying patch sizefrom 32 × 32 to 16× 16from 16 × 16 to 8 × 8</td><td rowspan=1 colspan=1>61.7363.75</td><td rowspan=1 colspan=1>53.8557.18</td></tr><tr><td rowspan=1 colspan=1>Number of input frames23</td><td rowspan=1 colspan=1>63.663.8</td><td rowspan=1 colspan=1>57.1157.4</td></tr><tr><td rowspan=1 colspan=1>Number of predicted frames24</td><td rowspan=1 colspan=1>64.164.54</td><td rowspan=1 colspan=1>57.557.88</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 CONCLUSIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we proposed a new approach to generative modeling of video sequences. This model does not make any assumption about the spatio-temporal resolution of video sequences nor about object categories. The key insight of our approach is to model in the space of transformations as opposed to raw pixel space. A priori we lack a good metric to measure how well a frame is reconstructed under uncertainty due to objects motion in natural scenes. Uncertainty about object motion and occlusions causes blurry generations when using MSE in pixel space. Instead, by operating in the space of transformations we aim at predicting how objects move, and estimation errors only yield a different, and possibly still plausible, motion. With this motivation we proposed a simple CNN operating in the space of affine transforms and we showed that it can generate sensible sequences up to about 4 frames. This model produces sequences that are both visually and quantitatively better than previously proposed approaches. ",
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+    {
+        "type": "text",
+        "text": "The second contribution of this work is the metric to compare generative models of video sequences. A good metric should not penalize a generative model for producing a sequence which is plausible but different from the ground truth. With this goal in mind and assuming we have at our disposal labeled sequences, we can first train a classifier using ground truth sequences. Next, the classifier is fed with sequences produced by our generative model for evaluation. A good generative model should produce sequences that still retain discriminative features. In other words, plausibility of generation is assessed in terms of how well inherent information is preserved during generation as opposed to necessarily and merely reproducing the ground truth sequences. ",
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+    {
+        "type": "text",
+        "text": "The proposed model is relatively simple; straightforward extensions that could improve its prediction accuracy are the use of a multi-scale architecture and the addition of recurrent units. These would enable a better modeling of objects of different sizes moving at varying speeds and to better capture complex temporal dynamics (e.g., cyclical movements like walking). A larger extension would be the addition of an appearance model, which together with our explicit transformation model could lead to learning better feature representations for classification. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In our view, the proposed approach should be considered as a stronger baseline for future research into next frame prediction. Even though our analysis shows improved performance and better looking generations, there are also obvious limitations. The first such limitation is the underestimation of transformations due to usage of the MSE as a criterion. We consider two main avenues worth pursuing in this space. First, we consider modelling a distribution of transformations and sampling one from it. The challenge of this approach is to sample a consistent trajectory. One could model the distribution of an entire trajectory, but that is a complex optimization problem. A second option is to use adversarial training to force the model to pick a plausible action. This option does not guarantee that underestimation of movement will be avoided. This will depend on the discriminator model accepting this as a plausible option. ",
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+    {
+        "type": "text",
+        "text": "Another limitation is that the current model does not factor out the “what” from the “where”, appearance from motion. The representation of two distinct objects subject to the same motion, as well as the representation of the same object subject to two different motion patterns are intrinsically different. Instead, it would be more powerful to learn models that can discover such factorization and leverage it to produce more efficient and compact representations. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Authors thank Camille Couprie and Michael Mathieu for discussions and helping with evaluation of their models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
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+    },
+    {
+        "type": "text",
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parse/train/Hyfg5o0qtm/Hyfg5o0qtm.md

@@ -0,0 +1,356 @@
+# TEMPORAL GAUSSIAN MIXTURE LAYER FOR VIDEOS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We introduce a new convolutional layer named the Temporal Gaussian Mixture (TGM) layer and present how it can be used to efficiently capture longer-term temporal information in continuous activity videos. The TGM layer is a temporal convolutional layer governed by a much smaller set of parameters (e.g., location/variance of Gaussians) that are fully differentiable. We present our fully convolutional video models with multiple TGM layers for activity detection. The experiments on multiple datasets including Charades and MultiTHUMOS confirm the effectiveness of TGM layers, outperforming the state-of-the-arts.
+
+# 1 INTRODUCTION
+
+Activity videos are spatio-temporal data: they are image frames with a specific width/height (XY) concatenated along time axis (T). Recognition from such videos requires capturing both spatial and temporal information in the videos, desirably using learned convolutional kernels. Temporal convolution is particularly beneficial in activity ‘detection’ tasks, which require making activity decisions at every frame given a continuous video (Sigurdsson et al., 2016b; Yeung et al., 2015). Previous methods investigated using 3-D XYT convolutional filters (Tran et al., 2014; Carreira & Zisserman, 2017) as well as the models with 2-D XY conv. layers followed by 1-D temporal conv. (Tran et al., 2018), pooling or attention layers (Piergiovanni et al., 2017).
+
+Understanding complex multi-activity videos requires capturing information in long-term time intervals. Different frames contain different information, and the model needs to learn to take advantage of as many frames as possible, while abstracting them efficiently. Previous attempts of simply pooling representations over time or learning temporal conv. filters with a small number of frames (e.g., 16 or 64) was thus often insufficient to fully consider rich long-term temporal context. Simultaneously, bruteforcely increasing the temporal filter length (to look at more frames) results more learnable parameters, requiring more training data, which can be expensive when activities are rare.
+
+In this paper, we introduce a new convolutional layer named the Temporal Gaussian Mixture (TGM) layer, and present how it can be used to efficiently capture longer-term temporal information in activity videos. Our temporal Gaussian mixture layer is a temporal convolutional layer, whose filters/kernels are controlled by a set of (temporal) Gaussian distribution parameters. Each of our temporal Gaussian distributions specify (temporally) ‘where’ the model should look, and our Gaussian mixture layer combines them as multiple convolutional filters to be applied on top of temporallycontinuous representations. This layer allows the video representation at each time step to be constructed while focusing on different neighboring temporal regions, instead of only focusing on its local segment. It is a convolutional layer governed by a much smaller set of parameters (i.e., locations/variances of the Gaussians as well as their mixture weights) that are fully differentiable.
+
+The motivation behind our temporal Gaussian mixture layer is to learn the temporal structure of an activity as a composition of temporal Gaussian regions/attentions. Such structure allows the model to obtain a compact spatio-temporal representation abstracting each (long-term) time interval, using multiple temporal conv. layers with far fewer parameters. It is also related to the previous temporal attention works (Piergiovanni et al., 2017), but our model is designed to be fully convolutional to handle continuous data and it learns more compositional structures with multiple layers.
+
+We present video-CNN models using our TGM layers for activity detection in continuous videos. Our model stacks TGM layers on top of several state-of-the-art CNNs such as I3D (Carreira & Zisserman, 2017). This enables our model to capture longer-term temporal information than what we use as base CNNs, compositionally modeling temporal structure with multiple TGM layers. Our model was evaluated on multiple public datasets including MultiTHUMOS and Charades, and was able to outperform the best previous activity detection CNNs by a meaningful margin.
+
+# 2 RELATED WORKS
+
+Learning video representations for human activity recognition has been successful. CNN methods allow end-to-end learning of video features and representations optimized for the training data, performing superior to traditional works (Aggarwal & Ryoo, 2011) for video understanding.
+
+Two-stream CNN models take a single RGB frame and a small number of optical flow frames as inputs to capture both motion and appearance information in videos (Simonyan & Zisserman, 2014; Feichtenhofer et al., 2016). Models learning 3-D spatio-temporal (XYT) convolutional filters were designed and applied to many activity recognition tasks as well (Tran et al., 2014; Carreira & Zisserman, 2017; Tran et al., 2017; Hara et al., 2017). Large scale datasets for activity detection, such as THUMOS (Jiang et al., 2014), ActivityNet (Heilbron et al., 2015), Kinetics (Kay et al., 2017), and Charades (Sigurdsson et al., 2016b) provided these approach the necessary training data to learn the models. Such 3-D XYT CNNs were also used to capture spatio-temporal information for activity detection (Xu et al., 2017; Shou et al., 2016; 2017; Zhao et al., 2017). However, all these CNNs were limited to the consideration of a fixed local video segment (e.g., 16 frames in (Tran et al., 2014) and 64-99 frames in (Carreira & Zisserman, 2017)) when making activity decisions.
+
+Some works studied combining representations over longer-term temporal intervals (Karpathy et al., 2014; $\mathrm { N g }$ et al., 2015; Varol et al., 2017), but it was generally done with a temporal pooling of local representations or (spatio-)temporal convolutions with a bit larger fixed intervals. Recurrent neural networks (RNNs) have also been used to model activity transitions between frames (Yeung et al., 2015; 2016; Escorcia et al., 2016), but they were strictly sequential and had limitations in maintaining temporal information over a longer temporal duration, particularly for videos with multiple complex activities. Recently, CNN models using temporal attention for activity videos (Piergiovanni et al., 2017; Piergiovanni & Ryoo, 2018b) were studied as well. However, a fully convolutional model to analyze continuous videos while efficiently representing information in long term intervals has been lacking.
+
+Our layer is different from the previous standard (spatio-)temporal convolutional layers in that it relies on significantly fewer parameters by forcing filter shapes to be Gaussian compositions. Our temporal layer is also different from previous Gaussian Mixture Model layers (Variani et al., 2015) in that our layer is convolutional while they are not.
+
+# 3 APPROACH
+
+In this section, we introduce a new convolutional layer named the Temporal Gaussian Mixture (TGM) layer, and present how it can be used for activity recognition. Our Temporal Gaussian Mixture layer is a temporal convolutional layer to be applied on top of a sequence of representations (usually from frame-level or segment-level CNNs), whose filters/kernels are controlled by a set of (temporal) Gaussian distribution parameters. The motivation is to make each temporal Gaussian distribution specify (temporally) ‘where to look’ with respect to the activity center, and represent the activity as a collection/mixture of such temporal Gaussians convolved with video features. Our layer is fully differentiable and trainable using standard backpropagation.
+
+Our TGM layer can be interpreted as a a form of 1-D convolution where the filters are determined by a mixture of Gaussians. However, our TGM layer differs from the standard temporal convolutional layers of learning 1-D (time) or 2-D (channel-by-time) filters in the following aspects:
+
+1. Our temporal Gaussian mixture layer handles multiple 3-D tensors internally to preserve channels from the frame-level CNN by adding a new temporal channel axis. Its input is 3-D (channel-by-channel-by-time), where one channel dimension is inherited from the frame-level CNN and this dimension size remains unchanged.   
+2. Instead of learning temporal convolution filters of any arbitrary values, our filter is forced to have the form of a temporal Gaussian mixture shared across all frame-level channels. This allows the layer to rely on significantly fewer number of (fully differentiable) parameters, while capturing the concept of temporal structure/attention.
+
+![](images/bb08fe1098ed6b3ebacab0ba55fa20f4c6eca53bc414048213d86cf9d3f10300.jpg)  
+Figure 1: Example illustrating how our Temporal Gaussian Mixture layer is computed. Multiple $( M )$ temporal Gaussian distributions are learned, and they are combined with the learned soft attention weights to form the $C$ temporal convolution filters. $L$ is the temporal length of the filter.
+
+# 3.1 TEMPORAL GAUSSIAN MIXTURE LAYER
+
+Our temporal Gaussian mixture layer takes a 3-D input with the dimensionality of $C _ { i n } \times D \times T$ , where $\dot { C } _ { i n }$ is the number of input channels, $D$ is the dimensionality of the representations from frame-level (or segment-level) CNNs, and $T$ is the time. Given such input, the TGM layer convolves it with $C _ { o u t }$ number of $1 \times L$ filters/kernels, generating a $C _ { o u t } \times D \times T$ -dim representation as an output. $L$ is the temporal length of the temporal Gaussian mixture filter. $D$ is usually 1K or 4K and $T$ is the number of time steps (frames) in each video (i.e., it varies per video). $C _ { o u t }$ is the number of different mixtures, corresponding to the number of output channels in standard convolution.
+
+Our layer is composed of a set of $M$ Gaussians. Each Gaussian has 2 parameters: a center $\hat { \mu }$ and a width $\hat { \sigma }$ . Each layer has additional hyper-parameters: $L$ , the temporal duration and $M$ , the number of Gaussians to learn. We force the learned center to be between ${ \bar { - } } { \frac { L } { 2 } }$ and $\begin{array} { l } { { \frac { L } { 2 } } } \end{array}$ and $\sigma$ to be positive:
+
+$$
+\mu = ( L - 1 ) \cdot \frac { \operatorname { t a n h } { ( \hat { \mu } + 1 ) } } { 2 } , \sigma ^ { 2 } = \exp { ( \hat { \sigma } ) } .
+$$
+
+We use the above $\mu$ and $\sigma$ to construct the temporal Gaussian kernels. This acts as a strong sparsity constraint on the convolutional kernel as well as a drastic reduction of the number of learnable parameters. We construct a temporal Gaussian mixture convolutional kernel as:
+
+$$
+\hat { K } _ { m , l } = \frac { 1 } { Z } \exp { - \frac { ( l - \mu _ { m } ) ^ { 2 } } { 2 \sigma _ { m } ^ { 2 } } }
+$$
+
+where $Z$ is a normalization constant such that $\begin{array} { r } { \sum _ { l } ^ { L } \hat { K } _ { m , l } = 1 } \end{array}$ , resulting in $\hat { K }$ being an $M \times L$ matrix.
+
+Instead of making the model learn a separate set of Gaussian distributions per activity class, we take the approach of maintaining multiple Gaussian distributions shared across classes and obtain a Gaussian ‘mixture’ filter by learning soft-attention weights. We learn a set of soft-attention weights per output channel $i$ , $\omega \in \overline { { \mathcal { R } } } ^ { C _ { o u t } \times \breve { M } }$ . We create the soft-attention weights by applying the softmax function over the $M$ Gaussians, enforcing each input channel weights sum to 1.
+
+$$
+a _ { i , m } = \frac { \exp \omega _ { i , m } } { \sum _ { j } \exp \omega _ { i , j } }
+$$
+
+Based on temporal Gaussian distributions $\hat { K } _ { i }$ and attention weights $a _ { i , m }$ , the temporal convolution filters our TGM layer is computed as:
+
+$$
+K _ { i } = \sum _ { m } a _ { i , m } \hat { K } _ { i } .
+$$
+
+This provides us convolutional filters having the form of a mixture of temporal Gaussians, controlled based on $2 \cdot M + C _ { i n } \cdot C _ { o u t } \cdot M$ parameters (instead of learning $D ^ { 2 } \cdot L$ parameters without any constraint, as in standard temporal convolution where $C < < D$ ). An overview of this process is shown in Fig. 1.
+
+# 3.1.1 SINGLE TGM LAYER - DIRECT PER-CLASS ACTIVITY MODELING
+
+The representation we obtain by applying our base CNNs to each frame (or local segment) has the dimensionality of $D$ , and stacking them along time axis provides us the representation with
+
+![](images/1e1b884ad92c56ef857b2a263d6acccb6a7a771dee8413a54c17dfc575c6b884.jpg)  
+Figure 2: Illustration of a TGM layer with grouped convolution. This layer learns a set of $C$ Gaussian mixtures that are convolved with the input channels.
+
+$1 \times D \times T$ -dim. That is, in the case of using only one TGM layer to capture activity representations, our $C _ { i n }$ is fixed to 1 and $C _ { o u t }$ is fixed to be the number of activity classes. This is the simplest case of our model, attaching one TGM layer on top of the $1 \times D \times T$ representation.
+
+Our convolutional kernel, $K$ , has a learned Gaussian mixture for each activity class. Let the video features $v$ be a $D \times T$ matrix. Each $K _ { i }$ is a 2-D convolutional filter with a size of $1 \times L$ , and convolving this with $v$ provides us a representation $S$ with $C _ { o u t }$ number of $D \times T$ responses since $C _ { i n }$ is 1 in this case. This per-class representation can then be used as input to a fully-connected layer for activity classification. For $i \in \mathsf { \bar { \{ 1 , 2 , \ldots , C _ { o u t } \} } }$ :
+
+$$
+s _ { i } = v * K _ { i } , \ S = [ s _ { 1 } , s _ { 2 } , \ldots , s _ { C _ { o u t } } ]
+$$
+
+Fig. 7 in the appendix visually illustrates how each TGM filter is convolved with the input (Fig. 7d), compared to the standard 1-D convolution (Fig. 7a) or other forms of the temporal layers (Fig. 7b-c).
+
+# 3.1.2 MULTIPLE TGM LAYERS - GROUPED CONVOLUTION
+
+We generalize the above formulation to allow the TGM layers to be sequentially applied. The idea is to enable our model to capture more complex, nonlinear temporal structure by having multiple levels of temporal layers. In this case, the input for each layer is $C _ { i n } \times D \times T$ dimensional (instead of $1 \times D \times T$ ), where the input channels are the number of output channels from the previous layer. Our kernels at each layer, $K _ { i }$ , are parameterized and learned as before.
+
+By using grouped convolution with the number of groups set to $C _ { i n }$ , we can efficiently separate the input into per-channel values and convolve each of them with the designated $K _ { i }$ kernel, as shown in Fig. 2. That is, we learn a filter $K _ { i }$ per channel by setting $C _ { i n } = C _ { o u t }$ . For $i \in [ 1 , C _ { o u t } ]$ ,
+
+$$
+s _ { i } = f _ { i } * K _ { i } , ~ S = [ s _ { 1 } , s _ { 2 } , . ~ . ~ . s _ { C _ { o u t } } ]
+$$
+
+Here, $f$ is a $C _ { i n } \times D \times T$ tensor, where $D$ is the dimensionality of the feature and $T$ is the number of frames. The result of the per-channel convolution, $s _ { i }$ , is a $D \times T$ representation. We concatenate these representations along the channel axis, resulting in $S$ , a $C _ { o u t } \times D \times T$ representation. As this convolution results in the same output shape, we can stack these layers. Each layer is able to capture increasing temporal resolution, allowing the model to capture levels of abstractions.
+
+# 3.1.3 MULTIPLE TGM LAYERS - CHANNEL COMBINATION
+
+In the above subsection, we introduced an approach of stacking multiple TGM layers to model a hierarchical composition of temporal representations. However, in the grouped convolution case, each output channel of the layer is solely dependent on its corresponding input channel. That is, each kernel only considers information from a single output channel of the previous layer.
+
+Therefore, we further generalize our TGM layer so that the layer combines representations from multiple input channels for each output channel while using the learned temporal kernels. We learn a set of convolutional kernels $K \in \mathop { \mathcal { R } } ^ { C _ { o u t } \times C _ { i n } \times L }$ (i.e., we learn $C _ { o u t } \cdot C _ { i n }$ Gaussian mixtures). Given $f$ which is the $C _ { i n } \times D \times T$ representation, for each output channel $i \in [ 1 , C _ { o u t } ]$ and each input channel $j \in [ 1 , C _ { i n } ]$ pair, we convolve the associated filters with the input.
+
+$$
+G _ { i , j } = ( f _ { j } * K _ { i , j } )
+$$
+
+where each $G _ { i , j }$ is a $D \times T$ -dim representation.
+
+We then learn a 1x1 convolution followed by a ReLU activation function for each $i \in [ 1 , C _ { o u t } ]$ , which we call $w _ { i }$ , that maps from $C _ { i n }$ channels to 1 channel. The 1x1 convolution learns to combine
+
+![](images/aca85f15ca42c4a0cdcd7ee131a3bb28c18e0cc54f9ba616551cbb896c0860e5.jpg)  
+Figure 3: Illustration of a TGM layer with channel combination. The kernels are applied to each input channel, $C _ { i n }$ , and a 1x1 convolution is applied to combine the $C _ { i n }$ input channels for each output channel, $C _ { o u t }$ .
+
+the channels from the previous layer. By design, the TGM kernel is positive and sums to 1. Adding the unconstrained 1x1 convolution adds non-linearity (using the ReLU activation function) to our layer and only adds $C _ { o u t } \cdot C _ { i n }$ parameters.
+
+$$
+s _ { i } = G _ { i } * w _ { i } = ( f _ { j } * K _ { i , j } ) * w _ { i } , \ S = [ s _ { 1 } , s _ { 2 } \ldots , s _ { C _ { o u t } } ]
+$$
+
+We then stack the $s _ { i }$ representations along the channel axis to produce $S$ , the $C _ { o u t } \times D \times T$ -dim representation. This process is illustrated in Fig. 3. This method generalizes our approach to allow the layer to take input of $C _ { i n } \times D \times T$ and produce output of $C _ { o u t } \times D \times T$ . These layers can easily be stacked to learn a hierarchical representation.
+
+# 3.2 VIDEO CNN MODELS WITH TGM LAYERS
+
+Our goal is to do activity detection which we define as making a per-frame (or per-segment) classification. Given a video, at each time step $t$ , we want to make the model decide which activity the frame corresponds to (including no-activity). As a baseline, we train a fully-connected layer that classifies each per-frame $D$ -dimensional vector, $v _ { t }$ . As multiple activities can occur at the same time, or no activities at all, we treat this as a mutli-label classification task. We minimize binary cross entropy:
+
+$$
+L ( v ) = \sum _ { t , c } z _ { t , c } \log ( p ( c | v _ { t } ) ) + ( 1 - z _ { t , c } ) \log ( 1 - p ( c | v _ { t } ) )
+$$
+
+where $z _ { t , c }$ is the ground truth label, 1 if activity $c$ is occurring at time $t$ and $p ( c | v _ { t } )$ is the output of our model for class $c$ at time $t$ . Fig. 4 shows an example CNN.
+
+![](images/cdd6c2d13bb16728b0fe94e3796a0d17a7ed1459c289733ad68ae2e40ee7b924.jpg)  
+Figure 4: An overview of an example video CNN model with two TGM layers. It is able to handle videos with any length, because of its fully convolutional design.
+
+# 4 EXPERIMENTS
+
+# 4.1 IMPLEMENTATION AND BASELINES
+
+Implementation We used I3D (Carreira & Zisserman, 2017) and the two-stream version of InceptionV3 (Szegedy et al., 2016) pretrained on Imagenet and Kinetics as our base per-frame CNNs. Our default $L$ setting used for the TGM layers as well as the other baselines was as follows: when using I3D segment features (collected at 3fps), the 1 layer models used $L = 1 5$ and the 3 layer models used $L = 5$ . When using InceptionV3 frame feature (collected at 8fps), the 1 layer models used $L = 3 0$ and the 3 layer models used $L = 1 0$ . These layers were attached on top of the base CNN, as described in Subsection 3.2. Please check the appendix for implementation and training details and results on other datasets.
+
+Baselines In order to confirm the advantages of our TGM layers, particularly against previous temporal models, we implemented several baselines. The first is (i) a standard per-frame classifier in which the prediction at each time-step only depends on a single feature vector with no contextual temporal information. We also used (ii) LSTMs on top of per-frame representations, which were popularly used to capture temporal information (Donahue et al., 2015). We train a bi-directional LSTM with 512 hidden units to make per-frame predictions. We also tried (iii) the fixed pyramid temporal max-pooling of level 3 (Ryoo et al., 2015). Finally, we compare our model against (iv) the model with standard temporal convolutional layers (i.e., 1-D convolution with a $D \times L$ kernel) on top of per-frame representations. This is similar to the temporal conv. used in (Tran et al., 2018). Temporal lengths (i.e., $L$ ) of the 1-D conv. filters and the pooling windows were set to be identical to the TGM filters. That is, they capture the same temporal duration as TGMs. In all our experiments, we follow the standard evaluation setting of computing per-frame mean average precision (mAP) and report those values. We also compare to different versions of the TGM layer, (v) with a learned mixture of random temporal filters and (vi) with a learned mixture of fixed Gaussians.
+
+In addition, we also tried the approach of combining our TGM layers with the recent super-event representations (Piergiovanni & Ryoo, 2018b). We concatenated the learned super-event representation with our representations from TGM layers.
+
+# 4.2 MULTITHUMOS
+
+Dataset MultiTHUMOS (Yeung et al., 2015) is an extended version of the THUMOS (Jiang et al., 2014) dataset that densely annotates the continuous videos. The dataset consists of 65 different classes, compared to 20 in THUMOS, and contains on average 10.5 activities per video and 1.5 labels per frame and up to 25 activity instances in each video. This is in contrast to many other activity detection dataset such as ActivityNet (Heilbron et al., 2015), which only has on average ${ \sim } 1$ activity per video. MultiTHUMOS consists of YouTube videos of various sport activities such as basketball games, volleyball games, weight lifting, and track and field.
+
+We followed the standard MultiTHUMOS evaluation setting of measuring mAP based on per-frame annotations. There are 1010 validation videos and 1574 test videos. We used these continuous validation videos for the training of our models. We did not need to take advantage of the separate training set with segmented videos; even without them, we outperformed the state-of-the-arts.
+
+Results We compared baselines as well as multiple different versions of our architectures, shown in Table 1. The model with our TGM layers consistently outperformed baseline I3D (or InceptionV3) while using the same per-segment representations. Learning 3 TGM layers further improved the performances. On the other hand, we found that stacking multiple standard temporal convolutional layers does not improve performance, often performing worse than the baseline. While a single standard temporal conv. layer improves over the baseline, having multiple of them significantly increases the number of parameters to learn (Table 2) and we suspect that this was causing the overfitting with the limited amount of samples in the dataset. In Table 3, we compare the results of using a LSTM or temporal conv. with a similar number of parameters. This was done by making their temporal conv. filters to share values across multiple channels. These models result in nearly random performance, as they were not designed to cope with a small number of parameters. We also show results with a mixture of random (fixed) temporal filters and with a mixture of fixed Gaussians. These results confirm that (i) modeling the temporal structure as a learned Gaussian mixture is beneficial and that (ii) further learning the Gaussian distribution parameters is important.
+
+Table 1: Comparison of various architectures on MultiTHUMOS using both I3D per-segment and InceptionV3 per-frame features. We found that TGM layers with 1x1 convolution channel combination performed the best. Results are in mAP $\%$ . Note that we use the same filter length for “Temporal Conv” and “TGM” models, as described in Section 4.1.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">13D</td><td colspan="3">InceptionV3</td></tr><tr><td>Spatial</td><td>Temporal</td><td>Two-Stream</td><td>Spatial</td><td>Temporal</td><td>Two-Stream</td></tr><tr><td>Baseline</td><td>22.3</td><td>25.0</td><td>29.7</td><td>13.6</td><td>14.1</td><td>15.2</td></tr><tr><td>Temporal Conv</td><td>32.5</td><td>35.5</td><td>38.4</td><td>15.2</td><td>15.5</td><td>15.8</td></tr><tr><td>3 Temporal Conv</td><td>20.4</td><td>23.4</td><td>24.4</td><td>5.3</td><td>6.1</td><td>6.5</td></tr><tr><td colspan="7">TGM layers with grouped convolution</td></tr><tr><td>1 TGM</td><td>35.1</td><td>37.8</td><td>40.5</td><td>16.3</td><td>17.5</td><td>18.0</td></tr><tr><td>3 TGM</td><td>36.4</td><td>42.3</td><td>43.5</td><td>17.5</td><td>18.3</td><td>19.2</td></tr><tr><td colspan="7">TGM layers with channel combination</td></tr><tr><td>1 TGM (soft)</td><td>35.2</td><td>37.9</td><td>40.2</td><td>17.2</td><td>17.6</td><td>18.4</td></tr><tr><td>1 TGM (1x1)</td><td>36.1</td><td>38.2</td><td>40.8</td><td>17.2</td><td>17.7</td><td>18.4</td></tr><tr><td>3 TGM (soft)</td><td>36.2</td><td>40.1</td><td>42.3</td><td>17.5</td><td>19.1</td><td>21.2</td></tr><tr><td>3 TGM (1x1)</td><td>37.2</td><td>42.1</td><td>44.3</td><td>17.9</td><td>19.3</td><td>22.2</td></tr></table>
+
+Table 2: Additional number of parameters for models when added to the base architecture (e.g., I3D or Inception V3).   
+
+<table><tr><td>Model</td><td> # of parameters</td></tr><tr><td>LSTM</td><td>10.5M</td></tr><tr><td>1 Temporal Conv</td><td>10.5M</td></tr><tr><td>3 Temporal Conv</td><td>31.5M</td></tr><tr><td>1 TGM Layer</td><td>10K</td></tr><tr><td>3 TGMLayers</td><td>100K</td></tr></table>
+
+Table 3: Comparison of previous methods with comparable number of parameters and random forms of our TGM layer.   
+
+<table><tr><td>Model</td><td>mAP</td></tr><tr><td>LSTM with 100k parameters</td><td>6.5</td></tr><tr><td>Temporal Conv. with 1OOk parameters</td><td>7.3</td></tr><tr><td>TGM with random temporal filters</td><td>34.5</td></tr><tr><td>TGM with fixed Gaussians</td><td>38.5</td></tr><tr><td>Full TGM</td><td>44.3</td></tr></table>
+
+Learning multiple TGM layers with channel combination outperforms the grouped convolution version of TGM and all the baselines. We also experimented with a version using soft-attention weights to combine the TGM layer channels, in addition to our method (Fig. 3) of using 1x1 convolution followed by a ReLU (to gain non-linearity). We found that the 1x1 convolution performed better. We tested various number of Gaussian mixtures (i.e., output channels) and found that using 80 for the first and second layer and using 65 (i.e., number of classes) for the final layer performs best.
+
+Table 4 compares our model using TGM layers with multiple previous state-of-the-art approaches and baselines such as LSTM. Our approach meaningfully outperforms all previous approaches. Importantly, we are comparing our approach with different methods of capturing temporal information such as LSTMs and fixed temporal pyramid pooling while making them use the exactly same per-frame representations. We found that while all these methods capture some temporal information, the TGM layers provide the best performance. Further, combining the super-event representation (Piergiovanni & Ryoo, 2018b) with our TGM feature also benefited detection, confirming that our TGMs and super-events capture different aspects of the activity videos. In Fig. 5, we show an example of the various models predictions on a basketball video. We outperform the previous state-of-the-art performance (mAP) by $10 \%$ (36.4 vs. 46.4).
+
+# 4.3 CHARADES
+
+Dataset Charades (Sigurdsson et al., 2016b) is a large scale dataset with 9848 videos across 157 activity classes. These videos were recorded in home environments of the participants based on provided scripts. Each video contains on an average of 6.8 activity instances, and there are often complex activities co-occurring. The activities were mainly performed at home. For example, some activity classes are ‘preparing a meal’, ‘eating’, ‘sitting’, ‘cleaning’, etc.
+
+In our experiments, we follow the original Charades detection setting (i.e., Charades v1 localize evaluation), which is the setting used in many previous approaches (Sigurdsson et al., 2016a; Xu
+
+![](images/c0d256bcd8222f7ad03ebc79eb263ced37551f8c7850c775d1e2e262f032da84.jpg)  
+Figure 5: Illustration of the temporal regions classified as various basketball activities from a basketball game video in MultiTHUMOS. Our TGM layers greatly improve performance.
+
+Table 4: Performances of the state-of-the-art methods and our approach on MultiTHUMOS. Our approach meaningfully outperforms all previous results.
+
+<table><tr><td></td><td>mAP</td></tr><tr><td>Two-stream (Yeung et al., 2015)</td><td>27.6</td></tr><tr><td>Two-stream + LSTM (Yeung et al., 2015)</td><td>28.1</td></tr><tr><td>Multi-LSTM (Yeung et al., 2015)</td><td>29.6</td></tr><tr><td>Predictive-corrective (Dave et al., 2017)</td><td>29.7</td></tr><tr><td>I3D baseline</td><td>29.7</td></tr><tr><td>I3D+LSTM</td><td>29.9</td></tr><tr><td>I3D + temporal pyramid</td><td>31.2</td></tr><tr><td>I3D + super-events (Piergiovanni &amp; Ryoo, 2018b)</td><td>36.4</td></tr><tr><td>I3D+our TGMs</td><td>44.3</td></tr><tr><td>I3D + super-events (Piergiovanni &amp; Ryoo,2018b) + our TGMs</td><td>46.4</td></tr></table>
+
+et al., 2017; Piergiovanni & Ryoo, 2018b). This is the original setting more challenging than the Charades Challenge 2017 setting (whose evaluation server was no longer approving new account access), in the aspect that it uses less amount of training videos.
+
+Results We compare our results with the state-of-the-arts in Table 5. To our knowledge, our method is obtaining the best known performance in the original localization setting of the Charades dataset. Notably, it is performing better than I3D that obtained the best competition performance, while using the same feature. Our method also outperforms standard temporal convolution, LSTMs, and fixed pyramid pooling, as well as the use of latent super-events. When setting $L = 3 0$ and using 3 TGM layers, our model is able to capture around 800 frames (about $\pm 1 5$ seconds from each frame) of temporal information, significantly more than previous works (e.g., I3D only captures $\pm 2$ seconds).
+
+# 5 CONCLUSIONS
+
+We newly introduced the Temporal Gaussian Mixture (TGM) layer and demonstrated its effectiveness for multi-activity detection in continuous videos. Our layer is fully differentiable and trainable using standard backpropagation, designed to learn temporal structure. We were able to confirm that our layer performs superior to state-of-the-art methods on activity detection datasets including MultiTHUMOS and Charades, obtaining the best known performance. We also tested our approach with two more public video datasets, MLB-YouTube (Piergiovanni & Ryoo, 2018a) and AVA (Gu et al., 2017), and confirmed its advantage over the previous works in Appendix.
+
+# REFERENCES
+
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+
+Table 5: Per-frame mAP on Charades, evaluated with the ‘Charades v1 localize’ setting. I3D models are two-stream, using both RGB and optical flow inputs.   
+
+<table><tr><td></td><td>mAP</td></tr><tr><td>Predictive-corrective (Dave et al., 2017) Two-stream (Sigurdsson et al., 2016a) Two-stream+LSTM (Sigurdsson et al., 2016a)</td><td>8.9 8.94 9.6</td></tr><tr><td>R-C3D (Xu et al., 2017) Sigurdsson et al. (Sigurdsson et al., 2016a)</td><td>12.7 12.8</td></tr><tr><td>I3D baseline</td><td>17.2</td></tr><tr><td>I3D + 3 temporal conv.layers (L = 5) I3D + 3 temporal conv. layers (L = 30)</td><td>17.5</td></tr><tr><td>I3D +LSTM</td><td>12.5</td></tr><tr><td>I3D + fixed temporal pyramid</td><td>18.1</td></tr><tr><td></td><td>18.2</td></tr><tr><td>I3D + super-events (Piergiovanni &amp; Ryoo,2018b)</td><td>19.4</td></tr><tr><td>I3D +3 TGMs (L = 5)</td><td></td></tr><tr><td></td><td>20.6</td></tr><tr><td>I3D +3 TGMs (L = 30)</td><td>21.5</td></tr><tr><td>I3D +3 TGMs (L = 5) + super-events</td><td>21.8</td></tr><tr><td>I3D +3 TGMs (L = 3O) + super-events</td><td>22.3</td></tr></table>
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+Y.-G. Jiang, J. Liu, A. Roshan Zamir, G. Toderici, I. Laptev, M. Shah, and R. Sukthankar. THUMOS challenge: Action recognition with a large number of classes. http://crcv.ucf.edu/THUMOS14/, 2014.
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+AJ Piergiovanni and Michael S. Ryoo. Fine-grained activity recognition in baseball videos. In CVPR Workshop on Computer Vision in Sports, 2018a.
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+AJ Piergiovanni and Michael S. Ryoo. Learning latent super-events to detect multiple activities in videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018b.
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+AJ Piergiovanni, Chenyou Fan, and Michael S Ryoo. Learning latent sub-events in activity videos using temporal attention filters. In Proceedings of the American Association for Artificial Intelligence (AAAI), 2017.
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+
+# A IMPLEMENTATION DETAILS
+
+As our base per-segment CNN, we use the I3D (Carreira & Zisserman, 2017) network pretrained on the ImageNet and Kinetics (Kay et al., 2017) datasets. I3D obtained state-of-the-art results on segmented video tasks, and this allows us to obtain reliable $v _ { t }$ . We also use two-stream version of InceptionV3 (Szegedy et al., 2016) pretrained on Imagenet and Kinetics as our base per-frame CNN, and compared them. We chose InceptionV3 as it is deeper than previous two-stream CNNs such as (Simonyan & Zisserman, 2014; Feichtenhofer et al., 2016). We extracted frames from the videos at 25 fps, computed TVL1 (Zach et al., 2007) optical flow, clipped to $[ - 2 0 , 2 0 ]$ . For InceptionV3, we computed features for every 3 frames (8 fps). For I3D, every frame was used as the input. I3D has a temporal stride of 8, resulting in 3 features per second (3 fps).
+
+We implemented our TGM layers as well as other baseline layers in PyTorch. Our default setting was as follows: for 3-layer models, we set $L = 1 0$ for frame-based features (i.e., InceptionV3) and $L = 5$ for segment-based features (i.e., I3D), as each segment already contains some temporal information. For 1-layer models, we set $L = 3 0$ for frame-based features and $L = 1 5$ for segmentbased features. We set $M = 1 6$ and $C _ { o u t } = 8 0 $ and $C _ { o u t } = 6 5$ for the last TGM layer. We found these values to work well on a held out portion of the training set of MultiTHUMOS. In all models, we used one fully-connected layer at the end to make the per-frame or per-segment classification.
+
+We trained our models using the Adam (Kingma & Ba, 2014) optimizer with the learning rate set to 0.01. We decayed the learning rate by a factor of 10 after every 10 training epochs. We trained our models for 50 epochs. We plan to make all our source code and trained models publicly available once the paper is published.
+
+# B HYPERPARAMETER EXPERIMENTS
+
+We conducted a set of experiments to compare the effects of the temporal duration, $L$ , number of Gaussians, $M$ , and the number of output channels, $C _ { o u t }$ . For these experiments, we only used the one-stream version of I3D with RGB inputs.
+
+Effect of $L$ : In Table 6, we compare different values of $L$ . For these experiments, we use $M = 1 6$ and $C _ { o u t } = 1 6$ . We find that the 3-layer model with $L = 5$ performs the best. With I3D features, this allows the model to capture up to 8 seconds of information. The average activity in MultiTHUMOS is 3.3 seconds long and the maximum is 14.7 seconds long, and with this setting, the model is able to capture enough temporal context to perform well. Larger values of $L$ capture too much temporal information, but due to the Gaussian structure, it does not drastically harm performance. Figure 6 shows that even with longer kernels, the Gaussians learn to focus mostly on the center of the interval and capture the rough duration of the activities. Thus, having too long intervals does not drastically harm performance, which is in contrast to the standard 1-D convolution. Note that for Charades, the temporal kernels are learned to capture much longer temporal duration, as the average activity in charades is 12.8 seconds and larger values of $L$ perform better.
+
+Figure 6 illustrates examples of the learned TGM kernels of various lengths. The figure shows that the kernels focus on short temporal intervals on MultiTHUMOS even if we make the filters longer, as the activities are an average of 3.3 seconds long. On Charades, the TGM kernels learn to capture much longer intervals, as the activities are an average of 12.8 seconds long. We believe that this suggests TGMs are learning to capture information from the important necessary intervals.
+
+In Table 6, we also report the results of using a standard 1-D conv. layer with different $L$ values. The number of parameters in our TGM layer is independent of $L$ , however, with the standard 1-D conv. layer, the number of parameters increases as $L$ increases. We find that increasing $L$ with 1-D convolution helps for small values of $L$ , but for $L > 1 5$ , the performance drastically drops, while TGM layers only show a small decrease.
+
+Effect of $M$ : In Table 7, we compare different values of $M$ . For these experiments, we set $L = 1 5$ and $C _ { o u t } = 1 6$ . We find that $M = 1 6$ performs best, suggesting that smaller values of $M$ restrict the possible temporal kernels too much. We also observe that larger values of $M$ performs slightly worse than $M = 1 6$ (but not much), likely because they introduce more parameters than needed. When $M$ and $L$ have similar values, it allows the model to learn a sufficient number of Gaussians and create a diverse range of temporal kernels. When $M$ is larger than $L$ , it results in learning a kernel similar to standard 1-D convolution.
+
+Table 6: Effect of $L$ on MultiTHUMOS and Charades using only RGB I3D features. Note that the 3 TGM layer models have larger temporal resolution than the 1 TGM layer models for the same values of $L$ . We also compare to using standard one-layer 1-D conv layer with different values of $L$ .   
+
+<table><tr><td rowspan="2"></td><td colspan="3">MultiTHUMOS</td><td colspan="3">Charades</td></tr><tr><td>1 Layer</td><td>3 Layers</td><td>1-D Conv</td><td>1 Layer</td><td>3 Layers</td><td>1-D Conv</td></tr><tr><td>I3DBaseline</td><td>22.3</td><td>=</td><td>=</td><td>15.3</td><td>=</td><td>=</td></tr><tr><td>L=3</td><td>30.2</td><td>31.7</td><td>26.6</td><td>15.5</td><td>16.1</td><td>15.5</td></tr><tr><td>L=5</td><td>32.5</td><td>37.2</td><td>28.3</td><td>15.7</td><td>17.8</td><td>16.3</td></tr><tr><td>L=10</td><td>34.5</td><td>35.4</td><td>31.7</td><td>16.1</td><td>18.2</td><td>16.6</td></tr><tr><td>L=15</td><td>36.1</td><td>34.1</td><td>32.5</td><td>17.5</td><td>18.6</td><td>16.8</td></tr><tr><td>L=30</td><td>32.5</td><td>33.9</td><td>26.5</td><td>18.1</td><td>18.9</td><td>12.1</td></tr><tr><td>L= 50</td><td>32.1</td><td>33.7</td><td>15.4</td><td>18.3</td><td>18.8</td><td>6.7</td></tr></table>
+
+Table 7: Comparison of various values of $M$ on MultiTHUMOS and Charades using RGB I3D features. For these experiments, 1 layer was used with $L = 1 5$ and $C _ { o u t } = 1 6$ .   
+
+<table><tr><td></td><td>MultiTHUMOS</td><td>Charades</td></tr><tr><td>M=2</td><td>27.8</td><td>15.5</td></tr><tr><td>M=4</td><td>33.1</td><td>16.2</td></tr><tr><td>M=8</td><td>34.8</td><td>17.5</td></tr><tr><td>M=16</td><td>36.1</td><td>17.5</td></tr><tr><td>M= 32</td><td>35.7</td><td>17.1</td></tr><tr><td>M= 64</td><td>35.8</td><td>17.3</td></tr></table>
+
+Table 8: Comparison of values of $C _ { o u t }$ on MultiTHUMOS and Charades using RGB I3D features. For these experiments, 1 layer was used with $L = 1 5$ and $M = 1 6$ .   
+
+<table><tr><td></td><td>MultiTHUMOS</td><td>Charades</td></tr><tr><td>Cout 1</td><td>33.5</td><td>16.2</td></tr><tr><td>Cout 4</td><td>34.2</td><td>17.4</td></tr><tr><td>Cout 8</td><td>35.5</td><td>17.5</td></tr><tr><td>Cout 16</td><td>36.1</td><td>17.5</td></tr><tr><td>Cout 32</td><td>36.0</td><td>17.2</td></tr><tr><td>Cout 64</td><td>36.1</td><td>17.4</td></tr><tr><td>Cout = 80</td><td>36.1</td><td>17.5</td></tr></table>
+
+Effect of $C _ { o u t }$ : In Table 8, we compare different values of $C _ { o u t }$ . For these experiments, $L =$ 15, we used 1-layer and $M = 1 6$ . We find that $C _ { o u t }$ performs best when set to 16 or larger on these datasets. Larger values of $C _ { o u t }$ seem to capture redundant information, as it does not lower performance.
+
+![](images/744a3159af7040ba3ca43621d47d5b12f250bda0c5cfd68bf678d41847d83116.jpg)  
+Figure 6: Illustration of several learned TGM kernels. On MultiTHUMOS, it learns to focus on shorter intervals to capture shorter events. On Charades, the Gaussians have a larger $\sigma$ value, resulting in filters that attend to longer temporal durations.
+
+![](images/b998f17e5d56ec0eab5405aa1323e9b9a726ecf049f10c25969a91979fea53b8.jpg)  
+Figure 7: (a-c) Different forms of 1-D temporal convolutions which take a $D \times T$ input and produces a $C \times T$ output based on $C$ number of $D \times L$ kernels: (a) the standard 1-D convolution, $\mathbf { ( b ) }$ using Gaussian mixtures for 1-D convolution while sharing Gaussian mixtures across input channels, and (c) using $D$ different Gaussian mixtures for 1-D convolution. (d) Our TGM layer in its simplest form (i.e., 1-layer case) applying the $1 \times L$ temporal kernel in a 2-D convolutional fashion, maintaining both time and feature axis.
+
+![](images/1a1a1342d875bc0c72fde175610ed02c50531ca14855a426d92b84f4adcfae2a.jpg)  
+Figure 8: A temporal convolutional layer with channel combination similar to Fig. 3. The difference is that this layer does not learn Gaussian mixtures, but unconstrained 1-D temporal kernels.
+
+# C COMPARISON OF DIFFERENT LAYER FORMS
+
+To confirm the various aspects of our design, we conducted experiments comparing different types of temporal convolution. In Fig. 7a we illustrate the standard 1-D convolution, taking $D \times T$ input and producing a $C \times T$ output, where $D$ is the number of input channels and $C$ is the number of output channels. In Fig. 7b, we illustrate the method of applying a Gaussian mixture kernel as 1-D convolution. Here, the Gaussian mixture kernel is shared by all $D$ input channels and we learn a $C$ number of such kernels. In Fig. 7c, we illustrate the approach of applying a Gaussian mixture kernel as 1-D convolution while learning $D$ different Gaussian mixtures. This is very similar to the standard 1-D convolution, except that the filter values are constrained to have the shape of Gaussian mixtures.
+
+Fig. 8 illustrates one more baseline. This is similar to our full TGM layer with the channelcombination described Fig. 3. However, in this baseline, instead of learning Gaussian mixtures, we learn $C _ { i n } \cdot C _ { o u t }$ number of $1 \times L$ kernels. The kernel values are left unconstrained. While the TGM layer has $2 \cdot M + C _ { i n } \cdot C _ { o u t } \cdot M + C _ { i n } \cdot C _ { o u t }$ parameters, this layer has $L \cdot C _ { i n } \cdot C _ { o u t } \cdot M + C _ { i n } \cdot C _ { o u t }$ , which is more than the TGM layer.
+
+In Table 9, we compare the results of the various above-mentioned layers on MultiTHUMOS using RGB I3D features. We find that the Fig. 7b method performs poorly, while the Fig. 7c method slightly outperforms the standard 1-D convolution. The Fig. 8 method is slightly better than the standard 1-D convolution, but performs worse than Fig. 7c. However, none of these layers perform as well as our TGM layer, confirming that both the design of learning Gaussian mixtures and maintaining temporal channel axis are important for activity detection.
+
+Table 9: Comparison of the different forms of temporal convolution on MultiTHUMOS using RGB I3D features. We set $L = 1 5$ and used 1 layer models for these experiments.   
+
+<table><tr><td></td><td>MultiTHUMOS</td></tr><tr><td>Standard 1-D Convolution (Fig. 7a)</td><td>32.5</td></tr><tr><td>The layer described in Fig.7b</td><td>28.6</td></tr><tr><td>The layer described in Fig. 7c</td><td>33.2</td></tr><tr><td>The layer described in Fig. 8</td><td>32.8</td></tr><tr><td>Our TGM Layer</td><td>36.1</td></tr></table>
+
+![](images/99b2898871e544d67d45bf16b97e65b0e77b16df16b200a2ad0c06f579924c1f.jpg)  
+Figure 9: Examples of several of the activities in the MLB-YouTube dataset: (a) Pitch, (b) Hit, (c) Bunt, (d) Hit by pitch, (e) No activity. This shows the difficulty of this dataset, as the difference between hit and bunt, swing and no swing are very small.
+
+# D EXPERIMENTS ON ADDITIONAL DATASETS
+
+# D.1 MLB-YOUTUBE DATASET
+
+# D.1.1 DATASET
+
+The MLB-YouTube dataset (Piergiovanni & Ryoo, 2018a) consists of 20 baseball games from the 2017 MLB post-season available on YouTube. This dataset consists of over 42 hours of video. For these experiments, we used the continuous video setting which have 2,126 1-2 minute long clips. Each clip is densely annotated with the baseball activities that occur. There are 8 activity classes: pitch, strike, ball, swing, hit, foul, hit by pitch, and bunt. Examples of some of these classes are shown in Fig. 9. Each continuous clip contains on average of 7.2 activities, giving a total of over 15,000 activity instances in the dataset.
+
+What makes this dataset challenging is that the variation between classes is very small. In ActivityNet (Heilbron et al., 2015), for example, the difference between swimming and brushing hair is drastic. The background, motion, and even size of the person in the video is different. However, in broadcast baseball videos, the difference between a ball and a strike, or a swing and a bunt, are small. All actions are recorded from the same camera angle as we can confirm from Fig. 9.
+
+# D.1.2 RESULTS
+
+In Table 10, we compare various approaches on this dataset. Our TGM layers improve over the baseline by ${ \sim } 6 \%$ (40.1 vs. 34.2). Additionally, we compare to methods using the super-event representation (Piergiovanni & Ryoo, 2018b), which previously achieved state-of-the-art performance on several activity detection datasets. On this dataset, our approach outperforms the super-event representation, and further the concatenation of our TGM representation with such super-event representation performs best by a significant margin $\sim 1 3 \%$ compared to the baseline). This suggests that TGMs and super-event capture different temporal information and are both useful to the detection task.
+
+We further find that using multiple, standard temporal convolution layers leads to worse performance, likely due to overfitting from the large number of parameters. While using multiple TGM layers improves performance, confirming that the Gaussian structure and sparsity constraint benefits model learning.
+
+Table 10: Result mAP on the MLB-YouTube dataset using InceptionV3 and I3D to obtain features. Our TGM layers significantly outperform the baseline models.   
+
+<table><tr><td>Model</td><td>Spatial</td><td>Temporal</td><td>Two-stream</td></tr><tr><td>Random</td><td>13.4</td><td>13.4</td><td>13.4</td></tr><tr><td>InceptionV3</td><td>31.2</td><td>31.8</td><td>31.9</td></tr><tr><td>InceptionV3 +LSTM</td><td>32.1</td><td>33.5</td><td>34.1</td></tr><tr><td>InceptionV3 +1 temporal conv</td><td>32.8</td><td>34.4</td><td>35.2</td></tr><tr><td>InceptionV3 + 3 temporal conv</td><td>28.4</td><td>29.8</td><td>30.1</td></tr><tr><td>InceptionV3 + super-events</td><td>31.5</td><td>36.2</td><td>39.6</td></tr><tr><td>InceptionV3 +1TGM</td><td>32.4</td><td>36.3</td><td>37.4</td></tr><tr><td>InceptionV3+3 TGM</td><td>33.2</td><td>38.2</td><td>38.2</td></tr><tr><td>InceptionV3 + 3 TGM+super-events</td><td>34.6</td><td>42.4</td><td>42.9</td></tr><tr><td>I3D</td><td>33.8</td><td>35.1</td><td>34.2</td></tr><tr><td>I3D + LSTM</td><td>36.2</td><td>37.3</td><td>39.4</td></tr><tr><td>I3D +1 temporal conv</td><td>37.3</td><td>38.6</td><td>39.9</td></tr><tr><td>I3D + 3 temporal conv</td><td>32.4</td><td>34.6</td><td>35.6</td></tr><tr><td>I3D + super-events</td><td>38.7</td><td>38.6</td><td>39.1</td></tr><tr><td>I3D+1TGM</td><td>35.5</td><td>37.5</td><td>38.5</td></tr><tr><td>I3D+3 TGM</td><td>36.5</td><td>38.4</td><td>40.1</td></tr><tr><td>I3D +3 TGM+super-events</td><td>39.4</td><td>46.0</td><td>47.1</td></tr></table>
+
+Table 11: Results on AVA dataset with the temporal annotation-only setting (i.e., frame classification without using bounding box training labels).   
+
+<table><tr><td></td><td>mAP</td></tr><tr><td>Random</td><td>2.65</td></tr><tr><td>I3D baseline</td><td>7.5</td></tr><tr><td>I3D + 3 temporal conv. layers</td><td>7.9</td></tr><tr><td>I3D+LSTM</td><td>7.8</td></tr><tr><td>I3D + super-events(Piergiovanni &amp; Ryoo,2018b)</td><td>9.8</td></tr><tr><td>I3D+1TGMs</td><td>11.2</td></tr><tr><td>I3D +3 TGMs</td><td>14.5</td></tr><tr><td>I3D +3 TGMs + super-events</td><td>14.9</td></tr></table>
+
+# D.2 AVA
+
+# D.2.1 DATASET
+
+AVA (Gu et al., 2017) is a large-scale video dataset containing of 80 atomic action classes in $5 7 \mathrm { k }$ video clips. These clips are drawn from movies. Existing datasets, such as Charades, have very specific actions that depend on objects, such as holding a cup vs. holding a picture. In AVA, the actions are intentionally generic, such as sit, stand, hold, carry, etc. Further, the AVA dataset is annotated with both spatial and temporal locations of activities. Since we are interested in temporal activity detection, we follow the setting of Piergiovanni & Ryoo (2018b) and label each frame with the occurring activities while ignoring the spatial location. We evaluate performance following the same method as MultiTHUMOS, Charades and MLB-YouTube by measuring per-frame mAP.
+
+# D.2.2 RESULTS
+
+In Table 11, we present the results of our model. We again find that temporal convolution and LSTMs provide some benefit over the baseline, but TGM layers further improve performance. Again, combining the TGM, which captures local temporal structure, with super-events which capture global temporal structure, provides the best performance by $\sim 7 . 4 \%$ .

parse/train/Hyfg5o0qtm/Hyfg5o0qtm_content_list.json

@@ -0,0 +1,1829 @@
+[
+    {
+        "type": "text",
+        "text": "TEMPORAL GAUSSIAN MIXTURE LAYER FOR VIDEOS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            816,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            209,
+            544,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We introduce a new convolutional layer named the Temporal Gaussian Mixture (TGM) layer and present how it can be used to efficiently capture longer-term temporal information in continuous activity videos. The TGM layer is a temporal convolutional layer governed by a much smaller set of parameters (e.g., location/variance of Gaussians) that are fully differentiable. We present our fully convolutional video models with multiple TGM layers for activity detection. The experiments on multiple datasets including Charades and MultiTHUMOS confirm the effectiveness of TGM layers, outperforming the state-of-the-arts. ",
+        "bbox": [
+            232,
+            243,
+            764,
+            352
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            385,
+            334,
+            400
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Activity videos are spatio-temporal data: they are image frames with a specific width/height (XY) concatenated along time axis (T). Recognition from such videos requires capturing both spatial and temporal information in the videos, desirably using learned convolutional kernels. Temporal convolution is particularly beneficial in activity ‘detection’ tasks, which require making activity decisions at every frame given a continuous video (Sigurdsson et al., 2016b; Yeung et al., 2015). Previous methods investigated using 3-D XYT convolutional filters (Tran et al., 2014; Carreira & Zisserman, 2017) as well as the models with 2-D XY conv. layers followed by 1-D temporal conv. (Tran et al., 2018), pooling or attention layers (Piergiovanni et al., 2017). ",
+        "bbox": [
+            174,
+            417,
+            823,
+            527
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding complex multi-activity videos requires capturing information in long-term time intervals. Different frames contain different information, and the model needs to learn to take advantage of as many frames as possible, while abstracting them efficiently. Previous attempts of simply pooling representations over time or learning temporal conv. filters with a small number of frames (e.g., 16 or 64) was thus often insufficient to fully consider rich long-term temporal context. Simultaneously, bruteforcely increasing the temporal filter length (to look at more frames) results more learnable parameters, requiring more training data, which can be expensive when activities are rare. ",
+        "bbox": [
+            174,
+            534,
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+            630
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduce a new convolutional layer named the Temporal Gaussian Mixture (TGM) layer, and present how it can be used to efficiently capture longer-term temporal information in activity videos. Our temporal Gaussian mixture layer is a temporal convolutional layer, whose filters/kernels are controlled by a set of (temporal) Gaussian distribution parameters. Each of our temporal Gaussian distributions specify (temporally) ‘where’ the model should look, and our Gaussian mixture layer combines them as multiple convolutional filters to be applied on top of temporallycontinuous representations. This layer allows the video representation at each time step to be constructed while focusing on different neighboring temporal regions, instead of only focusing on its local segment. It is a convolutional layer governed by a much smaller set of parameters (i.e., locations/variances of the Gaussians as well as their mixture weights) that are fully differentiable. ",
+        "bbox": [
+            174,
+            636,
+            823,
+            772
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The motivation behind our temporal Gaussian mixture layer is to learn the temporal structure of an activity as a composition of temporal Gaussian regions/attentions. Such structure allows the model to obtain a compact spatio-temporal representation abstracting each (long-term) time interval, using multiple temporal conv. layers with far fewer parameters. It is also related to the previous temporal attention works (Piergiovanni et al., 2017), but our model is designed to be fully convolutional to handle continuous data and it learns more compositional structures with multiple layers. ",
+        "bbox": [
+            174,
+            780,
+            823,
+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present video-CNN models using our TGM layers for activity detection in continuous videos. Our model stacks TGM layers on top of several state-of-the-art CNNs such as I3D (Carreira & Zisserman, 2017). This enables our model to capture longer-term temporal information than what we use as base CNNs, compositionally modeling temporal structure with multiple TGM layers. Our model was evaluated on multiple public datasets including MultiTHUMOS and Charades, and was able to outperform the best previous activity detection CNNs by a meaningful margin. ",
+        "bbox": [
+            176,
+            868,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            151,
+            354,
+            167
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning video representations for human activity recognition has been successful. CNN methods allow end-to-end learning of video features and representations optimized for the training data, performing superior to traditional works (Aggarwal & Ryoo, 2011) for video understanding. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            224
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Two-stream CNN models take a single RGB frame and a small number of optical flow frames as inputs to capture both motion and appearance information in videos (Simonyan & Zisserman, 2014; Feichtenhofer et al., 2016). Models learning 3-D spatio-temporal (XYT) convolutional filters were designed and applied to many activity recognition tasks as well (Tran et al., 2014; Carreira & Zisserman, 2017; Tran et al., 2017; Hara et al., 2017). Large scale datasets for activity detection, such as THUMOS (Jiang et al., 2014), ActivityNet (Heilbron et al., 2015), Kinetics (Kay et al., 2017), and Charades (Sigurdsson et al., 2016b) provided these approach the necessary training data to learn the models. Such 3-D XYT CNNs were also used to capture spatio-temporal information for activity detection (Xu et al., 2017; Shou et al., 2016; 2017; Zhao et al., 2017). However, all these CNNs were limited to the consideration of a fixed local video segment (e.g., 16 frames in (Tran et al., 2014) and 64-99 frames in (Carreira & Zisserman, 2017)) when making activity decisions. ",
+        "bbox": [
+            174,
+            231,
+            825,
+            381
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Some works studied combining representations over longer-term temporal intervals (Karpathy et al., 2014; $\\mathrm { N g }$ et al., 2015; Varol et al., 2017), but it was generally done with a temporal pooling of local representations or (spatio-)temporal convolutions with a bit larger fixed intervals. Recurrent neural networks (RNNs) have also been used to model activity transitions between frames (Yeung et al., 2015; 2016; Escorcia et al., 2016), but they were strictly sequential and had limitations in maintaining temporal information over a longer temporal duration, particularly for videos with multiple complex activities. Recently, CNN models using temporal attention for activity videos (Piergiovanni et al., 2017; Piergiovanni & Ryoo, 2018b) were studied as well. However, a fully convolutional model to analyze continuous videos while efficiently representing information in long term intervals has been lacking. ",
+        "bbox": [
+            174,
+            387,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our layer is different from the previous standard (spatio-)temporal convolutional layers in that it relies on significantly fewer parameters by forcing filter shapes to be Gaussian compositions. Our temporal layer is also different from previous Gaussian Mixture Model layers (Variani et al., 2015) in that our layer is convolutional while they are not. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
+        "bbox": [
+            176,
+            606,
+            299,
+            622
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce a new convolutional layer named the Temporal Gaussian Mixture (TGM) layer, and present how it can be used for activity recognition. Our Temporal Gaussian Mixture layer is a temporal convolutional layer to be applied on top of a sequence of representations (usually from frame-level or segment-level CNNs), whose filters/kernels are controlled by a set of (temporal) Gaussian distribution parameters. The motivation is to make each temporal Gaussian distribution specify (temporally) ‘where to look’ with respect to the activity center, and represent the activity as a collection/mixture of such temporal Gaussians convolved with video features. Our layer is fully differentiable and trainable using standard backpropagation. ",
+        "bbox": [
+            174,
+            637,
+            825,
+            747
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our TGM layer can be interpreted as a a form of 1-D convolution where the filters are determined by a mixture of Gaussians. However, our TGM layer differs from the standard temporal convolutional layers of learning 1-D (time) or 2-D (channel-by-time) filters in the following aspects: ",
+        "bbox": [
+            176,
+            753,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Our temporal Gaussian mixture layer handles multiple 3-D tensors internally to preserve channels from the frame-level CNN by adding a new temporal channel axis. Its input is 3-D (channel-by-channel-by-time), where one channel dimension is inherited from the frame-level CNN and this dimension size remains unchanged.   \n2. Instead of learning temporal convolution filters of any arbitrary values, our filter is forced to have the form of a temporal Gaussian mixture shared across all frame-level channels. This allows the layer to rely on significantly fewer number of (fully differentiable) parameters, while capturing the concept of temporal structure/attention. ",
+        "bbox": [
+            212,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb08fe1098ed6b3ebacab0ba55fa20f4c6eca53bc414048213d86cf9d3f10300.jpg",
+        "image_caption": [
+            "Figure 1: Example illustrating how our Temporal Gaussian Mixture layer is computed. Multiple $( M )$ temporal Gaussian distributions are learned, and they are combined with the learned soft attention weights to form the $C$ temporal convolution filters. $L$ is the temporal length of the filter. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 TEMPORAL GAUSSIAN MIXTURE LAYER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            291,
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+            306
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our temporal Gaussian mixture layer takes a 3-D input with the dimensionality of $C _ { i n } \\times D \\times T$ , where $\\dot { C } _ { i n }$ is the number of input channels, $D$ is the dimensionality of the representations from frame-level (or segment-level) CNNs, and $T$ is the time. Given such input, the TGM layer convolves it with $C _ { o u t }$ number of $1 \\times L$ filters/kernels, generating a $C _ { o u t } \\times D \\times T$ -dim representation as an output. $L$ is the temporal length of the temporal Gaussian mixture filter. $D$ is usually 1K or 4K and $T$ is the number of time steps (frames) in each video (i.e., it varies per video). $C _ { o u t }$ is the number of different mixtures, corresponding to the number of output channels in standard convolution. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our layer is composed of a set of $M$ Gaussians. Each Gaussian has 2 parameters: a center $\\hat { \\mu }$ and a width $\\hat { \\sigma }$ . Each layer has additional hyper-parameters: $L$ , the temporal duration and $M$ , the number of Gaussians to learn. We force the learned center to be between ${ \\bar { - } } { \\frac { L } { 2 } }$ and $\\begin{array} { l } { { \\frac { L } { 2 } } } \\end{array}$ and $\\sigma$ to be positive: ",
+        "bbox": [
+            174,
+            420,
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+            464
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/061d96df08ede9f55df2f0b42f97903024b5f3507c7b77a47b4fab7a7628de77.jpg",
+        "text": "$$\n\\mu = ( L - 1 ) \\cdot \\frac { \\operatorname { t a n h } { ( \\hat { \\mu } + 1 ) } } { 2 } , \\sigma ^ { 2 } = \\exp { ( \\hat { \\sigma } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            469,
+            648,
+            501
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We use the above $\\mu$ and $\\sigma$ to construct the temporal Gaussian kernels. This acts as a strong sparsity constraint on the convolutional kernel as well as a drastic reduction of the number of learnable parameters. We construct a temporal Gaussian mixture convolutional kernel as: ",
+        "bbox": [
+            173,
+            503,
+            825,
+            546
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c3e3835081e1d08d978cc191ddccbc3b8831754743b39be969867ed1ed6fa300.jpg",
+        "text": "$$\n\\hat { K } _ { m , l } = \\frac { 1 } { Z } \\exp { - \\frac { ( l - \\mu _ { m } ) ^ { 2 } } { 2 \\sigma _ { m } ^ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            549,
+            594,
+            584
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $Z$ is a normalization constant such that $\\begin{array} { r } { \\sum _ { l } ^ { L } \\hat { K } _ { m , l } = 1 } \\end{array}$ , resulting in $\\hat { K }$ being an $M \\times L$ matrix. ",
+        "bbox": [
+            173,
+            589,
+            825,
+            608
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Instead of making the model learn a separate set of Gaussian distributions per activity class, we take the approach of maintaining multiple Gaussian distributions shared across classes and obtain a Gaussian ‘mixture’ filter by learning soft-attention weights. We learn a set of soft-attention weights per output channel $i$ , $\\omega \\in \\overline { { \\mathcal { R } } } ^ { C _ { o u t } \\times \\breve { M } }$ . We create the soft-attention weights by applying the softmax function over the $M$ Gaussians, enforcing each input channel weights sum to 1. ",
+        "bbox": [
+            174,
+            613,
+            825,
+            683
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d820ecbe31a841ba5af9a245709bff7676ef0ec79faf2dfeae74aa73118270b0.jpg",
+        "text": "$$\na _ { i , m } = \\frac { \\exp \\omega _ { i , m } } { \\sum _ { j } \\exp \\omega _ { i , j } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            685,
+            568,
+            719
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on temporal Gaussian distributions $\\hat { K } _ { i }$ and attention weights $a _ { i , m }$ , the temporal convolution filters our TGM layer is computed as: ",
+        "bbox": [
+            174,
+            731,
+            825,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b959d0f554ae800598e65bf0f1f846ece267562150e1e75f8b8791e70fcc32f4.jpg",
+        "text": "$$\nK _ { i } = \\sum _ { m } a _ { i , m } \\hat { K } _ { i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            765,
+            562,
+            799
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This provides us convolutional filters having the form of a mixture of temporal Gaussians, controlled based on $2 \\cdot M + C _ { i n } \\cdot C _ { o u t } \\cdot M$ parameters (instead of learning $D ^ { 2 } \\cdot L$ parameters without any constraint, as in standard temporal convolution where $C < < D$ ). An overview of this process is shown in Fig. 1. ",
+        "bbox": [
+            173,
+            801,
+            825,
+            858
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 SINGLE TGM LAYER - DIRECT PER-CLASS ACTIVITY MODELING",
+        "text_level": 1,
+        "bbox": [
+            173,
+            872,
+            674,
+            887
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The representation we obtain by applying our base CNNs to each frame (or local segment) has the dimensionality of $D$ , and stacking them along time axis provides us the representation with ",
+        "bbox": [
+            173,
+            895,
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/1e1b884ad92c56ef857b2a263d6acccb6a7a771dee8413a54c17dfc575c6b884.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of a TGM layer with grouped convolution. This layer learns a set of $C$ Gaussian mixtures that are convolved with the input channels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            101,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$1 \\times D \\times T$ -dim. That is, in the case of using only one TGM layer to capture activity representations, our $C _ { i n }$ is fixed to 1 and $C _ { o u t }$ is fixed to be the number of activity classes. This is the simplest case of our model, attaching one TGM layer on top of the $1 \\times D \\times T$ representation. ",
+        "bbox": [
+            173,
+            250,
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+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our convolutional kernel, $K$ , has a learned Gaussian mixture for each activity class. Let the video features $v$ be a $D \\times T$ matrix. Each $K _ { i }$ is a 2-D convolutional filter with a size of $1 \\times L$ , and convolving this with $v$ provides us a representation $S$ with $C _ { o u t }$ number of $D \\times T$ responses since $C _ { i n }$ is 1 in this case. This per-class representation can then be used as input to a fully-connected layer for activity classification. For $i \\in \\mathsf { \\bar { \\{ 1 , 2 , \\ldots , C _ { o u t } \\} } }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/16628b09023f7e0dfb9cebed4d7dbd2e5f99a293768b84effe86d5bb0b519c3e.jpg",
+        "text": "$$\ns _ { i } = v * K _ { i } , \\ S = [ s _ { 1 } , s _ { 2 } , \\ldots , s _ { C _ { o u t } } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fig. 7 in the appendix visually illustrates how each TGM filter is convolved with the input (Fig. 7d), compared to the standard 1-D convolution (Fig. 7a) or other forms of the temporal layers (Fig. 7b-c). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1.2 MULTIPLE TGM LAYERS - GROUPED CONVOLUTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We generalize the above formulation to allow the TGM layers to be sequentially applied. The idea is to enable our model to capture more complex, nonlinear temporal structure by having multiple levels of temporal layers. In this case, the input for each layer is $C _ { i n } \\times D \\times T$ dimensional (instead of $1 \\times D \\times T$ ), where the input channels are the number of output channels from the previous layer. Our kernels at each layer, $K _ { i }$ , are parameterized and learned as before. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By using grouped convolution with the number of groups set to $C _ { i n }$ , we can efficiently separate the input into per-channel values and convolve each of them with the designated $K _ { i }$ kernel, as shown in Fig. 2. That is, we learn a filter $K _ { i }$ per channel by setting $C _ { i n } = C _ { o u t }$ . For $i \\in [ 1 , C _ { o u t } ]$ , ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/6201967b323c2264d03925ffaf57d0654af05f03d742733e045b4e7ae588f077.jpg",
+        "text": "$$\ns _ { i } = f _ { i } * K _ { i } , ~ S = [ s _ { 1 } , s _ { 2 } , . ~ . ~ . s _ { C _ { o u t } } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $f$ is a $C _ { i n } \\times D \\times T$ tensor, where $D$ is the dimensionality of the feature and $T$ is the number of frames. The result of the per-channel convolution, $s _ { i }$ , is a $D \\times T$ representation. We concatenate these representations along the channel axis, resulting in $S$ , a $C _ { o u t } \\times D \\times T$ representation. As this convolution results in the same output shape, we can stack these layers. Each layer is able to capture increasing temporal resolution, allowing the model to capture levels of abstractions. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.1.3 MULTIPLE TGM LAYERS - CHANNEL COMBINATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the above subsection, we introduced an approach of stacking multiple TGM layers to model a hierarchical composition of temporal representations. However, in the grouped convolution case, each output channel of the layer is solely dependent on its corresponding input channel. That is, each kernel only considers information from a single output channel of the previous layer. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Therefore, we further generalize our TGM layer so that the layer combines representations from multiple input channels for each output channel while using the learned temporal kernels. We learn a set of convolutional kernels $K \\in \\mathop { \\mathcal { R } } ^ { C _ { o u t } \\times C _ { i n } \\times L }$ (i.e., we learn $C _ { o u t } \\cdot C _ { i n }$ Gaussian mixtures). Given $f$ which is the $C _ { i n } \\times D \\times T$ representation, for each output channel $i \\in [ 1 , C _ { o u t } ]$ and each input channel $j \\in [ 1 , C _ { i n } ]$ pair, we convolve the associated filters with the input. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/461f79080e99e2cac6a46597cfc9ef8b1eae5a1731cafef9a1f421ff6046bd52.jpg",
+        "text": "$$\nG _ { i , j } = ( f _ { j } * K _ { i , j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            852,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where each $G _ { i , j }$ is a $D \\times T$ -dim representation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We then learn a 1x1 convolution followed by a ReLU activation function for each $i \\in [ 1 , C _ { o u t } ]$ , which we call $w _ { i }$ , that maps from $C _ { i n }$ channels to 1 channel. The 1x1 convolution learns to combine ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/aca85f15ca42c4a0cdcd7ee131a3bb28c18e0cc54f9ba616551cbb896c0860e5.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of a TGM layer with channel combination. The kernels are applied to each input channel, $C _ { i n }$ , and a 1x1 convolution is applied to combine the $C _ { i n }$ input channels for each output channel, $C _ { o u t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "the channels from the previous layer. By design, the TGM kernel is positive and sums to 1. Adding the unconstrained 1x1 convolution adds non-linearity (using the ReLU activation function) to our layer and only adds $C _ { o u t } \\cdot C _ { i n }$ parameters. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fb72fdcdc270fde16eaa3ca0acd2b8ef4086171f297567ba689c79ba911da4c7.jpg",
+        "text": "$$\ns _ { i } = G _ { i } * w _ { i } = ( f _ { j } * K _ { i , j } ) * w _ { i } , \\ S = [ s _ { 1 } , s _ { 2 } \\ldots , s _ { C _ { o u t } } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            362,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We then stack the $s _ { i }$ representations along the channel axis to produce $S$ , the $C _ { o u t } \\times D \\times T$ -dim representation. This process is illustrated in Fig. 3. This method generalizes our approach to allow the layer to take input of $C _ { i n } \\times D \\times T$ and produce output of $C _ { o u t } \\times D \\times T$ . These layers can easily be stacked to learn a hierarchical representation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 VIDEO CNN MODELS WITH TGM LAYERS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to do activity detection which we define as making a per-frame (or per-segment) classification. Given a video, at each time step $t$ , we want to make the model decide which activity the frame corresponds to (including no-activity). As a baseline, we train a fully-connected layer that classifies each per-frame $D$ -dimensional vector, $v _ { t }$ . As multiple activities can occur at the same time, or no activities at all, we treat this as a mutli-label classification task. We minimize binary cross entropy: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/689f1a007120579a5dfd7551883d7131e1dc9204d9e21883a09a0757412356c8.jpg",
+        "text": "$$\nL ( v ) = \\sum _ { t , c } z _ { t , c } \\log ( p ( c | v _ { t } ) ) + ( 1 - z _ { t , c } ) \\log ( 1 - p ( c | v _ { t } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            571,
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+            607
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { t , c }$ is the ground truth label, 1 if activity $c$ is occurring at time $t$ and $p ( c | v _ { t } )$ is the output of our model for class $c$ at time $t$ . Fig. 4 shows an example CNN. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/cdd6c2d13bb16728b0fe94e3796a0d17a7ed1459c289733ad68ae2e40ee7b924.jpg",
+        "image_caption": [
+            "Figure 4: An overview of an example video CNN model with two TGM layers. It is able to handle videos with any length, because of its fully convolutional design. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 IMPLEMENTATION AND BASELINES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation We used I3D (Carreira & Zisserman, 2017) and the two-stream version of InceptionV3 (Szegedy et al., 2016) pretrained on Imagenet and Kinetics as our base per-frame CNNs. Our default $L$ setting used for the TGM layers as well as the other baselines was as follows: when using I3D segment features (collected at 3fps), the 1 layer models used $L = 1 5$ and the 3 layer models used $L = 5$ . When using InceptionV3 frame feature (collected at 8fps), the 1 layer models used $L = 3 0$ and the 3 layer models used $L = 1 0$ . These layers were attached on top of the base CNN, as described in Subsection 3.2. Please check the appendix for implementation and training details and results on other datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Baselines In order to confirm the advantages of our TGM layers, particularly against previous temporal models, we implemented several baselines. The first is (i) a standard per-frame classifier in which the prediction at each time-step only depends on a single feature vector with no contextual temporal information. We also used (ii) LSTMs on top of per-frame representations, which were popularly used to capture temporal information (Donahue et al., 2015). We train a bi-directional LSTM with 512 hidden units to make per-frame predictions. We also tried (iii) the fixed pyramid temporal max-pooling of level 3 (Ryoo et al., 2015). Finally, we compare our model against (iv) the model with standard temporal convolutional layers (i.e., 1-D convolution with a $D \\times L$ kernel) on top of per-frame representations. This is similar to the temporal conv. used in (Tran et al., 2018). Temporal lengths (i.e., $L$ ) of the 1-D conv. filters and the pooling windows were set to be identical to the TGM filters. That is, they capture the same temporal duration as TGMs. In all our experiments, we follow the standard evaluation setting of computing per-frame mean average precision (mAP) and report those values. We also compare to different versions of the TGM layer, (v) with a learned mixture of random temporal filters and (vi) with a learned mixture of fixed Gaussians. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition, we also tried the approach of combining our TGM layers with the recent super-event representations (Piergiovanni & Ryoo, 2018b). We concatenated the learned super-event representation with our representations from TGM layers. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 MULTITHUMOS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Dataset MultiTHUMOS (Yeung et al., 2015) is an extended version of the THUMOS (Jiang et al., 2014) dataset that densely annotates the continuous videos. The dataset consists of 65 different classes, compared to 20 in THUMOS, and contains on average 10.5 activities per video and 1.5 labels per frame and up to 25 activity instances in each video. This is in contrast to many other activity detection dataset such as ActivityNet (Heilbron et al., 2015), which only has on average ${ \\sim } 1$ activity per video. MultiTHUMOS consists of YouTube videos of various sport activities such as basketball games, volleyball games, weight lifting, and track and field. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We followed the standard MultiTHUMOS evaluation setting of measuring mAP based on per-frame annotations. There are 1010 validation videos and 1574 test videos. We used these continuous validation videos for the training of our models. We did not need to take advantage of the separate training set with segmented videos; even without them, we outperformed the state-of-the-arts. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results We compared baselines as well as multiple different versions of our architectures, shown in Table 1. The model with our TGM layers consistently outperformed baseline I3D (or InceptionV3) while using the same per-segment representations. Learning 3 TGM layers further improved the performances. On the other hand, we found that stacking multiple standard temporal convolutional layers does not improve performance, often performing worse than the baseline. While a single standard temporal conv. layer improves over the baseline, having multiple of them significantly increases the number of parameters to learn (Table 2) and we suspect that this was causing the overfitting with the limited amount of samples in the dataset. In Table 3, we compare the results of using a LSTM or temporal conv. with a similar number of parameters. This was done by making their temporal conv. filters to share values across multiple channels. These models result in nearly random performance, as they were not designed to cope with a small number of parameters. We also show results with a mixture of random (fixed) temporal filters and with a mixture of fixed Gaussians. These results confirm that (i) modeling the temporal structure as a learned Gaussian mixture is beneficial and that (ii) further learning the Gaussian distribution parameters is important. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3ee7541ec5fe0cfc304befd8d4baafe39f90c2c466c427c9319575a708552eb9.jpg",
+        "table_caption": [
+            "Table 1: Comparison of various architectures on MultiTHUMOS using both I3D per-segment and InceptionV3 per-frame features. We found that TGM layers with 1x1 convolution channel combination performed the best. Results are in mAP $\\%$ . Note that we use the same filter length for “Temporal Conv” and “TGM” models, as described in Section 4.1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">13D</td><td colspan=\"3\">InceptionV3</td></tr><tr><td>Spatial</td><td>Temporal</td><td>Two-Stream</td><td>Spatial</td><td>Temporal</td><td>Two-Stream</td></tr><tr><td>Baseline</td><td>22.3</td><td>25.0</td><td>29.7</td><td>13.6</td><td>14.1</td><td>15.2</td></tr><tr><td>Temporal Conv</td><td>32.5</td><td>35.5</td><td>38.4</td><td>15.2</td><td>15.5</td><td>15.8</td></tr><tr><td>3 Temporal Conv</td><td>20.4</td><td>23.4</td><td>24.4</td><td>5.3</td><td>6.1</td><td>6.5</td></tr><tr><td colspan=\"7\">TGM layers with grouped convolution</td></tr><tr><td>1 TGM</td><td>35.1</td><td>37.8</td><td>40.5</td><td>16.3</td><td>17.5</td><td>18.0</td></tr><tr><td>3 TGM</td><td>36.4</td><td>42.3</td><td>43.5</td><td>17.5</td><td>18.3</td><td>19.2</td></tr><tr><td colspan=\"7\">TGM layers with channel combination</td></tr><tr><td>1 TGM (soft)</td><td>35.2</td><td>37.9</td><td>40.2</td><td>17.2</td><td>17.6</td><td>18.4</td></tr><tr><td>1 TGM (1x1)</td><td>36.1</td><td>38.2</td><td>40.8</td><td>17.2</td><td>17.7</td><td>18.4</td></tr><tr><td>3 TGM (soft)</td><td>36.2</td><td>40.1</td><td>42.3</td><td>17.5</td><td>19.1</td><td>21.2</td></tr><tr><td>3 TGM (1x1)</td><td>37.2</td><td>42.1</td><td>44.3</td><td>17.9</td><td>19.3</td><td>22.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/cbe109a470a95d9b3b855dc56211db11ad9f33880fd71c3c75b7b657d7ef4c0d.jpg",
+        "table_caption": [
+            "Table 2: Additional number of parameters for models when added to the base architecture (e.g., I3D or Inception V3). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td> # of parameters</td></tr><tr><td>LSTM</td><td>10.5M</td></tr><tr><td>1 Temporal Conv</td><td>10.5M</td></tr><tr><td>3 Temporal Conv</td><td>31.5M</td></tr><tr><td>1 TGM Layer</td><td>10K</td></tr><tr><td>3 TGMLayers</td><td>100K</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/21a4edaa6f9113adff4bc63cde6845095b552a1d34563610ba7abe3e85574dec.jpg",
+        "table_caption": [
+            "Table 3: Comparison of previous methods with comparable number of parameters and random forms of our TGM layer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>mAP</td></tr><tr><td>LSTM with 100k parameters</td><td>6.5</td></tr><tr><td>Temporal Conv. with 1OOk parameters</td><td>7.3</td></tr><tr><td>TGM with random temporal filters</td><td>34.5</td></tr><tr><td>TGM with fixed Gaussians</td><td>38.5</td></tr><tr><td>Full TGM</td><td>44.3</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Learning multiple TGM layers with channel combination outperforms the grouped convolution version of TGM and all the baselines. We also experimented with a version using soft-attention weights to combine the TGM layer channels, in addition to our method (Fig. 3) of using 1x1 convolution followed by a ReLU (to gain non-linearity). We found that the 1x1 convolution performed better. We tested various number of Gaussian mixtures (i.e., output channels) and found that using 80 for the first and second layer and using 65 (i.e., number of classes) for the final layer performs best. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4 compares our model using TGM layers with multiple previous state-of-the-art approaches and baselines such as LSTM. Our approach meaningfully outperforms all previous approaches. Importantly, we are comparing our approach with different methods of capturing temporal information such as LSTMs and fixed temporal pyramid pooling while making them use the exactly same per-frame representations. We found that while all these methods capture some temporal information, the TGM layers provide the best performance. Further, combining the super-event representation (Piergiovanni & Ryoo, 2018b) with our TGM feature also benefited detection, confirming that our TGMs and super-events capture different aspects of the activity videos. In Fig. 5, we show an example of the various models predictions on a basketball video. We outperform the previous state-of-the-art performance (mAP) by $10 \\%$ (36.4 vs. 46.4). ",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 CHARADES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Dataset Charades (Sigurdsson et al., 2016b) is a large scale dataset with 9848 videos across 157 activity classes. These videos were recorded in home environments of the participants based on provided scripts. Each video contains on an average of 6.8 activity instances, and there are often complex activities co-occurring. The activities were mainly performed at home. For example, some activity classes are ‘preparing a meal’, ‘eating’, ‘sitting’, ‘cleaning’, etc. ",
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+    {
+        "type": "text",
+        "text": "In our experiments, we follow the original Charades detection setting (i.e., Charades v1 localize evaluation), which is the setting used in many previous approaches (Sigurdsson et al., 2016a; Xu ",
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+        "page_idx": 6
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+    {
+        "type": "image",
+        "img_path": "images/c0d256bcd8222f7ad03ebc79eb263ced37551f8c7850c775d1e2e262f032da84.jpg",
+        "image_caption": [
+            "Figure 5: Illustration of the temporal regions classified as various basketball activities from a basketball game video in MultiTHUMOS. Our TGM layers greatly improve performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 4: Performances of the state-of-the-art methods and our approach on MultiTHUMOS. Our approach meaningfully outperforms all previous results. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/39e1554da1ac99c74b3ef2bc9480a3af509a1aeab04f60cb98b4b7c40164e493.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>mAP</td></tr><tr><td>Two-stream (Yeung et al., 2015)</td><td>27.6</td></tr><tr><td>Two-stream + LSTM (Yeung et al., 2015)</td><td>28.1</td></tr><tr><td>Multi-LSTM (Yeung et al., 2015)</td><td>29.6</td></tr><tr><td>Predictive-corrective (Dave et al., 2017)</td><td>29.7</td></tr><tr><td>I3D baseline</td><td>29.7</td></tr><tr><td>I3D+LSTM</td><td>29.9</td></tr><tr><td>I3D + temporal pyramid</td><td>31.2</td></tr><tr><td>I3D + super-events (Piergiovanni &amp; Ryoo, 2018b)</td><td>36.4</td></tr><tr><td>I3D+our TGMs</td><td>44.3</td></tr><tr><td>I3D + super-events (Piergiovanni &amp; Ryoo,2018b) + our TGMs</td><td>46.4</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "et al., 2017; Piergiovanni & Ryoo, 2018b). This is the original setting more challenging than the Charades Challenge 2017 setting (whose evaluation server was no longer approving new account access), in the aspect that it uses less amount of training videos. ",
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+    },
+    {
+        "type": "text",
+        "text": "Results We compare our results with the state-of-the-arts in Table 5. To our knowledge, our method is obtaining the best known performance in the original localization setting of the Charades dataset. Notably, it is performing better than I3D that obtained the best competition performance, while using the same feature. Our method also outperforms standard temporal convolution, LSTMs, and fixed pyramid pooling, as well as the use of latent super-events. When setting $L = 3 0$ and using 3 TGM layers, our model is able to capture around 800 frames (about $\\pm 1 5$ seconds from each frame) of temporal information, significantly more than previous works (e.g., I3D only captures $\\pm 2$ seconds). ",
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We newly introduced the Temporal Gaussian Mixture (TGM) layer and demonstrated its effectiveness for multi-activity detection in continuous videos. Our layer is fully differentiable and trainable using standard backpropagation, designed to learn temporal structure. We were able to confirm that our layer performs superior to state-of-the-art methods on activity detection datasets including MultiTHUMOS and Charades, obtaining the best known performance. We also tested our approach with two more public video datasets, MLB-YouTube (Piergiovanni & Ryoo, 2018a) and AVA (Gu et al., 2017), and confirmed its advantage over the previous works in Appendix. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "J. K. Aggarwal and M. S. Ryoo. Human activity analysis: A review. ACM Computing Surveys, 43: 16:1–16:43, April 2011. ",
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+    {
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+        "table_caption": [
+            "Table 5: Per-frame mAP on Charades, evaluated with the ‘Charades v1 localize’ setting. I3D models are two-stream, using both RGB and optical flow inputs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>mAP</td></tr><tr><td>Predictive-corrective (Dave et al., 2017) Two-stream (Sigurdsson et al., 2016a) Two-stream+LSTM (Sigurdsson et al., 2016a)</td><td>8.9 8.94 9.6</td></tr><tr><td>R-C3D (Xu et al., 2017) Sigurdsson et al. (Sigurdsson et al., 2016a)</td><td>12.7 12.8</td></tr><tr><td>I3D baseline</td><td>17.2</td></tr><tr><td>I3D + 3 temporal conv.layers (L = 5) I3D + 3 temporal conv. layers (L = 30)</td><td>17.5</td></tr><tr><td>I3D +LSTM</td><td>12.5</td></tr><tr><td>I3D + fixed temporal pyramid</td><td>18.1</td></tr><tr><td></td><td>18.2</td></tr><tr><td>I3D + super-events (Piergiovanni &amp; Ryoo,2018b)</td><td>19.4</td></tr><tr><td>I3D +3 TGMs (L = 5)</td><td></td></tr><tr><td></td><td>20.6</td></tr><tr><td>I3D +3 TGMs (L = 30)</td><td>21.5</td></tr><tr><td>I3D +3 TGMs (L = 5) + super-events</td><td>21.8</td></tr><tr><td>I3D +3 TGMs (L = 3O) + super-events</td><td>22.3</td></tr></table>",
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As our base per-segment CNN, we use the I3D (Carreira & Zisserman, 2017) network pretrained on the ImageNet and Kinetics (Kay et al., 2017) datasets. I3D obtained state-of-the-art results on segmented video tasks, and this allows us to obtain reliable $v _ { t }$ . We also use two-stream version of InceptionV3 (Szegedy et al., 2016) pretrained on Imagenet and Kinetics as our base per-frame CNN, and compared them. We chose InceptionV3 as it is deeper than previous two-stream CNNs such as (Simonyan & Zisserman, 2014; Feichtenhofer et al., 2016). We extracted frames from the videos at 25 fps, computed TVL1 (Zach et al., 2007) optical flow, clipped to $[ - 2 0 , 2 0 ]$ . For InceptionV3, we computed features for every 3 frames (8 fps). For I3D, every frame was used as the input. I3D has a temporal stride of 8, resulting in 3 features per second (3 fps). ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We implemented our TGM layers as well as other baseline layers in PyTorch. Our default setting was as follows: for 3-layer models, we set $L = 1 0$ for frame-based features (i.e., InceptionV3) and $L = 5$ for segment-based features (i.e., I3D), as each segment already contains some temporal information. For 1-layer models, we set $L = 3 0$ for frame-based features and $L = 1 5$ for segmentbased features. We set $M = 1 6$ and $C _ { o u t } = 8 0 $ and $C _ { o u t } = 6 5$ for the last TGM layer. We found these values to work well on a held out portion of the training set of MultiTHUMOS. In all models, we used one fully-connected layer at the end to make the per-frame or per-segment classification. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We trained our models using the Adam (Kingma & Ba, 2014) optimizer with the learning rate set to 0.01. We decayed the learning rate by a factor of 10 after every 10 training epochs. We trained our models for 50 epochs. We plan to make all our source code and trained models publicly available once the paper is published. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B HYPERPARAMETER EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We conducted a set of experiments to compare the effects of the temporal duration, $L$ , number of Gaussians, $M$ , and the number of output channels, $C _ { o u t }$ . For these experiments, we only used the one-stream version of I3D with RGB inputs. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Effect of $L$ : In Table 6, we compare different values of $L$ . For these experiments, we use $M = 1 6$ and $C _ { o u t } = 1 6$ . We find that the 3-layer model with $L = 5$ performs the best. With I3D features, this allows the model to capture up to 8 seconds of information. The average activity in MultiTHUMOS is 3.3 seconds long and the maximum is 14.7 seconds long, and with this setting, the model is able to capture enough temporal context to perform well. Larger values of $L$ capture too much temporal information, but due to the Gaussian structure, it does not drastically harm performance. Figure 6 shows that even with longer kernels, the Gaussians learn to focus mostly on the center of the interval and capture the rough duration of the activities. Thus, having too long intervals does not drastically harm performance, which is in contrast to the standard 1-D convolution. Note that for Charades, the temporal kernels are learned to capture much longer temporal duration, as the average activity in charades is 12.8 seconds and larger values of $L$ perform better. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 illustrates examples of the learned TGM kernels of various lengths. The figure shows that the kernels focus on short temporal intervals on MultiTHUMOS even if we make the filters longer, as the activities are an average of 3.3 seconds long. On Charades, the TGM kernels learn to capture much longer intervals, as the activities are an average of 12.8 seconds long. We believe that this suggests TGMs are learning to capture information from the important necessary intervals. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Table 6, we also report the results of using a standard 1-D conv. layer with different $L$ values. The number of parameters in our TGM layer is independent of $L$ , however, with the standard 1-D conv. layer, the number of parameters increases as $L$ increases. We find that increasing $L$ with 1-D convolution helps for small values of $L$ , but for $L > 1 5$ , the performance drastically drops, while TGM layers only show a small decrease. ",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Effect of $M$ : In Table 7, we compare different values of $M$ . For these experiments, we set $L = 1 5$ and $C _ { o u t } = 1 6$ . We find that $M = 1 6$ performs best, suggesting that smaller values of $M$ restrict the possible temporal kernels too much. We also observe that larger values of $M$ performs slightly worse than $M = 1 6$ (but not much), likely because they introduce more parameters than needed. When $M$ and $L$ have similar values, it allows the model to learn a sufficient number of Gaussians and create a diverse range of temporal kernels. When $M$ is larger than $L$ , it results in learning a kernel similar to standard 1-D convolution. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/496248dff304ce2ec268cfe0b46b066bd66a0fc5ac12d42056caf34981b030b3.jpg",
+        "table_caption": [
+            "Table 6: Effect of $L$ on MultiTHUMOS and Charades using only RGB I3D features. Note that the 3 TGM layer models have larger temporal resolution than the 1 TGM layer models for the same values of $L$ . We also compare to using standard one-layer 1-D conv layer with different values of $L$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">MultiTHUMOS</td><td colspan=\"3\">Charades</td></tr><tr><td>1 Layer</td><td>3 Layers</td><td>1-D Conv</td><td>1 Layer</td><td>3 Layers</td><td>1-D Conv</td></tr><tr><td>I3DBaseline</td><td>22.3</td><td>=</td><td>=</td><td>15.3</td><td>=</td><td>=</td></tr><tr><td>L=3</td><td>30.2</td><td>31.7</td><td>26.6</td><td>15.5</td><td>16.1</td><td>15.5</td></tr><tr><td>L=5</td><td>32.5</td><td>37.2</td><td>28.3</td><td>15.7</td><td>17.8</td><td>16.3</td></tr><tr><td>L=10</td><td>34.5</td><td>35.4</td><td>31.7</td><td>16.1</td><td>18.2</td><td>16.6</td></tr><tr><td>L=15</td><td>36.1</td><td>34.1</td><td>32.5</td><td>17.5</td><td>18.6</td><td>16.8</td></tr><tr><td>L=30</td><td>32.5</td><td>33.9</td><td>26.5</td><td>18.1</td><td>18.9</td><td>12.1</td></tr><tr><td>L= 50</td><td>32.1</td><td>33.7</td><td>15.4</td><td>18.3</td><td>18.8</td><td>6.7</td></tr></table>",
+        "bbox": [
+            212,
+            154,
+            785,
+            292
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/5901547c2929b615c8bd5fc6732a59595b3ed3a1b209319ebe0be4d9a5947c37.jpg",
+        "table_caption": [
+            "Table 7: Comparison of various values of $M$ on MultiTHUMOS and Charades using RGB I3D features. For these experiments, 1 layer was used with $L = 1 5$ and $C _ { o u t } = 1 6$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MultiTHUMOS</td><td>Charades</td></tr><tr><td>M=2</td><td>27.8</td><td>15.5</td></tr><tr><td>M=4</td><td>33.1</td><td>16.2</td></tr><tr><td>M=8</td><td>34.8</td><td>17.5</td></tr><tr><td>M=16</td><td>36.1</td><td>17.5</td></tr><tr><td>M= 32</td><td>35.7</td><td>17.1</td></tr><tr><td>M= 64</td><td>35.8</td><td>17.3</td></tr></table>",
+        "bbox": [
+            187,
+            380,
+            473,
+            491
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/7dc521f452b9ce7c13d0e999b91925c57b7ad91123b45e675542632467e4a602.jpg",
+        "table_caption": [
+            "Table 8: Comparison of values of $C _ { o u t }$ on MultiTHUMOS and Charades using RGB I3D features. For these experiments, 1 layer was used with $L = 1 5$ and $M = 1 6$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MultiTHUMOS</td><td>Charades</td></tr><tr><td>Cout 1</td><td>33.5</td><td>16.2</td></tr><tr><td>Cout 4</td><td>34.2</td><td>17.4</td></tr><tr><td>Cout 8</td><td>35.5</td><td>17.5</td></tr><tr><td>Cout 16</td><td>36.1</td><td>17.5</td></tr><tr><td>Cout 32</td><td>36.0</td><td>17.2</td></tr><tr><td>Cout 64</td><td>36.1</td><td>17.4</td></tr><tr><td>Cout = 80</td><td>36.1</td><td>17.5</td></tr></table>",
+        "bbox": [
+            516,
+            372,
+            816,
+            497
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            525,
+            825,
+            553
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Effect of $C _ { o u t }$ : In Table 8, we compare different values of $C _ { o u t }$ . For these experiments, $L =$ 15, we used 1-layer and $M = 1 6$ . We find that $C _ { o u t }$ performs best when set to 16 or larger on these datasets. Larger values of $C _ { o u t }$ seem to capture redundant information, as it does not lower performance. ",
+        "bbox": [
+            173,
+            577,
+            825,
+            633
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/744a3159af7040ba3ca43621d47d5b12f250bda0c5cfd68bf678d41847d83116.jpg",
+        "image_caption": [
+            "Figure 6: Illustration of several learned TGM kernels. On MultiTHUMOS, it learns to focus on shorter intervals to capture shorter events. On Charades, the Gaussians have a larger $\\sigma$ value, resulting in filters that attend to longer temporal durations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            659,
+            823,
+            863
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b998f17e5d56ec0eab5405aa1323e9b9a726ecf049f10c25969a91979fea53b8.jpg",
+        "image_caption": [
+            "Figure 7: (a-c) Different forms of 1-D temporal convolutions which take a $D \\times T$ input and produces a $C \\times T$ output based on $C$ number of $D \\times L$ kernels: (a) the standard 1-D convolution, $\\mathbf { ( b ) }$ using Gaussian mixtures for 1-D convolution while sharing Gaussian mixtures across input channels, and (c) using $D$ different Gaussian mixtures for 1-D convolution. (d) Our TGM layer in its simplest form (i.e., 1-layer case) applying the $1 \\times L$ temporal kernel in a 2-D convolutional fashion, maintaining both time and feature axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            99,
+            813,
+            203
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a1a1342d875bc0c72fde175610ed02c50531ca14855a426d92b84f4adcfae2a.jpg",
+        "image_caption": [
+            "Figure 8: A temporal convolutional layer with channel combination similar to Fig. 3. The difference is that this layer does not learn Gaussian mixtures, but unconstrained 1-D temporal kernels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            340,
+            826,
+            482
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C COMPARISON OF DIFFERENT LAYER FORMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            579,
+            576,
+            595
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To confirm the various aspects of our design, we conducted experiments comparing different types of temporal convolution. In Fig. 7a we illustrate the standard 1-D convolution, taking $D \\times T$ input and producing a $C \\times T$ output, where $D$ is the number of input channels and $C$ is the number of output channels. In Fig. 7b, we illustrate the method of applying a Gaussian mixture kernel as 1-D convolution. Here, the Gaussian mixture kernel is shared by all $D$ input channels and we learn a $C$ number of such kernels. In Fig. 7c, we illustrate the approach of applying a Gaussian mixture kernel as 1-D convolution while learning $D$ different Gaussian mixtures. This is very similar to the standard 1-D convolution, except that the filter values are constrained to have the shape of Gaussian mixtures. ",
+        "bbox": [
+            173,
+            636,
+            825,
+            760
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Fig. 8 illustrates one more baseline. This is similar to our full TGM layer with the channelcombination described Fig. 3. However, in this baseline, instead of learning Gaussian mixtures, we learn $C _ { i n } \\cdot C _ { o u t }$ number of $1 \\times L$ kernels. The kernel values are left unconstrained. While the TGM layer has $2 \\cdot M + C _ { i n } \\cdot C _ { o u t } \\cdot M + C _ { i n } \\cdot C _ { o u t }$ parameters, this layer has $L \\cdot C _ { i n } \\cdot C _ { o u t } \\cdot M + C _ { i n } \\cdot C _ { o u t }$ , which is more than the TGM layer. ",
+        "bbox": [
+            173,
+            765,
+            825,
+            835
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Table 9, we compare the results of the various above-mentioned layers on MultiTHUMOS using RGB I3D features. We find that the Fig. 7b method performs poorly, while the Fig. 7c method slightly outperforms the standard 1-D convolution. The Fig. 8 method is slightly better than the standard 1-D convolution, but performs worse than Fig. 7c. However, none of these layers perform as well as our TGM layer, confirming that both the design of learning Gaussian mixtures and maintaining temporal channel axis are important for activity detection. ",
+        "bbox": [
+            173,
+            842,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/e45aa46cb538ad26d05b75cf673ca7b1029e8448f398c1b9e7188e0c3fe811e1.jpg",
+        "table_caption": [
+            "Table 9: Comparison of the different forms of temporal convolution on MultiTHUMOS using RGB I3D features. We set $L = 1 5$ and used 1 layer models for these experiments. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MultiTHUMOS</td></tr><tr><td>Standard 1-D Convolution (Fig. 7a)</td><td>32.5</td></tr><tr><td>The layer described in Fig.7b</td><td>28.6</td></tr><tr><td>The layer described in Fig. 7c</td><td>33.2</td></tr><tr><td>The layer described in Fig. 8</td><td>32.8</td></tr><tr><td>Our TGM Layer</td><td>36.1</td></tr></table>",
+        "bbox": [
+            308,
+            140,
+            689,
+            238
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/99b2898871e544d67d45bf16b97e65b0e77b16df16b200a2ad0c06f579924c1f.jpg",
+        "image_caption": [
+            "Figure 9: Examples of several of the activities in the MLB-YouTube dataset: (a) Pitch, (b) Hit, (c) Bunt, (d) Hit by pitch, (e) No activity. This shows the difficulty of this dataset, as the difference between hit and bunt, swing and no swing are very small. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            253,
+            823,
+            344
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D EXPERIMENTS ON ADDITIONAL DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            428,
+            566,
+            443
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.1 MLB-YOUTUBE DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
+            401,
+            479
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.1.1 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            496,
+            295,
+            511
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The MLB-YouTube dataset (Piergiovanni & Ryoo, 2018a) consists of 20 baseball games from the 2017 MLB post-season available on YouTube. This dataset consists of over 42 hours of video. For these experiments, we used the continuous video setting which have 2,126 1-2 minute long clips. Each clip is densely annotated with the baseball activities that occur. There are 8 activity classes: pitch, strike, ball, swing, hit, foul, hit by pitch, and bunt. Examples of some of these classes are shown in Fig. 9. Each continuous clip contains on average of 7.2 activities, giving a total of over 15,000 activity instances in the dataset. ",
+        "bbox": [
+            174,
+            525,
+            825,
+            621
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "What makes this dataset challenging is that the variation between classes is very small. In ActivityNet (Heilbron et al., 2015), for example, the difference between swimming and brushing hair is drastic. The background, motion, and even size of the person in the video is different. However, in broadcast baseball videos, the difference between a ball and a strike, or a swing and a bunt, are small. All actions are recorded from the same camera angle as we can confirm from Fig. 9. ",
+        "bbox": [
+            174,
+            627,
+            825,
+            695
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.1.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            723,
+            295,
+            738
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Table 10, we compare various approaches on this dataset. Our TGM layers improve over the baseline by ${ \\sim } 6 \\%$ (40.1 vs. 34.2). Additionally, we compare to methods using the super-event representation (Piergiovanni & Ryoo, 2018b), which previously achieved state-of-the-art performance on several activity detection datasets. On this dataset, our approach outperforms the super-event representation, and further the concatenation of our TGM representation with such super-event representation performs best by a significant margin $\\sim 1 3 \\%$ compared to the baseline). This suggests that TGMs and super-event capture different temporal information and are both useful to the detection task. ",
+        "bbox": [
+            174,
+            752,
+            825,
+            862
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We further find that using multiple, standard temporal convolution layers leads to worse performance, likely due to overfitting from the large number of parameters. While using multiple TGM layers improves performance, confirming that the Gaussian structure and sparsity constraint benefits model learning. ",
+        "bbox": [
+            176,
+            868,
+            823,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/16813ccfc3d26c5b3fb17c95c20dfe0de98b9da80a158b4d18f1a3f02ca22604.jpg",
+        "table_caption": [
+            "Table 10: Result mAP on the MLB-YouTube dataset using InceptionV3 and I3D to obtain features. Our TGM layers significantly outperform the baseline models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Spatial</td><td>Temporal</td><td>Two-stream</td></tr><tr><td>Random</td><td>13.4</td><td>13.4</td><td>13.4</td></tr><tr><td>InceptionV3</td><td>31.2</td><td>31.8</td><td>31.9</td></tr><tr><td>InceptionV3 +LSTM</td><td>32.1</td><td>33.5</td><td>34.1</td></tr><tr><td>InceptionV3 +1 temporal conv</td><td>32.8</td><td>34.4</td><td>35.2</td></tr><tr><td>InceptionV3 + 3 temporal conv</td><td>28.4</td><td>29.8</td><td>30.1</td></tr><tr><td>InceptionV3 + super-events</td><td>31.5</td><td>36.2</td><td>39.6</td></tr><tr><td>InceptionV3 +1TGM</td><td>32.4</td><td>36.3</td><td>37.4</td></tr><tr><td>InceptionV3+3 TGM</td><td>33.2</td><td>38.2</td><td>38.2</td></tr><tr><td>InceptionV3 + 3 TGM+super-events</td><td>34.6</td><td>42.4</td><td>42.9</td></tr><tr><td>I3D</td><td>33.8</td><td>35.1</td><td>34.2</td></tr><tr><td>I3D + LSTM</td><td>36.2</td><td>37.3</td><td>39.4</td></tr><tr><td>I3D +1 temporal conv</td><td>37.3</td><td>38.6</td><td>39.9</td></tr><tr><td>I3D + 3 temporal conv</td><td>32.4</td><td>34.6</td><td>35.6</td></tr><tr><td>I3D + super-events</td><td>38.7</td><td>38.6</td><td>39.1</td></tr><tr><td>I3D+1TGM</td><td>35.5</td><td>37.5</td><td>38.5</td></tr><tr><td>I3D+3 TGM</td><td>36.5</td><td>38.4</td><td>40.1</td></tr><tr><td>I3D +3 TGM+super-events</td><td>39.4</td><td>46.0</td><td>47.1</td></tr></table>",
+        "bbox": [
+            245,
+            138,
+            753,
+            412
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/21025fd127391862b655a328f8a33cb2e284c7a676da3889cef1be5f3ae4b399.jpg",
+        "table_caption": [
+            "Table 11: Results on AVA dataset with the temporal annotation-only setting (i.e., frame classification without using bounding box training labels). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>mAP</td></tr><tr><td>Random</td><td>2.65</td></tr><tr><td>I3D baseline</td><td>7.5</td></tr><tr><td>I3D + 3 temporal conv. layers</td><td>7.9</td></tr><tr><td>I3D+LSTM</td><td>7.8</td></tr><tr><td>I3D + super-events(Piergiovanni &amp; Ryoo,2018b)</td><td>9.8</td></tr><tr><td>I3D+1TGMs</td><td>11.2</td></tr><tr><td>I3D +3 TGMs</td><td>14.5</td></tr><tr><td>I3D +3 TGMs + super-events</td><td>14.9</td></tr></table>",
+        "bbox": [
+            299,
+            458,
+            697,
+            597
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D.2 AVA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            616,
+            251,
+            630
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D.2.1 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            641,
+            297,
+            655
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "AVA (Gu et al., 2017) is a large-scale video dataset containing of 80 atomic action classes in $5 7 \\mathrm { k }$ video clips. These clips are drawn from movies. Existing datasets, such as Charades, have very specific actions that depend on objects, such as holding a cup vs. holding a picture. In AVA, the actions are intentionally generic, such as sit, stand, hold, carry, etc. Further, the AVA dataset is annotated with both spatial and temporal locations of activities. Since we are interested in temporal activity detection, we follow the setting of Piergiovanni & Ryoo (2018b) and label each frame with the occurring activities while ignoring the spatial location. We evaluate performance following the same method as MultiTHUMOS, Charades and MLB-YouTube by measuring per-frame mAP. ",
+        "bbox": [
+            174,
+            665,
+            825,
+            775
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D.2.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            790,
+            295,
+            804
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Table 11, we present the results of our model. We again find that temporal convolution and LSTMs provide some benefit over the baseline, but TGM layers further improve performance. Again, combining the TGM, which captures local temporal structure, with super-events which capture global temporal structure, provides the best performance by $\\sim 7 . 4 \\%$ . ",
+        "bbox": [
+            174,
+            814,
+            823,
+            869
+        ],
+        "page_idx": 15
+    }
+]

parse/train/OJLaKwiXSbx/OJLaKwiXSbx.md

@@ -0,0 +1,318 @@
+# Align before Fuse: Vision and Language Representation Learning with Momentum Distillation
+
+Junnan Li, Ramprasaath R. Selvaraju, Akhilesh D. Gotmare Shafiq Joty, Caiming Xiong, Steven C.H. Hoi Salesforce Research {junnan.li,rselvaraju,akhilesh.gotmare,sjoty,shoi}@salesforce.com
+
+# Abstract
+
+Large-scale vision and language representation learning has shown promising improvements on various vision-language tasks. Most existing methods employ a transformer-based multimodal encoder to jointly model visual tokens (region-based image features) and word tokens. Because the visual tokens and word tokens are unaligned, it is challenging for the multimodal encoder to learn image-text interactions. In this paper, we introduce a contrastive loss to ALign the image and text representations BEfore Fusing (ALBEF) them through cross-modal attention, which enables more grounded vision and language representation learning. Unlike most existing methods, our method does not require bounding box annotations nor high-resolution images. To improve learning from noisy web data, we propose momentum distillation, a self-training method which learns from pseudo-targets produced by a momentum model. We provide a theoretical analysis of ALBEF from a mutual information maximization perspective, showing that different training tasks can be interpreted as different ways to generate views for an image-text pair. ALBEF achieves state-of-the-art performance on multiple downstream visionlanguage tasks. On image-text retrieval, ALBEF outperforms methods that are pre-trained on orders of magnitude larger datasets. On VQA and $\mathrm { \Delta N L V R ^ { 2 } }$ , ALBEF achieves absolute improvements of $2 . 3 7 \%$ and $3 . 8 4 \%$ compared to the state-ofthe-art, while enjoying faster inference speed. Code and models are available at https://github.com/salesforce/ALBEF.
+
+# 1 Introduction
+
+Vision-and-Language Pre-training (VLP) aims to learn multimodal representations from large-scale image-text pairs that can improve downstream Vision-and-Language $( \mathrm { V } { + } \mathrm { L } )$ tasks. Most existing VLP methods (e.g. LXMERT [1], UNITER [2], OSCAR $\pmb { \| 3 \| }$ ) rely on pre-trained object detectors to extract region-based image features, and employ a multimodal encoder to fuse the image features with word tokens. The multimodal encoder is trained to solve tasks that require joint understanding of image and text, such as masked language modeling (MLM) and image-text matching (ITM).
+
+While effective, this VLP framework suffers from several key limitations: (1) The image features and the word token embeddings reside in their own spaces, which makes it challenging for the multimodal encoder to learn to model their interactions; (2) The object detector is both annotation-expensive and compute-expensive, because it requires bounding box annotations during pre-training, and highresolution (e.g. $6 0 0 \times 1 0 0 0 )$ images during inference; (3) The widely used image-text datasets [4, 5] are collected from the web and are inherently noisy, and existing pre-training objectives such as MLM may overfit to the noisy text and degrade the model’s generalization performance.
+
+We propose ALign BEfore Fuse (ALBEF), a new VLP framework to address these limitations. We first encode the image and text independently with a detector-free image encoder and a text encoder. Then we use a multimodal encoder to fuse the image features with the text features through crossmodal attention. We introduce an intermediate image-text contrastive (ITC) loss on representations from the unimodal encoders, which serves three purposes: (1) it aligns the image features and the text features, making it easier for the multimodal encoder to perform cross-modal learning; (2) it improves the unimodal encoders to better understand the semantic meaning of images and texts; (3) it learns a common low-dimensional space to embed images and texts, which enables the image-text matching objective to find more informative samples through our contrastive hard negative mining.
+
+To improve learning under noisy supervision, we propose Momentum Distillation (MoD), a simple method which enables the model to leverage a larger uncurated web dataset. During training, we keep a momentum version of the model by taking the moving-average of its parameters, and use the momentum model to generate pseudo-targets as additional supervision. With MoD, the model is not penalized for producing other reasonable outputs that are different from the web annotation. We show that MoD not only improves pre-training, but also downstream tasks with clean annotations.
+
+We provide theoretical justifications on ALBEF from the perspective of mutual information maximization. Specifically, we show that ITC and MLM maximize a lower bound on the mutual information between different views of an image-text pair, where the views are generated by taking partial information from each pair. From this perspective, our momentum distillation can be interpreted as generating new views with semantically similar samples. Therefore, ALBEF learns vision-language representations that are invariant to semantic-preserving transformations.
+
+We demonstrate the effectiveness of ALBEF on various downstream $_ { \mathrm { V + L } }$ tasks including image-text retrieval, visual question answering, visual reasoning, visual entailment, and weakly-supervised visual grounding. ALBEF achieves substantial improvements over existing state-of-the-art methods. On image-text retrieval, it outperforms methods that are pre-trained on orders of magnitude larger datasets (CLIP $\pmb { \Vert 6 \Vert }$ and ALIGN $\bar { \mathbb { Z } } \bar { \mathbb { I } }$ ). On VQA and $\mathrm { \Delta N L V R ^ { 2 } }$ , it achieves absolute improvements of $\mathrm { \bar { 2 . 3 7 \% } }$ and $3 . 8 4 \%$ compared to the state-of-the-art method VILLA $\textcircled { 8 }$ , while enjoying much faster inference speed. We also provide quantitative and qualitative analysis on ALBEF using Grad-CAM $\bigstar \bigstar$ , which reveals its ability to perform accurate object, attribute and relationship grounding implicitly.
+
+# 2 Related Work
+
+# 2.1 Vision-Language Representation Learning
+
+Most existing work on vision-language representation learning fall into two categories. The first category focuses on modelling the interactions between image and text features with transformerbased multimodal encoders [10, 11, 12, 13, 1, 14, 15, 2, 3, 16, 8, 17, 18]. Methods in this category achieve superior performance on downstream $_ { \mathrm { V + L } }$ tasks that require complex reasoning over image and text (e.g. NLVR2 [19], VQA $\pmb { \mathbb { D } } \pmb { \mathbb { O } } \Vert$ ), but most of them require high-resolution input images and pre-trained object detectors. A recent method $\mathbb { \left| \mathbb { Z } \right\| }$ improves inference speed by removing the object detector, but results in lower performance. The second category focuses on learning separate unimodal encoders for image and text [22, 23, 6, 7]. The recent CLIP $\boxed { 6 }$ and ALIGN [7] perform pre-training on massive noisy web data using a contrastive loss, one of the most effective loss for representation learning [24, 25, 26, 27]. They achieve remarkable performance on image-text retrieval tasks, but lack the ability to model more complex interactions between image and text for other $_ { \mathrm { V + L } }$ tasks $\scriptstyle { \left[ \left[ 2 1 \right] \right] }$ .
+
+ALBEF unifies the two categories, leading to strong unimodal and multimodal representations with superior performance on both retrieval and reasoning tasks. Furthermore, ALBEF does not require object detectors, a major computation bottleneck for many existing methods [1, 2, 3, 8, 17].
+
+# 2.2 Knowledge Distillation
+
+Knowledge distillation $\pmb { \left[ \widetilde { \left| 2 8 \right| } \right] }$ aims to improve a student model’s performance by distilling knowledge from a teacher model, usually through matching the student’s prediction with the teacher’s. While most methods focus on distilling knowledge from a pre-trained teacher model [28, 29, 30, 31, 32], online distillation [33, 34] simultaneously trains multiple models and use their ensemble as the teacher. Our momentum distillation can be interpreted as a form of online self-distillation, where a temporal ensemble of the student model is used as the teacher. Similar ideas have been explored in semi-supervised learning $\pmb { \Vert 3 5 \Vert }$ , label noise learning $\textcircled { \left| 3 6 \right| }$ , and very recently in contrastive learning $\pmb { \mathbb { B 7 } }$ . Different from existing studies, we theoretically and experimentally show that momentum distillation is a generic learning algorithm that can improve the model’s performance on many $_ { \mathrm { V + L } }$ tasks.
+
+# 3 ALBEF Pre-training
+
+In this section, we first introduce the model architecture (Section $\textcircled { 3 . 1 }$ . Then we delineate the pretraining objectives (Section $\boxed { 3 . 2 }$ , followed by the proposed momentum distillation (Section $\bar { 3 } . 3 )$ Lastly we describe the pre-training datasets (Section $3 . { \overset { \cdot } { 4 } } )$ and implementation details (Section $\underline { { \vert 3 . 5 \vert } }$
+
+![](images/f52f65848bf1ec30d6820d94a14836f967aa2f875accbf396dc0167dd6581d06.jpg)  
+Figure 1: Illustration of ALBEF. It consists of an image encoder, a text encoder, and a multimodal encoder. We propose an image-text contrastive loss to align the unimodal representations of an image-text pair before fusion. An image-text matching loss (using in-batch hard negatives mined through contrastive similarity) and a masked-language-modeling loss are applied to learn multimodal interactions between image and text. In order to improve learning with noisy data, we generate pseudo-targets using the momentum model (a moving-average version of the base model) as additional supervision during training.
+
+# 3.1 Model Architecture
+
+As illustrated in Figure $\mathbb { L } ,$ ALBEF contains an image encoder, a text encoder, and a multimodal encoder. We use a 12-layer visual transformer ViT-B/16 $\mathbb { \left. 3 8 \right. }$ as the image encoder, and initialize it with weights pre-trained on ImageNet-1k from $\pmb { \mathbb { B } } \mathbf { \mathbb { 1 } }$ . An input image $I$ is encoded into a sequence of embeddings: $\{ \pmb { v } _ { \mathrm { c l s } } , \pmb { v } _ { 1 } , . . . , \pmb { v } _ { N } \}$ , where $v _ { \mathrm { c l s } }$ is the embedding of the [CLS] token. We use a 6-layer transformer $\textcircled { \ 3 9 } \textcircled { }$ for both the text encoder and the multimodal encoder. The text encoder is initialized using the first 6 layers of the $\mathbf { B E R T _ { b a s e } }$ $\textcircled { | 4 0 | }$ model, and the multimodal encoder is initialized using the last 6 layers of the $\mathbf { B E R T _ { b a s e } }$ . The text encoder transforms an input text $T$ into a sequence of embeddings $\{ \boldsymbol { w } _ { \mathrm { c l s } } , \boldsymbol { w } _ { 1 } , . . . , \boldsymbol { w } _ { N } \}$ , which is fed to the multimodal encoder. The image features are fused with the text features through cross attention at each layer of the multimodal encoder.
+
+# 3.2 Pre-training Objectives
+
+We pre-train ALBEF with three objectives: image-text contrastive learning (ITC) on the unimodal encoders, masked language modeling (MLM) and image-text matching (ITM) on the multimodal encoder. We improve ITM with online contrastive hard negative mining.
+
+Image-Text Contrastive Learning aims to learn better unimodal representations before fusion. It learns a similarity function $\boldsymbol { s } = \boldsymbol { g _ { v } } \big ( \boldsymbol { v } _ { \mathrm { c l s } } \big ) ^ { \top } \boldsymbol { g _ { w } } \big ( \boldsymbol { w } _ { \mathrm { c l s } } \big )$ , such that parallel image-text pairs have higher similarity scores. $g _ { v }$ and $g _ { w }$ are linear transformations that map the [CLS] embeddings to normalized lower-dimensional (256-d) representations. Inspired by MoCo $\pmb { \Vert 2 4 \Vert }$ , we maintain two queues to store the most recent $M$ image-text representations from the momentum unimodal encoders. The normalized features from the momentum encoders are denoted as $g _ { v } ^ { \prime } ( v _ { \mathrm { c l s } } ^ { \prime } )$ and $g _ { w } ^ { \prime } ( w _ { \mathrm { c l s } } ^ { \prime } )$ . We define $s ( I , T ) = g _ { v } ( \pmb { v } _ { \mathrm { c l s } } ) ^ { \top } g _ { w } ^ { \prime } ( \pmb { w } _ { \mathrm { c l s } } ^ { \prime } )$ and $s ( T , I ) = g _ { w } ( \pmb { w } _ { \mathrm { c l s } } ) ^ { \top } g _ { v } ^ { \prime } ( \pmb { v } _ { \mathrm { c l s } } ^ { \prime } )$ .
+
+For each image and text, we calculate the softmax-normalized image-to-text and text-to-image similarity as:
+
+$$
+p _ { m } ^ { \mathrm { i 2 t } } ( I ) = \frac { \exp ( s ( I , T _ { m } ) / \tau ) } { \sum _ { m = 1 } ^ { M } \exp ( s ( I , T _ { m } ) / \tau ) } , ~ p _ { m } ^ { \mathrm { t 2 i } } ( T ) = \frac { \exp ( s ( T , I _ { m } ) / \tau ) } { \sum _ { m = 1 } ^ { M } \exp ( s ( T , I _ { m } ) / \tau ) }
+$$
+
+where $\tau$ is a learnable temperature parameter. Let ${ \boldsymbol { y } } ^ { \mathrm { i 2 t } } ( I )$ and $\boldsymbol { y } ^ { \mathrm { t 2 i } } ( \boldsymbol { T } )$ denote the ground-truth one-hot similarity, where negative pairs have a probability of 0 and the positive pair has a probability of 1. The image-text contrastive loss is defined as the cross-entropy $\mathrm { H }$ between $\pmb { p }$ and $\textbf {  { y } }$ :
+
+$$
+\mathcal { L } _ { \mathrm { i t c } } = \frac { 1 } { 2 } \mathbb { E } _ { ( I , T ) \sim D } \big [ \mathrm { H } ( y ^ { \mathrm { i 2 t } } ( I ) , p ^ { \mathrm { i 2 t } } ( I ) ) + \mathrm { H } ( y ^ { \mathrm { t 2 i } } ( T ) , p ^ { \mathrm { t 2 i } } ( T ) ) \big ]
+$$
+
+![](images/9b40d579bf8714f23012f3ca9f7342a41936bb4b8471e3021de0d7abda97496a.jpg)  
+Figure 2: Examples of the pseudo-targets for MLM (1st row) and ITC (2nd row). The pseudo-targets can capture visual concepts that are not described by the ground-truth text (e.g. “beautiful waterfall”, “young woman”).
+
+Masked Language Modeling utilizes both the image and the contextual text to predict the masked words. We randomly mask out the input tokens with a probability of $15 \%$ and replace them with the special token [MASK]1. Let $\hat { T }$ denote a masked text, and $p ^ { \mathrm { m s k } } ( I , \hat { T } )$ denote the model’s predicted probability for a masked token. MLM minimizes a cross-entropy loss:
+
+$$
+\mathcal { L } _ { \mathrm { m l m } } = \mathbb { E } _ { ( I , \hat { T } ) \sim D } \mathrm { H } ( \pmb { y } ^ { \mathrm { m s k } } , \pmb { p } ^ { \mathrm { m s k } } ( I , \hat { T } ) )
+$$
+
+where $y ^ { \mathrm { m s k } }$ is a one-hot vocabulary distribution where the ground-truth token has a probability of 1.
+
+Image-Text Matching predicts whether a pair of image and text is positive (matched) or negative (not matched). We use the multimodal encoder’s output embedding of the [CLS] token as the joint representation of the image-text pair, and append a fully-connected (FC) layer followed by softmax to predict a two-class probability $p ^ { \mathrm { i t m } }$ . The ITM loss is:
+
+$$
+\mathcal { L } _ { \mathrm { i t m } } = \mathbb { E } _ { ( I , T ) \sim D } \mathrm { H } ( \pmb { y } ^ { \mathrm { i t m } } , \pmb { p } ^ { \mathrm { i t m } } ( I , T ) )
+$$
+
+where ${ \boldsymbol { y } } ^ { \mathrm { i t m } }$ is a 2-dimensional one-hot vector representing the ground-truth label.
+
+We propose a strategy to sample hard negatives for the ITM task with zero computational overhead. A negative image-text pair is hard if they share similar semantics but differ in fine-grained details. We use the contrastive similarity from Equation $\bigstar$ to find in-batch hard negatives. For each image in a mini-batch, we sample one negative text from the same batch following the contrastive similarity distribution, where texts that are more similar to the image have a higher chance to be sampled. Likewise, we also sample one hard negative image for each text.
+
+The full pre-training objective of ALBEF is:
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathrm { i t c } } + \mathcal { L } _ { \mathrm { m l m } } + \mathcal { L } _ { \mathrm { i t m } }
+$$
+
+# 3.3 Momentum Distillation
+
+The image-text pairs used for pre-training are mostly collected from the web and they tend to be noisy. Positive pairs are usually weakly-correlated: the text may contain words that are unrelated to the image, or the image may contain entities that are not described in the text. For ITC learning, negative texts for an image may also match the image’s content. For MLM, there may exist other words different from the annotation that describes the image equally well (or better). However, the one-hot labels for ITC and MLM penalize all negative predictions regardless of their correctness.
+
+To address this, we propose to learn from pseudo-targets generated by the momentum model. The momentum model is a continuously-evolving teacher which consists of exponential-moving-average versions of the unimodal and multimodal encoders. During training, we train the base model such that its predictions match the ones from the momentum model. Specifically, for ITC, we first compute the image-text similarity using features from the momentum unimodal encoders as $s ^ { \prime } ( I , T ) \stackrel { \cdot } { = } g _ { v } ^ { \prime } ( { \pmb v } _ { \mathrm { c l s } } ^ { \prime } ) ^ { \top } \breve { g } _ { w } ^ { \prime } ( { \pmb w } _ { \mathrm { c l s } } ^ { \prime } )$ and $s ^ { \prime } ( T , I ) \stackrel { \mathrm { ~ \tiny ~ = ~ } } { = } g _ { w } ^ { \prime } ( { \pmb w } _ { \mathrm { c l s } } ) ^ { \top } g _ { v } ^ { \prime } ( { \pmb v } _ { \mathrm { c l s } } ^ { \prime } )$ . Then we compute soft pseudotargets $q ^ { \mathrm { i 2 t } }$ and $q ^ { \mathrm { t 2 i } }$ by replacing $s$ with $s ^ { \prime }$ in Equation 1. The $\mathrm { I T C } _ { \mathrm { M o D } }$ loss is defined as:
+
+$$
+{ \mathcal { L } } _ { \mathrm { i t c } } ^ { \mathrm { m o d } } = ( 1 - \alpha ) { \mathcal { L } } _ { \mathrm { i t c } } + { \frac { \alpha } { 2 } } { \mathbb { E } } _ { ( I , T ) \sim D } \left[ \mathrm { K L } ( q ^ { \mathrm { i } 2 \mathrm { t } } ( I ) \parallel p ^ { \mathrm { i } 2 \mathrm { t } } ( I ) ) + \mathrm { K L } ( q ^ { \mathrm { t } 2 \mathrm { i } } ( T ) \parallel p ^ { \mathrm { t } 2 \mathrm { i } } ( T ) ) \right]
+$$
+
+Similarly, for MLM, let $\pmb q ^ { \mathrm { m s k } } ( I , \hat { T } )$ denote the momentum model’s prediction probability for the masked token, the $\mathbf { M L M } _ { \mathrm { M o D } }$ loss is:
+
+$$
+\mathcal { L } _ { \mathrm { m l m } } ^ { \mathrm { m o d } } = ( 1 - \alpha ) \mathcal { L } _ { \mathrm { m l m } } + \alpha \mathbb { E } _ { ( I , \hat { T } ) \sim D } \mathrm { K L } ( \pmb { q } ^ { \mathrm { m s k } } ( I , \hat { T } ) \parallel p ^ { \mathrm { m s k } } ( I , \hat { T } ) )
+$$
+
+In Figure $\bigstar$ we show examples of the top-5 candidates from the pseudo-targets, which effectively capture relevant words/texts for an image. More examples can be found in Appendix.
+
+We also apply MoD to the downstream tasks. The final loss for each task is a weighted combination of the original task’s loss and the KL-divergence between the model’s prediction and the pseudo-targets. For simplicity, we set the weight $\alpha = 0 . 4$ for all pre-training and downstream tasks 2.
+
+# 3.4 Pre-training Datasets
+
+Following UNITER [2], we construct our pre-training data using two web datasets (Conceptual Captions [4], SBU Captions $\pmb { \mathbb { B } } \mathbf { \| }$ ) and two in-domain datasets (COCO [41] and Visual Genome [42]). The total number of unique images is $4 . 0 \mathbf { M }$ , and the number of image-text pairs is 5.1M. To show that our method is scalable with larger-scale web data, we also include the much noisier Conceptual 12M dataset $\mathbb { \lVert \rVert 3 \rVert }$ , increasing the total number of images to $1 4 . 1 \mathrm { M } \big \sharp$ Details are in Appendix.
+
+# 3.5 Implementation Details
+
+Our model consists of a $\mathbf { B E R T _ { b a s e } }$ with 123.7M parameters and a ViT-B/16 with $8 5 . 8 \mathbf { M }$ parameters. We pre-train the model for 30 epochs using a batch size of 512 on 8 NVIDIA A100 GPUs. We use the AdamW $\pm \boxed { \boxed { 4 4 } }$ optimizer with a weight decay of 0.02. The learning rate is warmed-up to $1 e ^ { - 4 }$ in the first 1000 iterations, and decayed to $1 e ^ { - 5 }$ following a cosine schedule. During pre-training, we take random image crops of resolution $2 5 6 \times 2 5 6$ as input, and also apply RandAugment4 [45]. During fine-tuning, we increase the image resolution to $3 8 4 \times 3 8 4$ and interpolate the positional encoding of image patches following $\left[ \left[ 3 8 \right] \right]$ . The momentum parameter for updating the momentum model is set as 0.995, and the size of the queue used for image-text contrastive learning is set as 65,536. We linearly ramp-up the distillation weight $\alpha$ from 0 to 0.4 within the 1st epoch.
+
+# 4 A Mutual Information Maximization Perspective
+
+In this section, we provide an alternative perspective of ALBEF and show that it maximizes a lower bound on the mutual information (MI) between different “views” of an image-text pair. ITC, MLM, and MoD can be interpreted as different ways to generate the views.
+
+Formally, we define two random variables $a$ and $b$ as two different views of a data point. In selfsupervised learning $[ 1 2 4 , 1 2 5 , | 4 6 |$ , $a$ and $b$ are two augmentations of the same image. In vision-language representation learning, we consider $a$ and $b$ as different variations of an image-text pair that capture its semantic meaning. We aim to learn representations invariant to the change of view. This can be achieved by maximizing the MI between $a$ and $b$ . In practice, we maximize a lower bound on $\textstyle \mathbf { M } ( a , b )$ by minimizing the InfoNCE loss [47] defined as:
+
+$$
+\mathcal { L } _ { \mathrm { N C E } } = - \mathbb { E } _ { p ( a , b ) } \left[ \log \frac { \exp ( s ( a , b ) ) } { \sum _ { \hat { b } \in \hat { B } } \exp ( s ( a , \hat { b } ) ) } \right]
+$$
+
+where $s ( a , b )$ is a scoring function (e.g., a dot product between two representations), and $\hat { B }$ contains the positive sample $b$ and $| \hat { B } | - 1$ negative samples drawn from a proposal distribution.
+
+Our ITC loss with one-hot labels (Equation $^ { 2 ) }$ can be re-written as:
+
+$$
+\mathcal { L } _ { \mathrm { i t c } } = - \frac { 1 } { 2 } \mathbb { E } _ { p ( I , T ) } \big [ \log \frac { \exp ( s ( I , T ) / \tau ) } { \sum _ { m = 1 } ^ { M } \exp ( s ( I , T _ { m } ) / \tau ) } + \log \frac { \exp ( s ( T , I ) / \tau ) } { \sum _ { m = 1 } ^ { M } \exp ( s ( T , I _ { m } ) / \tau ) } \big ]
+$$
+
+Minimizing $\mathcal { L } _ { \mathrm { i t c } }$ can be seen as maximizing a symmetric version of InfoNCE. Hence, ITC considers the two individual modalities (i.e., $I$ and $T$ ) as the two views of an image-text pair, and trains the unimodal encoders to maximize the MI between the image and text views for the positive pairs.
+
+As shown in $[ \overline { { | 4 8 | } }$ , we can also interpret MLM as maximizing the MI between a masked word token and its masked context (i.e. image $^ +$ masked text). Specifically, we can re-write the MLM loss with one-hot labels (Equation $\textcircled { 3 }$ as
+
+$$
+\mathcal { L } _ { \mathrm { m l m } } = - \mathbb { E } _ { p ( I , \hat { T } ) } \big [ \log \frac { \exp ( \psi ( y ^ { \mathrm { m s k } } ) ^ { \top } f ( I , \hat { T } ) ) } { \sum _ { y \in \mathcal { V } } \exp ( \psi ( y ) ^ { \top } f ( I , \hat { T } ) ) } \big ]
+$$
+
+where $\psi ( y ) : \mathcal { V } \to \mathbb { R } ^ { d }$ is a lookup function in the multimodal encoder’s output layer that maps a word token $y$ into a vector and $\nu$ is the full vocabulary set, and $f ( I , { \hat { T } } )$ is a function that returns the final hidden state of the multimodal encoder corresponding to the masked context. Hence, MLM considers the two views of an image-text pair to be: (1) a randomly selected word token, and (2) the image $^ +$ the contextual text with that word masked.
+
+Both ITC and MLM generate views by taking partial information from an image-text pair, through either modality separation or word masking. Our momentum distillation can be considered as generating alternative views from the entire proposal distribution. Take $\mathrm { I T C } _ { \mathrm { M o D } }$ in Equation $6$ as an example, minimizing $\mathrm { K L } ( p ^ { \mathrm { i 2 t } } ( I ) , q ^ { \mathrm { i 2 t } } ( I ) )$ is equivalent to minimizing the following objective:
+
+$$
+- \sum _ { m } q _ { m } ^ { \mathrm { i } 2 \mathrm { t } } ( I ) \log p _ { m } ^ { \mathrm { i } 2 \mathrm { t } } ( I ) = - \sum _ { m } \frac { \exp ( s ^ { \prime } ( I , T _ { m } ) / \tau ) } { \sum _ { m = 1 } ^ { M } \exp ( s ^ { \prime } ( I , T _ { m } ) / \tau ) } \log \frac { \exp ( s ( I , T _ { m } ) / \tau ) } { \sum _ { m = 1 } ^ { M } \exp ( s ( I , T _ { m } ) / \tau ) }
+$$
+
+It maximizes $\mathbf { M I } ( I , T _ { m } )$ for texts that share similar semantic meaning with the image $I$ because those texts would have larger $q _ { m } ^ { \mathrm { i 2 t } } ( I )$ . Similarly, $\mathrm { I T C } _ { \mathrm { M o D } }$ also maximizes $\mathbf { M } \mathbf { I } ( I _ { m } , T )$ for images that are similar to $T$ . We can follow the same method to show that $\mathbf { M L M } _ { \mathrm { M o D } }$ generates alternative views $y ^ { \prime } \in \mathcal { V }$ for the masked word $y ^ { \mathrm { m s k } }$ , and maximizes the MI between $y ^ { \prime }$ and $( I , { \hat { T } } )$ . Therefore, our momentum distillation can be considered as performing data augmentation to the original views. The momentum model generates a diverse set of views that are absent in the original image-text pairs, and encourages the base model to learn representations that capture view-invariant semantic information.
+
+# 5 Downstream $\mathbf { V } { + } \mathbf { L }$ Tasks
+
+We adapt the pre-trained model to five downstream $_ { \mathrm { V + L } }$ tasks. We introduce each task and our fine-tuning strategy below. Details of the datasets and fine-tuning hyperparameters are in Appendix.
+
+Image-Text Retrieval contains two subtasks: image-to-text retrieval (TR) and text-to-image retrieval (IR). We evaluate ALBEF on the Flickr30K $\bar { \mathbb { E 9 } } \bar { \mathbb { I } }$ and COCO benchmarks, and fine-tune the pretrained model using the training samples from each dataset. For zero-shot retrieval on Flickr30K, we evaluate with the model fine-tuned on COCO. During fine-tuning, we jointly optimize the ITC loss (Equation $2 )$ and the ITM loss (Equation $\textcircled{4}$ . ITC learns an image-text scoring function based on similarity of unimodal features, whereas ITM models the fine-grained interaction between image and text to predict a matching score. Since the downstream datasets contain multiple texts for each image, we change the ground-truth label of ITC to consider multiple positives in the queue, where each positive has a ground-truth probability of 1/#positives. During inference, we first compute the feature similarity score $s _ { \mathrm { i t c } }$ for all image-text pairs. Then we take the top- $k$ candidates and calculate their ITM score $s _ { \mathrm { i t m } }$ for ranking. Because $k$ can be set to be very small, our inference speed is much faster than methods that require computing the ITM score for all image-text pairs [2, 3, 8].
+
+Visual Entailment (SNLI-VE5 [51]) is a fine-grained visual reasoning task to predict whether the relationship between an image and a text is entailment, neutral, or contradictory. We follow UNITER $\left[ \left[ 2 \right] \right]$ and consider VE as a three-way classification problem, and predict the class probabilities using a multi-layer perceptron (MLP) on the multimodal encoder’s representation of the [CLS] token.
+
+Visual Question Answering (VQA $\pmb { \mathbb { B } 2 } \mathbf { l }$ ) requires the model to predict an answer given an image and a question. Different from existing methods that formulate VQA as a multi-answer classification problem [53, 2], we consider VQA as an answer generation problem, similar to [54]. Specifically, we use a 6-layer transformer decoder to generate the answer. As shown in Figure 3a, the auto-regressive answer decoder receives the multimodal embeddings through cross attention, and a start-of-sequence token ([CLS]) is used as the decoder’s initial input token. Likewise, an end-of-sequence token ([SEP]) is appended to the end of decoder outputs which indicates the completion of generation.
+
+![](images/0b0dc7535563e91da490a221fda745edf632ed8624683f45f89b326da9d67c5b.jpg)  
+Figure 3: The model architecture for VQA and $\mathrm { \tt N L V R } ^ { 2 }$ . For VQA, we append an auto-regressive decoder to generate the answer given the image-question embeddings. For $\mathrm { \dot { N L V R } ^ { 2 } }$ , we replicate the transformer block within each layer of multimodal encoder to enable reasoning over two images.
+
+The answer decoder is initialized using the pre-trained weights from the multimodal encoder, and finetuned with a conditional language-modeling loss. For a fair comparison with existing methods, we constrain the decoder to only generate from the 3,128 candidate answers $ { \Vert 5 5 \Vert }$ during inference.
+
+Natural Language for Visual Reasoning (NLVR2 [19]) requires the model to predict whether a text describes a pair of images. We extend our multimodal encoder to enable reasoning over two images. As shown in Figure $3 { \mathrm { b } }$ , each layer of the multimodal encoder is replicated to have two consecutive transformer blocks, where each block contains a self-attention layer, a cross-attention layer, and a feed-forward layer (see Figure 1). The two blocks within each layer are initialized using the same pre-trained weights, and the two cross-attention layers share the same linear projection weights for the keys and values. During training, the two blocks receive two sets of image embeddings for the image pair. We append a MLP classifier on the multimodal encoder’s [CLS] representation for prediction.
+
+For $\mathrm { \Delta N L V R ^ { 2 } }$ , we perform an additional pre-training step to prepare the new multimodal encoder for encoding an image-pair. We design a text-assignment (TA) task as follows: given a pair of images and a text, the model needs to assign the text to either the first image, the second image, or none of them. We consider it as a three-way classification problem, and use a FC layer on the [CLS] representation to predict the assignment. We pre-train with TA for only 1 epoch using the 4M images (Section 3.4).
+
+Visual Grounding aims to localize the region in an image that corresponds to a specific textual description. We study the weakly-supervised setting, where no bounding box annotations are available. We perform experiments on the $\operatorname { R e f C O C O + } \mathbb { I }$ 56] dataset, and fine-tune the model using only imagetext supervision following the same strategy as image-text retrieval. During inference, we extend Grad-CAM $\pmb { \mathbb { Q } } \mathbf { \| }$ to acquire heatmaps, and use them to rank the detected proposals provided by $\mathbb { \lVert 5 3 \rVert }$
+
+# 6 Experiments
+
+# 6.1 Evaluation on the Proposed Methods
+
+First, we evaluate the effectiveness of the proposed methods (i.e. image-text contrastive learning, contrastive hard negative mining, and momentum distillation). Table $\bar { \mathbb { \perp } }$ shows the performance of the downstream tasks with different variants of our method. Compared to the baseline pre-training tasks $( \mathbf { M L M + I T M } )$ ), adding ITC substantially improves the pre-trained model’s performance across all tasks. The proposed hard negative mining improves ITM by finding more informative training samples. Furthermore, adding momentum distillation improves learning for both ITC (row 4), MLM (row 5), and on all downstream tasks (row 6). In the last row, we show that ALBEF can effectively leverage more noisy web data to improve the pre-training performance.
+
+Table 1: Evaluation of the proposed methods on four downstream $_ { \mathrm { V + L } }$ tasks. For text-retrieval (TR) and image-retrieval (IR), we report the average of $\mathbf { R } \ @ 1$ , $\mathbf { R } @ 5$ and $\mathrm { R @ 1 0 }$ . ITC: image-text contrastive learning. MLM: masked language modeling. $\mathrm { I T M } _ { \mathrm { h a r d } }$ : image-text matching with contrastive hard negative mining. MoD: momentum distillation. MoDDownstream: momentum distillation on downstream tasks.   
+
+<table><tr><td>#Pre-train Images</td><td>Training tasks</td><td>TR IR (flickr test)</td><td>SNLI-VE (test)</td><td>NLVR² (test-P)</td><td>VQA (test-dev)</td></tr><tr><td rowspan="6">4M</td><td>MLM+ ITM</td><td>93.96 88.55</td><td>77.06</td><td>77.51</td><td>71.40</td></tr><tr><td>ITC +MLM+ ITM</td><td>96.55 91.69</td><td>79.15</td><td>79.88</td><td>73.29</td></tr><tr><td>ITC + MLM + ITMhard</td><td>97.01 92.16</td><td>79.77</td><td>80.35</td><td>73.81</td></tr><tr><td>ITCMoD +MLM+ ITMhard</td><td>97.33 92.43</td><td>79.99</td><td>80.34</td><td>74.06</td></tr><tr><td>Full (ITCMoD + MLMMoD + ITMhard)</td><td>97.47 92.58</td><td>80.12</td><td>80.44</td><td>74.42</td></tr><tr><td>ALBEF (Full + MoDDownstream)</td><td>97.83 92.65</td><td>80.30</td><td>80.50</td><td>74.54</td></tr><tr><td>14M</td><td>ALBEF</td><td>98.70 94.07</td><td>80.91</td><td>83.14</td><td>75.84</td></tr></table>
+
+Table 2: Fine-tuned image-text retrieval results on Flickr30K and COCO datasets.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">#Pre-train Images</td><td colspan="6">Flickr30K (1K test set)</td><td colspan="6">MSCOCO (5K test set)</td></tr><tr><td colspan="2">TR</td><td colspan="4"></td><td colspan="2">TR</td><td colspan="2"></td><td colspan="2">IR</td></tr><tr><td>UNITER</td><td>4M</td><td>R@1 87.3</td><td>R@5 98.0</td><td>R@10 99.2</td><td>R@1 75.6</td><td>R@5</td><td>R@10 96.8</td><td>R@1 65.7</td><td>R@5</td><td>R@10 93.8</td><td>R@1 52.9</td><td>R@5 79.9</td><td>R@10</td></tr><tr><td>VILLA</td><td>4M</td><td>87.9</td><td>97.5</td><td>98.8</td><td>76.3</td><td>94.1</td><td>96.8</td><td>-</td><td>88.6 -</td><td></td><td></td><td></td><td>88.0</td></tr><tr><td>OSCAR</td><td>4M</td><td></td><td></td><td></td><td></td><td>94.2</td><td></td><td>70.0</td><td></td><td>- 95.5</td><td>- 54.0</td><td>-</td><td>- 88.5</td></tr><tr><td>ALIGN</td><td>1.2B</td><td>- 95.3</td><td>-</td><td>1</td><td>1</td><td>-</td><td>-</td><td>77.0</td><td>91.1</td><td></td><td></td><td>80.8</td><td></td></tr><tr><td></td><td></td><td></td><td>99.8</td><td>100.0</td><td>84.9</td><td>97.4</td><td>98.6</td><td></td><td>93.5</td><td>96.9</td><td>59.9</td><td>83.3</td><td>89.8</td></tr><tr><td>ALBEF</td><td>4M</td><td>94.3</td><td>99.4</td><td>99.8</td><td>82.8</td><td>96.7</td><td>98.4</td><td>73.1</td><td>91.4</td><td>96.0</td><td>56.8</td><td>81.5</td><td>89.2</td></tr><tr><td>ALBEF</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>14M</td><td>95.9</td><td>99.8</td><td></td><td></td><td></td><td></td><td></td><td></td><td>97.2</td><td>60.7</td><td>84.3</td><td>90.5</td></tr><tr><td></td><td></td><td></td><td></td><td>100.0</td><td>85.6</td><td>97.5</td><td>98.9</td><td>77.6</td><td>94.3</td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 3: Zero-shot image-text retrieval results on Flickr30K.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">#Pre-train Images</td><td colspan="6">Flickr30K (1K test set)</td></tr><tr><td colspan="3">TR</td><td colspan="3">IR</td></tr><tr><td></td><td></td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td>UNITER 四</td><td>4M</td><td>83.6</td><td>95.7</td><td>97.7</td><td>68.7</td><td>89.2</td><td>93.9</td></tr><tr><td>CLIP 回</td><td>400M</td><td>88.0</td><td>98.7</td><td>99.4</td><td>68.7</td><td>90.6</td><td>95.2</td></tr><tr><td>ALIGN </td><td>1.2B</td><td>88.6</td><td>98.7</td><td>99.7</td><td>75.7</td><td>93.8</td><td>96.8</td></tr><tr><td>ALBEF</td><td>4M</td><td>90.5</td><td>98.8</td><td>99.7</td><td>76.8</td><td>93.7</td><td>96.7</td></tr><tr><td>ALBEF</td><td>14M</td><td>94.1</td><td>99.5</td><td>99.7</td><td>82.8</td><td>96.3</td><td>98.1</td></tr></table>
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">VQA</td><td colspan="2">NLVR²</td><td colspan="2">SNLI-VE</td></tr><tr><td>test-dev</td><td>test-std</td><td>dev</td><td>test-P</td><td>val</td><td>test</td></tr><tr><td>VisualBERT[13]</td><td>70.80</td><td>71.00</td><td>67.40</td><td>67.00</td><td>-</td><td>-</td></tr><tr><td>VL-BERT 目</td><td>71.16</td><td>=</td><td>=</td><td>=</td><td>-</td><td></td></tr><tr><td>LXMERT[</td><td>72.42</td><td>72.54</td><td>74.90</td><td>74.50</td><td>=</td><td>=</td></tr><tr><td>12-in-1 目</td><td>73.15</td><td>-</td><td>-</td><td>78.87</td><td>=</td><td>76.95</td></tr><tr><td>UNITER [2]</td><td>72.70</td><td>72.91</td><td>77.18</td><td>77.85</td><td>78.59</td><td>78.28</td></tr><tr><td>VL-BART/T5 54</td><td>-</td><td>71.3</td><td>=</td><td>73.6</td><td>-</td><td>1</td></tr><tr><td>ViLT 四</td><td>70.94</td><td>-</td><td>75.24</td><td>76.21</td><td>-</td><td>-</td></tr><tr><td>OSCAR [3</td><td>73.16</td><td>73.44</td><td>78.07</td><td>78.36</td><td>=</td><td>1</td></tr><tr><td>VILLA 图</td><td>73.59</td><td>73.67</td><td>78.39</td><td>79.30</td><td>79.47</td><td>79.03</td></tr><tr><td>ALBEF (4M)</td><td>74.54</td><td>74.70</td><td>80.24</td><td>80.50</td><td>80.14</td><td>80.30</td></tr><tr><td>ALBEF (14M)</td><td>75.84</td><td>76.04</td><td>82.55</td><td>83.14</td><td>80.80</td><td>80.91</td></tr></table>
+
+Table 4: Comparison with state-of-the-art methods on downstream vision-language tasks.
+
+# 6.2 Evaluation on Image-Text Retrieval
+
+Table 2 and Table 3 report results on fine-tuned and zero-shot image-text retrieval, respectively. Our ALBEF achieves state-of-the-art performance, outperforming CLIP $\pmb { \Vert 6 \Vert }$ and ALIGN $[ [ 7 ]$ which are trained on orders of magnitude larger datasets. Given the considerable amount of improvement of ALBEF when the number of training images increases from 4M to 14M, we hypothesize that it has potential to further grow by training on larger-scale web image-text pairs.
+
+# 6.3 Evaluation on VQA, NLVR, and VE
+
+Table $\sharp$ reports the comparison with existing methods on other $_ { \mathrm { V + L } }$ understanding tasks. With 4M pre-training images, ALBEF already achieves state-of-the-art performance. With 14M pre-training images, ALBEF substantially outperforms existing methods, including methods that additionally use object tags $\pmb { \mathbb { B } } \|$ or adversarial data augmentation $\pmb { \mathbb { B } } ] \mathbf  l$ . Compared to VILLA $\pmb { \mathbb { B } } ] \mathbf  l$ , ALBEF achieves absolute improvements of $2 . 3 7 \%$ on VQA test-std, $3 . 8 4 \%$ on $\mathrm { \bar { N L V R ^ { 2 } } }$ test-P, and $1 . 8 8 \%$ on SNLI-VE test. Because ALBEF is detector-free and requires lower resolution images, it also enjoys much faster inference speed compared to most existing methods ${ \tt > } 1 0$ times faster than VILLA on ${ \mathrm { N L V R } } ^ { 2 }$ ).
+
+# 6.4 Weakly-supervised Visual Grounding
+
+Table 5 shows the results on $\operatorname { R e f C O C O + }$ , where ALBEF substantially outperforms existing methods [57, 58] (which use weaker text embeddings). The $\mathbf { A L B E F _ { i t c } }$ variant computes Grad-CAM
+
+<table><tr><td>Method</td><td>Val</td><td>TestA</td><td>TestB</td></tr><tr><td>ARN 四</td><td>32.78</td><td>34.35</td><td>32.13</td></tr><tr><td>CCL[ 国</td><td>34.29</td><td>36.91</td><td>33.56</td></tr><tr><td>ALBEFitc</td><td>51.58</td><td>60.09</td><td>40.19</td></tr><tr><td>ALBEFitm</td><td>58.46</td><td>65.89</td><td>46.25</td></tr></table>
+
+Table 5: Weakly-supervised visual grounding on $\mathrm { R e f C O C O + }$ [56] dataset.
+
+![](images/4251bcd75e9054e149f7bf987e5ff45a120e0a49fcf551018cb3e8b81dc823a8.jpg)  
+Figure 4: Grad-CAM visualization on the cross-attention maps in the 3rd layer of the multimodal encoder.
+
+Q: is this rice noodle soup? Q: what is to the right of A: yes the soup? A: chopsticks
+
+Q: what does the truck on Q: what is the man doing in the street? A: walking the left sell? A: ice cream
+
+![](images/73781611098cb0fe1f63a3e0377c69a21f1e4b8ed04a2ab157bfdfaac6789207.jpg)
+
+![](images/ceb6f744b4870218abb363b8856d9195fd5ee3876c9a2f27efa97528688f2950.jpg)  
+Figure 5: Grad-CAM visualizations on the cross-attention maps of the multimodal encoder for the VQA model. “a little girl holding a kitten next to a blue fence”   
+Figure 6: Grad-CAM visualizations on the cross-attention maps corresponding to individual words.
+
+visualizations on the self-attention maps in the last layer of the image encoder, where the gradients are acquired by maximizing the image-text similarity $s _ { \mathrm { i t c } }$ . The $\mathbf { A L B E F _ { i t m } }$ variant computes Grad-CAM on the cross-attention maps in the 3rd layer of the multimodal encoder (which is a layer specialized in grounding), where the gradients are acquired by maximizing the image-text matching score $s _ { \mathrm { i t m } }$ Figure 4 provides a few visualizations. More analysis is in Appendix.
+
+We provide the Grad-CAM visualizations for VQA in Figure $\textcircled{5}$ As can be seen in Appendix, the Grad-CAM visualizations from ALBEF are highly correlated with where humans would look when making decisions. In Figure $6 ,$ we show per-word visualizations for COCO. Notice how our model not only grounds objects, but also their attributes and relationships.
+
+# 6.5 Ablation Study
+
+Table $\boxed { 6 }$ studies the effect of various design choices on image-text retrieval. Since we use $s _ { \mathrm { i t c } }$ to filter top- $k$ candidates during inference, we vary $k$ and report its effect. In general, the ranking result acquired by $s _ { \mathrm { i t m } }$ is not sensitive to changes
+
+<table><tr><td rowspan="2">Flickr30K</td><td colspan="4">w/ hard negs</td><td rowspan="2">w/o hard negs k =128</td></tr><tr><td>Sitc</td><td>k =16</td><td>k =128</td><td>k=256</td></tr><tr><td>TR</td><td>97.30</td><td>98.60</td><td>98.57</td><td>98.57</td><td>98.22 (-0.35)</td></tr><tr><td>IR</td><td>90.95</td><td>93.64</td><td>93.99</td><td>93.95</td><td>93.68 (-0.31)</td></tr></table>
+
+Table 6: Ablation study on fine-tuned image-text retrieval. The average recall on the test set is reported. We use $s _ { \mathrm { i t c } }$ to filter top- $k$ candidates and calculate their $s _ { \mathrm { i t m } }$ score for ranking.
+
+in $k$ . We also validate the effect of hard negative mining in the last column.
+
+Table $^ { 7 }$ studies the effect of textassignment (TA) pre-training and parameter sharing on $\mathrm { \tt N L V R } ^ { \mathrm { \bar { 2 } } }$ . We examine three strategies: (1) the two mutimodal blocks share all parameters, (2) only the cross
+
+<table><tr><td>NLVR²</td><td colspan="3">w/TA share all share CA</td><td colspan="3">w/o TA share all share CA</td></tr><tr><td>dev</td><td>82.13</td><td>82.55</td><td>no share 81.93</td><td>80.52</td><td>80.28</td><td>no share 77.84</td></tr><tr><td>test-P</td><td>82.36</td><td>83.14</td><td>82.85</td><td>81.29</td><td>80.45</td><td>77.58</td></tr></table>
+
+Table 7: Ablation study on NLVR2.
+
+attention (CA) layers are shared, (3) no sharing. Without TA, sharing the entire block has better performance. With TA to pre-train the model for image-pair, sharing CA leads to the best performance.
+
+# 7 Conclusion and Social Impacts
+
+This paper proposes ALBEF, a new framework for vision-language representation learning. ALBEF first aligns the unimodal image representation and text representation before fusing them with a multimodal encoder. We theoretically and experimentally verify the effectiveness of the proposed image-text contrastive learning and momentum distillation. Compared to existing methods, ALBEF offers better performance and faster inference speed on multiple downstream $_ { \mathrm { V + L } }$ tasks.
+
+While our paper shows promising results on vision-language representation learning, additional analysis on the data and the model is necessary before deploying it in practice, because web data may contain unintended private information, unsuitable images, or harmful texts, and only optimizing accuracy may have unwanted social implications.
+
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parse/train/OJLaKwiXSbx/OJLaKwiXSbx_content_list.json

@@ -0,0 +1,1356 @@
+[
+    {
+        "type": "text",
+        "text": "Align before Fuse: Vision and Language Representation Learning with Momentum Distillation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Junnan Li, Ramprasaath R. Selvaraju, Akhilesh D. Gotmare Shafiq Joty, Caiming Xiong, Steven C.H. Hoi Salesforce Research {junnan.li,rselvaraju,akhilesh.gotmare,sjoty,shoi}@salesforce.com ",
+        "bbox": [
+            222,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            295,
+            535,
+            310
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Large-scale vision and language representation learning has shown promising improvements on various vision-language tasks. Most existing methods employ a transformer-based multimodal encoder to jointly model visual tokens (region-based image features) and word tokens. Because the visual tokens and word tokens are unaligned, it is challenging for the multimodal encoder to learn image-text interactions. In this paper, we introduce a contrastive loss to ALign the image and text representations BEfore Fusing (ALBEF) them through cross-modal attention, which enables more grounded vision and language representation learning. Unlike most existing methods, our method does not require bounding box annotations nor high-resolution images. To improve learning from noisy web data, we propose momentum distillation, a self-training method which learns from pseudo-targets produced by a momentum model. We provide a theoretical analysis of ALBEF from a mutual information maximization perspective, showing that different training tasks can be interpreted as different ways to generate views for an image-text pair. ALBEF achieves state-of-the-art performance on multiple downstream visionlanguage tasks. On image-text retrieval, ALBEF outperforms methods that are pre-trained on orders of magnitude larger datasets. On VQA and $\\mathrm { \\Delta N L V R ^ { 2 } }$ , ALBEF achieves absolute improvements of $2 . 3 7 \\%$ and $3 . 8 4 \\%$ compared to the state-ofthe-art, while enjoying faster inference speed. Code and models are available at https://github.com/salesforce/ALBEF. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            604,
+            312,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Vision-and-Language Pre-training (VLP) aims to learn multimodal representations from large-scale image-text pairs that can improve downstream Vision-and-Language $( \\mathrm { V } { + } \\mathrm { L } )$ tasks. Most existing VLP methods (e.g. LXMERT [1], UNITER [2], OSCAR $\\pmb { \\| 3 \\| }$ ) rely on pre-trained object detectors to extract region-based image features, and employ a multimodal encoder to fuse the image features with word tokens. The multimodal encoder is trained to solve tasks that require joint understanding of image and text, such as masked language modeling (MLM) and image-text matching (ITM). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While effective, this VLP framework suffers from several key limitations: (1) The image features and the word token embeddings reside in their own spaces, which makes it challenging for the multimodal encoder to learn to model their interactions; (2) The object detector is both annotation-expensive and compute-expensive, because it requires bounding box annotations during pre-training, and highresolution (e.g. $6 0 0 \\times 1 0 0 0 )$ images during inference; (3) The widely used image-text datasets [4, 5] are collected from the web and are inherently noisy, and existing pre-training objectives such as MLM may overfit to the noisy text and degrade the model’s generalization performance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose ALign BEfore Fuse (ALBEF), a new VLP framework to address these limitations. We first encode the image and text independently with a detector-free image encoder and a text encoder. Then we use a multimodal encoder to fuse the image features with the text features through crossmodal attention. We introduce an intermediate image-text contrastive (ITC) loss on representations from the unimodal encoders, which serves three purposes: (1) it aligns the image features and the text features, making it easier for the multimodal encoder to perform cross-modal learning; (2) it improves the unimodal encoders to better understand the semantic meaning of images and texts; (3) it learns a common low-dimensional space to embed images and texts, which enables the image-text matching objective to find more informative samples through our contrastive hard negative mining. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To improve learning under noisy supervision, we propose Momentum Distillation (MoD), a simple method which enables the model to leverage a larger uncurated web dataset. During training, we keep a momentum version of the model by taking the moving-average of its parameters, and use the momentum model to generate pseudo-targets as additional supervision. With MoD, the model is not penalized for producing other reasonable outputs that are different from the web annotation. We show that MoD not only improves pre-training, but also downstream tasks with clean annotations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We provide theoretical justifications on ALBEF from the perspective of mutual information maximization. Specifically, we show that ITC and MLM maximize a lower bound on the mutual information between different views of an image-text pair, where the views are generated by taking partial information from each pair. From this perspective, our momentum distillation can be interpreted as generating new views with semantically similar samples. Therefore, ALBEF learns vision-language representations that are invariant to semantic-preserving transformations. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate the effectiveness of ALBEF on various downstream $_ { \\mathrm { V + L } }$ tasks including image-text retrieval, visual question answering, visual reasoning, visual entailment, and weakly-supervised visual grounding. ALBEF achieves substantial improvements over existing state-of-the-art methods. On image-text retrieval, it outperforms methods that are pre-trained on orders of magnitude larger datasets (CLIP $\\pmb { \\Vert 6 \\Vert }$ and ALIGN $\\bar { \\mathbb { Z } } \\bar { \\mathbb { I } }$ ). On VQA and $\\mathrm { \\Delta N L V R ^ { 2 } }$ , it achieves absolute improvements of $\\mathrm { \\bar { 2 . 3 7 \\% } }$ and $3 . 8 4 \\%$ compared to the state-of-the-art method VILLA $\\textcircled { 8 }$ , while enjoying much faster inference speed. We also provide quantitative and qualitative analysis on ALBEF using Grad-CAM $\\bigstar \\bigstar$ , which reveals its ability to perform accurate object, attribute and relationship grounding implicitly. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Vision-Language Representation Learning ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Most existing work on vision-language representation learning fall into two categories. The first category focuses on modelling the interactions between image and text features with transformerbased multimodal encoders [10, 11, 12, 13, 1, 14, 15, 2, 3, 16, 8, 17, 18]. Methods in this category achieve superior performance on downstream $_ { \\mathrm { V + L } }$ tasks that require complex reasoning over image and text (e.g. NLVR2 [19], VQA $\\pmb { \\mathbb { D } } \\pmb { \\mathbb { O } } \\Vert$ ), but most of them require high-resolution input images and pre-trained object detectors. A recent method $\\mathbb { \\left| \\mathbb { Z } \\right\\| }$ improves inference speed by removing the object detector, but results in lower performance. The second category focuses on learning separate unimodal encoders for image and text [22, 23, 6, 7]. The recent CLIP $\\boxed { 6 }$ and ALIGN [7] perform pre-training on massive noisy web data using a contrastive loss, one of the most effective loss for representation learning [24, 25, 26, 27]. They achieve remarkable performance on image-text retrieval tasks, but lack the ability to model more complex interactions between image and text for other $_ { \\mathrm { V + L } }$ tasks $\\scriptstyle { \\left[ \\left[ 2 1 \\right] \\right] }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "ALBEF unifies the two categories, leading to strong unimodal and multimodal representations with superior performance on both retrieval and reasoning tasks. Furthermore, ALBEF does not require object detectors, a major computation bottleneck for many existing methods [1, 2, 3, 8, 17]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 Knowledge Distillation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation $\\pmb { \\left[ \\widetilde { \\left| 2 8 \\right| } \\right] }$ aims to improve a student model’s performance by distilling knowledge from a teacher model, usually through matching the student’s prediction with the teacher’s. While most methods focus on distilling knowledge from a pre-trained teacher model [28, 29, 30, 31, 32], online distillation [33, 34] simultaneously trains multiple models and use their ensemble as the teacher. Our momentum distillation can be interpreted as a form of online self-distillation, where a temporal ensemble of the student model is used as the teacher. Similar ideas have been explored in semi-supervised learning $\\pmb { \\Vert 3 5 \\Vert }$ , label noise learning $\\textcircled { \\left| 3 6 \\right| }$ , and very recently in contrastive learning $\\pmb { \\mathbb { B 7 } }$ . Different from existing studies, we theoretically and experimentally show that momentum distillation is a generic learning algorithm that can improve the model’s performance on many $_ { \\mathrm { V + L } }$ tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 ALBEF Pre-training ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            845,
+            379,
+            863
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first introduce the model architecture (Section $\\textcircled { 3 . 1 }$ . Then we delineate the pretraining objectives (Section $\\boxed { 3 . 2 }$ , followed by the proposed momentum distillation (Section $\\bar { 3 } . 3 )$ Lastly we describe the pre-training datasets (Section $3 . { \\overset { \\cdot } { 4 } } )$ and implementation details (Section $\\underline { { \\vert 3 . 5 \\vert } }$ ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/f52f65848bf1ec30d6820d94a14836f967aa2f875accbf396dc0167dd6581d06.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of ALBEF. It consists of an image encoder, a text encoder, and a multimodal encoder. We propose an image-text contrastive loss to align the unimodal representations of an image-text pair before fusion. An image-text matching loss (using in-batch hard negatives mined through contrastive similarity) and a masked-language-modeling loss are applied to learn multimodal interactions between image and text. In order to improve learning with noisy data, we generate pseudo-targets using the momentum model (a moving-average version of the base model) as additional supervision during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Model Architecture ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            420,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure $\\mathbb { L } ,$ ALBEF contains an image encoder, a text encoder, and a multimodal encoder. We use a 12-layer visual transformer ViT-B/16 $\\mathbb { \\left. 3 8 \\right. }$ as the image encoder, and initialize it with weights pre-trained on ImageNet-1k from $\\pmb { \\mathbb { B } } \\mathbf { \\mathbb { 1 } }$ . An input image $I$ is encoded into a sequence of embeddings: $\\{ \\pmb { v } _ { \\mathrm { c l s } } , \\pmb { v } _ { 1 } , . . . , \\pmb { v } _ { N } \\}$ , where $v _ { \\mathrm { c l s } }$ is the embedding of the [CLS] token. We use a 6-layer transformer $\\textcircled { \\ 3 9 } \\textcircled { }$ for both the text encoder and the multimodal encoder. The text encoder is initialized using the first 6 layers of the $\\mathbf { B E R T _ { b a s e } }$ $\\textcircled { | 4 0 | }$ model, and the multimodal encoder is initialized using the last 6 layers of the $\\mathbf { B E R T _ { b a s e } }$ . The text encoder transforms an input text $T$ into a sequence of embeddings $\\{ \\boldsymbol { w } _ { \\mathrm { c l s } } , \\boldsymbol { w } _ { 1 } , . . . , \\boldsymbol { w } _ { N } \\}$ , which is fed to the multimodal encoder. The image features are fused with the text features through cross attention at each layer of the multimodal encoder. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Pre-training Objectives ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            585,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We pre-train ALBEF with three objectives: image-text contrastive learning (ITC) on the unimodal encoders, masked language modeling (MLM) and image-text matching (ITM) on the multimodal encoder. We improve ITM with online contrastive hard negative mining. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Image-Text Contrastive Learning aims to learn better unimodal representations before fusion. It learns a similarity function $\\boldsymbol { s } = \\boldsymbol { g _ { v } } \\big ( \\boldsymbol { v } _ { \\mathrm { c l s } } \\big ) ^ { \\top } \\boldsymbol { g _ { w } } \\big ( \\boldsymbol { w } _ { \\mathrm { c l s } } \\big )$ , such that parallel image-text pairs have higher similarity scores. $g _ { v }$ and $g _ { w }$ are linear transformations that map the [CLS] embeddings to normalized lower-dimensional (256-d) representations. Inspired by MoCo $\\pmb { \\Vert 2 4 \\Vert }$ , we maintain two queues to store the most recent $M$ image-text representations from the momentum unimodal encoders. The normalized features from the momentum encoders are denoted as $g _ { v } ^ { \\prime } ( v _ { \\mathrm { c l s } } ^ { \\prime } )$ and $g _ { w } ^ { \\prime } ( w _ { \\mathrm { c l s } } ^ { \\prime } )$ . We define $s ( I , T ) = g _ { v } ( \\pmb { v } _ { \\mathrm { c l s } } ) ^ { \\top } g _ { w } ^ { \\prime } ( \\pmb { w } _ { \\mathrm { c l s } } ^ { \\prime } )$ and $s ( T , I ) = g _ { w } ( \\pmb { w } _ { \\mathrm { c l s } } ) ^ { \\top } g _ { v } ^ { \\prime } ( \\pmb { v } _ { \\mathrm { c l s } } ^ { \\prime } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For each image and text, we calculate the softmax-normalized image-to-text and text-to-image similarity as: ",
+        "bbox": [
+            173,
+            763,
+            823,
+            792
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d6be1e6b539c4157190671e41f194b968b754886e3bef27e4fdd8a675cd5928.jpg",
+        "text": "$$\np _ { m } ^ { \\mathrm { i 2 t } } ( I ) = \\frac { \\exp ( s ( I , T _ { m } ) / \\tau ) } { \\sum _ { m = 1 } ^ { M } \\exp ( s ( I , T _ { m } ) / \\tau ) } , ~ p _ { m } ^ { \\mathrm { t 2 i } } ( T ) = \\frac { \\exp ( s ( T , I _ { m } ) / \\tau ) } { \\sum _ { m = 1 } ^ { M } \\exp ( s ( T , I _ { m } ) / \\tau ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            248,
+            796,
+            750,
+            835
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is a learnable temperature parameter. Let ${ \\boldsymbol { y } } ^ { \\mathrm { i 2 t } } ( I )$ and $\\boldsymbol { y } ^ { \\mathrm { t 2 i } } ( \\boldsymbol { T } )$ denote the ground-truth one-hot similarity, where negative pairs have a probability of 0 and the positive pair has a probability of 1. The image-text contrastive loss is defined as the cross-entropy $\\mathrm { H }$ between $\\pmb { p }$ and $\\textbf {  { y } }$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2189a82640615d35869d5aff34b6eee7569b7c300585c8320b8b7b3b386750d8.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { i t c } } = \\frac { 1 } { 2 } \\mathbb { E } _ { ( I , T ) \\sim D } \\big [ \\mathrm { H } ( y ^ { \\mathrm { i 2 t } } ( I ) , p ^ { \\mathrm { i 2 t } } ( I ) ) + \\mathrm { H } ( y ^ { \\mathrm { t 2 i } } ( T ) , p ^ { \\mathrm { t 2 i } } ( T ) ) \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b40d579bf8714f23012f3ca9f7342a41936bb4b8471e3021de0d7abda97496a.jpg",
+        "image_caption": [
+            "Figure 2: Examples of the pseudo-targets for MLM (1st row) and ITC (2nd row). The pseudo-targets can capture visual concepts that are not described by the ground-truth text (e.g. “beautiful waterfall”, “young woman”). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            89,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Masked Language Modeling utilizes both the image and the contextual text to predict the masked words. We randomly mask out the input tokens with a probability of $15 \\%$ and replace them with the special token [MASK]1. Let $\\hat { T }$ denote a masked text, and $p ^ { \\mathrm { m s k } } ( I , \\hat { T } )$ denote the model’s predicted probability for a masked token. MLM minimizes a cross-entropy loss: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b44d3f223a60a0a0f0308f63fbad4d934249d8f57a34c0b2c65ea041ded7fc40.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { m l m } } = \\mathbb { E } _ { ( I , \\hat { T } ) \\sim D } \\mathrm { H } ( \\pmb { y } ^ { \\mathrm { m s k } } , \\pmb { p } ^ { \\mathrm { m s k } } ( I , \\hat { T } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            363,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $y ^ { \\mathrm { m s k } }$ is a one-hot vocabulary distribution where the ground-truth token has a probability of 1. ",
+        "bbox": [
+            176,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Image-Text Matching predicts whether a pair of image and text is positive (matched) or negative (not matched). We use the multimodal encoder’s output embedding of the [CLS] token as the joint representation of the image-text pair, and append a fully-connected (FC) layer followed by softmax to predict a two-class probability $p ^ { \\mathrm { i t m } }$ . The ITM loss is: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92d904631f26e7eccfc67d62b85a2b0e413d468492c2ede127243cff1768d28f.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { i t m } } = \\mathbb { E } _ { ( I , T ) \\sim D } \\mathrm { H } ( \\pmb { y } ^ { \\mathrm { i t m } } , \\pmb { p } ^ { \\mathrm { i t m } } ( I , T ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            467,
+            620,
+            487
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\boldsymbol { y } } ^ { \\mathrm { i t m } }$ is a 2-dimensional one-hot vector representing the ground-truth label. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose a strategy to sample hard negatives for the ITM task with zero computational overhead. A negative image-text pair is hard if they share similar semantics but differ in fine-grained details. We use the contrastive similarity from Equation $\\bigstar$ to find in-batch hard negatives. For each image in a mini-batch, we sample one negative text from the same batch following the contrastive similarity distribution, where texts that are more similar to the image have a higher chance to be sampled. Likewise, we also sample one hard negative image for each text. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The full pre-training objective of ALBEF is: ",
+        "bbox": [
+            176,
+            601,
+            464,
+            614
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d152ceb8758891ce6dabd6bcf6a5e738044a533fbbfd13b89894c0b1cdd12bf.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { \\mathrm { i t c } } + \\mathcal { L } _ { \\mathrm { m l m } } + \\mathcal { L } _ { \\mathrm { i t m } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 Momentum Distillation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The image-text pairs used for pre-training are mostly collected from the web and they tend to be noisy. Positive pairs are usually weakly-correlated: the text may contain words that are unrelated to the image, or the image may contain entities that are not described in the text. For ITC learning, negative texts for an image may also match the image’s content. For MLM, there may exist other words different from the annotation that describes the image equally well (or better). However, the one-hot labels for ITC and MLM penalize all negative predictions regardless of their correctness. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To address this, we propose to learn from pseudo-targets generated by the momentum model. The momentum model is a continuously-evolving teacher which consists of exponential-moving-average versions of the unimodal and multimodal encoders. During training, we train the base model such that its predictions match the ones from the momentum model. Specifically, for ITC, we first compute the image-text similarity using features from the momentum unimodal encoders as $s ^ { \\prime } ( I , T ) \\stackrel { \\cdot } { = } g _ { v } ^ { \\prime } ( { \\pmb v } _ { \\mathrm { c l s } } ^ { \\prime } ) ^ { \\top } \\breve { g } _ { w } ^ { \\prime } ( { \\pmb w } _ { \\mathrm { c l s } } ^ { \\prime } )$ and $s ^ { \\prime } ( T , I ) \\stackrel { \\mathrm { ~ \\tiny ~ = ~ } } { = } g _ { w } ^ { \\prime } ( { \\pmb w } _ { \\mathrm { c l s } } ) ^ { \\top } g _ { v } ^ { \\prime } ( { \\pmb v } _ { \\mathrm { c l s } } ^ { \\prime } )$ . Then we compute soft pseudotargets $q ^ { \\mathrm { i 2 t } }$ and $q ^ { \\mathrm { t 2 i } }$ by replacing $s$ with $s ^ { \\prime }$ in Equation 1. The $\\mathrm { I T C } _ { \\mathrm { M o D } }$ loss is defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1c968814e0958bf0e11029a46b2f0a3a1a075a176fab8849389b809ffc5030a9.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { i t c } } ^ { \\mathrm { m o d } } = ( 1 - \\alpha ) { \\mathcal { L } } _ { \\mathrm { i t c } } + { \\frac { \\alpha } { 2 } } { \\mathbb { E } } _ { ( I , T ) \\sim D } \\left[ \\mathrm { K L } ( q ^ { \\mathrm { i } 2 \\mathrm { t } } ( I ) \\parallel p ^ { \\mathrm { i } 2 \\mathrm { t } } ( I ) ) + \\mathrm { K L } ( q ^ { \\mathrm { t } 2 \\mathrm { i } } ( T ) \\parallel p ^ { \\mathrm { t } 2 \\mathrm { i } } ( T ) ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similarly, for MLM, let $\\pmb q ^ { \\mathrm { m s k } } ( I , \\hat { T } )$ denote the momentum model’s prediction probability for the masked token, the $\\mathbf { M L M } _ { \\mathrm { M o D } }$ loss is: ",
+        "bbox": [
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+            89,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae1b2659258cce633d9384d7569c980637e0399d397560d3cc211f1ef5981811.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { m l m } } ^ { \\mathrm { m o d } } = ( 1 - \\alpha ) \\mathcal { L } _ { \\mathrm { m l m } } + \\alpha \\mathbb { E } _ { ( I , \\hat { T } ) \\sim D } \\mathrm { K L } ( \\pmb { q } ^ { \\mathrm { m s k } } ( I , \\hat { T } ) \\parallel p ^ { \\mathrm { m s k } } ( I , \\hat { T } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            121,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Figure $\\bigstar$ we show examples of the top-5 candidates from the pseudo-targets, which effectively capture relevant words/texts for an image. More examples can be found in Appendix. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We also apply MoD to the downstream tasks. The final loss for each task is a weighted combination of the original task’s loss and the KL-divergence between the model’s prediction and the pseudo-targets. For simplicity, we set the weight $\\alpha = 0 . 4$ for all pre-training and downstream tasks 2. ",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 Pre-training Datasets ",
+        "text_level": 1,
+        "bbox": [
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+            233,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following UNITER [2], we construct our pre-training data using two web datasets (Conceptual Captions [4], SBU Captions $\\pmb { \\mathbb { B } } \\mathbf { \\| }$ ) and two in-domain datasets (COCO [41] and Visual Genome [42]). The total number of unique images is $4 . 0 \\mathbf { M }$ , and the number of image-text pairs is 5.1M. To show that our method is scalable with larger-scale web data, we also include the much noisier Conceptual 12M dataset $\\mathbb { \\lVert \\rVert 3 \\rVert }$ , increasing the total number of images to $1 4 . 1 \\mathrm { M } \\big \\sharp$ Details are in Appendix. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.5 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our model consists of a $\\mathbf { B E R T _ { b a s e } }$ with 123.7M parameters and a ViT-B/16 with $8 5 . 8 \\mathbf { M }$ parameters. We pre-train the model for 30 epochs using a batch size of 512 on 8 NVIDIA A100 GPUs. We use the AdamW $\\pm \\boxed { \\boxed { 4 4 } }$ optimizer with a weight decay of 0.02. The learning rate is warmed-up to $1 e ^ { - 4 }$ in the first 1000 iterations, and decayed to $1 e ^ { - 5 }$ following a cosine schedule. During pre-training, we take random image crops of resolution $2 5 6 \\times 2 5 6$ as input, and also apply RandAugment4 [45]. During fine-tuning, we increase the image resolution to $3 8 4 \\times 3 8 4$ and interpolate the positional encoding of image patches following $\\left[ \\left[ 3 8 \\right] \\right]$ . The momentum parameter for updating the momentum model is set as 0.995, and the size of the queue used for image-text contrastive learning is set as 65,536. We linearly ramp-up the distillation weight $\\alpha$ from 0 to 0.4 within the 1st epoch. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 A Mutual Information Maximization Perspective ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide an alternative perspective of ALBEF and show that it maximizes a lower bound on the mutual information (MI) between different “views” of an image-text pair. ITC, MLM, and MoD can be interpreted as different ways to generate the views. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Formally, we define two random variables $a$ and $b$ as two different views of a data point. In selfsupervised learning $[ 1 2 4 , 1 2 5 , | 4 6 |$ , $a$ and $b$ are two augmentations of the same image. In vision-language representation learning, we consider $a$ and $b$ as different variations of an image-text pair that capture its semantic meaning. We aim to learn representations invariant to the change of view. This can be achieved by maximizing the MI between $a$ and $b$ . In practice, we maximize a lower bound on $\\textstyle \\mathbf { M } ( a , b )$ by minimizing the InfoNCE loss [47] defined as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c1df4d0eb6e5673caba54781b3283e6c64d001bc5d8795e5e4b9e4928178e4b.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { N C E } } = - \\mathbb { E } _ { p ( a , b ) } \\left[ \\log \\frac { \\exp ( s ( a , b ) ) } { \\sum _ { \\hat { b } \\in \\hat { B } } \\exp ( s ( a , \\hat { b } ) ) } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $s ( a , b )$ is a scoring function (e.g., a dot product between two representations), and $\\hat { B }$ contains the positive sample $b$ and $| \\hat { B } | - 1$ negative samples drawn from a proposal distribution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our ITC loss with one-hot labels (Equation $^ { 2 ) }$ can be re-written as: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/298f854462bbc8f03bfe330c83e6f285f3fa9aec5157e5aae99cc337d007c44e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { i t c } } = - \\frac { 1 } { 2 } \\mathbb { E } _ { p ( I , T ) } \\big [ \\log \\frac { \\exp ( s ( I , T ) / \\tau ) } { \\sum _ { m = 1 } ^ { M } \\exp ( s ( I , T _ { m } ) / \\tau ) } + \\log \\frac { \\exp ( s ( T , I ) / \\tau ) } { \\sum _ { m = 1 } ^ { M } \\exp ( s ( T , I _ { m } ) / \\tau ) } \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Minimizing $\\mathcal { L } _ { \\mathrm { i t c } }$ can be seen as maximizing a symmetric version of InfoNCE. Hence, ITC considers the two individual modalities (i.e., $I$ and $T$ ) as the two views of an image-text pair, and trains the unimodal encoders to maximize the MI between the image and text views for the positive pairs. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in $[ \\overline { { | 4 8 | } }$ , we can also interpret MLM as maximizing the MI between a masked word token and its masked context (i.e. image $^ +$ masked text). Specifically, we can re-write the MLM loss with one-hot labels (Equation $\\textcircled { 3 }$ as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d54f0ba96e2babeea4b9f64d321ff3a4ea922b83e250db1941aa32728f127988.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { m l m } } = - \\mathbb { E } _ { p ( I , \\hat { T } ) } \\big [ \\log \\frac { \\exp ( \\psi ( y ^ { \\mathrm { m s k } } ) ^ { \\top } f ( I , \\hat { T } ) ) } { \\sum _ { y \\in \\mathcal { V } } \\exp ( \\psi ( y ) ^ { \\top } f ( I , \\hat { T } ) ) } \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            140,
+            671,
+            181
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\psi ( y ) : \\mathcal { V } \\to \\mathbb { R } ^ { d }$ is a lookup function in the multimodal encoder’s output layer that maps a word token $y$ into a vector and $\\nu$ is the full vocabulary set, and $f ( I , { \\hat { T } } )$ is a function that returns the final hidden state of the multimodal encoder corresponding to the masked context. Hence, MLM considers the two views of an image-text pair to be: (1) a randomly selected word token, and (2) the image $^ +$ the contextual text with that word masked. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Both ITC and MLM generate views by taking partial information from an image-text pair, through either modality separation or word masking. Our momentum distillation can be considered as generating alternative views from the entire proposal distribution. Take $\\mathrm { I T C } _ { \\mathrm { M o D } }$ in Equation $6$ as an example, minimizing $\\mathrm { K L } ( p ^ { \\mathrm { i 2 t } } ( I ) , q ^ { \\mathrm { i 2 t } } ( I ) )$ is equivalent to minimizing the following objective: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca3ef3c3db95acf8048ef17e96071edf75a8e8873da35d848d600aaaa47cb03b.jpg",
+        "text": "$$\n- \\sum _ { m } q _ { m } ^ { \\mathrm { i } 2 \\mathrm { t } } ( I ) \\log p _ { m } ^ { \\mathrm { i } 2 \\mathrm { t } } ( I ) = - \\sum _ { m } \\frac { \\exp ( s ^ { \\prime } ( I , T _ { m } ) / \\tau ) } { \\sum _ { m = 1 } ^ { M } \\exp ( s ^ { \\prime } ( I , T _ { m } ) / \\tau ) } \\log \\frac { \\exp ( s ( I , T _ { m } ) / \\tau ) } { \\sum _ { m = 1 } ^ { M } \\exp ( s ( I , T _ { m } ) / \\tau ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "It maximizes $\\mathbf { M I } ( I , T _ { m } )$ for texts that share similar semantic meaning with the image $I$ because those texts would have larger $q _ { m } ^ { \\mathrm { i 2 t } } ( I )$ . Similarly, $\\mathrm { I T C } _ { \\mathrm { M o D } }$ also maximizes $\\mathbf { M } \\mathbf { I } ( I _ { m } , T )$ for images that are similar to $T$ . We can follow the same method to show that $\\mathbf { M L M } _ { \\mathrm { M o D } }$ generates alternative views $y ^ { \\prime } \\in \\mathcal { V }$ for the masked word $y ^ { \\mathrm { m s k } }$ , and maximizes the MI between $y ^ { \\prime }$ and $( I , { \\hat { T } } )$ . Therefore, our momentum distillation can be considered as performing data augmentation to the original views. The momentum model generates a diverse set of views that are absent in the original image-text pairs, and encourages the base model to learn representations that capture view-invariant semantic information. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Downstream $\\mathbf { V } { + } \\mathbf { L }$ Tasks ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We adapt the pre-trained model to five downstream $_ { \\mathrm { V + L } }$ tasks. We introduce each task and our fine-tuning strategy below. Details of the datasets and fine-tuning hyperparameters are in Appendix. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Image-Text Retrieval contains two subtasks: image-to-text retrieval (TR) and text-to-image retrieval (IR). We evaluate ALBEF on the Flickr30K $\\bar { \\mathbb { E 9 } } \\bar { \\mathbb { I } }$ and COCO benchmarks, and fine-tune the pretrained model using the training samples from each dataset. For zero-shot retrieval on Flickr30K, we evaluate with the model fine-tuned on COCO. During fine-tuning, we jointly optimize the ITC loss (Equation $2 )$ and the ITM loss (Equation $\\textcircled{4}$ . ITC learns an image-text scoring function based on similarity of unimodal features, whereas ITM models the fine-grained interaction between image and text to predict a matching score. Since the downstream datasets contain multiple texts for each image, we change the ground-truth label of ITC to consider multiple positives in the queue, where each positive has a ground-truth probability of 1/#positives. During inference, we first compute the feature similarity score $s _ { \\mathrm { i t c } }$ for all image-text pairs. Then we take the top- $k$ candidates and calculate their ITM score $s _ { \\mathrm { i t m } }$ for ranking. Because $k$ can be set to be very small, our inference speed is much faster than methods that require computing the ITM score for all image-text pairs [2, 3, 8]. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Visual Entailment (SNLI-VE5 [51]) is a fine-grained visual reasoning task to predict whether the relationship between an image and a text is entailment, neutral, or contradictory. We follow UNITER $\\left[ \\left[ 2 \\right] \\right]$ and consider VE as a three-way classification problem, and predict the class probabilities using a multi-layer perceptron (MLP) on the multimodal encoder’s representation of the [CLS] token. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Visual Question Answering (VQA $\\pmb { \\mathbb { B } 2 } \\mathbf { l }$ ) requires the model to predict an answer given an image and a question. Different from existing methods that formulate VQA as a multi-answer classification problem [53, 2], we consider VQA as an answer generation problem, similar to [54]. Specifically, we use a 6-layer transformer decoder to generate the answer. As shown in Figure 3a, the auto-regressive answer decoder receives the multimodal embeddings through cross attention, and a start-of-sequence token ([CLS]) is used as the decoder’s initial input token. Likewise, an end-of-sequence token ([SEP]) is appended to the end of decoder outputs which indicates the completion of generation. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/0b0dc7535563e91da490a221fda745edf632ed8624683f45f89b326da9d67c5b.jpg",
+        "image_caption": [
+            "Figure 3: The model architecture for VQA and $\\mathrm { \\tt N L V R } ^ { 2 }$ . For VQA, we append an auto-regressive decoder to generate the answer given the image-question embeddings. For $\\mathrm { \\dot { N L V R } ^ { 2 } }$ , we replicate the transformer block within each layer of multimodal encoder to enable reasoning over two images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The answer decoder is initialized using the pre-trained weights from the multimodal encoder, and finetuned with a conditional language-modeling loss. For a fair comparison with existing methods, we constrain the decoder to only generate from the 3,128 candidate answers $ { \\Vert 5 5 \\Vert }$ during inference. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Natural Language for Visual Reasoning (NLVR2 [19]) requires the model to predict whether a text describes a pair of images. We extend our multimodal encoder to enable reasoning over two images. As shown in Figure $3 { \\mathrm { b } }$ , each layer of the multimodal encoder is replicated to have two consecutive transformer blocks, where each block contains a self-attention layer, a cross-attention layer, and a feed-forward layer (see Figure 1). The two blocks within each layer are initialized using the same pre-trained weights, and the two cross-attention layers share the same linear projection weights for the keys and values. During training, the two blocks receive two sets of image embeddings for the image pair. We append a MLP classifier on the multimodal encoder’s [CLS] representation for prediction. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For $\\mathrm { \\Delta N L V R ^ { 2 } }$ , we perform an additional pre-training step to prepare the new multimodal encoder for encoding an image-pair. We design a text-assignment (TA) task as follows: given a pair of images and a text, the model needs to assign the text to either the first image, the second image, or none of them. We consider it as a three-way classification problem, and use a FC layer on the [CLS] representation to predict the assignment. We pre-train with TA for only 1 epoch using the 4M images (Section 3.4). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Visual Grounding aims to localize the region in an image that corresponds to a specific textual description. We study the weakly-supervised setting, where no bounding box annotations are available. We perform experiments on the $\\operatorname { R e f C O C O + } \\mathbb { I }$ 56] dataset, and fine-tune the model using only imagetext supervision following the same strategy as image-text retrieval. During inference, we extend Grad-CAM $\\pmb { \\mathbb { Q } } \\mathbf { \\| }$ to acquire heatmaps, and use them to rank the detected proposals provided by $\\mathbb { \\lVert 5 3 \\rVert }$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.1 Evaluation on the Proposed Methods ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "First, we evaluate the effectiveness of the proposed methods (i.e. image-text contrastive learning, contrastive hard negative mining, and momentum distillation). Table $\\bar { \\mathbb { \\perp } }$ shows the performance of the downstream tasks with different variants of our method. Compared to the baseline pre-training tasks $( \\mathbf { M L M + I T M } )$ ), adding ITC substantially improves the pre-trained model’s performance across all tasks. The proposed hard negative mining improves ITM by finding more informative training samples. Furthermore, adding momentum distillation improves learning for both ITC (row 4), MLM (row 5), and on all downstream tasks (row 6). In the last row, we show that ALBEF can effectively leverage more noisy web data to improve the pre-training performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/305ca6c9a3f0fc0d5a55206bf9789c66cd7d7c45b55d6413524c2d44dc3fc209.jpg",
+        "table_caption": [
+            "Table 1: Evaluation of the proposed methods on four downstream $_ { \\mathrm { V + L } }$ tasks. For text-retrieval (TR) and image-retrieval (IR), we report the average of $\\mathbf { R } \\ @ 1$ , $\\mathbf { R } @ 5$ and $\\mathrm { R @ 1 0 }$ . ITC: image-text contrastive learning. MLM: masked language modeling. $\\mathrm { I T M } _ { \\mathrm { h a r d } }$ : image-text matching with contrastive hard negative mining. MoD: momentum distillation. MoDDownstream: momentum distillation on downstream tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#Pre-train Images</td><td>Training tasks</td><td>TR IR (flickr test)</td><td>SNLI-VE (test)</td><td>NLVR² (test-P)</td><td>VQA (test-dev)</td></tr><tr><td rowspan=\"6\">4M</td><td>MLM+ ITM</td><td>93.96 88.55</td><td>77.06</td><td>77.51</td><td>71.40</td></tr><tr><td>ITC +MLM+ ITM</td><td>96.55 91.69</td><td>79.15</td><td>79.88</td><td>73.29</td></tr><tr><td>ITC + MLM + ITMhard</td><td>97.01 92.16</td><td>79.77</td><td>80.35</td><td>73.81</td></tr><tr><td>ITCMoD +MLM+ ITMhard</td><td>97.33 92.43</td><td>79.99</td><td>80.34</td><td>74.06</td></tr><tr><td>Full (ITCMoD + MLMMoD + ITMhard)</td><td>97.47 92.58</td><td>80.12</td><td>80.44</td><td>74.42</td></tr><tr><td>ALBEF (Full + MoDDownstream)</td><td>97.83 92.65</td><td>80.30</td><td>80.50</td><td>74.54</td></tr><tr><td>14M</td><td>ALBEF</td><td>98.70 94.07</td><td>80.91</td><td>83.14</td><td>75.84</td></tr></table>",
+        "bbox": [
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+            728,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/78ea2426ebebef0eb7ef7266792282451a64e4ef9a8bd05030122bad14960416.jpg",
+        "table_caption": [
+            "Table 2: Fine-tuned image-text retrieval results on Flickr30K and COCO datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">#Pre-train Images</td><td colspan=\"6\">Flickr30K (1K test set)</td><td colspan=\"6\">MSCOCO (5K test set)</td></tr><tr><td colspan=\"2\">TR</td><td colspan=\"4\"></td><td colspan=\"2\">TR</td><td colspan=\"2\"></td><td colspan=\"2\">IR</td></tr><tr><td>UNITER</td><td>4M</td><td>R@1 87.3</td><td>R@5 98.0</td><td>R@10 99.2</td><td>R@1 75.6</td><td>R@5</td><td>R@10 96.8</td><td>R@1 65.7</td><td>R@5</td><td>R@10 93.8</td><td>R@1 52.9</td><td>R@5 79.9</td><td>R@10</td></tr><tr><td>VILLA</td><td>4M</td><td>87.9</td><td>97.5</td><td>98.8</td><td>76.3</td><td>94.1</td><td>96.8</td><td>-</td><td>88.6 -</td><td></td><td></td><td></td><td>88.0</td></tr><tr><td>OSCAR</td><td>4M</td><td></td><td></td><td></td><td></td><td>94.2</td><td></td><td>70.0</td><td></td><td>- 95.5</td><td>- 54.0</td><td>-</td><td>- 88.5</td></tr><tr><td>ALIGN</td><td>1.2B</td><td>- 95.3</td><td>-</td><td>1</td><td>1</td><td>-</td><td>-</td><td>77.0</td><td>91.1</td><td></td><td></td><td>80.8</td><td></td></tr><tr><td></td><td></td><td></td><td>99.8</td><td>100.0</td><td>84.9</td><td>97.4</td><td>98.6</td><td></td><td>93.5</td><td>96.9</td><td>59.9</td><td>83.3</td><td>89.8</td></tr><tr><td>ALBEF</td><td>4M</td><td>94.3</td><td>99.4</td><td>99.8</td><td>82.8</td><td>96.7</td><td>98.4</td><td>73.1</td><td>91.4</td><td>96.0</td><td>56.8</td><td>81.5</td><td>89.2</td></tr><tr><td>ALBEF</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>14M</td><td>95.9</td><td>99.8</td><td></td><td></td><td></td><td></td><td></td><td></td><td>97.2</td><td>60.7</td><td>84.3</td><td>90.5</td></tr><tr><td></td><td></td><td></td><td></td><td>100.0</td><td>85.6</td><td>97.5</td><td>98.9</td><td>77.6</td><td>94.3</td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            173,
+            73,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/3dd999bfbbc13378ebced4ad22307027f1e86beab56912eed83a7f8f56c8e3f2.jpg",
+        "table_caption": [
+            "Table 3: Zero-shot image-text retrieval results on Flickr30K. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">#Pre-train Images</td><td colspan=\"6\">Flickr30K (1K test set)</td></tr><tr><td colspan=\"3\">TR</td><td colspan=\"3\">IR</td></tr><tr><td></td><td></td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@1</td><td>R@5</td><td>R@10</td></tr><tr><td>UNITER 四</td><td>4M</td><td>83.6</td><td>95.7</td><td>97.7</td><td>68.7</td><td>89.2</td><td>93.9</td></tr><tr><td>CLIP 回</td><td>400M</td><td>88.0</td><td>98.7</td><td>99.4</td><td>68.7</td><td>90.6</td><td>95.2</td></tr><tr><td>ALIGN </td><td>1.2B</td><td>88.6</td><td>98.7</td><td>99.7</td><td>75.7</td><td>93.8</td><td>96.8</td></tr><tr><td>ALBEF</td><td>4M</td><td>90.5</td><td>98.8</td><td>99.7</td><td>76.8</td><td>93.7</td><td>96.7</td></tr><tr><td>ALBEF</td><td>14M</td><td>94.1</td><td>99.5</td><td>99.7</td><td>82.8</td><td>96.3</td><td>98.1</td></tr></table>",
+        "bbox": [
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+            220,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f1b8e1a70ca08da555a2f5c10100ce8aaa7a1664b04f626da78483800cd87e84.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Comparison with state-of-the-art methods on downstream vision-language tasks. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">VQA</td><td colspan=\"2\">NLVR²</td><td colspan=\"2\">SNLI-VE</td></tr><tr><td>test-dev</td><td>test-std</td><td>dev</td><td>test-P</td><td>val</td><td>test</td></tr><tr><td>VisualBERT[13]</td><td>70.80</td><td>71.00</td><td>67.40</td><td>67.00</td><td>-</td><td>-</td></tr><tr><td>VL-BERT 目</td><td>71.16</td><td>=</td><td>=</td><td>=</td><td>-</td><td></td></tr><tr><td>LXMERT[</td><td>72.42</td><td>72.54</td><td>74.90</td><td>74.50</td><td>=</td><td>=</td></tr><tr><td>12-in-1 目</td><td>73.15</td><td>-</td><td>-</td><td>78.87</td><td>=</td><td>76.95</td></tr><tr><td>UNITER [2]</td><td>72.70</td><td>72.91</td><td>77.18</td><td>77.85</td><td>78.59</td><td>78.28</td></tr><tr><td>VL-BART/T5 54</td><td>-</td><td>71.3</td><td>=</td><td>73.6</td><td>-</td><td>1</td></tr><tr><td>ViLT 四</td><td>70.94</td><td>-</td><td>75.24</td><td>76.21</td><td>-</td><td>-</td></tr><tr><td>OSCAR [3</td><td>73.16</td><td>73.44</td><td>78.07</td><td>78.36</td><td>=</td><td>1</td></tr><tr><td>VILLA 图</td><td>73.59</td><td>73.67</td><td>78.39</td><td>79.30</td><td>79.47</td><td>79.03</td></tr><tr><td>ALBEF (4M)</td><td>74.54</td><td>74.70</td><td>80.24</td><td>80.50</td><td>80.14</td><td>80.30</td></tr><tr><td>ALBEF (14M)</td><td>75.84</td><td>76.04</td><td>82.55</td><td>83.14</td><td>80.80</td><td>80.91</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.2 Evaluation on Image-Text Retrieval ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 and Table 3 report results on fine-tuned and zero-shot image-text retrieval, respectively. Our ALBEF achieves state-of-the-art performance, outperforming CLIP $\\pmb { \\Vert 6 \\Vert }$ and ALIGN $[ [ 7 ]$ which are trained on orders of magnitude larger datasets. Given the considerable amount of improvement of ALBEF when the number of training images increases from 4M to 14M, we hypothesize that it has potential to further grow by training on larger-scale web image-text pairs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.3 Evaluation on VQA, NLVR, and VE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table $\\sharp$ reports the comparison with existing methods on other $_ { \\mathrm { V + L } }$ understanding tasks. With 4M pre-training images, ALBEF already achieves state-of-the-art performance. With 14M pre-training images, ALBEF substantially outperforms existing methods, including methods that additionally use object tags $\\pmb { \\mathbb { B } } \\|$ or adversarial data augmentation $\\pmb { \\mathbb { B } } ] \\mathbf  l$ . Compared to VILLA $\\pmb { \\mathbb { B } } ] \\mathbf  l$ , ALBEF achieves absolute improvements of $2 . 3 7 \\%$ on VQA test-std, $3 . 8 4 \\%$ on $\\mathrm { \\bar { N L V R ^ { 2 } } }$ test-P, and $1 . 8 8 \\%$ on SNLI-VE test. Because ALBEF is detector-free and requires lower resolution images, it also enjoys much faster inference speed compared to most existing methods ${ \\tt > } 1 0$ times faster than VILLA on ${ \\mathrm { N L V R } } ^ { 2 }$ ). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.4 Weakly-supervised Visual Grounding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            862,
+            473,
+            877
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 5 shows the results on $\\operatorname { R e f C O C O + }$ , where ALBEF substantially outperforms existing methods [57, 58] (which use weaker text embeddings). The $\\mathbf { A L B E F _ { i t c } }$ variant computes Grad-CAM ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a734e17611d02ed4e676d4d4d8d49efa22fc0dbbdb2a79f5566cbeeda7d81fd3.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 5: Weakly-supervised visual grounding on $\\mathrm { R e f C O C O + }$ [56] dataset. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Val</td><td>TestA</td><td>TestB</td></tr><tr><td>ARN 四</td><td>32.78</td><td>34.35</td><td>32.13</td></tr><tr><td>CCL[ 国</td><td>34.29</td><td>36.91</td><td>33.56</td></tr><tr><td>ALBEFitc</td><td>51.58</td><td>60.09</td><td>40.19</td></tr><tr><td>ALBEFitm</td><td>58.46</td><td>65.89</td><td>46.25</td></tr></table>",
+        "bbox": [
+            179,
+            70,
+            403,
+            145
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/4251bcd75e9054e149f7bf987e5ff45a120e0a49fcf551018cb3e8b81dc823a8.jpg",
+        "image_caption": [
+            "Figure 4: Grad-CAM visualization on the cross-attention maps in the 3rd layer of the multimodal encoder. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            424,
+            65,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Q: is this rice noodle soup? Q: what is to the right of A: yes the soup? A: chopsticks ",
+        "bbox": [
+            181,
+            184,
+            478,
+            207
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Q: what does the truck on Q: what is the man doing in the street? A: walking the left sell? A: ice cream ",
+        "bbox": [
+            498,
+            184,
+            808,
+            208
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/73781611098cb0fe1f63a3e0377c69a21f1e4b8ed04a2ab157bfdfaac6789207.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            209,
+            816,
+            287
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ceb6f744b4870218abb363b8856d9195fd5ee3876c9a2f27efa97528688f2950.jpg",
+        "image_caption": [
+            "Figure 5: Grad-CAM visualizations on the cross-attention maps of the multimodal encoder for the VQA model. “a little girl holding a kitten next to a blue fence” ",
+            "Figure 6: Grad-CAM visualizations on the cross-attention maps corresponding to individual words. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            319,
+            818,
+            410
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "visualizations on the self-attention maps in the last layer of the image encoder, where the gradients are acquired by maximizing the image-text similarity $s _ { \\mathrm { i t c } }$ . The $\\mathbf { A L B E F _ { i t m } }$ variant computes Grad-CAM on the cross-attention maps in the 3rd layer of the multimodal encoder (which is a layer specialized in grounding), where the gradients are acquired by maximizing the image-text matching score $s _ { \\mathrm { i t m } }$ Figure 4 provides a few visualizations. More analysis is in Appendix. ",
+        "bbox": [
+            173,
+            439,
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+            508
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We provide the Grad-CAM visualizations for VQA in Figure $\\textcircled{5}$ As can be seen in Appendix, the Grad-CAM visualizations from ALBEF are highly correlated with where humans would look when making decisions. In Figure $6 ,$ we show per-word visualizations for COCO. Notice how our model not only grounds objects, but also their attributes and relationships. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.5 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            580,
+            318,
+            594
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table $\\boxed { 6 }$ studies the effect of various design choices on image-text retrieval. Since we use $s _ { \\mathrm { i t c } }$ to filter top- $k$ candidates during inference, we vary $k$ and report its effect. In general, the ranking result acquired by $s _ { \\mathrm { i t m } }$ is not sensitive to changes ",
+        "bbox": [
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+            601,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/569338ef9e212108cdfde60bc9757da949676722a3a94db99f77ebeff5279bad.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Flickr30K</td><td colspan=\"4\">w/ hard negs</td><td rowspan=\"2\">w/o hard negs k =128</td></tr><tr><td>Sitc</td><td>k =16</td><td>k =128</td><td>k=256</td></tr><tr><td>TR</td><td>97.30</td><td>98.60</td><td>98.57</td><td>98.57</td><td>98.22 (-0.35)</td></tr><tr><td>IR</td><td>90.95</td><td>93.64</td><td>93.99</td><td>93.95</td><td>93.68 (-0.31)</td></tr></table>",
+        "bbox": [
+            413,
+            590,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Ablation study on fine-tuned image-text retrieval. The average recall on the test set is reported. We use $s _ { \\mathrm { i t c } }$ to filter top- $k$ candidates and calculate their $s _ { \\mathrm { i t m } }$ score for ranking. ",
+        "bbox": [
+            413,
+            655,
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+            694
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "in $k$ . We also validate the effect of hard negative mining in the last column. ",
+        "bbox": [
+            174,
+            698,
+            666,
+            712
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table $^ { 7 }$ studies the effect of textassignment (TA) pre-training and parameter sharing on $\\mathrm { \\tt N L V R } ^ { \\mathrm { \\bar { 2 } } }$ . We examine three strategies: (1) the two mutimodal blocks share all parameters, (2) only the cross",
+        "bbox": [
+            173,
+            718,
+            400,
+            801
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/821711171c8c80af8e5884c06903997fb99ad472af08d713e9b02b2a76048bae.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 7: Ablation study on NLVR2. "
+        ],
+        "table_body": "<table><tr><td>NLVR²</td><td colspan=\"3\">w/TA share all share CA</td><td colspan=\"3\">w/o TA share all share CA</td></tr><tr><td>dev</td><td>82.13</td><td>82.55</td><td>no share 81.93</td><td>80.52</td><td>80.28</td><td>no share 77.84</td></tr><tr><td>test-P</td><td>82.36</td><td>83.14</td><td>82.85</td><td>81.29</td><td>80.45</td><td>77.58</td></tr></table>",
+        "bbox": [
+            415,
+            723,
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+            784
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "attention (CA) layers are shared, (3) no sharing. Without TA, sharing the entire block has better performance. With TA to pre-train the model for image-pair, sharing CA leads to the best performance. ",
+        "bbox": [
+            179,
+            801,
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+            829
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion and Social Impacts ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "text": "This paper proposes ALBEF, a new framework for vision-language representation learning. ALBEF first aligns the unimodal image representation and text representation before fusing them with a multimodal encoder. We theoretically and experimentally verify the effectiveness of the proposed image-text contrastive learning and momentum distillation. Compared to existing methods, ALBEF offers better performance and faster inference speed on multiple downstream $_ { \\mathrm { V + L } }$ tasks. ",
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+        "text": "While our paper shows promising results on vision-language representation learning, additional analysis on the data and the model is necessary before deploying it in practice, because web data may contain unintended private information, unsuitable images, or harmful texts, and only optimizing accuracy may have unwanted social implications. ",
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+        "text": "References ",
+        "text_level": 1,
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parse/train/SJfZKiC5FX/SJfZKiC5FX.md

@@ -0,0 +1,272 @@
+# DYNAMICALLY UNFOLDING RECURRENT RESTORER:A MOVING ENDPOINT CONTROL METHOD FOR IMAGERESTORATION
+
+Xiaoshuai Zhang∗   
+Institute of Computer Science and Technology,   
+Peking University   
+jet@pku.edu.cn Yiping Lu∗   
+School Of Mathmatical Science, Peking university   
+luyiping9712@pku.edu.cn   
+Jiaying Liu   
+Institute of Computer Science and Technology,   
+Peking University   
+liujiaying@pku.edu.cn   
+Bin Dong   
+Beijing International Center for Mathematical Research, Peking University   
+Center for Data Science, Peking University   
+Beijing Institute of Big Data Research,   
+Beijing, China   
+dongbin@math.pku.edu.cn
+
+# ABSTRACT
+
+In this paper, we propose a new control framework called the moving endpoint control to restore images corrupted by different degradation levels using a single model. The proposed control problem contains an image restoration dynamic which is modeled by a convolutional RNN. The moving endpoint, which is essentially the terminal time of the associated dynamic, is determined by a policy network. We call the proposed model the dynamically unfolding recurrent restorer (DURR). Numerical experiments show that DURR is able to achieve state-of-the-art performances on blind image denoising and JPEG image deblocking. Furthermore, DURR can well generalize to images with higher degradation levels that are not included in the training stage.1
+
+# 1 INTRODUCTION
+
+Image restoration, including image denoising, deblurring, inpainting, etc., is one of the most important areas in imaging science. Its major purpose is to obtain high quality reconstructions of images corrupted in various ways during imaging, acquisiting, and storing, and enable us to see crucial but subtle objects that reside in the images. Image restoration has been an active research area. Numerous models and algorithms have been developed for the past few decades. Before the uprise of deep learning methods, there were two classes of image restoration approaches that were widely adopted in the field: transformation based approach and PDE approach. The transformation based approach includes wavelet and wavelet frame based methods (Elad et al., 2005; Starck et al., 2005; Daubechies et al., 2007; Cai et al., 2009), dictionary learning based methods (Aharon et al., 2006), similarity based methods (Buades et al., 2005; Dabov et al., 2007), low-rank models (Ji et al., 2010; Gu et al., 2014), etc. The PDE approach includes variational models (Mumford & Shah, 1989; Rudin et al., 1992; Bredies et al., 2010), nonlinear diffusions (Perona & Malik, 1990; Catté et al., 1992; Weickert, 1998), nonlinear hyperbolic equations (Osher & Rudin, 1990), etc. More recently, deep connections
+
+between wavelet frame based methods and PDE approach were established (Cai et al., 2012; 2016;   
+Dong et al., 2017).
+
+One of the greatest challenge for image restoration is to properly handle image degradations of different levels. In the existing transformation based or PDE based methods, there is always at least one tuning parameter (e.g. the regularization parameter for variational models and terminal time for nonlinear diffusions) that needs to be manually selected. The choice of the parameter heavily relies on the degradation level.
+
+Recent years, deep learning models for image restoration tasks have significantly advanced the state-of-the-art of the field. Jain & Seung (2009) proposed a convolutional neural network (CNN) for image denoising which has better expressive power than the MRF models by Lan et al. (2006). Inspired by nonlinear diffusions, Chen & Pock (2017) designed a deep neural network for image denoising and Zhang et al. (2017a) improves the capacity by introducing a deeper neural network with residual connections. Chen et al. (2017) use the CNN to simulate a wide variety of image processing operators, achieving high efficiencies with little accuracy drop. However, these models cannot gracefully handle images with varied degradation levels. Although one may train different models for images with different levels, this may limit the application of these models in practice due to lack of flexibility.
+
+Taking blind image denoising for example. Zhang et al. (2017a) designed a 20-layer neural network for the task, called DnCNN-B, which had a huge number of parameters. To reduce number of parameters, Lefkimmiatis (2017) proposed the UNLNet5, by unrolling a projection gradient algorithm for a constrained optimization model. However, Lefkimmiatis (2017) also observed a drop in PSNR comparing to DnCNN. Therefore, the design of a light-weighted and yet effective model for blind image denoising remains a challenge. Moreover, deep learning based models trained on simulated gaussian noise images usually fail to handle real world noise, as will be illustrated in later sections.
+
+Another example is JPEG image deblocking. JPEG is the most commonly used lossy image compression method. However, this method tend to introduce undesired artifacts as the compression rate increases. JPEG image deblocking aims to eliminate the artifacts and improve the image quality. Recently, deep learning based methods were proposed for JPEG deblocking (Dong et al., 2015; Zhang et al., 2017a; 2018). However, most of their models are trained and evaluated on a given quality factor. Thus it would be hard for these methods to apply to Internet images, where the quality factors are usually unknown.
+
+In this paper, we propose a single image restoration model that can robustly restore images with varied degradation levels even when the degradation level is well outside of that of the training set. Our proposed model for image restoration is inspired by the recent development on the relation between deep learning and optimal control. The relation between supervised deep learning methods and optimal control has been discovered and exploited by Weinan (2017); Lu et al. (2018); Chang et al. (2017); Fang et al. (2017). The key idea is to consider the residual block $x _ { n + 1 } = x _ { n } + f ( x _ { n } )$ as an approximation to the continuous dynamics ${ \dot { X } } = f ( X )$ . In particular, Lu et al. (2018); Fang et al. (2017) demonstrated that the training process of a class of deep models (e.g. ResNet by He et al. (2016), PolyNet by Zhang et al. (2017b), etc.) can be understood as solving the following control problem:
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { w } \bigg ( L ( X ( T ) , y ) + \int _ { 0 } ^ { \tau } R ( w ( t ) , t ) d t \bigg ) } \\ { \displaystyle s . t . \dot { X } = f ( X ( t ) , w ( t ) ) , t \in ( 0 , \tau ) } \\ { \displaystyle X ( 0 ) = x _ { 0 } . } \end{array}
+$$
+
+Here $x _ { 0 }$ is the input, $y$ is the regression target or label, $\dot { \boldsymbol X } = f ( \boldsymbol X , \boldsymbol w )$ is the deep neural network with parameter $w ( t )$ , $R$ is the regularization term and $L$ can be any loss function to measure the difference between the reconstructed images and the ground truths.
+
+In the context of image restoration, the control dynamic $\dot { X } = f ( X ( t ) , \omega ( t ) ) , t \in ( 0 , \tau )$ can be, for example, a diffusion process learned using a deep neural network. The terminal time $\tau$ of the diffusion corresponds to the depth of the neural network. Previous works simply fixed the depth of the network, i.e. the terminal time, as a fixed hyper-parameter. However Mrázek & Navara (2003) showed that the optimal terminal time of diffusion differs from image to image. Furthermore, when an image is corrupted by higher noise levels, the optimal terminal time for a typical noise removal diffusion should be greater than when a less noisy image is being processed. This is the main reason why current deep models are not robust enough to handle images with varied noise levels. In this paper, we no longer treat the terminal time as a hyper-parameter. Instead, we design a new architecture (see Fig. 3) that contains both a deep diffusion-like network and another network that determines the optimal terminal time for each input image. We propose a novel moving endpoint control model to train the aforementioned architecture. We call the proposed architecture the dynamically unfolding recurrent restorer (DURR).
+
+We first cast the model in the continuum setting. Let $x _ { 0 }$ be an observed degraded image and $y$ be its corresponding damage-free counterpart. We want to learn a time-independent dynamic system $\dot { X } = f ( X ( t ) , w )$ with parameters $w$ so that $X ( 0 ) = x$ and $X ( \tau ) \approx y$ for some $\tau > 0$ . See Fig. 2 for an illustration of our idea. The reason that we do not require $X ( \tau ) = y$ is to avoid over-fitting. For varied degradation levels and different images, the optimal terminal time $\tau$ of the dynamics may vary. Therefore, we need to include the variable $\tau$ in the learning process as well. The learning of the dynamic system and the terminal time can be gracefully casted as the following moving endpoint control problem:
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { w , \tau ( x ) } L ( X ( \tau ) , y ) + \int _ { 0 } ^ { \tau ( x ) } R ( w ( t ) , t ) d t } \\ { \displaystyle s . t . \dot { X } = f ( X ( t ) , w ( t ) ) , t \in ( 0 , \tau ( x ) ) } \\ { \displaystyle X ( 0 ) = x . } \end{array}
+$$
+
+Different from the previous control problem, in our model the terminal time $\tau$ is also a parameter to be optimized and it depends on the data $x$ . The dynamic system $\dot { X } = f ( X ( t ) , w )$ is modeled by a recurrent neural network (RNN) with a residual connection, which can be understood as a residual network with shared weights (Liao & Poggio, 2016). We shall refer to this RNN as the restoration unit. In order to learn the terminal time of the dynamics, we adopt a policy network to adaptively determine an optimal stopping time. Our learning framework is demonstrated in Fig. 3. We note that the above moving endpoint control problem can be regarded as the penalized version of the well-known fixed endpoint control problem in optimal control (Evans, 2005), where instead of penalizing the difference between $X ( \tau )$ and $y$ , the constraint $X ( \tau ) = y$ is strictly enforced.
+
+In short, we summarize our contribution as following:
+
+• We are the first to use convolutional RNN for image restoration with unknown degradation levels, where the unfolding time of the RNN is determined dynamically at run-time by a policy unit (could be either handcrafted or RL-based).   
+• The proposed model achieves state-of-the-art performances with significantly less parameters and better running efficiencies than some of the state-of-the-art models.   
+We reveal the relationship between the generalization power and unfolding time of the RNN by extensive experiments. The proposed model, DURR, has strong generalization to images with varied degradation levels and even to the degradation level that is unseen by the model during training (Fig. 1).   
+The DURR is able to well handle real image denoising without further modification. Qualitative results have shown that our processed images have better visual quality, especially sharper details compared to others.
+
+# 2 METHOD
+
+The proposed architecture, i.e. DURR, contains an RNN (called the restoration unit) imitating a nonlinear diffusion for image restoration, and a deep policy network (policy unit) to determine the terminal time of the RNN. In this section, we discuss the training of the two components based on our moving endpoint control formulation. As will be elaborated, we first train the restoration unit to determine $\omega$ , and then train the policy unit to estimate $\tau ( x )$ .
+
+![](images/b9994af11293ec16af192ae7ba007b3f5e40a1eac061fe66610f8cf89429001c.jpg)  
+Figure 1: Denoising results of images from BSD68 under extreme noise conditions not seen in training data $( \sigma = 9 5 $ ).
+
+![](images/1230eb448c1f9a9db33389714ffbb02e617438d1676bcef3d0ffcd6a69481f05.jpg)  
+Figure 2: The proposed moving endpoint control model: evolving a learned reconstruction dynamics and ending at high-quality images.
+
+# 2.1 TRAINING THE RESTORATION UNIT
+
+If the terminal time $\tau$ for every input $x _ { i }$ is given (i.e. given a certain policy), the restoration unit can be optimized accordingly. We would like to show in this section that the policy used during training greatly influences the performance and the generalization ability of the restoration unit. More specifically, a restoration unit can be better trained by a good policy.
+
+The simplest policy is to fix the loop time $\tau$ as a constant for every input. We name such policy as “naive policy”. A more reasonable policy is to manually assign an unfolding time for each degradation level during training. We shall call this policy the “refined policy”. Since we have not trained the policy unit yet, to evaluate the performance of the trained restoration units, we manually pick the output image with the highest PSNR (i.e. the peak PSNR).
+
+We take denoising as an example here. The peak PSNRs of the restoration unit trained with different policies are listed in Table. 1. Fig. 4 illustrates the average loop times when the peak PSNRs appear. The training is done on both single noise level $\sigma = 4 0$ ) and multiple noise levels $( \sigma = 3 5 , 4 5 $ ). For the refined policy, the noise levels and the associated loop times are (35, 6), (45, 9). For the naive policy, we always fix the loop times to 8.
+
+![](images/4dcb6e57d263bea9e5c4beaf15bcc3cbc502e0f8678ca1d807e74f31359f2aea.jpg)  
+Figure 3: Pipeline of the dynamically unfolding recurrent restorer (DURR).
+
+Table 1: Average peak PSNR on BSD68 with different training strategies.   
+
+<table><tr><td colspan="2">Strategy</td><td colspan="7">Noise Level</td></tr><tr><td> Training Noise</td><td>Policy</td><td>25</td><td>30</td><td>35</td><td>40</td><td>45</td><td>50</td><td>55</td></tr><tr><td>40</td><td>Naive</td><td>28.61</td><td>28.13</td><td>27.62</td><td>27.19</td><td>26.57</td><td>26.17</td><td>24.00</td></tr><tr><td>35,45</td><td>Naive</td><td>27.74</td><td>27.17</td><td>26.66</td><td>26.24</td><td>26.75</td><td>25.61</td><td>24.75</td></tr><tr><td>35,45</td><td>Refined</td><td>29.14</td><td>28.33</td><td>27.67</td><td>27.19</td><td>27.69</td><td>26.61</td><td>25.88</td></tr></table>
+
+As we can see, the refined policy brings the best performance on all the noise levels including 40. The restoration unit trained for specific noise level (i.e. $\sigma = 4 0 ^ { \cdot }$ ) is only comparable to the one with refined policy on noise level 40. The restoration unit trained on multiple noise levels with naive policy has the worst performance.
+
+These results indicate that the restoration unit has the potential to generalize on unseen degradation levels when trained with good policies. According to Fig. 4, the generalization reflects on the loop times of the restoration unit. It can be observed that the model with steeper slopes have stronger ability to generalize as well as better performances.
+
+According to these results, the restoration unit we used in DURR is trained using the refined policy. More specifically, for image denoising, the noise level and the associated loop times are set to (25, 4), (35, 6), (45, 9), and (55, 12). For JPEG image deblocking, the quality factor (QF) and the associated loop times are set to (20, 6) and (30, 4).
+
+![](images/11cb76c9d0c212380465da4678d763e9c72c6a3be2231e08908fffeed1d04edc.jpg)  
+Figure 4: Average peak time on BSD68 with different training strategies.
+
+# 2.2 TRAINING THE POLICY UNIT
+
+We discuss two approaches that can be used as policy unit:
+
+Handcraft policy: Previous work (Mrázek & Navara, 2003) has proposed a handcraft policy that selects a terminal time which optimizes the correlation of the signal and noise in the filtered image. This criterion can be used directly as our policy unit, but the independency of signal and noise may not hold for some restoration tasks such as real image denoising, which has higher noise level in the low-light regions, and JPEG image deblocking, in which artifacts are highly related to the original image. Another potential stopping criterion of the diffusion is no-reference image quality assessment (Mittal et al., 2012), which can provide quality assessment to a processed image without the ground truth image. However, to the best of our knowledge, the performances of these assessments are still far from satisfactory. Because of the limitations of the handcraft policies, we will not include them in our experiments.
+
+Reinforcement learning based policy: We start with a discretization of the moving endpoint problem (1) on the dataset $\{ ( x _ { i } , \bar { y _ { i } } ) | i = 1 , 2 , \cdot \cdot \cdot , d \}$ , where $\{ x _ { i } \}$ are degraded observations of the damage-free images $\{ y _ { i } \}$ . The discrete moving endpoint control problem is given as follows:
+
+$$
+\begin{array} { r l r } {  { \operatorname* { m i n } _ { w , \{ N _ { i } \} _ { i = 1 } ^ { d } } r ( w ) + \sum _ { i = 1 } ^ { d } L ( X _ { N _ { i } } ^ { i } , y _ { i } ) } } \\ & { } & { s . t . X _ { n } ^ { i } = X _ { n - 1 } ^ { i } + \Delta t f ( X _ { n - 1 } ^ { i } , w ) , n = 1 , 2 , \cdots , N _ { i } , ( i = 1 , 2 , \cdots , d ) } \\ & { } & { X _ { 0 } ^ { i } = x _ { i } , i = 1 , 2 , \cdots , d . } \end{array}
+$$
+
+Here, $X _ { n } ^ { i } = X _ { n - 1 } ^ { i } + \Delta t f ( X _ { n - 1 } ^ { i } , w )$ is the forward Euler approximation of the dynamics $\dot { X } =$ $f ( X ( t ) , w )$ . The terminal time $\{ N _ { i } \}$ is determined by a policy network $P ( x , \theta )$ , where $x$ is the output of the restoration unit at each iteration and $\theta$ the set of weights. In our experiment, we simply set $r = 0$ , i.e. doesn’t introduce any regularization which might bring further benefit but is beyond this paper’s scope of discussion. In other words, the role of the policy network is to stop the iteration of the restoration unit when an ideal image restoration result is achieved. The reward function of the policy unit can be naturally defined by
+
+$$
+r ( \{ X _ { n } ^ { i } \} ) = { \left\{ \begin{array} { l l } { \lambda \left( L ( x _ { n - 1 } , y _ { i } ) - L ( x _ { n } , y _ { i } ) \right) } & { { \mathrm { I f ~ c h o o s e ~ t o ~ c o n t i n u e } } } \\ { 0 } & { { \mathrm { O t h e r w i s e } } } \end{array} \right. }
+$$
+
+In order to solve the problem (2.2), we need to optimize two networks simultaneously, i.e. the restoration unit and the policy unit. The first is an restoration unit which approximates the controlled dynamics and the other is the policy unit to give the optimized terminating conditions. The objective function we use to optimize the policy network can be written as
+
+$$
+J = \mathbb { E } _ { X \sim \pi _ { \theta } } \sum _ { n } ^ { N _ { i } } [ r ( \{ X _ { n } ^ { i } , w \} ) ] ,
+$$
+
+where $\pi _ { \theta }$ denotes the distribution of the trajectories $X = \{ X _ { n } ^ { i } , n = 1 , \ldots , N _ { i } , i = 1 , \ldots , d \}$ under the policy network $P ( \cdot , \theta )$ . Thus, reinforcement learning techniques can be used here to learn a neural network to work as a policy unit. We utilize Deep Q-learning (Mnih et al., 2015) as our learning strategy and denote this approach simply as DURR. However, different learning strategies can be used (e.g. the Policy Gradient).
+
+# 3 EXPERIMENTS
+
+# 3.1 EXPERIMENT SETTINGS
+
+In all denoising experiments, we follow the same settings as in Chen & Pock (2017); Zhang et al. (2017a); Lefkimmiatis (2017). All models are evaluated using the mean PSNR as the quantitative metric on the BSD68 (Martin et al., 2001). The training set and test set of BSD500 (400 images) are used for training. Six gaussian noise levels are evaluated, namely $\sigma = 2 5$ , 35, 45, 55, 65 and 75. Additive noise are applied to the image on the fly during training and testing. Both the training and evaluation process are done on gray-scale images.
+
+The restoration unit is a simple U-Net (Ronneberger et al., 2015) style fully convolutional neural network. For the training process of the restoration unit, the noise levels of 25, 35, 45 and 55 are used. Images are cut into $6 4 \times 6 4$ patches, and the batch-size is set to 24. The Adam optimizer with the learning rate 1e-3 is adopted and the learning rate is scaled down by a factor of 10 on training plateaux.
+
+The policy unit is composed of two ResUnit and an LSTM cell. For the policy unit training, we utilize the reward function in Eq.4. For training the policy unit, an RMSprop optimizer with learning rate 1e-4 is adopted. We’ve also tested other network structures, these tests and the detailed network structures of our model are demonstrated in the supplementary materials.
+
+In all JPEG deblocking experiments, we follow the settings as in Zhang et al. (2017a; 2018). All models are evaluated using the mean PSNR as the quantitative metric on the LIVE1 dataset (Sheikh, 2005). Both the training and evaluation processes are done on the Y channel (the luminance channel) of the YCbCr color space. The PIL module of python is applied to generate JPEG-compressed images. The module produces numerically identical images as the commonly used MATLAB JPEG encoder after setting the quantization tables manually. The images with quality factors 20 and 30 are used during training. De-blocking performances are evaluated on four quality factors, namely $\mathrm { Q F = 1 0 }$ , 20, 30, and 40. All other parameter settings are the same as in the denoising experiments.
+
+# 3.2 IMAGE DENOISING
+
+We select DnCNN-B(Zhang et al., 2017a) and UNLNet5 (Lefkimmiatis, 2017) for comparisons since these models are designed for blind image denoising. Moreover, we also compare our model with non-learning-based algorithms BM3D (Dabov et al., 2007) and WNNM (Gu et al., 2014). The noise levels are assumed known for BM3D and WNNM due to their requirements. Comparison results are shown in Table 2.
+
+Despite the fact that the parameters of our model $( 1 . 8 \times 1 0 ^ { 5 }$ for the restoration unit and $1 . 0 \times 1 0 ^ { 5 }$ for the policy unit) is less than the DnCNN (approximately $7 . 0 \times 1 0 ^ { 5 }$ ), one can see that DURR outperforms DnCNN on most of the noise-levels. More interestingly, DURR does not degrade too much when the the noise level goes beyond the level we used during training. The noise level $\sigma = 6 5$ , 75 is not included in the training set of both DnCNN and DURR. DnCNN reports notable drops of PSNR when evaluated on the images with such noise levels, while DURR only reports small drops of PSNR (see the last row of Table 2 and Fig. 6). Note that the reason we do not provide the results of UNLNet5 in Table 2 is because the authors of Lefkimmiatis (2017) has not released their codes yet, and they only reported the noise levels from 15 to 55 in their paper. We also want to emphasize that they trained two networks, one for the low noise level $( 5 \leq \sigma \leq 2 9 )$ and one for higher noise level $3 0 \leq \sigma \leq 5 5$ ). The reason is that due to the use of the constraint $| | y - x | | _ { 2 } \leq \epsilon$ by Lefkimmiatis (2017), we should not expect the model generalizes well to the noise levels surpasses the noise level of the training set.
+
+For qualitative comparisons, some restored images of different models on the BSD68 dataset are presented in Fig. 5 and Fig. 6. As can be seen, more details are preserved in DURR than other models. It is worth noting that the noise level of the input image in Fig. 6 is 65, which is unseen by both DnCNN and DURR during training. Nonetheless, DURR achieves a significant gain of nearly 1 dB than DnCNN. Moreover, the texture on the cameo is very well restored by DURR. These results clearly indicate the strong generalization ability of our model.
+
+More interestingly, due to the generalization ability in denoising, DURR is able to handle the problem of real image denoising without additional training. For testing, we test the images obtained from Lebrun et al. (2015). We present the representative results in Fig. 7 and more results are listed in the supplementary materials.
+
+We also train our model for blind color image denoising, please refer to the supplementary materials for more details.
+
+# 3.3 JPEG IMAGE DEBLOCKING
+
+For deep learning based models, we select DnCNN-3 (Zhang et al., 2017a) for comparisons since it is the only known deep model for multiple QFs deblocking. As the AR-CNN (Dong et al., 2015) is a commonly used baseline, we re-train the AR-CNN on a training set with mixed QFs and denote this model as AR-CNN-B. Original AR-CNN as well as a non-learning-based method SA-DCT (Foi et al., 2007) are also tested. The quality factors are assumed known for these models.
+
+Table 2: Average PSNR (dB) results for gray image denoising on the BSD68 dataset. Values with ∗ means the corresponding noise level is not present in the training data of the model. The best results are indicated in red.   
+
+<table><tr><td></td><td>BM3D</td><td>WNNM</td><td>DnCNN-B</td><td>UNLNet5</td><td>DURR</td></tr><tr><td>σ=25</td><td>28.55</td><td>28.73</td><td>29.16</td><td>28.96</td><td>29.16</td></tr><tr><td>9 二 35</td><td>27.07</td><td>27.28</td><td>27.66</td><td>27.50</td><td>27.72</td></tr><tr><td>g= 45</td><td>25.99</td><td>26.26</td><td>26.62</td><td>26.48</td><td>26.71</td></tr><tr><td>0 = ：55</td><td>25.26</td><td>25.49</td><td>25.80</td><td>25.64</td><td>25.91</td></tr><tr><td>g= 65</td><td>24.69</td><td>24.51</td><td>23.40*</td><td>1</td><td>25.26*</td></tr><tr><td>σ=75</td><td>22.63</td><td>22.71</td><td>18.73*</td><td>=</td><td>24.71*</td></tr></table>
+
+![](images/730e2ff7c56f5b60b821847bc21e00371b2073b674705befc58d3d5e514a40b6.jpg)  
+Figure 5: Denoising results of an image from BSD68 with noise level 35.
+
+![](images/f144ebee89bc37164941004353b3fc95d7782634d8ba2eacdb315062efcd2c5b.jpg)  
+Figure 6: Denoising results of an image from BSD68 with noise level 65 (unseen by both DnCNN and DURR in their training sets).
+
+Quantitative results are shown in Table 3. Though the number of parameters of DURR is significantly less than the DnCNN-3, the proposed DURR outperforms DnCNN-3 in most cases. Specifically, considerable gains can be observed for our model on seen QFs, and the performances are comparable on unseen QFs. A representative result on the LIVE1 dataset is presented in Fig. 8. Our model generates the most clean and accurate details. More experiment details are given in the supplementary materials.
+
+Table 3: The average PSNR(dB) on the LIVE1 dataset. Values with ∗ means the corresponding QF is not present in the training data of the model. The best results are indicated in red and the second best results are indicated in blue.   
+
+<table><tr><td>QF</td><td>JPEG</td><td>SA-DCT</td><td>AR-CNN</td><td>AR-CNN-B</td><td>DnCNN-3</td><td>DURR</td></tr><tr><td>10</td><td>27.77</td><td>28.65</td><td>28.98</td><td>28.53</td><td>29.40</td><td>29.23*</td></tr><tr><td>20</td><td>30.07</td><td>30.81</td><td>31.29</td><td>30.88</td><td>31.59</td><td>31.68</td></tr><tr><td>30</td><td>31.41</td><td>32.08</td><td>32.69</td><td>32.31</td><td>32.98</td><td>33.05</td></tr><tr><td>40</td><td>32.45</td><td>32.99</td><td>33.63</td><td>33.39</td><td>33.96</td><td>34.01*</td></tr></table>
+
+![](images/b7d8f79a5dabb17ddd97b6885d14884e9c6a18a074d11c3c74bfea9f7dbc46e7.jpg)  
+Figure 7: Denoising results on a real image from Lebrun et al. (2015).
+
+# 3.4 OTHER APPLICATIONS
+
+Our model can be easily extended to other applications such as deraining, dehazing and deblurring. In all these applications, there are images corrupted at different levels. Rainfall intensity, haze density and different blur kernels will all effect the image quality.
+
+# 4 CONCLUSIONS
+
+In this paper, we proposed a novel image restoration model based on the moving endpoint control in order to handle varied noise levels using a single model. The problem was solved by jointly optimizing two units: restoration unit and policy unit. The restoration unit used an RNN to realize the dynamics in the control problem. A policy unit was proposed for the policy unit to determine the loop times of the restoration unit for optimal results. Our model achieved the state-of-the-art results in blind image denoising and JPEG deblocking. Moreover, thanks to the flexibility of the given policy, DURR has shown strong abilities of generalization in our experiments.
+
+# ACKNOWLEDGMENTS
+
+Bin Dong is supported in part by Beijing Natural Science Foundation (Z180001).Yiping Lu is supported by the Elite Undergraduate Training Program of the School of Mathematical Sciences at Peking University.
+
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+        "type": "text",
+        "text": "DYNAMICALLY UNFOLDING RECURRENT RESTORER:A MOVING ENDPOINT CONTROL METHOD FOR IMAGERESTORATION",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Xiaoshuai Zhang∗   \nInstitute of Computer Science and Technology,   \nPeking University   \njet@pku.edu.cn Yiping Lu∗   \nSchool Of Mathmatical Science, Peking university   \nluyiping9712@pku.edu.cn   \nJiaying Liu   \nInstitute of Computer Science and Technology,   \nPeking University   \nliujiaying@pku.edu.cn   \nBin Dong   \nBeijing International Center for Mathematical Research, Peking University   \nCenter for Data Science, Peking University   \nBeijing Institute of Big Data Research,   \nBeijing, China   \ndongbin@math.pku.edu.cn ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
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+            484
+        ],
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+    },
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+        "type": "text",
+        "text": "In this paper, we propose a new control framework called the moving endpoint control to restore images corrupted by different degradation levels using a single model. The proposed control problem contains an image restoration dynamic which is modeled by a convolutional RNN. The moving endpoint, which is essentially the terminal time of the associated dynamic, is determined by a policy network. We call the proposed model the dynamically unfolding recurrent restorer (DURR). Numerical experiments show that DURR is able to achieve state-of-the-art performances on blind image denoising and JPEG image deblocking. Furthermore, DURR can well generalize to images with higher degradation levels that are not included in the training stage.1 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            336,
+            686
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Image restoration, including image denoising, deblurring, inpainting, etc., is one of the most important areas in imaging science. Its major purpose is to obtain high quality reconstructions of images corrupted in various ways during imaging, acquisiting, and storing, and enable us to see crucial but subtle objects that reside in the images. Image restoration has been an active research area. Numerous models and algorithms have been developed for the past few decades. Before the uprise of deep learning methods, there were two classes of image restoration approaches that were widely adopted in the field: transformation based approach and PDE approach. The transformation based approach includes wavelet and wavelet frame based methods (Elad et al., 2005; Starck et al., 2005; Daubechies et al., 2007; Cai et al., 2009), dictionary learning based methods (Aharon et al., 2006), similarity based methods (Buades et al., 2005; Dabov et al., 2007), low-rank models (Ji et al., 2010; Gu et al., 2014), etc. The PDE approach includes variational models (Mumford & Shah, 1989; Rudin et al., 1992; Bredies et al., 2010), nonlinear diffusions (Perona & Malik, 1990; Catté et al., 1992; Weickert, 1998), nonlinear hyperbolic equations (Osher & Rudin, 1990), etc. More recently, deep connections ",
+        "bbox": [
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+            885
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "between wavelet frame based methods and PDE approach were established (Cai et al., 2012; 2016;   \nDong et al., 2017). ",
+        "bbox": [
+            173,
+            103,
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+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One of the greatest challenge for image restoration is to properly handle image degradations of different levels. In the existing transformation based or PDE based methods, there is always at least one tuning parameter (e.g. the regularization parameter for variational models and terminal time for nonlinear diffusions) that needs to be manually selected. The choice of the parameter heavily relies on the degradation level. ",
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+        "type": "text",
+        "text": "Recent years, deep learning models for image restoration tasks have significantly advanced the state-of-the-art of the field. Jain & Seung (2009) proposed a convolutional neural network (CNN) for image denoising which has better expressive power than the MRF models by Lan et al. (2006). Inspired by nonlinear diffusions, Chen & Pock (2017) designed a deep neural network for image denoising and Zhang et al. (2017a) improves the capacity by introducing a deeper neural network with residual connections. Chen et al. (2017) use the CNN to simulate a wide variety of image processing operators, achieving high efficiencies with little accuracy drop. However, these models cannot gracefully handle images with varied degradation levels. Although one may train different models for images with different levels, this may limit the application of these models in practice due to lack of flexibility. ",
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+        "type": "text",
+        "text": "Taking blind image denoising for example. Zhang et al. (2017a) designed a 20-layer neural network for the task, called DnCNN-B, which had a huge number of parameters. To reduce number of parameters, Lefkimmiatis (2017) proposed the UNLNet5, by unrolling a projection gradient algorithm for a constrained optimization model. However, Lefkimmiatis (2017) also observed a drop in PSNR comparing to DnCNN. Therefore, the design of a light-weighted and yet effective model for blind image denoising remains a challenge. Moreover, deep learning based models trained on simulated gaussian noise images usually fail to handle real world noise, as will be illustrated in later sections. ",
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+    {
+        "type": "text",
+        "text": "Another example is JPEG image deblocking. JPEG is the most commonly used lossy image compression method. However, this method tend to introduce undesired artifacts as the compression rate increases. JPEG image deblocking aims to eliminate the artifacts and improve the image quality. Recently, deep learning based methods were proposed for JPEG deblocking (Dong et al., 2015; Zhang et al., 2017a; 2018). However, most of their models are trained and evaluated on a given quality factor. Thus it would be hard for these methods to apply to Internet images, where the quality factors are usually unknown. ",
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+        "type": "text",
+        "text": "In this paper, we propose a single image restoration model that can robustly restore images with varied degradation levels even when the degradation level is well outside of that of the training set. Our proposed model for image restoration is inspired by the recent development on the relation between deep learning and optimal control. The relation between supervised deep learning methods and optimal control has been discovered and exploited by Weinan (2017); Lu et al. (2018); Chang et al. (2017); Fang et al. (2017). The key idea is to consider the residual block $x _ { n + 1 } = x _ { n } + f ( x _ { n } )$ as an approximation to the continuous dynamics ${ \\dot { X } } = f ( X )$ . In particular, Lu et al. (2018); Fang et al. (2017) demonstrated that the training process of a class of deep models (e.g. ResNet by He et al. (2016), PolyNet by Zhang et al. (2017b), etc.) can be understood as solving the following control problem: ",
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+        "img_path": "images/30386db8f325d4f10b2ffb94c1937ad4c9b086dc83bcb4b948ba70ba89fb8177.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { w } \\bigg ( L ( X ( T ) , y ) + \\int _ { 0 } ^ { \\tau } R ( w ( t ) , t ) d t \\bigg ) } \\\\ { \\displaystyle s . t . \\dot { X } = f ( X ( t ) , w ( t ) ) , t \\in ( 0 , \\tau ) } \\\\ { \\displaystyle X ( 0 ) = x _ { 0 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "Here $x _ { 0 }$ is the input, $y$ is the regression target or label, $\\dot { \\boldsymbol X } = f ( \\boldsymbol X , \\boldsymbol w )$ is the deep neural network with parameter $w ( t )$ , $R$ is the regularization term and $L$ can be any loss function to measure the difference between the reconstructed images and the ground truths. ",
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+        "type": "text",
+        "text": "In the context of image restoration, the control dynamic $\\dot { X } = f ( X ( t ) , \\omega ( t ) ) , t \\in ( 0 , \\tau )$ can be, for example, a diffusion process learned using a deep neural network. The terminal time $\\tau$ of the diffusion corresponds to the depth of the neural network. Previous works simply fixed the depth of the network, i.e. the terminal time, as a fixed hyper-parameter. However Mrázek & Navara (2003) showed that the optimal terminal time of diffusion differs from image to image. Furthermore, when an image is corrupted by higher noise levels, the optimal terminal time for a typical noise removal diffusion should be greater than when a less noisy image is being processed. This is the main reason why current deep models are not robust enough to handle images with varied noise levels. In this paper, we no longer treat the terminal time as a hyper-parameter. Instead, we design a new architecture (see Fig. 3) that contains both a deep diffusion-like network and another network that determines the optimal terminal time for each input image. We propose a novel moving endpoint control model to train the aforementioned architecture. We call the proposed architecture the dynamically unfolding recurrent restorer (DURR). ",
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+        "type": "text",
+        "text": "We first cast the model in the continuum setting. Let $x _ { 0 }$ be an observed degraded image and $y$ be its corresponding damage-free counterpart. We want to learn a time-independent dynamic system $\\dot { X } = f ( X ( t ) , w )$ with parameters $w$ so that $X ( 0 ) = x$ and $X ( \\tau ) \\approx y$ for some $\\tau > 0$ . See Fig. 2 for an illustration of our idea. The reason that we do not require $X ( \\tau ) = y$ is to avoid over-fitting. For varied degradation levels and different images, the optimal terminal time $\\tau$ of the dynamics may vary. Therefore, we need to include the variable $\\tau$ in the learning process as well. The learning of the dynamic system and the terminal time can be gracefully casted as the following moving endpoint control problem: ",
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+        "img_path": "images/0e29ac0d65669d7658fffc8837268796e5879edefaa1a778a88fe7434159199a.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { w , \\tau ( x ) } L ( X ( \\tau ) , y ) + \\int _ { 0 } ^ { \\tau ( x ) } R ( w ( t ) , t ) d t } \\\\ { \\displaystyle s . t . \\dot { X } = f ( X ( t ) , w ( t ) ) , t \\in ( 0 , \\tau ( x ) ) } \\\\ { \\displaystyle X ( 0 ) = x . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Different from the previous control problem, in our model the terminal time $\\tau$ is also a parameter to be optimized and it depends on the data $x$ . The dynamic system $\\dot { X } = f ( X ( t ) , w )$ is modeled by a recurrent neural network (RNN) with a residual connection, which can be understood as a residual network with shared weights (Liao & Poggio, 2016). We shall refer to this RNN as the restoration unit. In order to learn the terminal time of the dynamics, we adopt a policy network to adaptively determine an optimal stopping time. Our learning framework is demonstrated in Fig. 3. We note that the above moving endpoint control problem can be regarded as the penalized version of the well-known fixed endpoint control problem in optimal control (Evans, 2005), where instead of penalizing the difference between $X ( \\tau )$ and $y$ , the constraint $X ( \\tau ) = y$ is strictly enforced. ",
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+        "type": "text",
+        "text": "In short, we summarize our contribution as following: ",
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+        "text": "• We are the first to use convolutional RNN for image restoration with unknown degradation levels, where the unfolding time of the RNN is determined dynamically at run-time by a policy unit (could be either handcrafted or RL-based).   \n• The proposed model achieves state-of-the-art performances with significantly less parameters and better running efficiencies than some of the state-of-the-art models.   \nWe reveal the relationship between the generalization power and unfolding time of the RNN by extensive experiments. The proposed model, DURR, has strong generalization to images with varied degradation levels and even to the degradation level that is unseen by the model during training (Fig. 1).   \nThe DURR is able to well handle real image denoising without further modification. Qualitative results have shown that our processed images have better visual quality, especially sharper details compared to others. ",
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The proposed architecture, i.e. DURR, contains an RNN (called the restoration unit) imitating a nonlinear diffusion for image restoration, and a deep policy network (policy unit) to determine the terminal time of the RNN. In this section, we discuss the training of the two components based on our moving endpoint control formulation. As will be elaborated, we first train the restoration unit to determine $\\omega$ , and then train the policy unit to estimate $\\tau ( x )$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/b9994af11293ec16af192ae7ba007b3f5e40a1eac061fe66610f8cf89429001c.jpg",
+        "image_caption": [
+            "Figure 1: Denoising results of images from BSD68 under extreme noise conditions not seen in training data $( \\sigma = 9 5 $ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1230eb448c1f9a9db33389714ffbb02e617438d1676bcef3d0ffcd6a69481f05.jpg",
+        "image_caption": [
+            "Figure 2: The proposed moving endpoint control model: evolving a learned reconstruction dynamics and ending at high-quality images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 TRAINING THE RESTORATION UNIT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If the terminal time $\\tau$ for every input $x _ { i }$ is given (i.e. given a certain policy), the restoration unit can be optimized accordingly. We would like to show in this section that the policy used during training greatly influences the performance and the generalization ability of the restoration unit. More specifically, a restoration unit can be better trained by a good policy. ",
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+        "type": "text",
+        "text": "The simplest policy is to fix the loop time $\\tau$ as a constant for every input. We name such policy as “naive policy”. A more reasonable policy is to manually assign an unfolding time for each degradation level during training. We shall call this policy the “refined policy”. Since we have not trained the policy unit yet, to evaluate the performance of the trained restoration units, we manually pick the output image with the highest PSNR (i.e. the peak PSNR). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We take denoising as an example here. The peak PSNRs of the restoration unit trained with different policies are listed in Table. 1. Fig. 4 illustrates the average loop times when the peak PSNRs appear. The training is done on both single noise level $\\sigma = 4 0$ ) and multiple noise levels $( \\sigma = 3 5 , 4 5 $ ). For the refined policy, the noise levels and the associated loop times are (35, 6), (45, 9). For the naive policy, we always fix the loop times to 8. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/4dcb6e57d263bea9e5c4beaf15bcc3cbc502e0f8678ca1d807e74f31359f2aea.jpg",
+        "image_caption": [
+            "Figure 3: Pipeline of the dynamically unfolding recurrent restorer (DURR). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/1a674c26d63578219a36e53b31f70334181348c727a135f4c84eadcb8eafe1cd.jpg",
+        "table_caption": [
+            "Table 1: Average peak PSNR on BSD68 with different training strategies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Strategy</td><td colspan=\"7\">Noise Level</td></tr><tr><td> Training Noise</td><td>Policy</td><td>25</td><td>30</td><td>35</td><td>40</td><td>45</td><td>50</td><td>55</td></tr><tr><td>40</td><td>Naive</td><td>28.61</td><td>28.13</td><td>27.62</td><td>27.19</td><td>26.57</td><td>26.17</td><td>24.00</td></tr><tr><td>35,45</td><td>Naive</td><td>27.74</td><td>27.17</td><td>26.66</td><td>26.24</td><td>26.75</td><td>25.61</td><td>24.75</td></tr><tr><td>35,45</td><td>Refined</td><td>29.14</td><td>28.33</td><td>27.67</td><td>27.19</td><td>27.69</td><td>26.61</td><td>25.88</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "As we can see, the refined policy brings the best performance on all the noise levels including 40. The restoration unit trained for specific noise level (i.e. $\\sigma = 4 0 ^ { \\cdot }$ ) is only comparable to the one with refined policy on noise level 40. The restoration unit trained on multiple noise levels with naive policy has the worst performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These results indicate that the restoration unit has the potential to generalize on unseen degradation levels when trained with good policies. According to Fig. 4, the generalization reflects on the loop times of the restoration unit. It can be observed that the model with steeper slopes have stronger ability to generalize as well as better performances. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "According to these results, the restoration unit we used in DURR is trained using the refined policy. More specifically, for image denoising, the noise level and the associated loop times are set to (25, 4), (35, 6), (45, 9), and (55, 12). For JPEG image deblocking, the quality factor (QF) and the associated loop times are set to (20, 6) and (30, 4). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/11cb76c9d0c212380465da4678d763e9c72c6a3be2231e08908fffeed1d04edc.jpg",
+        "image_caption": [
+            "Figure 4: Average peak time on BSD68 with different training strategies. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 TRAINING THE POLICY UNIT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We discuss two approaches that can be used as policy unit: ",
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+    {
+        "type": "text",
+        "text": "Handcraft policy: Previous work (Mrázek & Navara, 2003) has proposed a handcraft policy that selects a terminal time which optimizes the correlation of the signal and noise in the filtered image. This criterion can be used directly as our policy unit, but the independency of signal and noise may not hold for some restoration tasks such as real image denoising, which has higher noise level in the low-light regions, and JPEG image deblocking, in which artifacts are highly related to the original image. Another potential stopping criterion of the diffusion is no-reference image quality assessment (Mittal et al., 2012), which can provide quality assessment to a processed image without the ground truth image. However, to the best of our knowledge, the performances of these assessments are still far from satisfactory. Because of the limitations of the handcraft policies, we will not include them in our experiments. ",
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+        "type": "text",
+        "text": "Reinforcement learning based policy: We start with a discretization of the moving endpoint problem (1) on the dataset $\\{ ( x _ { i } , \\bar { y _ { i } } ) | i = 1 , 2 , \\cdot \\cdot \\cdot , d \\}$ , where $\\{ x _ { i } \\}$ are degraded observations of the damage-free images $\\{ y _ { i } \\}$ . The discrete moving endpoint control problem is given as follows: ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/fa4f4ef969c68957da033dd7230aefe7532b7587177a67105dd980f159ed45e4.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\operatorname* { m i n } _ { w , \\{ N _ { i } \\} _ { i = 1 } ^ { d } } r ( w ) + \\sum _ { i = 1 } ^ { d } L ( X _ { N _ { i } } ^ { i } , y _ { i } ) } } \\\\ & { } & { s . t . X _ { n } ^ { i } = X _ { n - 1 } ^ { i } + \\Delta t f ( X _ { n - 1 } ^ { i } , w ) , n = 1 , 2 , \\cdots , N _ { i } , ( i = 1 , 2 , \\cdots , d ) } \\\\ & { } & { X _ { 0 } ^ { i } = x _ { i } , i = 1 , 2 , \\cdots , d . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here, $X _ { n } ^ { i } = X _ { n - 1 } ^ { i } + \\Delta t f ( X _ { n - 1 } ^ { i } , w )$ is the forward Euler approximation of the dynamics $\\dot { X } =$ $f ( X ( t ) , w )$ . The terminal time $\\{ N _ { i } \\}$ is determined by a policy network $P ( x , \\theta )$ , where $x$ is the output of the restoration unit at each iteration and $\\theta$ the set of weights. In our experiment, we simply set $r = 0$ , i.e. doesn’t introduce any regularization which might bring further benefit but is beyond this paper’s scope of discussion. In other words, the role of the policy network is to stop the iteration of the restoration unit when an ideal image restoration result is achieved. The reward function of the policy unit can be naturally defined by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/26f38e0eef942a8163562669d706f48bdba93a7eaf103ab6712007e3b218fe13.jpg",
+        "text": "$$\nr ( \\{ X _ { n } ^ { i } \\} ) = { \\left\\{ \\begin{array} { l l } { \\lambda \\left( L ( x _ { n - 1 } , y _ { i } ) - L ( x _ { n } , y _ { i } ) \\right) } & { { \\mathrm { I f ~ c h o o s e ~ t o ~ c o n t i n u e } } } \\\\ { 0 } & { { \\mathrm { O t h e r w i s e } } } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to solve the problem (2.2), we need to optimize two networks simultaneously, i.e. the restoration unit and the policy unit. The first is an restoration unit which approximates the controlled dynamics and the other is the policy unit to give the optimized terminating conditions. The objective function we use to optimize the policy network can be written as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48bac04faca34f06316467fceec937099c603b9c8b840a8d3674e28ef616dbbc.jpg",
+        "text": "$$\nJ = \\mathbb { E } _ { X \\sim \\pi _ { \\theta } } \\sum _ { n } ^ { N _ { i } } [ r ( \\{ X _ { n } ^ { i } , w \\} ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi _ { \\theta }$ denotes the distribution of the trajectories $X = \\{ X _ { n } ^ { i } , n = 1 , \\ldots , N _ { i } , i = 1 , \\ldots , d \\}$ under the policy network $P ( \\cdot , \\theta )$ . Thus, reinforcement learning techniques can be used here to learn a neural network to work as a policy unit. We utilize Deep Q-learning (Mnih et al., 2015) as our learning strategy and denote this approach simply as DURR. However, different learning strategies can be used (e.g. the Policy Gradient). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 EXPERIMENT SETTINGS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In all denoising experiments, we follow the same settings as in Chen & Pock (2017); Zhang et al. (2017a); Lefkimmiatis (2017). All models are evaluated using the mean PSNR as the quantitative metric on the BSD68 (Martin et al., 2001). The training set and test set of BSD500 (400 images) are used for training. Six gaussian noise levels are evaluated, namely $\\sigma = 2 5$ , 35, 45, 55, 65 and 75. Additive noise are applied to the image on the fly during training and testing. Both the training and evaluation process are done on gray-scale images. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The restoration unit is a simple U-Net (Ronneberger et al., 2015) style fully convolutional neural network. For the training process of the restoration unit, the noise levels of 25, 35, 45 and 55 are used. Images are cut into $6 4 \\times 6 4$ patches, and the batch-size is set to 24. The Adam optimizer with the learning rate 1e-3 is adopted and the learning rate is scaled down by a factor of 10 on training plateaux. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The policy unit is composed of two ResUnit and an LSTM cell. For the policy unit training, we utilize the reward function in Eq.4. For training the policy unit, an RMSprop optimizer with learning rate 1e-4 is adopted. We’ve also tested other network structures, these tests and the detailed network structures of our model are demonstrated in the supplementary materials. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In all JPEG deblocking experiments, we follow the settings as in Zhang et al. (2017a; 2018). All models are evaluated using the mean PSNR as the quantitative metric on the LIVE1 dataset (Sheikh, 2005). Both the training and evaluation processes are done on the Y channel (the luminance channel) of the YCbCr color space. The PIL module of python is applied to generate JPEG-compressed images. The module produces numerically identical images as the commonly used MATLAB JPEG encoder after setting the quantization tables manually. The images with quality factors 20 and 30 are used during training. De-blocking performances are evaluated on four quality factors, namely $\\mathrm { Q F = 1 0 }$ , 20, 30, and 40. All other parameter settings are the same as in the denoising experiments. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.2 IMAGE DENOISING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We select DnCNN-B(Zhang et al., 2017a) and UNLNet5 (Lefkimmiatis, 2017) for comparisons since these models are designed for blind image denoising. Moreover, we also compare our model with non-learning-based algorithms BM3D (Dabov et al., 2007) and WNNM (Gu et al., 2014). The noise levels are assumed known for BM3D and WNNM due to their requirements. Comparison results are shown in Table 2. ",
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+    },
+    {
+        "type": "text",
+        "text": "Despite the fact that the parameters of our model $( 1 . 8 \\times 1 0 ^ { 5 }$ for the restoration unit and $1 . 0 \\times 1 0 ^ { 5 }$ for the policy unit) is less than the DnCNN (approximately $7 . 0 \\times 1 0 ^ { 5 }$ ), one can see that DURR outperforms DnCNN on most of the noise-levels. More interestingly, DURR does not degrade too much when the the noise level goes beyond the level we used during training. The noise level $\\sigma = 6 5$ , 75 is not included in the training set of both DnCNN and DURR. DnCNN reports notable drops of PSNR when evaluated on the images with such noise levels, while DURR only reports small drops of PSNR (see the last row of Table 2 and Fig. 6). Note that the reason we do not provide the results of UNLNet5 in Table 2 is because the authors of Lefkimmiatis (2017) has not released their codes yet, and they only reported the noise levels from 15 to 55 in their paper. We also want to emphasize that they trained two networks, one for the low noise level $( 5 \\leq \\sigma \\leq 2 9 )$ and one for higher noise level $3 0 \\leq \\sigma \\leq 5 5$ ). The reason is that due to the use of the constraint $| | y - x | | _ { 2 } \\leq \\epsilon$ by Lefkimmiatis (2017), we should not expect the model generalizes well to the noise levels surpasses the noise level of the training set. ",
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+    {
+        "type": "text",
+        "text": "For qualitative comparisons, some restored images of different models on the BSD68 dataset are presented in Fig. 5 and Fig. 6. As can be seen, more details are preserved in DURR than other models. It is worth noting that the noise level of the input image in Fig. 6 is 65, which is unseen by both DnCNN and DURR during training. Nonetheless, DURR achieves a significant gain of nearly 1 dB than DnCNN. Moreover, the texture on the cameo is very well restored by DURR. These results clearly indicate the strong generalization ability of our model. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "More interestingly, due to the generalization ability in denoising, DURR is able to handle the problem of real image denoising without additional training. For testing, we test the images obtained from Lebrun et al. (2015). We present the representative results in Fig. 7 and more results are listed in the supplementary materials. ",
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+    },
+    {
+        "type": "text",
+        "text": "We also train our model for blind color image denoising, please refer to the supplementary materials for more details. ",
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+    {
+        "type": "text",
+        "text": "3.3 JPEG IMAGE DEBLOCKING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For deep learning based models, we select DnCNN-3 (Zhang et al., 2017a) for comparisons since it is the only known deep model for multiple QFs deblocking. As the AR-CNN (Dong et al., 2015) is a commonly used baseline, we re-train the AR-CNN on a training set with mixed QFs and denote this model as AR-CNN-B. Original AR-CNN as well as a non-learning-based method SA-DCT (Foi et al., 2007) are also tested. The quality factors are assumed known for these models. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c26d68118544016535c3019b6db245fb91f953aa319ee3df63a5652b45adcfc6.jpg",
+        "table_caption": [
+            "Table 2: Average PSNR (dB) results for gray image denoising on the BSD68 dataset. Values with ∗ means the corresponding noise level is not present in the training data of the model. The best results are indicated in red. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>BM3D</td><td>WNNM</td><td>DnCNN-B</td><td>UNLNet5</td><td>DURR</td></tr><tr><td>σ=25</td><td>28.55</td><td>28.73</td><td>29.16</td><td>28.96</td><td>29.16</td></tr><tr><td>9 二 35</td><td>27.07</td><td>27.28</td><td>27.66</td><td>27.50</td><td>27.72</td></tr><tr><td>g= 45</td><td>25.99</td><td>26.26</td><td>26.62</td><td>26.48</td><td>26.71</td></tr><tr><td>0 = ：55</td><td>25.26</td><td>25.49</td><td>25.80</td><td>25.64</td><td>25.91</td></tr><tr><td>g= 65</td><td>24.69</td><td>24.51</td><td>23.40*</td><td>1</td><td>25.26*</td></tr><tr><td>σ=75</td><td>22.63</td><td>22.71</td><td>18.73*</td><td>=</td><td>24.71*</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/730e2ff7c56f5b60b821847bc21e00371b2073b674705befc58d3d5e514a40b6.jpg",
+        "image_caption": [
+            "Figure 5: Denoising results of an image from BSD68 with noise level 35. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f144ebee89bc37164941004353b3fc95d7782634d8ba2eacdb315062efcd2c5b.jpg",
+        "image_caption": [
+            "Figure 6: Denoising results of an image from BSD68 with noise level 65 (unseen by both DnCNN and DURR in their training sets). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Quantitative results are shown in Table 3. Though the number of parameters of DURR is significantly less than the DnCNN-3, the proposed DURR outperforms DnCNN-3 in most cases. Specifically, considerable gains can be observed for our model on seen QFs, and the performances are comparable on unseen QFs. A representative result on the LIVE1 dataset is presented in Fig. 8. Our model generates the most clean and accurate details. More experiment details are given in the supplementary materials. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/f771c5f4b796ce5d612f74ae9fef533bf66be14bd2a7fc918137b381133df246.jpg",
+        "table_caption": [
+            "Table 3: The average PSNR(dB) on the LIVE1 dataset. Values with ∗ means the corresponding QF is not present in the training data of the model. The best results are indicated in red and the second best results are indicated in blue. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>QF</td><td>JPEG</td><td>SA-DCT</td><td>AR-CNN</td><td>AR-CNN-B</td><td>DnCNN-3</td><td>DURR</td></tr><tr><td>10</td><td>27.77</td><td>28.65</td><td>28.98</td><td>28.53</td><td>29.40</td><td>29.23*</td></tr><tr><td>20</td><td>30.07</td><td>30.81</td><td>31.29</td><td>30.88</td><td>31.59</td><td>31.68</td></tr><tr><td>30</td><td>31.41</td><td>32.08</td><td>32.69</td><td>32.31</td><td>32.98</td><td>33.05</td></tr><tr><td>40</td><td>32.45</td><td>32.99</td><td>33.63</td><td>33.39</td><td>33.96</td><td>34.01*</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/b7d8f79a5dabb17ddd97b6885d14884e9c6a18a074d11c3c74bfea9f7dbc46e7.jpg",
+        "image_caption": [
+            "Figure 7: Denoising results on a real image from Lebrun et al. (2015). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.4 OTHER APPLICATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our model can be easily extended to other applications such as deraining, dehazing and deblurring. In all these applications, there are images corrupted at different levels. Rainfall intensity, haze density and different blur kernels will all effect the image quality. ",
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+    {
+        "type": "text",
+        "text": "4 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a novel image restoration model based on the moving endpoint control in order to handle varied noise levels using a single model. The problem was solved by jointly optimizing two units: restoration unit and policy unit. The restoration unit used an RNN to realize the dynamics in the control problem. A policy unit was proposed for the policy unit to determine the loop times of the restoration unit for optimal results. Our model achieved the state-of-the-art results in blind image denoising and JPEG deblocking. Moreover, thanks to the flexibility of the given policy, DURR has shown strong abilities of generalization in our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Bin Dong is supported in part by Beijing Natural Science Foundation (Z180001).Yiping Lu is supported by the Elite Undergraduate Training Program of the School of Mathematical Sciences at Peking University. ",
+        "bbox": [
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+    },
+    {
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parse/train/SJl2niR9KQ/SJl2niR9KQ.md

@@ -0,0 +1,629 @@
+# BEYOND PIXEL NORM-BALLS: PARAMETRIC ADVERSARIES USING AN ANALYTICALLY DIFFERENTIABLE RENDERER
+
+Hsueh-Ti Derek Liu University of Toronto hsuehtil@cs.toronto.edu
+
+Michael Tao University of Toronto mtao@dgp.toronto.edu
+
+Chun-Liang Li Carnegie Mellon University chunlial@cs.cmu.edu
+
+Derek Nowrouzezahrai McGill University derek@cim.mcgill.ca
+
+Alec Jacobson University of Toronto jacobson@cs.toronto.edu
+
+# ABSTRACT
+
+Many machine learning image classifiers are vulnerable to adversarial attacks, inputs with perturbations designed to intentionally trigger misclassification. Current adversarial methods directly alter pixel colors and evaluate against pixel norm-balls: pixel perturbations smaller than a specified magnitude, according to a measurement norm. This evaluation, however, has limited practical utility since perturbations in the pixel space do not correspond to underlying real-world phenomena of image formation that lead to them and has no security motivation attached. Pixels in natural images are measurements of light that has interacted with the geometry of a physical scene. As such, we propose a novel evaluation measure, parametric normballs, by directly perturbing physical parameters that underly image formation. One enabling contribution we present is a physically-based differentiable renderer that allows us to propagate pixel gradients to the parametric space of lighting and geometry. Our approach enables physically-based adversarial attacks, and our differentiable renderer leverages models from the interactive rendering literature to balance the performance and accuracy trade-offs necessary for a memory-efficient and scalable adversarial data augmentation workflow.
+
+# 1 INTRODUCTION
+
+Research in adversarial examples continues to contribute to the development of robust (semi-)supervised learning (Miyato et al., 2018), data augmentation (Goodfellow et al., 2015; Sun et al., 2018), and machine learning understanding (Kanbak et al., 2018). One important caveat of the approach pursued by much of the literature in adversarial machine learning, as discussed recently (Goodfellow,
+
+![](images/6b3cf82b25aae0a2ddeeb7f9414d2d23acd8eac439fd20752d04391cd224023a.jpg)  
+Figure 1: Traditional pixel-based adversarial attacks yield unrealistic images under a larger perturbation $( L ^ { \infty } \mathrm { - n o r m \approx 0 . 8 2 } )$ , however our parametric lighting and geometry perturbations output more realistic images under the same norm (more results in Appendix A).
+
+![](images/0225063af5b871e53180a24d9f50d54d9370c22a3f1a7fb0f1ae5c6b058dfdd8.jpg)  
+Figure 2: Parametrically-perturbed images remain natural, whereas pixel-perturbed ones do not.
+
+2018; Gilmer et al., 2018), is the reliance on overly simplified attack metrics: namely, the use of pixel value differences between an adversary and an input image, also referred to as the pixel norm-balls.
+
+The pixel norm-balls game considers pixel perturbations of norm-constrained magnitude (Goodfellow et al., 2015), and is used to develop adversarial attackers, defenders and training strategies. The pixel norm-ball game is attractive from a research perspective due to its simplicity and well-posedness: no knowledge of image formation is required and any arbitrary pixel perturbation remains eligible (so long as it is “small”, in the perceptual sense). Although the pixel norm-ball is useful for research purposes, it only captures limited real-world security scenarios.
+
+Despite the ability to devise effective adversarial methods through the direct employment of optimizations using the pixel norm-balls measure, the pixel manipulations they promote are divorced from the types of variations present in the real world, limiting their usefulness “in the wild”. Moreover, this methodology leads to defenders that are only effective when defending against unrealistic images/attacks, not generalizing outside of the space constrained by pixel norm-balls. In order to consider conditions that enable adversarial attacks in the real world, we advocate for a new measurement norm that is rooted in the physical processes that underly realistic image synthesis, moving away from overly simplified metrics, e.g., pixel norm-balls.
+
+Our proposed solution – parametric norm-balls – rely on perturbations of physical parameters of a synthetic image formation model, instead of pixel color perturbations (Figure 2). To achieve this, we use a physically-based differentiable renderer which allows us to perturb the underlying parameters of the image formation process. Since these parameters indirectly control pixel colors, perturbations in this parametric space implicitly span the space of natural images. We will demonstrate two advantages that fall from considering perturbations in this parametric space: (1) they enable adversarial approaches that more readily apply to real-world applications, and (2) they permit the use of much more significant perturbations (compared to pixel norms), without invalidating the realism of the resulting image (Figure 1). We validate that parametric norm-balls game playing is critical for a variety of important adversarial tasks, such as building defenders robust to perturbations that can occur naturally in the real world.
+
+We perform perturbations in the underlying image formation parameter space using a novel physicallybased differentiable renderer. Our renderer analytically computes the derivatives of pixel color with respect to these physical parameters, allowing us to extend traditional pixel norm-balls to physicallyvalid parametric norm-balls. Notably, we demonstrate perturbations on an environment’s lighting and on the shape of the 3D geometry it shades. Our differentiable renderer achieves state-of-the-art performance in speed and scalability (Section 3) and is fast enough for rendered adversarial data augmentation (Section 5): training augmented with adversarial images generated with a renderer.
+
+Existing differentiable renders are slow and do not scalable to the volume of high-quality, highresolutions images needed to make adversarial data augmentation tractable (Section 2). Given our analytically-differentiable renderer (Section 3), we are able to demonstrate the efficacy of parametric space perturbations for generating adversarial examples. These adversaries are based on a substantially different phenomenology than their pixel norm-balls counterparts (Section 4). Ours is among the first steps towards the deployment of rendered adversarial data augmentation in real-world applications: we train a classifier with computer-generated adversarial images, evaluating the performance of the training against real photographs (i.e., captured using cameras; Section 5). We test on real photos to show the parametric adversarial data augmentation increases the classifier’s robustness to “deformations” happened in the real world. Our evaluation differs from the majority of existing literature which evaluates against computer-generated adversarial images, since our parametric space perturbation is no-longer a wholly idealized representation of the image formation model but, instead, modeled against of theory of realistic image generation.
+
+# 2 RELATED WORK
+
+Our work is built upon the fact that simulated or rendered images can participate in computer vision and machine learning on real-world tasks. Many previous works use rendered (simulated) data to train deep networks, and those networks can be deployed to real-world or even outperform the state-of-the-art networks trained on real photos (Movshovitz-Attias et al., 2016; Chen et al., 2016; Varol et al., 2017; Su et al., 2015; Johnson-Roberson et al., 2017; Veeravasarapu et al., 2017b; Sadeghi & Levine, 2016; James & Johns, 2016). For instance, Veeravasarapu et al. (2017a) show that training with $1 0 \%$ real-world data and $9 0 \%$ simulation data can reach the level of training with full real data. Tremblay et al. (2018) even demonstrate that the network trained on synthetic data yields a better performance than using real data alone. As rendering can cheaply provide a theoretically infinite supply of annotated input data, it can generate data which is orders of magnitude larger than existing datasets. This emerging trend of training on synthetic data provides an exciting direction for future machine learning development. Our work complements these works. We demonstrate the utility of rendering can be used to study the potential danger lurking in misclassification due to subtle changes to geometry and lighting. This provides a future direction of combining with synthetic data generation pipelines to perform physically based adversarial training on synthetic data.
+
+Adversarial Examples Szegedy et al. (2014) expose the vulnerability of modern deep neural nets using purposefully-manipulated images with human-imperceptible misclassification-inducing noise. Goodfellow et al. (2015) introduce a fast method to harness adversarial examples, leading to the idea of pixel norm-balls for evaluating adversarial attackers/defenders. Since then, many significant developments in adversarial techniques have been proposed (Akhtar & Mian, 2018; Szegedy et al., 2014; Rozsa et al., 2016; Kurakin et al., 2017; Moosavi Dezfooli et al., 2016; Dong et al., 2018; Papernot et al., 2017; Moosavi-Dezfooli et al., 2017; Chen et al., 2017; Su et al., 2017). Our work extends this progression in constructing adversarial examples, a problem that lies at the foundation of adversarial machine learning. Kurakin et al. (2016) study the transferability of attacks to the physical world by printing then photographing adversarial images. Athalye et al. (2017) and Eykholt et al. (2018) propose extensions to non-planar (yet, still fixed) geometry and multiple viewing angles. These works still rely fundamentally on the direct pixel or texture manipulation on physical objects. Since these methods assume independence between pixels in the image or texture space they remain variants of pixel norm-balls. This leads to unrealistic attack images that cannot model real-world scenarios (Goodfellow, 2018; Hendrycks & Dietterich, 2018; Gilmer et al., 2018). Zeng et al. (2017) generate adversarial examples by altering physical parameters using a rendering network (Liu et al., 2017) trained to approximate the physics of realistic image formation. This data-driven approach leads to an image formation model biased towards the rendering style present in the training data. This method also relies on differentiation through the rendering network in order to compute adversaries, which requires high-quality training on a large amount of data. Even with perfect training, in their reported performance, it still requires 12 minutes on average to find new adversaries, we only take a few seconds Section 4.1. Our approach is based on a differentiable physically-based renderer that directly (and, so, more convincingly) models the image formation process, allowing us to alter physical parameters – like geometry and lighting – and compute
+
+derivatives (and adversarial examples) much more rapidly compared to the (Zeng et al., 2017). We summarize the difference between our approach and the previous non-image adversarial attacks in Table 1.
+
+Table 1: Previous non-pixel attacks fall short in either the parameter range they can take derivatives or the performance.   
+
+<table><tr><td>Methods</td><td>Perf.</td><td>Color</td><td>Normal</td><td>Material</td><td>Light</td><td>Geo.</td></tr><tr><td>Athalye 17</td><td>√</td><td>√</td><td></td><td></td><td></td><td></td></tr><tr><td>Zeng 17</td><td></td><td></td><td>√</td><td>√</td><td></td><td></td></tr><tr><td>Ours</td><td>√</td><td>√</td><td></td><td></td><td>√</td><td>√</td></tr></table>
+
+Differentiable Renderer Applying parametric norm-balls requires that we differentiate the image formation model with respect to the physical parameters of the image formation model. Modern realistic computer graphics models do not expose facilities to directly accommodate the computation of derivatives or automatic differentiation of pixel colors with respect to geometry and lighting variables. A physically-based
+
+Table 2: Previous differentiable renderers fall short in one way or another among Performance, Bias, or Accuracy.   
+
+<table><tr><td>Methods</td><td>Perf.</td><td>Unbias</td><td>Accu.</td></tr><tr><td>NN proxy (Liu 17)</td><td></td><td></td><td></td></tr><tr><td>Approx. (Kato 18)</td><td>?</td><td>√</td><td></td></tr><tr><td>Autodiff (Loper 14)</td><td></td><td></td><td>&lt;</td></tr><tr><td>Analytical (Ours)</td><td>√</td><td>&lt;</td><td></td></tr></table>
+
+differentiable renderer is fundamental to computing derivative of pixel colors with respect to scene parameters and can benefit machine learning in several ways, including promoting the development of novel network architectures (Liu et al., 2017), in computing adversarial examples (Athalye et al., 2017; Zeng et al., 2017), and in generalizing neural style transfer to a 3D context (Kato et al., 2018; Liu et al., 2018). Recently, various techniques have been proposed to obtain these derivatives: Wu et al. (2017); Liu et al. (2017); Eslami et al. (2016) use neural networks to learn the image formation process provided a large amount of input/output pairs. This introduces unnecessary bias in favor of the training data distribution, leading to inaccurate derivatives due to imperfect learning. Kato et al. (2018) propose a differentiable renderer based on a simplified image formation model and an underlying linear approximation. Their approach requires no training and is unbiased, but their approximation of the image formation and the derivatives introduce more errors. Loper & Black (2014); Genova et al. (2018) use automatic differentiation to build fully differentiable renderers. These renderers, however, are expensive to evaluate, requiring orders of magnitude more computation and much larger memory footprints compared to our method.
+
+Our novel differentiable renderer overcomes these limitations by efficiently computing analytical derivatives of a physically-based image formation model. The key idea is that the non-differentiable visibility change can be ignored when considering infinitesimal perturbations. We model image variations by changing geometry and realistic lighting conditions in an analytically differentiable manner, relying on an accurate model of diffuse image formation that extend spherical harmonicsbased shading methods (Appendix C). Our analytic derivatives are efficient to evaluate, have scalable memory consumption, are unbiased, and are accurate by construction (Table 2). Our renderer explicitly models the physics of the image formation processes, and so the images it generates are realistic enough to illicit correct classifications from networks trained on real-world photographs.
+
+# 3 ADVERSARIAL ATTACKS IN PARAMETRIC SPACES
+
+Adversarial attacks based on pixel norm-balls typically generate adversarial examples by defining a cost function over the space of images $\mathcal { C } : I  \mathbb { R }$ that enforces some intuition of what failure should look like, typically using variants of gradient descent where the gradient $\partial { \mathcal { C } } / \partial I$ is accessible by differentiating through networks (Szegedy et al., 2014; Goodfellow et al., 2015; Rozsa et al., 2016; Kurakin et al., 2017; Moosavi Dezfooli et al., 2016; Dong et al., 2018).
+
+The choices for $\mathcal { C }$ include increasing the cross-entropy loss of the correct class (Goodfellow et al., 2015), decreasing the cross-entropy loss of the least-likely class (Kurakin et al., 2017), using a combination of cross-entropies (Moosavi Dezfooli et al., 2016), and more (Szegedy et al., 2014; Rozsa et al., 2016; Dong et al., 2018; Tramèr et al., 2017). We combine of cross-entropies to provide flexibility for choosing untargeted and targeted attacks by specifying a different set of labels:
+
+$$
+\mathcal { C } \big ( I ( U , V ) \big ) = - \mathrm { C r o s s E n t r o p y } \big ( f ( I ( U , V ) ) , L _ { d } \big ) + \mathrm { C r o s s E n t r o p y } \big ( f ( I ( U , V ) ) , L _ { i } \big ) ,
+$$
+
+where $I$ is the image, $f ( I )$ is the output of the classifier, $L _ { d } , L _ { i }$ are labels which a user wants to decrease and increase the predicted confidences respectively. In our experiments, $L _ { d }$ is the correct class and $L _ { i }$ is either ignored or chosen according to user preference. Our adversarial attacks in the parametric space consider an image $I ( U , V )$ is the function of physical parameters of the image formation model, including the lighting $U$ and the geometry $V$ . Adversarial examples constructed by perturbing physical parameters can then be computed via the chain rule
+
+$$
+\frac { \partial \mathcal { C } } { \partial U } = \frac { \partial \mathcal { C } } { \partial I } \frac { \partial I } { \partial U } \qquad \frac { \partial \mathcal { C } } { \partial V } = \frac { \partial \mathcal { C } } { \partial I } \frac { \partial U } { \partial V } ,
+$$
+
+where $\partial I / \partial U , \partial I / \partial V$ are derivatives with respect to the physical parameters and we evaluate using our physically based differentiable renderer. In our experiments, we use gradient descent for finding parametric adversarial examples where the gradient is the direction of ${ \partial \bar { I } } / { \partial U } , { \partial I } / { \partial V }$ .
+
+# 3.1 PHYSICALLY BASED DIFFERENTIABLE RENDERER
+
+Rendering is the process of generating a 2D image from a 3D scene by simulating the physics of light. Light sources in the scene emit photons that then interact with objects in the scene. At each interaction, photons are either reflected, transmitted or absorbed, changing trajectory and repeating until arriving at a sensor such as a camera. A physically based renderer models the interactions mathematically (Pharr et al., 2016), and our task is to analytically differentiate the physical process.
+
+Top 5:   
+miniskirt $2 8 \%$ t-shirt $21 \%$   
+boot $6 \%$   
+crutch $5 \%$   
+sweatshirt $5 \%$
+
+![](images/99a7b0f58a8f92b000edf89fdac5b41abd221b3c48647a1e98c23f3399c3138a.jpg)  
+Figure 4: By changing the lighting, we fool the classifier into seeing miniskirt and water tower, demonstrating the existence of adversarial lighting.
+
+![](images/73c054c91ae8c0e6721840b84bede10f2930523f75290ced6cd9f23fb0ee7df7.jpg)  
+street sign $5 7 \%$   
+Top 5: water tower $4 8 \%$ street sign $1 8 \%$ mailbox $9 \%$ gas pump $3 \%$ barn $3 \%$
+
+![](images/a0ea287bfa9d740500a85109ceb40b6401ddf5f903a83e4fc23bc0cf16bcbe2f.jpg)  
+Figure 5: We construct a single lighting condition that can simultaneously fool the classifier viewing from different angles.
+
+We develop our differentiable renderer with common assumptions in real-time rendering (AkenineMoller et al., 2008) – diffuse material, local illumination, and distant light sources. Our diffuse material assumption considers materials which reflect lights uniformly for all directions, equivalent to considering non-specular objects. We assume that variations in the material (texture) are piece-wise constant with respect to our triangle mesh discretization. The local illumination assumption only considers lights that bounce directly from the light source to the camera. Lastly, we assume light sources are far away from the scene, allowing us to represent lighting with one spherical function. For a more detailed rationale of our assumptions, we refer readers to Appendix B).
+
+These assumptions simplify the complicated integral required for rendering (Kajiya, 1986) and allow us to represent lighting in terms of spherical harmonics, an orthonormal basis for spherical functions analogous to Fourier transformation. Thus, we can analytically differentiate the rendering equation to acquire derivatives with respect to lighting, geometry, and texture (derivations found in Appendix C).
+
+Using analytical derivatives avoids pitfalls of previous differentiable renderers (see Section 2) and make our differentiable renderer orders of magnitude faster than the previous fully differentiable renderer OPENDR (Loper & Black, 2014) (see Figure 3). Our approach is scalable to handle problems with more than
+
+![](images/0ef74421c5313d152a0fb0fb832a4beabd5aca4caf080b787a2283fac7da9563.jpg)  
+Figure 3: Our differentiable renderer based on analytical derivatives is faster and more scalable than the previous method.
+
+100,000 variables, while OPENDR runs out of memory for problems with more than 3,500 variables.
+
+# 3.2 ADVERSARIAL LIGHTING AND GEOMETRY
+
+Adversarial lighting denotes adversarial examples generated by changing the spherical harmonics lighting coefficients $U$ (Green, 2003). As our differentiable renderer allows us to compute $\partial I / \partial U$ analytically (derivation is provided in Appendix C.4), we can simply apply the chain rule:
+
+$$
+U  U - \gamma \frac { \partial \mathcal { C } } { \partial I } \frac { \partial I } { \partial U } ,
+$$
+
+where $\partial { \mathcal { C } } / \partial I$ is the derivative of the cost function with respect to pixel colors and can be obtained by differentiating through the network. Spherical harmonics act as an implicit constraint to prevent unrealistic lighting because natural lighting environments everyday life are dominated by lowfrequency signals. For instance, rendering of diffuse materials can be approximated with only $1 \%$ pixel intensity error by the first 2 orders of spherical harmonics (Ramamoorthi & Hanrahan, 2001). As computers can only represent a finite number of coefficients, using spherical harmonics for lighting implicitly filters out high-frequency, unrealistic lightings. Thus, perturbing the parametric space of spherical harmonics lighting gives us more realistic compared to image-pixel perturbations Figure 1.
+
+![](images/7a791ae776c7c36d46fcaccd704fc5fc8d5565cb724173103d7bca7f6604db5b.jpg)  
+Figure 6: By specifying different target labels, we can create an optical illusion: a jaguar is classified as cat and dog from two different views after geometry perturbations.
+
+Adversarial geometry is an adversarial example computed by changes the position of the shape’s surface. The shape is encoded as a triangle mesh with $| V |$ vertices and $| F |$ faces, surface points are vertex positions $V \in \mathbb { R } ^ { | V | \times 3 }$ which determine per-face normals $N \in \mathbb { R } ^ { | F | \times 3 }$ which in turn determine the shading of the surface. We can compute adversarial shapes by applying the chain rule:
+
+$$
+V  V - \gamma \frac { \partial \mathcal { C } } { \partial I } \frac { \partial I } { \partial N } \frac { \partial N } { \partial V } ,
+$$
+
+where $\partial I / \partial N$ is computed via a derivation in Appendix E. Each triangle only has one normal on its face, making $\partial N / \partial V$ computable analytically. In particular, the $3 \times 3$ Jacobian of a unit face normal vector $\mathbf { n } _ { i } \in \mathbb { R } ^ { 3 }$ of the $j \mathrm { t h }$ face of the triangle mesh $V$ with respect to one of its corner vertices $\mathbf { v } _ { j } \in \mathbb { R } ^ { 3 }$ is
+
+![](images/60412e1329301d85c6ea801d96abc455c89dfbac30cf0aba4835469607bdbfdb.jpg)
+
+$$
+\frac { \partial { \mathbf n } _ { i } } { \partial { \mathbf v } _ { j } } = \frac { \mathbf h _ { i j } \mathbf n _ { i } ^ { \top } } { \| \mathbf h _ { i j } \| ^ { 2 } } ,
+$$
+
+where $\mathbf { h } _ { i j } \in \mathbb { R } ^ { 3 }$ is the height vector: the shortest vector to the corner $\mathbf { v } _ { j }$ from the opposite edge.
+
+# 4 RESULTS AND EVALUATION
+
+We have described how to compute adversarial examples by parametric perturbations, including lighting and geometry. In this section, we show that adversarial examples exist in the parametric spaces, then we analyze the characteristics of those adversaries and parametric norm-balls.
+
+We use $4 9 \times 3$ spherical harmonics coefficients to represent environment lighting, with an initial realworld lighting condition (Ramamoorthi & Hanrahan, 2001). Camera parameters and the background images are empirically chosen to have correct initial classifications and avoid synonym sets. In Figure 4 we show that single-view adversarial lighting attack can fool the classifier (pre-trained ResNet-101 on ImageNet (He et al., 2016)). Figure 5 shows multi-view adversarial lighting, which optimizes the summation of the cost functions for each view, thus the gradient is computed as the summation over all camera views:
+
+$$
+U  U - \sum _ { i \in \mathrm { c a m e r a s } } \gamma \frac { \partial \mathcal { C } } { \partial I _ { i } } \frac { \partial I _ { i } } { \partial U } .
+$$
+
+If one is interested in a more specific subspace, such as outdoor lighting conditions governed by sunlight and weather, our adversarial lighting can adapt to it. In Figure 7, we compute adversarial lights over the space of skylights by applying one more chain rule to the Preetham skylight parameters (Preetham et al., 1999; Habel et al., 2008). Details about taking these derivatives are provided in Appendix D. Although adversarial skylight exists, its low degrees of freedom (only three parameters) makes it more difficult to find adversaries.
+
+![](images/932d73e839210912e1e549eade83b667f44ed03f82a76d34555b2c36ac39727c.jpg)  
+Figure 7: Even if we further constrain to a lighting subspace, skylight, we can still find adversaries.
+
+In Figure 8 and Figure 9 we show the existence of adversarial geometry in both single-view and multi-view cases. Note that we upsample meshes to have ${ \displaystyle > 1 0 \mathrm { K } }$ vertices as a preprocessing step to increase the degrees of freedom available for perturbations. Multiview adversarial geometry enables us to perturb the same 3D shape from different viewing directions, which enables us to construct a deep optical illusion: The same 3D shape are classified differently from different angles. To create the optical illusion in Figure 6, we only need to specify the $L _ { i }$ in Equation (1) to be a dog and a cat for two different views.
+
+![](images/eb2652d2330248bb7e612c93b52fc8fd91cf7e84a550c4c9d2ffdab16533ea4d.jpg)  
+Top 3: assault rifleloggerhead $8 7 \%$ , military turtle $67 \%$ uniform $6 \%$ , six-gun $1 \%$
+
+![](images/ac7bbf96a26ee0573a4240ed716442e033ef0a38c193e02f33a1502d74d42148.jpg)
+
+Top 3: slug $91 \%$ , roundworm $3 \%$ , banana $1 \%$
+
+![](images/92a13a20e794393c90ea8b24c8cedab8be8a0a0d24ab12df79c0bb0e88c4094b.jpg)  
+Figure 8: Perturbing points on 3D shapes fools the classifier into seeing rifle/slug.   
+Figure 9: We construct a single adversarial geometry that fools the classifier seeing a mailbox from different angles.
+
+# 4.1 PROPERTIES OF PARAMETRIC NORM-BALLS AND ADVERSARIES
+
+To further understand parametric adversaries, we analyze how do parametric adversarial examples generalize to black-box models. In Table 3, we test 5,000 ResNet parametric adversaries on unseen networks including AlexNet (Krizhevsky et al., 2012), DenseNet (Huang et al., 2017), SqueezeNet (Iandola et al., 2016), and VGG (Simonyan & Zisserman, 2014). Our result shows that parametric adversarial examples also share across models.
+
+In addition to different models, we evaluate parametric adversaries on black-box viewing directions. This evaluation mimics the real-world scenario that a self-driving car would “see” a stop sign from different angles while driving. In Table 4, we randomly sample 500 correctly classified views for a given shape and perform adversarial lighting and geometry algorithms only on a subset of views, then evaluate the resulting adversarial lights/shapes on all the views. The results show that adversarial lights are more generalizable to fool unseen views; adversarial shapes, yet, are less generalizable.
+
+Switching from pixel norm-balls to parametric norm-balls only requires to change the normconstraint from the pixel color space to the parametric space. For instance, we can perform a quantitative comparison between parametric adversarial and random perturbations in Figure 10. We use $L ^ { \infty }  – n o r m \ = \ 0 . 1$ to constraint the perturbed magnitude of each lighting coefficient, and $L ^ { \infty } \mathbf { - } n o r m = 0 . 0 0 2$ to constrain the maximum displacement of surface points along each axis. The results show how many parametric adversaries can fool the classifier out of 10,000 adversarial lights and shapes respectively. Not only do the
+
+![](images/407b46907c2ca9ab28fda6cee03cdac6f9fe323d682cebbc24bbbec2e2b4f120.jpg)
+
+parametric norm-balls show the effectiveness of adversarial perturbation, evaluating robustness using parametric norm-balls has real-world implications.
+
+Table 3: We evaluate ResNet adversaries on unseen models and show that parametric adversarial examples also share across models. The table shows the success rate of attacks $( \% )$ .   
+
+<table><tr><td></td><td>Alex</td><td>VGG</td><td>Squeeze</td><td>Dense</td></tr><tr><td>Lighting</td><td>81.2%</td><td>65.0%</td><td>78.6%</td><td>43.5%</td></tr><tr><td>Geometry</td><td>70.3%</td><td>58.9%</td><td>71.1%</td><td>40.1%</td></tr></table>
+
+Figure 10: A quantitative comparison using parametric norm-balls shows the fact that adversarial lighting/geometry perturbations have a higher success rate $( \% )$ in fooling classifiers comparing to random perturbations in the parametric spaces.   
+Table 4: We compute parametric adversaries using a subset of views (#Views) and evaluate the success rates $( \% )$ of attacks on unseen views.   
+
+<table><tr><td>#Views</td><td>0</td><td>1</td><td>5</td></tr><tr><td>Lighting</td><td>0.0%</td><td>29.4%</td><td>64.2%</td></tr><tr><td>Geometry</td><td>0.0%</td><td>0.6%</td><td>3.6%</td></tr></table>
+
+Runtime The inset presents our runtime per iteration for computing derivatives. An adversary normally requires less than 10 iterations, thus takes a few seconds. We evaluate our CPU PYTHON implementation and the OPENGL rendering, on an Intel Xeon 3.5GHz CPU with 64GB of RAM and an NVIDIA
+
+![](images/3889c37489b42c5f646dc682574dcd9bb43a307cff9f5cc1a7283b0df6d12564.jpg)
+
+GeForce GTX 1080. Our runtime depends on the number of pixels requiring derivatives.
+
+# 5 RENDERED ADVERSARIAL DATA AUGMENTATION AGAINST REAL PHOTOS
+
+We inject adversarial examples, generated using our differentiable renderer, into the training process of modern image classifiers. Our goal is to increase the robustness of these classifiers to real-world perturbations. Traditionally, adversarial training is evaluated against computer-generated adversarial images (Kurakin et al., 2017; Madry et al., 2018; Tramèr et al., 2017). In contrast, our evaluation differs from the majority of the literature, as we evaluate performance against real photos (i.e., images captured using a camera), and not computer-generated images. This evaluation method is motivated by our goal of increasing a classifier’s robustness to “perturbations” that occur in the real world and result from the physical processes underlying real-world image formation. We present preliminary steps towards this objective, resolving the lack of realism of pixel norm-balls and evaluating our augmented classifiers (i.e., those trained using our rendered adversaries) against real photographs.
+
+Training We train the WideResNet (16 layers, 4 wide factor) (Zagoruyko & Komodakis, 2016) on CIFAR-100 (Krizhevsky & Hinton, 2009) augmented with adversarial lighting examples. We apply a common adversarial training method that adds a fixed number of adversarial examples each epoch (Goodfellow et al., 2015; Kurakin et al., 2017). We refer readers to Appendix F for the training detail. In our experiments, we compare three training scenarios: (1) CIFAR-100, (2) CIFAR-100 $+ ~ 1 0 0$ images under random lighting, and (3) CIFAR- $1 0 0 + 1 0 0$ images under adversarial lighting. Comparing to the accuracy reported in (Zagoruyko & Komodakis, 2016), WideResNets trained on these three cases all have comparable performance $( \approx 7 7 \% )$ ) on the CIFAR-100 test set.
+
+Testing We create a test set of real photos, captured in a laboratory setting with controlled lighting and camera parameters: we photographed oranges using a calibrated Prosilica GT 1920 camera under different lighting conditions, each generated by projecting different lighting patterns using an LG PH550 projector. This hardware lighting setup projects lighting patterns from a fixed solid angle of directions onto the scene objects. Figure 11 illustrates samples from the 500 real photographs of our dataset. We evaluate the robustness of our classifier models according to test accuracy. Of note, average prediction accuracies over five trained WideResNets on our test data under the three training cases are (1) $4 . 6 \%$ , (2) $4 0 . 4 \%$ , and (3) ${ \bf 6 5 . 8 \% }$ . This result supports the fact that training on rendered images can improve the networks’ performance on real photographs. Our preliminary experiments motivate the potential of relying on rendered adversarial training to increase the robustness to visual phenomena present in the real-world inputs.
+
+![](images/5f0ff22debeefd083577da700ac31248e64be1bde67de9f4c0f5960723304b10.jpg)  
+Figure 11: Unlike much of the literature on adversarial training, we evaluate against real photos (captured by a camera), not computergenerated images. This figure illustrates a subset of our test data.
+
+# 6 LIMITATIONS & FUTURE WORK
+
+Using parametric norm-balls to remove the lack of realism of pixel norm-balls is only the first step to bring adversarial machine learning to real-world. More evaluations beyond the lab experimental data could uncover the potential of the rendered adversarial data augmentation. Coupling the differentiable renderer with methods for reconstructing 3D scenes, such as (Veeravasarapu et al., 2017b; Tremblay et al., 2018), has the potential to develop a complete pipeline for rendered adversarial training. We can take a small set of real images, constructing 3D virtual scenes which have real image statistics, using our approach to manipulate the predicted parameters to construct the parametric adversarial examples, then perform rendered adversarial training. This direction has the potential to produce limitless simulated adversarial data augmentation for real-world tasks.
+
+Our differentiable renderer models the change of realistic environment lighting and geometry. Incorporating real-time rendering techniques from the graphics community could further improve the quality of rendering. Removing the locally constant texture assumption could improve our results. Extending the derivative computation to materials could enable “adversarial materials”. Incorporating derivatives of the visibility change and propagating gradient information to shape skeleton could also create “adversarial poses”. These extensions offer a set of tools for modeling real security scenarios. For instance, we can train a self-driving car classifier that can robustly recognize pedestrians under different poses, lightings, and cloth deformations.
+
+# ACKNOWLEDGMENTS
+
+This work is funded in part by NSERC Discovery Grants (RGPIN–2017–05235 & RGPAS–2017–507938), Connaught Funds (NR2016–17), the Canada Research Chairs Program, the Fields Institute, and gifts by Adobe Systems Inc., Autodesk Inc., MESH Inc. We thank members of Dynamic Graphics Project for feedback and draft reviews; Wenzheng Chen for photography equipments; Colin Raffel and David Duvenaud for discussions and feedback.
+
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+
+# Supplementary Material
+
+# A COMPARISON BETWEEN PERTURBATION SPACES
+
+We extend our comparisons against pixel norm-balls methods (Figure 1) by visualizing the results and the generated perturbations (Figure 12). We hope this figure elucidates that our parametric perturbation are more realistic several scales of perturbations.
+
+![](images/b894a79a226ab1495f0f4ea6defbcb1e1be53b54c99696b7a28842a29469730e.jpg)  
+Figure 12: We compare our parametric perturbations (the first two columns) with pixel/color perturbations under the same $L ^ { \infty }$ pixel norm (small: 0.12, medium: 0.53, large: 0.82). As changing physical parameters corresponds to real-world phenomena, our parametric perturbation are more realistic.
+
+# B PHYSICALLY BASED RENDERING
+
+Physically based rendering (PBR) seeks to model the flow of light, typically the assumption that there exists a collection of light sources that generate light; a camera that receives this light; and a scene that modulates the flow light between the light sources and camera (Pharr et al., 2016). What follows is a brief discussion of the general task of rendering an image from a scene description and the approximations we take in order to make our renderer efficient yet differentiable.
+
+Computer graphics has dedicated decades of effort into developing methods and technologies to enable PBR to synthesize of photorealistic images under a large gamut of performance requirements. Much of this work is focused around taking approximations of the cherished Rendering equation (Kajiya, 1986), which describes the propagation of light through a point in space. If we let $u _ { o }$ be the output radiance, $p$ be the point in space, $\omega _ { o }$ be the output direction, $u _ { e }$ be the emitted radiance, $u _ { i }$ be incoming radiance, $\omega _ { i }$ be the incoming angle, $f _ { r }$ be the way light be reflected off the material at that given point in space we have:
+
+![](images/24bf33f8388785b1f364a67a6807ac27cca24248a5695254915916f1f22e6a1b.jpg)  
+Figure 13: PBR models the physics of light that emitted from the light source, interact with the scene, then arrive a camera.
+
+$$
+u _ { o } ( p , \omega _ { o } ) = u _ { e } ( p , \omega _ { o } ) + \int _ { S ^ { 2 } } f _ { r } ( p , \omega _ { i } , \omega _ { o } ) u _ { i } ( p , \omega _ { i } ) ( \omega _ { i } \cdot \mathbf { n } ) d \omega _ { i } .
+$$
+
+From now on we will ignore the emission term $u _ { e }$ as it is not pertinent to our discussion. Furthermore, because the speed of light is substantially faster than the exposure time of our eyes, what we perceive is not the propagation of light at an instant, but the steady state solution to the rendering equation evaluated at every point in space. Explicitly computing this steady state is intractable for our applications and will mainly serve as a reference for which to place a plethora of assumptions and simplifications we will make for the sake of tractability. Many of these methods focus on ignoring light with nominal effects on the final rendered image vis a vis assumptions on the way light travels. For instance, light is usually assumed to have nominal interacts with air, which is described as the assumption that the space between objects is a vacuum, which constrains the interactions of light to the objects in a scene. Another common assumption is that light does not penetrate objects, which makes it difficult to render objects like milk and human $\mathrm { s k i n } ^ { \mathrm { 1 } }$ . This constrains the complexity of light propagation to the behavior of light bouncing off of object surfaces.
+
+# B.1 LOCAL ILLUMINATION
+
+It is common to see assumptions that limit number of bounces light is allowed.In our case we chose to assume that the steady state is sufficiently approximated by an extremely low number of iterations: one. This means that it seems sufficient to model the lighting of a point in space by the light sent to it directly by light sources. Working with such a strong simplification does, of course, lead to a few artifacts. For instance, light occluded by other objects is ignored so shadows disappear and auxiliary techniques are usually employed to evaluate shadows (Williams, 1978; Miller, 1994).
+
+![](images/977ee117acc76d365e2e90f830596a6e3286a31f711f8fd0ef1bb50de4bc5dbc.jpg)  
+Figure 14: Rasterization converts a 3D scene into pixels.
+
+When this assumption is coupled with a camera we approach what is used in standard rasterization systems such as OPENGL (Shreiner & Group, 2009), which is what we use. These systems compute the illumination of a single pixel by determining the fragment of an object visible through that pixel and only computing the light that traverses directly from the light sources, through that fragment, to that pixel. The lighting of a fragment is therefore determined by a point and the surface normal at that point, so we write the fragment’s radiance as $R ( p , { \bf n } , \omega _ { o } ) = \dot { u } _ { o } ( { p , \omega _ { o } } )$ :
+
+$$
+R ( p , \mathbf { n } , \omega _ { o } ) = \int _ { S ^ { 2 } } f _ { r } ( p , \omega _ { i } , \omega _ { o } ) u _ { i } ( p , \omega _ { i } ) ( \omega _ { i } \cdot \mathbf { n } ) d \omega _ { i } .
+$$
+
+# B.2 LAMBERTIAN MATERIAL
+
+Each point on an object has a model approximating the transfer of incoming light to a given output direction $f _ { r }$ , which is usually called the material. On a single object the material parameters may vary quite a bit and the correspondence between points and material parameters is usually called the texture map which forms the texture of an object. There exists a wide gamut of material models, from mirror materials that transport light from a single input direction to a single output direction, to materials that reflect light evenly in all directions, to materials liked brushed metal that reflect differently along different angles. For the sake f document we only consider diffuse materials, also called Lambertian materials, where we assume that incoming light is reflected uniformly, i.e $f _ { r }$ is a constant function with respect to angle, which we denote $f _ { r } ( p , \omega _ { i } , \omega _ { o } ) = \rho ( p )$ :
+
+![](images/9b793cf1efcea77b4038f9e7ea6c33b1379ee0a8f3a02f8ad9954f3f68c509b7.jpg)  
+Figure 15: We consider the Lambertian material (left) where lights get reflected uniformly in every direction.
+
+$$
+R ( p , { \bf n } ) = \rho ( p ) \int _ { \Omega ( { \bf n } ) } u ( p , \omega ) ( \omega \cdot { \bf n } ) d \omega .
+$$
+
+This function $\rho$ is usually called the albedo, which can be perceived as color on the surface for diffuse material, and we reduce our integration domain to the upper hemisphere $\Omega ( \mathbf { n } )$ in order to model light not bouncing through objects. Furthermore, since only the only $\omega$ and $u$ are the incoming ones we can now suppress the “incoming” in our notation and just use $\omega$ and $u$ respectively.
+
+# B.3 ENVIRONMENT MAPPING
+
+The illumination of static, distant objects such as the ground, the sky, or mountains do not change in any noticeable fashion when objects in a scene are moved around, so $u$ can be written entirely in terms of $\omega$ , $u ( p , \omega ) = u ( \omega )$ . If their illumination forms a constant it seems prudent to pre-compute or cache their contributions to the illumination of a scene. This is what is usually called environment mapping and they fit in the rendering equation as a representation for the total lighting of a scene, i.e the total incoming radiance $u _ { i }$ . Because the environment is distant, it is common to also assume that the position of the object receiving light from an environment map does not matter so this simplifies $u _ { i }$ to be independent of position:
+
+$$
+R ( p , \mathbf { n } ) = \rho ( p ) \int _ { \Omega ( \mathbf { n } ) } u ( \omega ) \left( { \boldsymbol { \omega } } \cdot \mathbf { n } \right) d \omega .
+$$
+
+# B.4 SPHERICAL HARMONICS
+
+Despite all of our simplifications, the inner integral is still a fairly generic function over $S ^ { 2 }$ . Many techniques for numerically integrating the rendering equation have emerged in the graphics community and we choose one which enables us to perform pre-computation and select a desired spectral accuracy: spherical harmonics. Spherical harmonics are a basis on $S ^ { 2 }$ so, given a spherical harmonics expansion of the integrand, the evaluation of the above integral can be reduced to a weighted product of coefficients. This particular basis is chosen because it acts as a sort of Fourier basis for functions on the sphere and so the bases are each associated with a frequency, which leads to a convenient multi-resolution structure. In fact, the rendering of diffuse objects under distant lighting can be $9 9 \%$ approximated by just the first few spherical harmonics bases (Ramamoorthi & Hanrahan, 2001).
+
+We will only need to note that the spherical harmonics bases $Y _ { l } ^ { m }$ are denoted with the subscript with $l$ as the frequency and that there are $2 l + 1$ functions per frequency, denoted by superscripts $m$ between $- l$ to $l$ inclusively. For further details on them please take a glance at Appendix C.
+
+If we approximate a function $f$ in terms of spherical harmonics coefficients $\begin{array} { r } { f \approx \sum _ { l m } f _ { l , m } Y _ { l } ^ { m } } \end{array}$ the integral can be precomputed as
+
+$$
+\int _ { S ^ { 2 } } { f } \approx \int _ { S ^ { 2 } } \sum _ { l m } f _ { l , m } Y _ { l } ^ { m } = \sum _ { l m } f _ { l , m } \int _ { S ^ { 2 } } Y _ { l } ^ { m } ,
+$$
+
+Thus we have defined a reduced rendering equation that can be efficiently evaluated using OPENGL while maintaining differentiability with respect to lighting and vertices. In the following appendix we will derive the derivatives necessary to implement our system.
+
+# C DIFFERENTIABLE RENDERER
+
+Rendering computes an image of a 3D shape given lighting conditions and the prescribed material properties on the surface of the shape. Our differentiable renderer assumes Lambertian reflectance, distant light sources, local illumination, and piece-wise constant textures. We will discuss how to explicitly compute the derivatives used in the main body of this text. Here we give a detailed discussion about spherical harmonics and their advantages.
+
+# C.1 SPHERICAL HARMONICS
+
+Spherical harmonics are usually defined in terms of the Legendre polynomials, which are a class of orthogonal polynomials defined by the recurrence relation
+
+$$
+\begin{array} { c } { { P _ { 0 } = 1 } } \\ { { P _ { 1 } = x } } \\ { { ( l + 1 ) P _ { l + 1 } ( x ) = ( 2 l + 1 ) x P _ { l } ( x ) - l P _ { l - 1 } ( x ) . } } \end{array}
+$$
+
+The associated Legendre polynomials are a generalization of the Legendre polynomials and can be fully defined by the relations
+
+$$
+\begin{array} { c } { { P _ { l } ^ { 0 } = P _ { l } } } \\ { { \ } } \\ { { ( l - m + 1 ) P _ { l + 1 } ^ { m } ( x ) = ( 2 l + 1 ) x P _ { l } ^ { m } ( x ) - ( l + m ) P _ { l - 1 } ^ { m } ( x ) } } \\ { { 2 m x P _ { l } ^ { m } ( x ) = - \sqrt { 1 - x ^ { 2 } } \left[ P _ { l } ^ { m + 1 } ( x ) + ( l + m ) ( l - m + 1 ) P _ { l } ^ { m - 1 } ( x ) \right] . } } \end{array}
+$$
+
+Using the associated Legendre polynomials $P _ { l } ^ { m }$ we can define the spherical harmonics basis as
+
+$$
+Y _ { l } ^ { m } ( \theta , \phi ) = K _ { l } ^ { m } \left\{ \begin{array} { l l } { ( - 1 ) ^ { m } \sqrt { 2 } P _ { l } ^ { - m } ( \cos \theta ) \sin ( - m \phi ) } & { \quad m < 0 } \\ { ( - 1 ) ^ { m } \sqrt { 2 } P _ { l } ^ { m } ( \cos \theta ) \cos ( m \phi ) } & { \quad m > 0 \ . } \\ { P _ { l } ^ { 0 } ( \cos \theta ) } & { \quad m = 0 } \end{array} \right.
+$$
+
+$$
+\mathrm { w h e r e } K _ { l } ^ { m } = \sqrt { \frac { ( 2 l + 1 ) ( l - | m | ) ! } { 4 \pi ( l + | m | ) ! } } .
+$$
+
+We will use the fact that the associated Legendre polynomials correspond to the spherical harmonics bases that are rotationally symmetric along the $z$ axis $( m = 0$ ).
+
+In order to incorporate spherical harmonics into Equation 8, we change the integral domain from the upper hemisphere $\Omega ( \mathbf { n } )$ back to $S ^ { 2 }$ via a max operation
+
+$$
+\begin{array} { l } { \displaystyle R ( p , { \bf n } ) = \rho ( p ) \int _ { \Omega ( { \bf n } ) } u ( \omega ) ( \omega \cdot { \bf n } ) d \omega } \\ { \displaystyle \quad = \rho ( p ) \int _ { S ^ { 2 } } u ( \omega ) \operatorname* { m a x } ( \omega \cdot { \bf n } , 0 ) d \omega . } \end{array}
+$$
+
+We see that the integral is comprised of two components: a lighting component $u ( \omega )$ and a component that depends on the normal $\operatorname* { m a x } ( \omega \cdot \mathbf n , 0 )$ . The strategy is to pre-compute the two components by projecting onto spherical harmonics, and evaluating the integral via a dot product at runtime, as we will now derive.
+
+# C.2 LIGHTING IN SPHERICAL HARMONICS
+
+Approximating the lighting component $u ( \omega )$ in Equation 19 using spherical harmonics $Y _ { l } ^ { m }$ up to band $n$ can be written as
+
+$$
+u ( \omega ) \approx \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } U _ { l , m } Y _ { l } ^ { m } ( \omega ) ,
+$$
+
+where $U _ { l , m } \in \mathbb { R }$ are coefficients. By using the orthogonality of spherical harmonics we can use evaluate these coefficients as an integral between $u ( \omega )$ and $Y _ { l } ^ { m } ( \omega )$
+
+$$
+U _ { l , m } = \langle u , Y _ { l } ^ { m } \rangle _ { S ^ { 2 } } = \int _ { S ^ { 2 } } u ( \omega ) Y _ { l } ^ { m } ( \omega ) d \omega ,
+$$
+
+which can be evaluated via quadrature.
+
+# C.3 CLAMPED COSINE IN SPHERICAL HARMONICS
+
+So far, we have projected the lighting term $u ( \omega )$ onto the spherical harmonics basis. To complete evaluating Equation 19 we also need to approximate the second component $\operatorname* { m a x } ( \omega \cdot \mathbf { n } , 0 )$ in spherical
+
+harmonics. This is the so-called the clamped cosine function.
+
+$$
+g ( \omega , { \bf n } ) = \mathrm { m a x } ( \omega \cdot { \bf n } , 0 ) = \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } G _ { l , m } ( { \bf n } ) Y _ { l } ^ { m } ( \omega ) ,
+$$
+
+where $G _ { l , m } ( { \mathbf n } ) \in \mathbb { R }$ can be computed by projecting $g ( \omega , \mathbf { n } )$ onto $Y _ { l } ^ { m } ( \omega )$
+
+$$
+G _ { l , m } ( \mathbf { n } ) = \int _ { S ^ { 2 } } \mathrm { m a x } ( \omega \cdot \mathbf { n } , 0 ) Y _ { l } ^ { m } ( \omega ) d \omega .
+$$
+
+Unfortunately, this formulation turns out to be tricky to compute. Instead, the common practice is to analytically compute the coefficients for unit $z$ direction $\tilde { G } _ { l , m } = G _ { l , m } ( \mathbf { n } _ { z } ) = G _ { l , m } ( [ 0 , 0 , 1 ] ^ { \boldsymbol { \mathsf { T } } } )$ and evaluate the coefficients for different normals $G _ { l , m } ( \mathbf { n } )$ by rotating $\tilde { G } _ { l , m }$ . This rotation, $\tilde { G } _ { l , m }$ , can be computed analytically:
+
+$$
+\begin{array} { r l } & { \tilde { G } _ { l , m } = \displaystyle \int _ { S ^ { 2 } } \operatorname* { m a x } ( \omega \cdot \mathbf { n } _ { z } , 0 ) Y _ { l } ^ { m } ( \omega ) d \omega } \\ & { \quad \quad = \displaystyle \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \operatorname* { m a x } ( [ \sin \theta \cos \phi , \sin \theta \sin \phi , \cos \theta ] [ 0 , 0 , 1 ] ^ { \tau } , 0 ) Y _ { l } ^ { m } ( \theta , \phi ) \sin \theta d \theta d \phi } \\ & { \quad \quad = \displaystyle \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \operatorname* { m a x } ( \cos \theta , 0 ) Y _ { l } ^ { m } ( \theta , \phi ) \sin \theta d \theta d \phi } \\ & { \quad \quad = \displaystyle \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi / 2 } \cos \theta Y _ { l } ^ { m } ( \theta , \phi ) \sin \theta d \theta d \phi . } \end{array}
+$$
+
+In fact, because $\operatorname* { m a x } ( \omega \cdot \mathbf { n } _ { z } , 0 )$ is rotationally symmetric around the $z$ -axis, its projection onto $Y _ { l } ^ { m } ( \omega )$ will have many zeros except the rotationally symmetric spherical harmonics $\bar { Y } _ { l } ^ { 0 }$ . In other words, $\tilde { G } _ { l , m }$ is non-zero only when $m = 0$ . So we can simplify Equation 20 to
+
+$$
+\tilde { G } _ { l } = \tilde { G } _ { l , 0 } = 2 \pi \int _ { 0 } ^ { \pi / 2 } \cos \theta Y _ { l } ^ { 0 } ( \theta ) \sin \theta d \theta .
+$$
+
+The evaluation of this integral can be found in Appendix A in (Basri & Jacobs, 2003). We provide this here as well:
+
+$$
+\tilde { G } _ { l } = \left\{ \begin{array} { l l } { \frac { \sqrt { \pi } } { 2 } } & { l = 0 } \\ { \sqrt { \frac { \pi } { 3 } } } & { l = 1 } \\ { ( - 1 ) ^ { \frac { l } { 2 } + 1 } \frac { ( l - 2 ) ! \sqrt { ( 2 l + 1 ) \pi } } { 2 ^ { l } ( \frac { l } { 2 } - 1 ) ! ( \frac { l } { 2 } + 1 ) ! } } & { l \geq 2 , \mathrm { e v e n } } \\ { 0 } & { l \geq 2 , \mathrm { o d d } } \end{array} \right. .
+$$
+
+The spherical harmonics coefficients $G _ { l , m } ( \mathbf { n } )$ of the clamped cosine function $g ( \omega , \mathbf { n } )$ can be computed by rotating $\tilde { G } _ { l }$ (Sloan et al., 2005) using this formula
+
+$$
+G _ { l , m } ( \mathbf { n } ) = \sqrt { \frac { 4 \pi } { 2 l + 1 } } \tilde { G } _ { l } Y _ { l } ^ { m } ( \mathbf { n } ) .
+$$
+
+So far we have projected the two terms in Equation 19 into the spherical harmonics basis. Orthogonality of spherical harmonics makes the evaluation of this integral straightforward:
+
+$$
+\begin{array} { l } { \displaystyle \int _ { { \mathcal { S } } ^ { 2 } } u ( \boldsymbol { \omega } ) \operatorname* { m a x } ( \boldsymbol { \omega } \cdot { \bf n } , 0 ) d \boldsymbol { \omega } = \int _ { { \mathcal { S } } ^ { 2 } } \left[ \sum _ { l , m } U _ { l , m } Y _ { l } ^ { m } ( \boldsymbol { \omega } ) \right] \left[ \sum _ { j , k } G _ { j , k } ( { \bf n } ) Y _ { j } ^ { k } ( \boldsymbol { \omega } ) \right] d \boldsymbol { \omega } } \\ { \displaystyle = \sum _ { j , k , l , m } U _ { l , m } G _ { j , k } ( { \bf n } ) \delta _ { j } ^ { l } \delta _ { k } ^ { m } } \\ { \displaystyle = \sum _ { l , m } U _ { l , m } G _ { l , m } ( { \bf n } ) . } \end{array}
+$$
+
+This, in conjunction with Equation 21allows us to derive the rendering equation using spherical harmonics lighting for Lambertian objects:
+
+$$
+R ( { \boldsymbol { p } } , \mathbf { n } ) = \rho ( { \boldsymbol { p } } ) \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } U _ { l , m } { \sqrt { \frac { 4 \pi } { 2 l + 1 } } } { \tilde { G } } _ { l } Y _ { l } ^ { m } ( \mathbf { n } ) .
+$$
+
+So far we have only considered the shading of a specific point $p$ with surface normal $\mathbf { n }$ . If we consider the rendered image $I$ given a shape $V$ , lighting $U$ , and camera parameters $\eta$ , the image $I$ is the evaluation of the rendering equation $R$ of each point in $V$ visible through each pixel in the image. This pixel to point mapping is determined by $\eta$ . Therefore, we can write $I$ as
+
+$$
+I ( V , U , \eta ) = \rho ( V , \eta ) \underbrace { \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } U _ { l , m } \sqrt { \frac { 4 \pi } { 2 l + 1 } } \tilde { G } _ { l } Y _ { l } ^ { m } ( N ( V ) ) } _ { F ( V , U ) } ,
+$$
+
+where $N ( V )$ is the surface normal. We exploit the notation and use $\rho ( V , \eta )$ to represent the texture of $V$ mapped to the image space through $\eta$ .
+
+# C.4 LIGHTING AND TEXTURE DERIVATIVES
+
+For our applications we must differentiate Equation 25 with respect to lighting and material parameters. The derivative with respect to the lighting coefficients $U$ can be obtained by
+
+$$
+\begin{array} { c } { \displaystyle \frac { \partial I } { \partial U } = \frac { \partial \rho } { \partial U } F + \rho \frac { \partial F } { \partial U } } \\ { = 0 + \rho \displaystyle \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } \frac { \partial F } { \partial U _ { l , m } } . } \end{array}
+$$
+
+This is the Jacobian matrix that maps from spherical harmonics coefficients to pixels. The term $\partial F / \partial U _ { l , m }$ can then be computed as
+
+$$
+\frac { \partial F } { \partial U _ { l , m } } = \sqrt { \frac { 4 \pi } { 2 l + 1 } } \tilde { G } _ { l } Y _ { l } ^ { m } ( N ( V ) ) .
+$$
+
+The derivative with respect to texture is defined by
+
+$$
+\frac { \partial I } { \partial \rho } = \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } U _ { l , m } \sqrt { \frac { 4 \pi } { 2 l + 1 } } \tilde { G } _ { l } Y _ { l } ^ { m } ( N ( V ) ) .
+$$
+
+Note that we assume texture variations are piece-wise constant with respect to our triangle mesh discretization.
+
+# D DIFFERENTIATING SKYLIGHT PARAMETERS
+
+To model possible outdoor daylight conditions, we use the analytical Preetham skylight model (Preetham et al., 1999). This model is calibrated by atmospheric data and parameterized by two intuitive parameters: turbidity $\tau$ , which describes the cloudiness of the atmosphere, and two polar angles $\theta _ { s } \in [ 0 , \pi / 2 ] , \phi _ { s } \in [ 0 , 2 \pi ]$ , which are encode the direction of the sun. Note that $\theta _ { s } , \phi _ { s }$ are not the polar angles $\theta , \phi$ for representing incoming light direction $\omega$ in $u ( \omega )$ . The spherical harmonics representation of the Preetham skylight is presented in (Habel et al., 2008) as
+
+$$
+u ( \omega ) = \sum _ { l = 0 } ^ { 6 } \sum _ { m = - l } ^ { l } U _ { l , m } ( \theta _ { s } , \phi _ { s } , \tau ) Y _ { l } ^ { m } ( \omega ) .
+$$
+
+This is derived by first performing a non-linear least squares fit to write $U _ { l , m }$ as a polynomial of $\theta _ { s }$ and $\tau$ which lets them solve for $\tilde { U } _ { l , m } ( \theta _ { s } , \tau ) = U _ { l , m } ( \theta _ { s } , 0 , \tau )$
+
+$$
+\tilde { U } _ { l , m } ( \theta _ { s } , \tau ) = \sum _ { i = 0 } ^ { 1 3 } \sum _ { j = 0 } ^ { 7 } ( p _ { l , m } ) _ { i , j } \theta _ { s } ^ { i } \tau ^ { j } ,
+$$
+
+where $( p _ { l , m } ) _ { i , j }$ are scalar coefficients, then $U _ { l , m } ( \theta _ { s } , \phi _ { s } , \tau )$ can be computed by applying a spherical harmonics rotation with $\phi _ { s }$ using
+
+$$
+U _ { l , m } ( \theta _ { s } , \phi _ { s } , \tau ) = \tilde { U } _ { l , m } ( \theta _ { s } , \tau ) \cos ( m \phi _ { s } ) + \tilde { U } _ { l , - m } ( \theta _ { s } , \tau ) \sin ( m \phi _ { s } ) .
+$$
+
+We refer the reader to (Preetham et al., 1999) for more detail. For the purposes of this article we just need the above form to compute the derivatives.
+
+# D.1 DERIVATIVES
+
+The derivatives of the lighting with respect to the skylight parameters $( \theta _ { s } , \phi _ { s } , \tau )$ are
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial \tilde { U } _ { l , m } ( \theta _ { s } , \phi _ { s } , \tau ) } { \partial \phi _ { s } } = - m \tilde { U } _ { l , m } ( \theta _ { s } , \tau ) \sin ( m \phi _ { s } ) + m \tilde { U } _ { l , - m } ( \theta _ { s } , \tau ) \cos ( m \phi _ { s } ) } \\ { \displaystyle \frac { \partial \tilde { U } _ { l , m } ( \theta _ { s } , \phi _ { s } , \tau ) } { \partial \theta _ { s } } = \frac { \partial \tilde { U } _ { l , m } ( \theta _ { s } , \tau ) \cos ( m \phi _ { s } ) + \tilde { U } _ { l , - m } ( \theta _ { s } , \tau ) \sin ( m \phi _ { s } ) } { \partial \theta _ { s } } } \\ { \displaystyle \qquad = \sum _ { i j } i \theta _ { s } ^ { i - 1 } \tau ^ { j } ( p _ { l , m } ) _ { i , j } \cos ( m \phi _ { s } ) + \sum _ { i j } i \theta _ { s } ^ { i - 1 } ( p _ { l , - m } ) _ { i , j } \sin ( m \phi _ { s } ) } \\ { \displaystyle \frac { \partial \tilde { U } _ { l , m } ( \theta _ { s } , \phi _ { s } , \tau ) } { \partial \tau } = \sum _ { i j } j \theta _ { s } ^ { i } \tau ^ { j - 1 } ( p _ { l , m } ) _ { i , j } \cos ( m \phi _ { s } ) + \sum _ { i j } j \theta _ { s } ^ { i } \tau ^ { j - 1 } ( p _ { l , - m } ) _ { i , j } \sin ( m \phi _ { s } ) } \end{array}
+$$
+
+# E DERIVATIVES OF SURFACE NORMALS
+
+Taking the derivative of the rendered image $I$ with respect to surface normals $N$ is an essential task for computing the derivative of $I$ with respect to the geometry $V$ . Specifically, the derivative of the rendering equation Equation 25 with respect to $V$ is
+
+$$
+\begin{array} { c } { \displaystyle { \frac { \partial I } { \partial V } = \frac { \partial \rho } { \partial V } F + \rho \frac { \partial F } { \partial V } } } \\ { \displaystyle { = \frac { \partial \rho } { \partial V } F + \rho \frac { \partial F } { \partial N } \frac { \partial N } { \partial V } } } \end{array}
+$$
+
+We assume the texture variations are piece-wise constant with respect to our triangle mesh discretization and omit the first term $\partial \rho / \partial V$ as the magnitude is zero. Computing ${ \partial \bar { N _ { \big / { \partial V } } } }$ is provided in Section 3.2. Computing $\partial F / \partial N _ { i }$ on face $i$ is
+
+$$
+\frac { \partial F } { \partial N _ { i } } = \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } U _ { l , m } \sqrt { \frac { 4 \pi } { 2 l + 1 } } \tilde { G } _ { l } \frac { \partial Y _ { l } ^ { m } } { \partial N _ { i } } ,
+$$
+
+where the ${ \partial Y _ { l } ^ { m } } / { \partial N _ { i } }$ is the derivative of the spherical harmonics with respect to the face normal $N _ { i }$ .
+
+To begin this derivation recall the relationship between a unit normal vector $\mathbf { n } = ( n _ { x } , n _ { y } , n _ { z } )$ and its corresponding polar angles $\theta , \phi$
+
+$$
+\theta = \cos ^ { - 1 } \bigg ( \frac { n _ { z } } { \sqrt { n _ { x } ^ { 2 } + n _ { y } ^ { 2 } + n _ { z } ^ { 2 } } } \bigg ) \qquad \phi = \tan ^ { - 1 } \bigg ( \frac { n _ { y } } { n _ { x } } \bigg ) ,
+$$
+
+we can compute the derivative of spherical harmonics with respect to the normal vector through
+
+$$
+\begin{array} { r l r } { \frac { \partial Y _ { i } ^ { n } ( t , \theta , \phi ) } { \partial \mathbf { n } } } & { } & { = 0 } \\ & { } & { \left[ ( 1 - 1 ) ^ { n } \sqrt { 2 } \left[ \frac { \partial P _ { i } ^ { n - 1 } ( c \times \theta ) \tilde { \theta } } { \partial \theta } \frac { \partial \theta } { \partial \mathbf { n } } \sin ( - m \phi ) + P _ { i } ^ { - n } ( c \times \theta ) \frac { \partial \sin ( - m \phi ) } { \partial \phi } \frac { \partial \phi } { \partial \mathbf { n } } \right] \right. \qquad m < 0 } \\ & { } & { = K _ { i } ^ { n } \left( \cdots \right) ^ { n } \sqrt { 2 } \left[ \frac { \partial P _ { i } ^ { n } ( c \times \theta ) } { \partial \theta } \frac { \partial \theta } { \partial \mathbf { n } } \cos ( m \phi ) + P _ { i } ^ { n } ( c \times ( \cos \theta ) \frac { \partial \cos ( \theta ) \phi } { \partial \phi } ) \frac { \partial \phi } { \partial \mathbf { n } } \right] \qquad m > 0 } \\ & { } & { \left. \qquad \left[ \frac { \partial P _ { i } ^ { n } ( c \times \theta ) } { \partial \theta } \frac { \partial \theta } { \partial \mathbf { n } } \right. \qquad \partial \qquad \left. \partial \qquad \sin ( - m \phi ) - m P _ { i } ^ { n - 1 } ( c \times \theta ) \cos ( - m \phi ) \frac { \partial \theta } { \partial \mathbf { n } } \right] \right. \qquad m = 0 } \\ & { } & { \left. \qquad \left( \cdots \right) ^ { n } \sqrt { 2 } \left[ \frac { \partial P _ { i } ^ { n - 1 } ( c \times \theta ) \tilde { \theta } } { \partial \theta } \frac { \partial \theta } { \partial \mathbf { n } } \sin ( - m \phi ) - m P _ { i } ^ { n - 1 } ( c \times \theta ) \cos ( - m \phi ) \frac { \partial \theta } { \partial \mathbf { n } } \right] \right. \qquad m < 0 } \\ & { } & { = K _ { i } ^ { n } \left( \cdots \right) ^ { n } \sqrt { 2 } \left[ \frac { \partial P _ { i } ^ { n } ( c \times \theta ) } { \partial \theta } \frac { \partial \theta } { \partial \mathbf { n } } \cos ( m \phi ) - m P _ { i } ^ { n } ( c \times \theta ) \sin ( m \phi ) \frac { \partial \theta } { \partial \mathbf { n } } \right] \qquad m > 0 } \\ & { } &  \qquad \left. \partial P _ { i } ^ { n } ( c \times \theta ) \right) \frac  \end{array}
+$$
+
+Note that the derivative of the associated Legendre polynomials $P _ { l } ^ { m } ( \cos \theta )$ can be computed by applying the recurrence formula Dunster (2010)
+
+$$
+\begin{array} { r l } & { \frac { \partial P _ { l } ^ { m } ( \cos \theta ) } { \partial \theta } = \frac { - \cos \theta ( l + 1 ) P _ { l } ^ { m } ( \cos \theta ) + ( l - m + 1 ) P _ { l + 1 } ^ { m } ( \cos \theta ) } { \cos ^ { 2 } \theta - 1 } \times ( - \sin \theta ) } \\ & { \qquad = \frac { - \cos \theta ( l + 1 ) P _ { l } ^ { m } ( \cos \theta ) + ( l - m + 1 ) P _ { l + 1 } ^ { m } ( \cos \theta ) } { \sin \theta } . } \end{array}
+$$
+
+Thus the derivatives of polar angles $( \theta , \phi )$ with respect to surface normals $\mathbf { n } = [ n _ { x } , n _ { y } , n _ { z } ]$ are
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial \theta } { \partial \mathbf { n } } = \Big [ \frac { \partial \theta } { \partial n _ { x } } , \frac { \partial \theta } { \partial n _ { y } } , \frac { \partial \theta } { \partial n _ { z } } \Big ] = \frac { \big [ n _ { x } n _ { z } , n _ { y } n _ { z } , - ( n _ { x } ^ { 2 } + n _ { y } ^ { 2 } ) \big ] } { ( n _ { x } ^ { 2 } + n _ { y } ^ { 2 } + n _ { z } ^ { 2 } ) \sqrt { n _ { x } ^ { 2 } + n _ { y } ^ { 2 } } } , } \\ { \displaystyle \frac { \partial \phi } { \partial \mathbf { n } } = \Big [ \frac { \partial \phi } { \partial n _ { x } } , \frac { \partial \phi } { \partial n _ { y } } , \frac { \partial \phi } { \partial n _ { z } } \Big ] = \Big [ \frac { - n _ { y } } { n _ { x } ^ { 2 } + n _ { y } ^ { 2 } } , \frac { n _ { x } } { n _ { x } ^ { 2 } + n _ { y } ^ { 2 } } , 0 \Big ] . } \end{array}
+$$
+
+In summary, the results of Equation 37, Equation 38, Equation 39, and Equation 40 tell us how to compute ${ \partial { Y _ { l } } ^ { m } } / { \partial { N _ { i } } }$ . Then the derivative of the pixel $j$ with respect to vertex $p$ which belongs to face $i$ can be computed as
+
+$$
+\begin{array} { r l r } {  { \frac { \partial I _ { j } } { \partial V _ { p } } \approx \rho _ { j } \frac { \partial F } { \partial N _ { i } } \frac { \partial N _ { i } } { \partial V _ { p } } } } \\ & { } & { = \rho _ { j } \sum _ { l = 0 } ^ { n } \sum _ { m = - l } ^ { l } U _ { l , m } \sqrt { \frac { 4 \pi } { 2 l + 1 } } \tilde { G } _ { l } \frac { \partial Y _ { l } ^ { m } ( \theta , \phi ) } { \partial N _ { i } } \frac { \partial N _ { i } } { \partial V _ { p } } . } \end{array}
+$$
+
+# F ADVERSARIAL TRAINING IMPLEMENTATION DETAIL
+
+Our adversarial training is based on the basic idea of injecting adversarial examples into the training set at each step and continuously updating the adversaries according to the current model parameters (Goodfellow et al., 2015; Kurakin et al., 2017). Our experiments inject 100 adversarial lighting examples to the CIFAR-100 data $( \approx 0 . 1 7 \%$ of the training set) and keep updating these adversaries at each epoch.
+
+We compute the adversarial lighting examples using the orange models collected from cgtrader.com and turbosquid.com. We uses five gray-scale background colors with intensities 0.0, 0.25, 0.5, 0.75, 1.0 to mimic images in the CIFAR-100 which contains many pure color backgrounds. Our orthographic cameras are placed at polar angle $\theta = \pi / 3$ with 10 uniformly sampled azimuthal angles ranging from $\phi = 0$ to $2 \pi$ . Our initial spherical harmonics lighting is the same as other experiments, using the real-world lighting data provided in (Ramamoorthi & Hanrahan, 2001). Our stepsize for computing adversaries is 0.05 along the direction of lighting gradients. We run our adversarial lighting iterations until fooling the network or reaching the maximum 30 iterations to avoid too extreme lighting conditions, such as turning the lights off.
+
+![](images/66de3929cf3e0142697900fd6687c94f96fb8dfdc7aa9d747c7bb4970e77138b.jpg)  
+Figure 16: This figure visualizes the images of oranges from CIFAR-100, random lighting, and adversarial lighting. In early training stage, small changes in lighting are sufficient to construct adversarial examples. In late training stage, we require more dramatic changes as the model is becoming robust to differ lightings.
+
+Our random lighting examples are constructed at each epoch by randomly perturb the lighting coefficients ranging from -0.5 to 0.5.
+
+When training the 16-layers WideResNet (Zagoruyko & Komodakis, 2016) with wide-factor 4, we use batch size 128, learning rate 0.125, dropout rate 0.3, and the standard cross entropy loss. We implement the training using PYTORCH (Paszke et al., 2017), with the SGD optimizer and set the Nesterov momentum 0.9, weight decay 5e-4. We train the model for 150 epochs and use the one with best accuracy on the validation set. Figure 16 shows examples of our adversarial lights at different training stages. In the early stages, the model is not robust to different lighting conditions, thus small lighting perturbations are sufficient to fool the model. In the late stages, the network becomes more robust to different lightings. Thus it requires dramatic changes to fool a model or even fail to fool the model within 30 iterations.
+
+# G EVALUATE RENDERING QUALITY
+
+We evaluated our rendering quality by whether our rendered images are recognizable by models trained on real photographs. Although large 3D shape datasets, such as ShapeNet (Chang et al., 2015), are available, they do not have have geometries or textures at the resolutions necessary to create realistic renderings. We collected 75 high-quality textured 3D shapes from cgtrader.com and turbosquid.com to evaluate our rendering quality. We augmented the shapes by changing the field of view, backgrounds, and viewing directions, then keep the configurations that were correctly classified by a pre-trained ResNet-101 on ImageNet. Specifically, we place the centroid, calculated as the weighted average of the mesh vertices where the weights are the vertex areas, at the origin and normalize shapes to range $^ { - 1 }$ to 1; the field of view is chosen to be 2 and 3 in the same unit with the normalized shape; background images include plain colors and real photos, which have small influence on model predictions; viewing directions are chosen to be 60 degree zenith and uniformly sampled 16 views from 0 to $2 \pi$ azimuthal angle. In Figure 17, we show that the histogram of model confidence on the correct labels over 10,000 correctly classified rendered images from our differentiable renderer. The confidence is computed using softmax function and the results show that our rendering quality is faithful enough to be recognized by models trained on natural images.
+
+![](images/2cd7e96dd9c8c33e64047cfcbfe5addec9c0509848fe8ec576684c235d66399d.jpg)  
+Figure 17: Prediction confidence on rendered images, showing our rendering quality is faithful enough to be confidently recognized by ImageNet models.

parse/train/SkHkeixAW/SkHkeixAW.md

@@ -0,0 +1,506 @@
+# Regularization for Deep Learning: A Taxonomy
+
+Anonymous authors
+
+Paper under double-blind review
+
+# Abstract
+
+Regularization is one of the crucial ingredients of deep learning, yet the term regularization has various definitions, and regularization methods are often studied separately from each other. In our work we present a novel, systematic, unifying taxonomy to categorize existing methods. We distinguish methods that affect data, network architectures, error terms, regularization terms, and optimization procedures. We identify the atomic building blocks of existing methods, and decouple the assumptions they enforce from the mathematical tools they rely on. We do not provide all details about the listed methods; instead, we present an overview of how the methods can be sorted into meaningful categories and sub-categories. This helps revealing links and fundamental similarities between them. Finally, we include practical recommendations both for users and for developers of new regularization methods.
+
+# 1 Introduction
+
+Regularization is one of the key elements of machine learning, particularly of deep learning (Goodfellow et al., 2016), allowing to generalize well to unseen data even when training on a finite training set or with an imperfect optimization procedure. In the traditional sense of optimization and also in older neural networks literature, the term “regularization” is reserved solely for a penalty term in the loss function (Bishop, 1995a). Recently, the term has adopted a broader meaning: Goodfellow et al. (2016, Chap. 5) loosely define it as “any modification we make to a learning algorithm that is intended to reduce its test error but not its training error”. We find this definition slightly restrictive and present our working definition of regularization, since many techniques considered as regularization do reduce the training error (e.g. weight decay in AlexNet (Krizhevsky et al., 2012)).
+
+Definition 1. Regularization is any supplementary technique that aims at making the model generalize better, i.e. produce better results on the test set.
+
+This can include various properties of the loss function, the loss optimization algorithm, or other techniques. Note that this definition is more in line with machine learning literature than with inverse problems literature, the latter using a more restrictive definition.
+
+In this work, we create a novel, systematic, unifying taxonomy of regularization methods for deep learning. We analyze existing methods and identify their atomic building blocks. This leads to decoupling of two important concepts: Which assumptions the methods rely on (and try to enforce), and which mathematical and algorithmic tools they use. In turn, this enables better understanding of existing methods and speeds up development of new ones: The researchers can focus either on finding new, better ways of enforcing existing assumptions, or focus on discovery of new assumptions that can be enforced in some existing way.
+
+Before we proceed to the presentation of our taxonomy, we revisit some basic machine learning theory in Section 2. This will provide a justification of the top level of the taxonomy. In Sections 3–7, we continue with a finer division of the individual classes of the regularization techniques, aiming at separating as many clearly separable concepts as possible and isolating atomic building blocks of individual methods. Finally, in Section 8 we present our practical recommendations for using existing methods and designing new methods. We are aware that the many research works discussed in this taxonomy cannot be summarized in a single sentence. For the sake of structuring the multitude of papers, we decided to merely describe a certain subset of their properties according to the focus of our taxonomy.
+
+# 2 Theoretical framework
+
+The central task of our interest is model fitting: finding a function $f$ that can well approximate a desired mapping from inputs $x$ to desired outputs $f ( x )$ . A given input $x$ can have an associated target $t$ which dictates the desired output $f ( x )$ directly (or in some applications indirectly (Ulyanov et al., 2016; Johnson et al., 2016)). A typical example of having available targets $t$ is supervised learning. Data samples $( x , t )$ then follow a ground truth probability distribution $P$ .
+
+In many applications, neural networks have proven to be a good family of functions to choose $f$ from. A neural network is a function $f _ { w } : x \mapsto y$ with trainable weights $w \in W$ . Training the network means finding a weight configuration $w ^ { * }$ , which is a result of performing a minimization procedure of a loss function $\mathcal { L } : W \to \mathbb { R }$ as follows:
+
+$$
+w ^ { * } = \mathrm { m i n i m i z e } ~ { \mathcal { L } } ( w ) .
+$$
+
+Usually the loss function takes the form of expected risk :
+
+$$
+\mathcal { L } = \mathbb { E } _ { ( x , t ) \sim P } \Big [ E \big ( f _ { w } ( x ) , t \big ) + R ( . . . ) \Big ] ,
+$$
+
+where we identify two parts, an error function $E$ and a regularization term $R$ . The error function depends on the targets and assigns a penalty to model predictions according to their consistency with the targets. The regularization term assigns a penalty to the model based on other criteria. It may depend on anything except the targets, for example on the weights (see Section 6).
+
+The expected risk cannot be minimized directly since the data distribution $P$ is unknown. Instead, a training set $\mathcal { D }$ sampled from the distribution is given. The minimization of the expected risk can be then approximated by (approximately) minimizing the empirical risk $\hat { \mathcal { L } }$ :
+
+$$
+\underset { w } { \mathrm { m i n i m i z e } } \frac { 1 } { | \mathscr { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } E \big ( f _ { w } ( x _ { i } ) , t _ { i } \big ) + R ( . . . )
+$$
+
+where $( x _ { i } , t _ { i } )$ are samples from $\mathcal { D }$ .
+
+Now we have the minimal background to formalize the division of regularization methods into a systematic taxonomy. In the minimization of the empirical risk, Eq. (3), we can identify the following elements that are responsible for the value of the learned weights, and thus can contribute to regularization:
+
+$\mathcal { D }$ : The training set, discussed in Section 3   
+$f$ : The selected model family, discussed in Section 4   
+$E$ : The error function, briefly discussed in Section 5   
+$R$ : The regularization term, discussed in Section 6   
+The optimization procedure itself, discussed in Section 7
+
+Ambiguity regarding the splitting of methods into these categories and their subcategories is discussed in Appendix A using notation from Section 3.
+
+# 3 Regularization via data
+
+The quality of a trained model depends largely on the training data. Apart from acquisition/selection of appropriate training data, it is possible to employ regularization via data. This is done by applying some transformation to the training set $\mathcal { D }$ , resulting in a new set $\mathcal { D } _ { R }$ . Some transformations perform feature extraction or pre-processing, modifying the feature space or the distribution of the data to some representation simplifying the learning task. Other methods allow generating new samples to create a larger, possibly infinite, augmented dataset. These two principles are somewhat independent and may be combined. The goal of regularization via data is either one of them, or the other, or both. They both rely on transformations with (stochastic) parameters:
+
+Definition 2. Transformation with stochastic parameters is a function $\tau _ { \theta }$ with parameters $\theta$ which follow some probability distribution.
+
+In this context we consider $^ Ḋ \prime \theta Ḍ$ which can operate on network inputs, activations in hidden layers, or targets. An example of a transformation with stochastic parameters is the corruption of inputs by Gaussian noise (Bishop, 1995b; An, 1996):
+
+$$
+\tau _ { \theta } ( x ) = x + \theta , \quad \theta \sim \mathcal { N } ( \mathbf { 0 } , \Sigma ) .
+$$
+
+The stochasticity of the transformation parameters is responsible for generating new samples, i.e. data augmentation. Note that the term data augmentation often refers specifically to transformations of inputs or hidden activations, but here we also list transformations of targets for completeness. The exception to the stochasticity is when $\theta$ follows a delta distribution, in which case the transformation parameters become deterministic and the dataset size is not augmented.
+
+We can categorize the data-based methods according to the properties of the used transformation and of the distribution of its parameters. We identify the following criteria for categorization (some of them later serve as columns in Tables 1–2):
+
+# Stochasticity of the transformation parameters $\theta$
+
+∙ Deterministic parameters: Parameters $\theta$ follow a delta distribution, size of the dataset remains unchanged
+
+∙ Stochastic parameters: Allow generation of a larger, possibly infinite, dataset. Various strategies for sampling of $\theta$ exist:
+
+– Random: Draw a random $\theta$ from the specified distribution   
+– Adaptive: Value of $\theta$ is the result of an optimization procedure, usually with the objective of maximizing the network error on the transformed sample (such “challenging” sample is considered to be the most informative one at current training stage), or minimizing the difference between the network prediction and a predefined fake target $t ^ { \prime }$ $^ *$ Constrained optimization: $\theta$ found by maximizing error under hard constraints (support of the distribution of $\theta$ controls the strongest allowed transformation) $^ *$ Unconstrained optimization: $\theta$ found by maximizing modified error function, using the distribution of $\theta$ as weighting (proposed herein for completeness, not yet tested) $^ *$ Stochastic: $\theta$ found by taking a fixed number of samples of $\theta$ and using the one yielding the highest error
+
+# Effect on the data representation
+
+∙ Representation-preserving transformations: Preserve the feature space and attempt to preserve the data distribution   
+∙ Representation-modifying transformations: Map the data to a different representation (different distribution or even new feature space) that may disentangle the underlying factors of the original representation and make the learning problem easier
+
+# Transformation space
+
+∙ Input: Transformation is applied to $x$
+
+∙ Hidden-feature space: Transformation is applied to some deep-layer representation of samples (this also uses parts of $f$ and $w$ to map the input into the hidden-feature space; such transformations act inside the network $f _ { w }$ and thus can be considered part of the architecture, additionally fitting Section 4)   
+∙ Target: Transformation is applied to $t$ (can only be used during the training phase since labels are not shown to the model at test time)
+
+# Universality
+
+∙ Generic: Applicable to all data domains   
+∙ Domain-specific: Specific (handcrafted) for the problem at hand, for example image rotations
+
+# Dependence of the distribution of $\theta$
+
+∙ $p ( \theta )$ : distribution of $\theta$ is the same for all samples   
+∙ $p ( \theta | t )$ : distribution of $\theta$ can be different for each target (class)   
+∙ $p ( \theta | t ^ { \prime } )$ : distribution of $\theta$ depends on desired (fake) target $t ^ { \prime }$   
+∙ $p ( \theta | x )$ : distribution of $\theta$ can be different for each input vector (with implicit dependence on $f$ and $w$ if the transformation is in hidden-feature space)   
+∙ $p ( \boldsymbol { \theta } | \mathcal { D } )$ : distribution of $\theta$ depends on the whole training dataset   
+∙ $p ( \boldsymbol { \theta } | \mathbf { x } )$ : distribution of $\theta$ depends on a batch of training inputs (for example (parts of) the current mini-batch, or also previous mini-batches)   
+∙ $p ( \theta | \mathrm { t i m e } )$ : distribution of $\theta$ depends on time (current training iteration)   
+∙ $p ( \theta | \pi )$ : distribution of $\theta$ depends on some trainable parameters $\pi$ subject to loss minimization (i.e. the parameters $\pi$ evolve during training along with the network weights $w$ )   
+∙ Combinations of the above, e.g. $p ( \theta | x , t )$ , $p ( \theta | x , \pi )$ , ??(??|??, ??′), ??(??|??, ??), ??(??|??, ??), $p ( \boldsymbol { \theta } | \boldsymbol { x } , t , \mathcal { D } )$
+
+# Phase
+
+∙ Training: Transformation of training samples ∙ Test: Transformation of test samples, for example multiple augmented variants of a sample are classified and the result is aggregated over them
+
+A review of existing methods that use generic transformations can be found in Table 1. Dropout in its original form (Hinton et al., 2012; Srivastava et al., 2014) is one of the most popular methods from the generic group, but also several variants of Dropout have been proposed that provide additional theoretical motivation and improved empirical results (Standout (Ba and Frey, 2013), Random dropout probability (Bouthillier et al., 2015), Bayesian dropout (Maeda, 2014), Test-time dropout (Gal and Ghahramani, 2016)).
+
+Table 2 contains a list of some domain-specific methods focused especially on the image domain. Here the most used method is rigid and elastic image deformation.
+
+Target-preserving data augmentation In the following, we discuss an important group of methods: target-preserving data augmentation. These methods use stochastic transformations in input and hidden-feature spaces, while preserving the original target $t$ . As can be seen in the respective two columns in Tables 1–2, most of the listed methods have exactly these properties. These methods transform the training set to a distribution $Q$ , which is used for training instead. In other words, the training samples $( x _ { i } , t _ { i } ) \in \mathcal { D }$ are replaced in the empirical risk loss function (Eq. (3)) by augmented training samples $( \tau _ { \theta } ( x _ { i } ) , t _ { i } ) \sim Q$ . By randomly sampling the transformation parameters $\theta$ and thus creating many new samples $( \tau _ { \theta } ( x _ { i } ) , t _ { i } )$ from each original training sample $( x _ { i } , t _ { i } )$ , data augmentation attempts to bridge the limited-data gap between the expected and the empirical risk, Eqs. (2)–(3). While unlimited sampling from $Q$ provides more data than the original dataset $\mathcal { D }$ , both of them usually are merely approximations of the ground truth data distribution or of an ideal training dataset; both $\mathcal { D }$ and $Q$ have their own distinct biases, advantages and disadvantages. For example, elastic image deformations result in images that are not perfectly realistic; this is not necessarily a disadvantage, but it is a bias compared to the ground truth data distribution; in any case, the advantages (having more training data) often prevail. In some cases, it may be even desired for $Q$ to be deliberately different from the ground truth data distribution. For example, in case of class imbalance (unbalanced abundance or importance of classes), a common regularization strategy is to undersample or oversample the data, sometimes leading to a less realistic $Q$ but better models. This is how an ideal training dataset may be different from the ground truth data distribution.
+
+Table 1: Existing generic data-based methods classified according to our taxonomy. Table columns are described in Section 3.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Dependence</td><td rowspan=1 colspan=1>Transformationspace</td><td rowspan=1 colspan=1>Stochasticity(0 sampling)</td><td rowspan=1 colspan=1>Phase</td></tr><tr><td rowspan=1 colspan=1>Gaussian noise on input(Bishop,1995a; An,1996)</td><td rowspan=1 colspan=1>p(）</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Gaussian noise on hidden units(DeVries and Taylor, 2017)</td><td rowspan=1 colspan=1>p(0)</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Dropout (Hinton et al., 2012; Srivastavaet al., 2014)</td><td rowspan=1 colspan=1>p(）</td><td rowspan=1 colspan=1>Input andhidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Random dropout probability(Bouthillier et al., 2015, Sec. 4)</td><td rowspan=1 colspan=1>p()</td><td rowspan=1 colspan=1>Input andhidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Curriculum dropout(Morerio et al., 2017)</td><td rowspan=1 colspan=1>p(0|time)</td><td rowspan=1 colspan=1>Input andhidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Bayesian dropout(Maeda,2014)</td><td rowspan=1 colspan=1>p(0|π)</td><td rowspan=1 colspan=1>Input andhidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Standout (adaptive dropout)(Ba and Frey, 2013)</td><td rowspan=1 colspan=1>p(0|x,π)</td><td rowspan=1 colspan=1>Input andhidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>&quot;Projection” of dropout noise into inputspace (Bouthillier et al., 2015, Sec. 3)</td><td rowspan=1 colspan=1>p(0lx,f,w)</td><td rowspan=1 colspan=1>InputUses auxiliary Tin hidden-featurespace.</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Approximation of Gaussian process bytest-time dropout(Gal and Ghahramani, 2016)</td><td rowspan=1 colspan=1>p(0)</td><td rowspan=1 colspan=1>Input andhidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=1 colspan=1>Stochastic depth (Huang et al., 2016b)</td><td rowspan=1 colspan=1>p（）</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Noisy activation functions(Nair and Hinton, 2010; Xu et al., 2015;Gülcehre et al., 2016a)</td><td rowspan=1 colspan=1>p(0x）</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Training with adversarial examples(Szegedy et al., 2014)</td><td rowspan=1 colspan=1>p(0|x,t&#x27;)</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>AdaptiveConstrained</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Network fooling (adversarial examples)(Szegedy et al., 2014)(Not for regularization)</td><td rowspan=1 colspan=1>p(0lx,t）</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>AdaptiveConstrained</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=1 colspan=1>Synthetic minority oversampling inhidden-feature space (Wong et al., 2016)</td><td rowspan=1 colspan=1>p(0x,t,D)</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Inter-and extrapolation in hidden-featurespace (DeVries and Taylor, 2017)</td><td rowspan=1 colspan=1>p（0|x,t,D）</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Batch normalization (Ioffe and Szegedy,2015),Ghost batch normalization (Hofferet al., 2017)</td><td rowspan=1 colspan=1>p(0|x)</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Deterministic</td><td rowspan=1 colspan=1>Trainingand test</td></tr><tr><td rowspan=1 colspan=1>Layer normalization(Ba et al., 2016)</td><td rowspan=1 colspan=1>p(0|x)</td><td rowspan=1 colspan=1>Hidden features</td><td rowspan=1 colspan=1>Deterministic</td><td rowspan=1 colspan=1>Trainingand test</td></tr><tr><td rowspan=1 colspan=1>Annealed noise on targets(Wang and Principe, 1999)</td><td rowspan=1 colspan=1>p(0|time)</td><td rowspan=1 colspan=1>Target</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Label smoothing (Szegedy et al., 2016,Sec. 7; Goodfellow et al.,2016, Chap.7)</td><td rowspan=1 colspan=1>p()</td><td rowspan=1 colspan=1>Target</td><td rowspan=1 colspan=1>Deterministic</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Model compression (mimic models,distilled models) (Bucila et al., 2006; Baand Caruana,2014; Hinton et al., 2015)</td><td rowspan=1 colspan=1>p（0|x,D)</td><td rowspan=1 colspan=1>Target</td><td rowspan=1 colspan=1>Deterministic</td><td rowspan=1 colspan=1>Training</td></tr></table>
+
+Table 2: Existing domain-specific data-based methods classified according to our taxonomy. Table columns are described in Section 3. Note that these methods are never applied on the hidden features, because domain knowledge cannot be applied on them.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Dependence</td><td rowspan=1 colspan=1>Transformationspace</td><td rowspan=1 colspan=1>Stochasticity(0 sampling)</td><td rowspan=1 colspan=1>Phase</td></tr><tr><td rowspan=1 colspan=1>Rigid and elastic image transformation(Baird,1990; Yaegger et al., 1996; Simardet al.,2003; Ciresan et al., 2010)</td><td rowspan=1 colspan=1>p(0）</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Test-time image transformations(Simonyan and Zisserman, 2015;Dielemanet al., 2015)</td><td rowspan=1 colspan=1>p(0)</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=1 colspan=1>Sound transformations(Salamon and Bello, 2017)</td><td rowspan=1 colspan=1>p(）</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Error-maximizing rigid imagetransformations(Loosli et al., 2007; Fawzi et al., 2016)</td><td rowspan=1 colspan=1>p(）</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Adaptivestochastic&amp;constrained,respectively</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Learning class-specific elasticimage-deformation fields(Hauberg et al., 2016)</td><td rowspan=1 colspan=1>p(0|t,D)</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Random</td><td rowspan=1 colspan=1>Training</td></tr><tr><td rowspan=1 colspan=1>Any handcrafted data preprocessing, forexample scale-invariant feature transform(SIFT) for images (Lowe,1999)</td><td rowspan=1 colspan=1>p(）</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Deterministic</td><td rowspan=1 colspan=1>Trainingand test</td></tr><tr><td rowspan=1 colspan=1>Overfeat (Sermanet et al., 2013)</td><td rowspan=1 colspan=1>p(0)</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>Deterministic</td><td rowspan=1 colspan=1>Trainingand test</td></tr></table>
+
+If the transformation is additionally representation-preserving, then the distribution $Q$ created by the transformation $\tau _ { \theta }$ attempts to mimic the ground truth data distribution $P$ . Otherwise, the notion of a “ground truth data distribution” in the modified representation may be vague. We provide more details about the transition from $\mathcal { D }$ to $Q$ in Appendix B.
+
+Summary of data-based methods Data-based regularization is a popular and very useful way to improve the results of deep learning. In this section we formalized this group of methods and showed that seemingly unrelated techniques such as Target-preserving data augmentation, Dropout, or Batch normalization are methodologically surprisingly close to each other. In Section 8 we discuss future directions that we find promising.
+
+# 4 Regularization via the network architecture
+
+A network architecture $f$ can be selected to have certain properties or match certain assumptions in order to have a regularizing effect.1
+
+Table 3: Methods based on network architecture, and rough description of assumptions that they encode. There are partial overlaps between some listed methods. For example, Residual learning uses Skip-connections. Many noise-based methods also fit Table 1 (cf. Appendix A).   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Method class</td><td rowspan=1 colspan=1>Assumptions about an appropriate learnable input-output mapping</td></tr><tr><td rowspan=1 colspan=1>Any chosen (not overlycomplex） architecture</td><td rowspan=1 colspan=1>*</td><td rowspan=1 colspan=1>Mapping can be well approximated by functions from the chosen familywhich are easily accessible by optimization.</td></tr><tr><td rowspan=1 colspan=1>Small network</td><td rowspan=1 colspan=1>*</td><td rowspan=1 colspan=1>Mapping is simple (complexity of the mapping depends on the number ofnetwork units and layers).</td></tr><tr><td rowspan=1 colspan=1>Deep network</td><td rowspan=1 colspan=1>*</td><td rowspan=1 colspan=1>The mapping is complex,but can be decomposed into a composition (orgenerally into a directed acyclic graph) of simple nonlinear transformations,e.g.affine transformation followed by simple nonlinearity(fully-connectedlayer),“multi-channel convolution”followed by simple nonlinearity (convo-lutional layer),etc.</td></tr><tr><td rowspan=1 colspan=1>Hard bottleneck (layer withfew neurons)；soft bottleneck(e.g.Jacobian penalty (Rifaiet al.,201lc),see Section 6)</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>Data concentrates around a lower-dimensional manifold; has few factors ofvariation.</td></tr><tr><td rowspan=1 colspan=1>Convolutional networks(Fukushima and Miyake,1982;Rumelhart et al.,1986,Pp.348-352; LeCun et al.,1989;Simard et al., 2003)</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>Spatially local and shift-equivariant feature extraction is all we need.</td></tr><tr><td rowspan=1 colspan=1>Dilated convolutions(Yu and Koltun,2015)</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>Like convolutional networks.Additionally:Sparse sampling of wide localneighborhoods provides relevant information,and better preserves rele-vant high-resolution information than architectures with downscaling andupsampling.</td></tr><tr><td rowspan=1 colspan=1>Strided convolutions (seeDumoulin and Visin,2016)</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>The mapping is reliable at reacting to features that do not vary tooabruptly in space,i.e.which are present in several neighboring pixels andcan be detected even if the filter center skips some of the pixels.The out-put is robust towards slight changes of the location of features,and changesof strength/presence of spatially strongly varying features.</td></tr><tr><td rowspan=1 colspan=1>Pooling</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>The output is invariant to slight spatial distortions of the input (slightchanges of the location of (deep) features).Features that are sensitive tosuch distortions can be discarded.</td></tr><tr><td rowspan=1 colspan=1>Stochastic pooling(Zeiler and Fergus,2013)</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>The output is robust towards slight changes of the location (like pooling)but also of the strength/presence of (deep) features.</td></tr><tr><td rowspan=1 colspan=1>Training with different kindsof noise (including Dropout;see Section 3)</td><td rowspan=1 colspan=1>Noise</td><td rowspan=1 colspan=1>The mapping is robust to noise:the given class of perturbations of theinput or deep features should not affect the output too much.</td></tr><tr><td rowspan=1 colspan=1>Dropout (Hinton et al., 2012;Srivastava etal.,2014),DropConnect (Wan et al.,2013),and related methods</td><td rowspan=1 colspan=1>Noise</td><td rowspan=1 colspan=1>Extracting complementary (non-coadapted） features is helpful.Non-coadapted features are more informative,better disentangle factors of vari-ation.(We want to disentangle factors of variation because they are en-tangled in different ways in inputs vs.in outputs.)When interpreted as ensemble learning:usual assumptions of ensemblelearning (predictions of weak learners have complementary info and can becombined to strong prediction).</td></tr><tr><td rowspan=1 colspan=1>Maxout units(Goodfellow et al., 2013)</td><td rowspan=1 colspan=1>Layer operation</td><td rowspan=1 colspan=1>Assumptions similar to Dropout,with more accurate approximation ofmodel averaging (when interpreted as ensemble learning)</td></tr><tr><td rowspan=1 colspan=1>Skip-connections (Long et al.,2015;Huang et al.,2016a)</td><td rowspan=1 colspan=1>Connections be-tween layers</td><td rowspan=1 colspan=1>Certain lower-level features can directly be reused in a meaningful way at(several) higher levels of abstraction</td></tr><tr><td rowspan=1 colspan=1>Linearly augmentedfeed-forward network (van derSmagt and Hirzinger,1998)</td><td rowspan=1 colspan=1>Connections be-tween layers</td><td rowspan=1 colspan=1>Skip-connections that share weights with the non-skip-connections.Helpsagainst vanishing gradients. Rather changes the learning algorithm thanthe network mapping.</td></tr><tr><td rowspan=1 colspan=1>Residual learning(He et al., 2016)</td><td rowspan=1 colspan=1>Connections be-tween layers</td><td rowspan=1 colspan=1>Learning additive difference of a mapping f (or its compositional parts)from the identity mapping is easier than learning f itself. Meaningful deepfeatures can be composed asa sum of lower-level and intermediate-levelfeatures.</td></tr><tr><td rowspan=1 colspan=1>Stochastic depth(Huang et al.,2016b),DropIn(Smith et al., 2015)</td><td rowspan=1 colspan=1>Connectionsbetween layers;noise</td><td rowspan=1 colspan=1>Similar to Dropout:extracting complementary (non-coadapted) featuresacross different levels of abstraction is helpful; implicit model ensem-ble.Similar to Residual learning: meaningful deep features can be com-posed as a sum of lower-level and intermediate-level features,with theintermediate-level ones being optional,and leaving them out beingmeaningful data augmentation. Similar to Mollifying networks: simpli-fying random parts of the mapping improves training.</td></tr><tr><td rowspan=1 colspan=1>Mollifying networks(Gülcehre et al., 2016b)</td><td rowspan=1 colspan=1>Connectionsbetweenlayers;noise</td><td rowspan=1 colspan=1>The mapping can be easier approximated by estimating its decreasinglylinear simplified version</td></tr><tr><td rowspan=1 colspan=1>Network information criterion(Murata et al.,1994),Networkgrowing and network pruning(see Bishop,1995a, Sec. 9.5)</td><td rowspan=1 colspan=1>Model selection</td><td rowspan=1 colspan=1>Optimal generalization is reached by a network that has the right numberof units (not too few,not too many)</td></tr><tr><td rowspan=1 colspan=1>Multi-task learning (seeCaruana,1998;Ruder,2017)</td><td rowspan=1 colspan=1>*</td><td rowspan=1 colspan=1>Several tasks can help each other to learn mutually useful feature extrac-tors,as long as the tasks do not compete for resources (network capacity)</td></tr></table>
+
+Assumptions about the mapping An input-output mapping $f _ { w }$ must have certain properties in order to fit the data $P$ well. Although it may be intractable to enforce the precise properties of an ideal mapping, it may be possible to approximate them by simplified assumptions about the mapping. These properties and assumptions can then be imposed upon model fitting in a hard or soft manner. This limits the search space of models and allows finding better solutions. An example is the decision about the number of layers and units, which allows the mapping to be neither too simple nor too complex (thus avoiding underfitting and overfitting). Another example are certain invariances of the mapping, such as locality and shift-equivariance of feature extraction hardwired in convolutional layers. Overall, the approach of imposing assumptions about the input-output mapping discussed in this section is the selection of the network architecture $f$ . The choice of architecture $f$ on the one hand hardwires certain properties of the mapping; additionally, in an interplay between $f$ and the optimization algorithm (Section 7), certain weight configurations are more likely accessible by optimization than others, further limiting the likely search space in a soft way. A complementary way of imposing certain assumptions about the mapping are regularization terms (Section 6), as well as invariances present in the (augmented) data set (Section 3).
+
+Assumptions can be hardwired into the definition of the operation performed by certain layers, and/or into the connections between layers. This distinction is made in Table 3, where these and other methods are listed.
+
+In Section 3 about data, we mentioned regularization methods that transform data in the hidden-feature space. They can be considered part of the architecture. In other words, they fit both Sections 3 (data) and 4 (architecture). These methods are listed in Table 1 with hidden features as their transformation space.
+
+Weight sharing Reusing a certain trainable parameter in several parts of the network is referred to as weight sharing. This usually makes the model less complex than using separately trainable parameters. An example are convolutional networks (LeCun et al., 1989). Here the weight sharing does not merely reduce the number of weights that need to be learned; it also encodes the prior knowledge about the shift-equivariance and locality of feature extraction. Another example is weight sharing in autoencoders.
+
+Activation functions Choosing the right activation function is quite important; for example, using Rectified linear units (ReLUs) improved the performance of many deep architectures both in the sense of training times and accuracy as well as overcoming the need for greedy layer-wise pre-training (Hahnloser et al., 2000; Jarrett et al., 2009; Nair and Hinton, 2010; Glorot et al., 2011). The success of ReLUs can be partially attributed to the fact that they provide more expressive families of mappings compared to sigmoid activations (in the sense that the classical sigmoid nonlinearity can be approximated very well $^ 2$ with only two ReLUs, but it takes an infinite number of sigmoid units to approximate a ReLU) and their affine extrapolation to unknown regions of data space seems to provide better generalization in practice than the “stagnating” extrapolation of sigmoid units. However, their hard negative cut-off and unbounded positive part are not always desired properties. Some activation functions were designed explicitly for regularization. For Dropout, Maxout units (Goodfellow et al., 2013) allow a more precise approximation of the geometric mean of the model ensemble predictions at test time. Stochastic pooling (Zeiler and Fergus, 2013), on the other hand, is a noisy version of max-pooling. The authors claim that this allows modelling distributions of activations instead of taking just the maximum.
+
+Noisy models Stochastic pooling was one example of a stochastic generalization of a deterministic model. Some models are stochastic by injecting random noise into various parts of the model. The most frequently used noisy model is Dropout (Hinton et al., 2012; Srivastava et al., 2014).
+
+Multi-task learning A special type of regularization is multi-task learning (see Caruana, 1998; Ruder, 2017), where the network is modified to predict targets for several tasks at once. It can be combined with semi-supervised learning to utilize unlabeled data on an auxiliary task (Rasmus et al., 2015). A similar concept of sharing knowledge between tasks is also utilized in meta-learning, where multiple tasks from the same domain are learned sequentially, using previously gained knowledge as bias for new tasks (Baxter, 2000); and transfer learning, where knowledge from one domain is transferred into another domain (Pan and Yang, 2010). These approaches differ from other methods in the sense that they require some additional target data, which are not always available.
+
+Model selection The best among several trained models (e.g. with different architectures) can be selected by evaluating the predictions on a validation set. It should be noted that this holds for selecting the best combination of all techniques (Sections 3–7), not just architecture; and that the validation set used for model selection in the “outer loop” should be different from the validation set used e.g. for Early stopping (Section 7), and different from the test set (Cawley and Talbot, 2010). However, there are also model selection methods that specifically target the selection of the number of units in a specific network architecture, e.g. using network growing and network pruning (see Bishop, 1995a, Sec. 9.5), or additionally do not require a validation set, e.g. the Network information criterion to compare models based on the training error and second derivatives of the loss function (Murata et al., 1994).
+
+# 5 Regularization via the error function
+
+Ideally, the error function $E$ reflects an appropriate notion of quality, and in some cases some assumptions about the data distribution. Typical examples are mean squared error or cross-entropy. The error function $E$ can also have a regularizing effect. An example is Dice coefficient optimization (Milletari et al., 2016) which is robust to class imbalance. Moreover, the overall form of the loss function can be different than Eq. (3). For example, in certain loss functions that are robust to class imbalance, the sum is taken over pairwise combinations $\mathcal { D } \times \mathcal { D }$ of training samples (Yan et al., 2003), rather than over training samples. But such alternatives to Eq. (3) are rather rare, and similar principles apply. If additional tasks are added for a regularizing effect (multi-task learning (see Caruana, 1998; Ruder, 2017)), then targets $t$ are modified to consist of several tasks, the mapping $f _ { w }$ is modified to produce an according output $y$ , and $E$ is modified to account for the modified $t$ and $y$ . Besides, there are regularization terms that depend on $\partial E / \partial x$ . They depend on $t$ and thus in our definition are considered part of $E$ rather than of $R$ , but they are listed in Section 6 among $R$ (rather than here) for a better overview.
+
+# 6 Regularization via the regularization term
+
+Regularization can be achieved by adding a regularizer $R$ into the loss function. Unlike the error function $E$ (which expresses consistency of outputs with targets), the regularization term is independent of the targets. Instead, it is used to encode other properties of the desired model, to provide inductive bias (i.e. assumptions about the mapping other than consistency of outputs with targets). The value of $R$ can thus be computed for an unlabeled test sample, whereas the value of $E$ cannot.
+
+The independence of $R$ from $t$ has an important implication: it allows additionally using unlabeled samples (semi-supervised learning) to improve the learned model based on its compliance with some desired properties (Sajjadi et al., 2016). For example, semi-supervised learning with ladder networks (Rasmus et al., 2015) combines a supervised task with an unsupervised auxiliary denoising task in a “multi-task” learning fashion. (For alternative interpretations, see Appendix A.) Unlabeled samples are extremely useful when labeled samples are scarce. A Bayesian perspective on the combination of labeled and unlabeled data in a semi-supervised manner is offered by Lasserre et al. (2006).
+
+A classical regularizer is weight decay (see Plaut et al., 1986; Lang and Hinton, 1990; Goodfellow et al., 2016, Chap. 7):
+
+$$
+R ( w ) = \lambda \frac 1 2 \| w \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\lambda$ is a weighting term controlling the importance of the regularization over the consistency. From the Bayesian perspective, weight decay corresponds to using a symmetric multivariate normal distribution as prior for the weights: $p ( w ) = \mathcal { N } ( w | \mathbf { 0 } , \lambda ^ { - 1 } \mathbf { I } )$ (Nowlan and Hinton, 1992). Indeed, $\begin{array} { r } { - \log \mathcal { N } ( w | \mathbf { 0 } , \lambda _ { \cdot } ^ { - 1 } \mathbf { I } ) \propto - \log \exp \left( - \frac { \lambda } { 2 } \| w \| _ { 2 } ^ { 2 } \right) = \frac { \lambda } { 2 } \| w \| _ { 2 } ^ { 2 } = R ( w ) } \end{array}$ . Weight decay has gained big popularity, and it is being successfully used; Krizhevsky et al. (2012) even observe reduction of the error on the training set.
+
+Another common prior assumption that can be expressed via the regularization term is “smoothness” of the learned mapping (see Bengio et al., 2013, Section 3.2): if $x _ { 1 } \approx x _ { 2 }$ , then $f _ { w } ( x _ { 1 } ) \approx f _ { w } ( x _ { 2 } )$ . It can be expressed by the following loss term:
+
+$$
+R ( f _ { w } , x ) = \left. J _ { f _ { w } } ( x ) \right. _ { F } ^ { 2 } ,
+$$
+
+where $\left\| \cdot \right\| _ { F }$ denotes the Frobenius norm, and $J _ { f _ { w } } ( x )$ is the Jacobian of the neural network input-to-output mapping $f _ { w }$ for some fixed network weights $w$ . This term penalizes mappings with large derivatives, and is used in contractive autoencoders (Rifai et al., 2011c).
+
+The domain of loss regularizers is very heterogeneous. We propose a natural way to categorize them by their dependence. We saw in Eq. (5) that weight decay depends on $w$ only, whereas the Jacobian penalty in Eq. (6) depends on $w$ , $f$ , and $x$ . More precisely, the Jacobian penalty uses the derivative $\partial y / \partial x$ of output $y = f _ { w } ( x )$ w.r.t. input $x$ . (We use vector-by-vector derivative notation from matrix calculus, i.e. $\partial y / \partial x = \partial f _ { w } ( x ) / \partial x = J _ { f _ { w } }$ is the Jacobian of $f _ { w }$ with fixed weights $w$ .) We identify the following dependencies of $R$ :
+
+∙ Dependence on the weights $w$   
+∙ Dependence on the network output $y = f _ { w } ( x )$   
+∙ Dependence on the derivative $\partial y / \partial w$ of the output $y = f _ { w } ( x )$ w.r.t. the weights $w$   
+∙ Dependence on the derivative $\partial y / \partial x$ of the output $y = f _ { w } ( x )$ w.r.t. the input $x$   
+∙ Dependence on the derivative $\partial E / \partial x$ of the error term $E$ w.r.t. the input $x$ ( $E$ depends on $t$ , and according to our definition such methods belong to Section 5, but they are listed here for overview)
+
+A review of existing methods can be found in Table 4. Weight decay seems to be still the most popular of the regularization terms. Some of the methods are equivalent or nearly equivalent to other methods from different taxonomy branches. For example, Tangent prop simulates minimal data augmentation (Simard et al., 1992); Injection of small-variance Gaussian noise (Bishop, 1995b; An, 1996) is an approximation of Jacobian penalty (Rifai et al., 2011c); and Fast dropout (Wang and Manning, 2013) is (in shallow networks) a deterministic approximation of Dropout. This is indicated in the Equivalence column in Table 4.
+
+# 7 Regularization via optimization
+
+The last class of the regularization methods according to our taxonomy is the regularization through optimization. While this may sound unusual, optimization and regularization cannot be clearly separated in the context of deep learning where it is not so crucial what the optimum of the empirical risk is (because it cannot be found exactly, and the ultimate goal is minimizing the expected risk anyway). Instead, the shape of the loss function and the optimization procedure play together to dictate how the training proceeds in the weight space and where it ends up. To demonstrate the overlap of regularization and optimization, we show in Figure 1 how one of the most prominent regularization methods, Dropout, can be seen as a modification of the optimization procedure.
+
+Stochastic gradient descent (SGD) (see Bottou, 1998) (along with its derivations) is the most frequently used optimization algorithm in the context of deep neural networks and is the center of our attention. We also list some alternative methods below.
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=2 colspan=1>Description</td><td rowspan=1 colspan=5>Dependency</td><td rowspan=2 colspan=1>Equivalence</td></tr><tr><td rowspan=1 colspan=1>w</td><td rowspan=1 colspan=1>y</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Weight decay (see Plaut et al.,1986;Lang and Hinton,1990;Goodfellow et al.,2016,Chap.7)</td><td rowspan=1 colspan=1>L² norm on network weights (notbiases). Favors smaller weights,thus for usual architectures tendsto make the mapping less“extreme&quot;,more robust to noise in the input.</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Early stopping (seeCollobert and Bengio,2004;Goodfellow et al., 2016,Chap.7)</td></tr><tr><td rowspan=1 colspan=1>Weight smoothing(Lang and Hinton,1990)</td><td rowspan=1 colspan=1>Penalizes L²norm of gradientsof learned filters,making themsmooth. Not beneficial in practice.</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Weight elimination(Weigend et al.,1991)</td><td rowspan=1 colspan=1>Similar to weight decay but favorsfew stronger connections over manyweak ones.</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Goal similar to Narrow andbroad Gaussians</td></tr><tr><td rowspan=1 colspan=1>Soft weight-sharing(Nowlan and Hinton,1992)</td><td rowspan=1 colspan=1>Mixture-of-Gaussians prior onweights.Generalization of weightdecay.Weights are pushed to forma predefined number of groups withsimilar values.</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Narrow and broad Gaussians(Nowlan and Hinton,1992;Blundell et al., 2015)</td><td rowspan=1 colspan=1>Weights come from two Gaussians,a narrow and a broad one.Specialcase of Soft weight-sharing.</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Goal similar to Weightelimination</td></tr><tr><td rowspan=1 colspan=1>Fast dropout approximation(Wang and Manning,2013)</td><td rowspan=1 colspan=1>Approximates the loss that dropoutminimizes. Weighted L2weightpenalty. Only for shallow networks.</td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1>x</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Dropout</td></tr><tr><td rowspan=1 colspan=1>Mutual exclusivity(Sajjadi et al., 2016)</td><td rowspan=1 colspan=1>Unlabeled samples push decisionboundaries to low-density regions ininput space,promoting sharp (con-fident） predictions.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Segmentation with binarypotentials (BenTaieb andHamarneh,2016)</td><td rowspan=1 colspan=1>Penalty on anatomically implausi-ble image segmentations.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Flat minima search(Hochreiter and Schmidhuber,1995)</td><td rowspan=1 colspan=1>Penalty for sharp minima,i.e. forweight configurations where smallweight perturbation leads to higherror increase.Flat minima havelow Minimum description length(i.e.exhibit ideal balance betweentraining error and model complex-ity)and thus should generalize bet-ter (Rissanen,1986).</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Tangent prop(Simard et al.,1992)</td><td rowspan=1 colspan=1>L² penalty on directional derivativeof mapping in the predefined tan-gent directions that correspond toknown input-space transformations.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Simple data augmentation</td></tr><tr><td rowspan=1 colspan=1>Jacobian penalty(Rifai et al., 2011c)</td><td rowspan=1 colspan=1>L2penalty on the Jacobian of(partsof） the network mapping-smoothness prior.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Noise on inputs injection(not exact (see An,1996))</td></tr><tr><td rowspan=1 colspan=1>Manifold tangent classifier(Rifai et al., 2011a)</td><td rowspan=1 colspan=1>Like tangent prop,but the input“tangent” directions are extractedfrom manifold learned by a stack ofcontractive autoencoders and thenperforming SVD of the Jacobian ateach input sample.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Hessian penalty(Rifai et al.,2011b)</td><td rowspan=1 colspan=1>Fast waytoapproximateL²penalty of thepenalizingJacobianwithnoisyinput.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>X</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Tikhonov regularizers(Bishop,1995b)</td><td rowspan=1 colspan=1>L² penalty on (up to) n-th deriva-tive of the learned mapping w.r.t.input.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Forpenalty on firstderivative: noise on inputsinjection (not exact (see An,1996)）</td></tr><tr><td rowspan=1 colspan=1>Loss-invariant backpropagation(Demyanov et al., 2015, Sec. 3.1;Lyu et al., 2015)</td><td rowspan=1 colspan=1>(L²)norm of gradient of loss w.r.t.input.Changes the mapping suchthat the loss becomes rather invari-ant to changes of the input.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>Adversarial training</td></tr><tr><td rowspan=1 colspan=1>Prediction-invariantbackpropagation(Demyanov et al.,2015, Sec. 3.2)</td><td rowspan=1 colspan=1>(L2） norm of directional derivativeof mapping w.r.t.input in the di-rection of x causing the largest in-crease in loss.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>Adversarial training</td></tr></table>
+
+Table 4: Regularization terms, with dependencies marked by $\pmb { * }$ . Methods that depend on $\partial E / \partial x$ implicitly depend on targets $t$ and thus can be considered part of the error function (Section 5) rather than regularization term (Section 6).
+
+Stochastic gradient descent is an iterative optimization algorithm using the following update rule:
+
+$$
+w _ { t + 1 } = w _ { t } - \eta _ { t } \nabla _ { w } \mathcal { L } ( w _ { t } , d _ { t } ) ,
+$$
+
+where $\nabla \mathcal { L } ( w _ { t } , d _ { t } )$ is the gradient of the loss $\mathcal { L }$ evaluated on a mini-batch $d _ { t }$ from the training set $\mathcal { D }$ . It is frequently used in combination with momentum and other tweaks improving the convergence speed (see Wilson et al., 2017). Moreover, the noise induced by the varying mini-batches helps the algorithm escape saddle points (Ge et al., 2015); this can be further reinforced by adding supplementary gradient noise (Neelakantan et al., 2015; Chaudhari and Soatto, 2015).
+
+If the algorithm reaches a low training error in a reasonable time (linear in the size of the training set, allowing multiple passes through $\mathcal { D }$ ), the solution generalizes well under certain mild assumptions; in that sense SGD works as an implicit regularizer : a short training time prevents overfitting even without any additional regularizer used (Hardt et al., 2016). This is in line with (Zhang et al., 2017) who find in a series of experiments that regularization (such as Dropout, data augmentation, and weight decay) is by itself neither necessary nor sufficient for good generalization.
+
+We divide the methods into three groups: initialization/warm-start methods, update methods, and termination methods, discussed in the following.
+
+Initialization and warm-start methods These methods affect the initial selection of the model weights. Currently the most frequently used method is sampling the initial weights from a carefully tuned distribution. There are multiple strategies based on the architecture choice, aiming at keeping the variance of activations in all layers around 1, thus preventing vanishing or exploding activations (and gradients) in deeper layers (Glorot and Bengio, 2010, Sec. 4.2; He et al., 2015).
+
+Another (complementary) option is pre-training on different data, or with a different objective, or with partially different architecture. This can prime the learning algorithm towards a good solution before the fine-tuning on the actual objective starts. Pre-training the model on a different task in the same domain may lead to learning useful features, making the primary task easier. However, pre-trained models are also often misused as a lazy approach to problems where training from scratch or using thorough domain adaptation, transfer learning, or multi-task learning methods would be worth trying. On the other hand, pre-training or similar techniques may be a useful part of such methods.
+
+Finally, with some methods such as Curriculum learning (Bengio et al., 2009), the transition between pre-training and fine-tuning is smooth. We refer to them as warm-start methods.
+
+∙ Initialization without pre-training
+
+– Random weight initialization (Rumelhart et al., 1986, p. 330; Glorot and Bengio, 2010; He et al., 2015; Hendrycks and Gimpel, 2016)   
+– Orthogonal weight matrices (Saxe et al., 2013)   
+– Data-dependent weight initialization (Krähenbühl et al., 2015)
+
+∙ Initialization with pre-training
+
+– Greedy layer-wise pre-training (Hinton et al., 2006; Bengio et al., 2007; Erhan et al., 2010) (has become less important due to advances (e.g. ReLUs) in effective end-to-end training that optimizes all parameters simultaneously)   
+– Curriculum learning (Bengio et al., 2009)   
+– Spatial contrasting (Hoffer et al., 2016)   
+– Subtask splitting (Gülçehre and Bengio, 2016)
+
+Update methods This class of methods affects individual weight updates. There are two complementary subgroups: Update rules modify the form of the update formula; Weight and gradient filters are methods that affect the value of the gradient or weights, which are used in the update formula, e.g. by injecting noise into the gradient (Neelakantan et al., 2015).
+
+![](images/2d3c043833918e9948dba4ab7bc13a7e06caab09934ecb7fb43bf136b98df82a.jpg)  
+Figure 1: Effect of Dropout on weight optimization. Starting from the current weight configuration (red dot), all weights of certain neurons are set to zero (black arrow), descent step is performed in that subspace (teal arrow), and then the discarded weight-space coordinates are restored (blue arrow).
+
+Again, it is not entirely clear which of the methods only speed up the optimization and which actually help the generalization. Wilson et al. (2017) show that some of the methods such as AdaGrad or Adam even lose the regularization abilities of SGD.
+
+∙ Update rules
+
+– Momentum, Nesterov’s accelerated gradient method, AdaGrad, AdaDelta, RMSProp, Adam—overview in (Wilson et al., 2017) Learning rate schedules (Girosi et al., 1995; Hoffer et al., 2017)   
+Online batch selection (Loshchilov and Hutter, 2015) SGD alternatives: L-BFGS (Liu and Nocedal, 1989; Le et al., 2011), Hessianfree methods (Martens, 2010), Sum-of-functions optimizer (Sohl-Dickstein et al., 2014), ProxProp (Frerix et al., 2017)
+
+∙ Gradient and weight filters
+
+– Annealed Langevin noise (Neelakantan et al., 2015)   
+– AnnealSGD (Chaudhari and Soatto, 2015) Dropout (Hinton et al., 2012; Srivastava et al., 2014) corresponds to optimization steps in subspaces of weight space, see Figure 1   
+Annealed noise on targets (Wang and Principe, 1999) (works as noise on gradient, but belongs rather to data-based methods, Section 3)
+
+Termination methods There are numerous possible stopping criteria and selecting the right moment to stop the optimization procedure may improve the generalization by reducing the error caused by the discrepancy between the minimizers of expected and empirical risk: The network first learns general concepts that work for all samples from the ground truth distribution $P$ before fitting the specific sample $\mathcal { D }$ and its noise (Krueger et al., 2017).
+
+The most successful and popular termination methods put a portion of the labeled data aside as a validation set and use it to evaluate performance (validation error ). The most prominent example is Early stopping (see Prechelt, 1998). Collobert and Bengio (2004) show that Early stopping has the same effect as Weight decay regularization penalty term in multi-layered perceptrons with linear output units; however, its hyperparameters are easier to tune.
+
+In scenarios where the training data are scarce it is possible to resort to termination methods that do not use a validation set. The simplest case is fixing the number of passes through the training set.
+
+∙ Termination using a validation set – Early stopping (see Morgan and Bourlard, 1990; Prechelt, 1998) – Choice of validation set size based on test set size (Amari et al., 1997)   
+∙ Termination without using a validation set – Fixed number of iterations – Optimized approximation algorithm (Liu et al., 2008)
+
+# 8 Recommendations, discussion, conclusions
+
+We see the main benefits of our taxonomy to be two-fold: Firstly, it provides an overview of the existing techniques to the users of regularization methods and gives them a better idea of how to choose the ideal combination of regularization techniques for their problem. Secondly, it is useful for development of new methods, as it gives a comprehensive overview of the main principles that can be exploited to regularize the models. We summarize our recommendations $^ 3$ in the following paragraphs:
+
+Recommendations for users of existing regularization methods Overall, using the information contained in data as well as prior knowledge as much as possible, and primarily starting with popular methods, the following procedure can be helpful:
+
+∙ Common recommendations for the first steps:
+
+– Deep learning is about disentangling the factors of variation. An appropriate data representation should be chosen; known meaningful data transformations should not be outsourced to the learning. Redundantly providing the same information in several representations is okay. Output nonlinearity and error function should reflect the learning goals.   
+– A good starting point are techniques that usually work well (e.g. ReLU, successful architectures). Hyperparameters (and architecture) can be tuned jointly, but “lazily” (interpolating/extrapolating from experience instead of trying too many combinations). Often it is helpful to start with a simplified dataset (e.g. fewer and/or easier samples) and a simple network, and after obtaining promising results gradually increasing the complexity of both data and network while tuning hyperparameters and trying regularization methods.
+
+∙ Regularization via data:
+
+– When not working with nearly infinite/abundant data: $^ *$ Gathering more real data (and using methods that take its properties into account) is advisable if possible: · Labeled samples are best, but unlabeled ones can also be helpful (compatible with semi-supervised learning). Samples from the same domain are best, but samples from similar domains can also be helpful (compatible with domain adaptation and transfer learning). · Reliable high-quality samples are best, but lower-quality ones can also be helpful (their confidence/importance can be adjusted accordingly). · Labels for an additional task can be helpful (compatible with multi-task learning). · Additional input features (from additional information sources) and/or data preprocessing (i.e. domain-specific data transformations) can be helpful (the network architecture needs to be adjusted accordingly).
+
+$^ *$ Data augmentation (e.g. target-preserving handcrafted domain-specific transformations) can well compensate for limited data. If natural ways to augment data (to mimic natural transformations sufficiently well) are known, they can be tried (and combined). \* If natural ways to augment data are unknown or turn out to be insufficient, it may be possible to infer the transformation from data (e.g. learning imagedeformation fields) if a sufficient amount of data is available for that.
+
+– Popular generic methods (e.g. advanced variants of Dropout) often also help.
+
+∙ Architecture and regularization terms:
+
+– Knowledge about possible meaningful properties of the mapping can be used to e.g. hardwire invariances (to certain transformations) into the architecture, or be formulated as regularization terms.
+
+– Popular methods may help as well (see Tables 3–4), but should be chosen to match the assumptions about the mapping (e.g. convolutional layers are fully appropriate only if local and shift-equivariant feature extraction on regular-grid data is desired).
+
+∙ Optimization:
+
+Initialization: Even though pre-trained ready-made models greatly speed up prototyping, training from a good random initialization should also be considered. – Optimizers: Trying a few different ones, including advanced ones (e.g. Nesterov momentum, Adam, ProxProp), may lead to improved results. Correctly chosen parameters, such as learning rate, usually make a big difference.
+
+Recommendations for developers of novel regularization methods Getting an overview and understanding the reasons for the success of the best methods is a great foundation. Promising empty niches (certain combinations of taxonomy properties) exist that can be addressed. The assumptions to be imposed upon the model can have a strong impact on most elements of the taxonomy. Data augmentation is more expressive than loss terms (loss terms enforce properties only in infinitesimally small neighborhood of the training samples; data augmentation can use rich transformation parameter distributions). Data and loss terms impose assumptions and invariances in a rather soft manner, and their influence can be tuned, whereas hardwiring the network architecture is a harsher way to impose assumptions. Different assumptions and options to impose them have different advantages and disadvantages.
+
+Future directions for data-based methods There are several promising directions that in our opinion require more investigation: Adaptive sampling of $\theta$ might lead to lower errors and shorter training times (Fawzi et al., 2016) (in turn, shorter training times may additionally work as implicit regularization (Hardt et al., 2016), see also Section 7). Secondly, learning class-dependent transformations (i.e. $p ( \theta | t )$ ) in our opinion might lead to more plausible samples. Furthermore, the field of adversarial examples (and network robustness to them) is gaining increased attention after the recently sparked discussion on real-world adversarial examples and their robustness/invariance to transformations such as the change of camera position (Lu et al., 2017; Athalye and Sutskever, 2017). Countering strong adversarial examples may require better regularization techniques.
+
+Summary In this work we proposed a broad definition of regularization for deep learning, identified five main elements of neural network training (data, architecture, error term, regularization term, optimization procedure), described regularization via each of them, including a further, finer taxonomy for each, and presented example methods from these subcategories. Instead of attempting to explain referenced works in detail, we merely pinpointed their properties relevant to our categorization. Our work demonstrates some links between existing methods. Moreover, our systematic approach enables the discovery of new, improved regularization methods by combining the best properties of the existing ones.
+
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+
+# A Ambiguities in the taxonomy
+
+Although our proposed taxonomy seems intuitive, there are some ambiguities: Certain methods have multiple interpretations matching various categories. Viewed from the exterior, a neural network maps inputs $x$ to outputs $y$ . We formulate this as $y = f _ { w } ( \tau _ { \theta } ( x ) )$ for transformations $\tau _ { \theta }$ in input space (and similarly for hidden-feature space, where $^ Ḋ \prime \theta Ḍ$ is applied in between layers of the network $f _ { w }$ ). However, how to split this $x$ -to- $y$ mapping into “the $\tau _ { \theta }$ part” and “the $f _ { w }$ part”, and thus into Section 3 vs. Section 4, is ambiguous and up to one’s taste and goals. In our choices (marked with “ ” below), we attempt to use common notions and Occam’s razor.
+
+Ambiguity of attributing noise to $f$ , or to $w$ , or to data transformations $\tau _ { \theta }$ : – Stochastic methods such as Stochastic depth (Huang et al., 2016b) can have several interpretations if stochastic transformations are allowed for $f$ or $w$ : Stochastic transformation of the architecture $f$ (randomly dropping some connections), Table 3 $\sqcup$ Stochastic transformation of the weights $w$ (setting some weights to $0$ in a certain random pattern) $\sqcup$ Stochastic transformation $\tau _ { \theta }$ of data in hidden-feature space; dependence is $p ( \theta )$ , described in Table 1 for completeness
+
+∙ Ambiguity of splitting $^ Ḋ \prime \theta Ḍ$ into $\tau$ and $\theta$ :
+
+– Dropout:
+
+Parameters $\theta$ are the dropout mask; dependence is $p ( \theta )$ ; transformation $\tau$ applies the dropout mask to the hidden features
+
+$\sqcup$ Parameters $\theta$ are the seed state of a pseudorandom number generator; dependence is $p ( \theta )$ ; transformation $\tau$ internally generates the random dropout mask from the random seed and applies it to the hidden features
+
+– Projecting dropout noise into input space (Bouthillier et al., 2015, Sec. 3) can fit our taxonomy in different ways by defining $\tau$ and $\theta$ accordingly. It can have similar interpretations as Dropout above (if $\tau$ is generalized to allow for dependence on $x , f , w )$ , but we prefer the third interpretation without such generalizations:
+
+$\sqcup$ Parameters $\theta$ are the dropout mask (to be applied in a hidden layer); dependence is $p ( \theta )$ ; transformation $\gamma$ transforms the input to mimic the effect of the mask   
+$\sqcup$ Parameters $\theta$ are the seed state of a pseudorandom number generator; dependence is $p ( \theta )$ ; transformation $\tau$ internally generates the random dropout mask from the random seed and transforms the input to mimic the effect of the mask   
+V Parameters $\theta$ describe the transformation of the input in any formulation; dependence is $p ( \theta | x , f , w )$ ; transformation $\tau$ merely applies the transformation in input space
+
+∙ Ambiguity of splitting the network operation $f _ { w }$ into layers: There are several possibilities to represent a function (neural network) as a composition (or directed acyclic graph) of functions (layers).
+
+∙ Many of the input and hidden-feature transformations (Section 3) can be considered layers of the network (Section 4). In fact, the term “layer” is not uncommon for Dropout or Batch normalization.
+
+∙ The usage of a trainable parameter in several parts of the network is called weight sharing. However, some mappings can be expressed with two equivalent formulas such that a parameter appears only once in one formulation, and several times in the other.
+
+∙ Ambiguity of $E$ vs. $R$ : Auxiliary denoising task in ladder networks (Rasmus et al., 2015) and similar autoencoder-style loss terms can be interpreted in different ways:
+
+V Regularization term $R$ without given auxiliary targets $t$   
+$\sqcup$ The ideal reconstructions can be considered as targets $t$ (if the definition of “targets” is slightly modified) and thus the denoising task becomes part of the error term $E$
+
+# B Data-augmented loss function
+
+To understand the success of target-preserving data augmentation methods, we consider the data-augmented loss function, which we obtain by replacing the training samples $( x _ { i } , t _ { i } ) \in \mathcal { D }$ in the empirical risk loss function (Eq. (3)) by augmented training samples $( \tau _ { \theta } ( x _ { i } ) , t _ { i } )$ :
+
+$$
+\begin{array} { r l } & { \hat { \mathcal { L } } _ { A } = \displaystyle \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } \mathbb { E } _ { \theta } \Big [ \ell \big ( \tau _ { \theta } ( x _ { i } ) , t _ { i } \big ) \Big ] } \\ & { \quad \quad = \displaystyle \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } \int \Big ( \ell \big ( \tau _ { \theta } ( x _ { i } ) , t _ { i } \big ) \Big ) p ( \theta ) \mathrm { d } \theta , } \end{array}
+$$
+
+where we have replaced the inner part ( $E$ and $R$ ) of the loss function by $\ell$ to simplify the notation. Moreover, $\hat { \mathcal { L } } _ { A }$ can be rewritten as
+
+$$
+\begin{array} { r l } & { \hat { \mathcal { L } } _ { A } = \displaystyle \iint \frac { 1 } { X , T } \displaystyle \sum _ { ( x , t _ { i } ) \in \mathcal { D } _ { \Theta } } \int \ell ( x , t ) \ p ( \theta ) \ \delta \big ( x - \tau _ { \theta } ( x _ { i } ) \big ) \ \delta ( t - t _ { i } ) \ \mathrm { d } \theta \mathrm { d } t \mathrm { d } x } \\ & { \quad = \displaystyle \iint \ell ( x , t ) \Bigg [ \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } _ { \Theta } } \int \delta \big ( x - \tau _ { \theta } ( x _ { i } ) \big ) \ \delta ( t - t _ { i } ) \ p ( \theta ) \mathrm { d } \theta \Bigg ] \mathrm { d } t \mathrm { d } x } \\ & { \quad = \displaystyle \iint \ell ( x , t ) \ q ( x , t ) \ \mathrm { d } t \mathrm { d } x , } \end{array}
+$$
+
+where $\delta ( x )$ is the Dirac delta function: $\delta ( x ) = 0 \forall x \neq 0$ and $\textstyle \int \delta ( x ) \mathrm { d } x = 1$ ; and $\boldsymbol { q } ( \boldsymbol { x } , t )$ is defined as
+
+$$
+q ( x , t ) = \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } \int _ { \Theta } \delta \big ( x - \tau _ { \theta } ( x _ { i } ) \big ) \delta ( t - t _ { i } ) p ( \theta ) \mathrm { d } \theta .
+$$
+
+Since $q$ is non-negative and $\int _ { \ a } ^ { \cdot } \int q ( x , t ) \mathrm { d } x \mathrm { d } t = 1$ , it is a valid probability density function inducing the distribution $Q$ of augmented data. Therefore,
+
+$$
+\begin{array} { r } { \hat { \mathcal { L } } _ { A } = \mathbb { E } _ { ( x , t ) \sim Q } \big [ \ell ( x , t ) \big ] . } \end{array}
+$$
+
+When $Q = P$ , Eq. (11) becomes the expected risk (2). We can show how this is related to importance sampling:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } = \mathbb { E } _ { ( x , t ) \sim P } \big [ \ell ( x , t ) \big ] } \\ { \displaystyle = \int _ { X , T } \ell ( x , t ) p ( x , t ) \mathrm { d } t \mathrm { d } x } \\ { \displaystyle \quad \times \int _ { X , T } \ell ( x , \ell ) \frac { p ( x , t ) } { q ( x , t ) } q ( x , \ell ) \mathrm { d } t \mathrm { d } x } \\ { \displaystyle \quad \times _ { X , T } } \\ { \displaystyle = \mathbb { E } _ { ( x , t ) \sim Q } \Big [ \ell ( x , t ) \frac { p ( x , t ) } { q ( x , t ) } \Big ] } \\ { \displaystyle \quad \neq \mathbb { E } _ { ( x , t ) \sim Q } \big [ \ell ( x , t ) \big ] } \\ { \displaystyle \quad = \hat { \ell } _ { A + } } \end{array}
+$$
+
+The difference between $\mathcal { L }$ and $\hat { \mathcal { L } } _ { A }$ is the re-weighting term $p ( x , t ) / q ( x , t )$ identical to the one known from importance sampling (see Bishop, 1995a). The more similar $Q$ is to $P$ (i.e. the closer $Q$ models the ground truth distribution $P$ ), the more similar the augmented-data loss $\hat { \mathcal { L } } _ { A }$ is to the expected loss $\mathcal { L }$ . We see that data augmentation tries to simulate the real distribution $P$ by creating new samples from the training set $\mathcal { D }$ , bridging the gap between the expected and the empirical risk.