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+# On Training Implicit Models
+
+Zhengyang Geng1,2∗ Xin-Yu Zhang2∗ Shaojie Bai4 Yisen Wang2,3 Zhouchen Lin2,3,5†
+
+1Zhejiang Lab, China 2Key Lab. of Machine Perception, School of AI, Peking University 3Institute for Artificial Intelligence, Peking University 4Carnegie Mellon University 5Pazhou Lab, China
+
+# Abstract
+
+This paper focuses on training implicit models of infinite layers. Specifically, previous works employ implicit differentiation and solve the exact gradient for the backward propagation. However, is it necessary to compute such an exact but expensive gradient for training? In this work, we propose a novel gradient estimate for implicit models, named phantom gradient, that 1) forgoes the costly computation of the exact gradient; and 2) provides an update direction empirically preferable to the implicit model training. We theoretically analyze the condition under which an ascent direction of the loss landscape could be found, and provide two specific instantiations of the phantom gradient based on the damped unrolling and Neumann series. Experiments on large-scale tasks demonstrate that these lightweight phantom gradients significantly accelerate the backward passes in training implicit models by roughly $1 . 7 \times$ , and even boost the performance over approaches based on the exact gradient on ImageNet.
+
+# 1 Introduction
+
+Conventional neural networks are typically constructed by explicitly stacking multiple linear and non-linear operators in a feed-forward manner. Recently, the implicitly-defined models [1, 2, 3, 4, 5] have attracted increasing attentions and are able to match the state-of-the-art results by explicit models on several vision [3, 6], language [2] and graph [4] tasks. These works treat the evolution of the intermediate hidden states as a certain form of dynamics, such as fixed-point equations [2, 3] or ordinary differential equations (ODEs) [1, 7], which represents infinite latent states. The forward passes of implicit models are therefore formulated as solving the underlying dynamics, by either black-box ODE solvers [1, 7] or root-finding algorithms [2, 3]. As for the backward passes, however, directly differentiating through the forward pass trajectories could induce a heavy memory overhead [8, 9]. To this end, researchers have developed memory-efficient backpropagation via implicit differentiation, such as solving a Jacobian-based linear fixed-point equation for the backward pass of deep equilibrium models (DEQs) [2], which eventually makes the backpropagation trajectories independent of the forward passes. This technique allows one to train implicit models with essentially constant memory consumption, as we only need to store the final output and the layer itself without saving any intermediate states. However, in order to estimate the exact gradient by implicit differentiation, implicit models have to rely on expensive black-box solvers for backward passes, e.g., ODE solvers or root-solving algorithms. These black-box solver usually makes the gradient computation very costly in practice, even taking weeks to train state-of-the-art implicit models on ImageNet [10] with 8 GPUs.
+
+This work investigates fast approximate gradients for training implicit models. We found that a firstorder oracle that produces good gradient estimates is enough to efficiently and effectively train implicit models, circumventing laboriously computing the exact gradient as in prior arts [2, 3, 4, 11, 12]. We develop a framework in which a balanced trade-off is made between the precision and conditioning of the gradient estimate. Specifically, we provide the general condition under which the phantom gradient can provide an ascent direction of the loss landscape. We further propose two instantiations of phantom gradients in the context of DEQ models, which are based on the the damped fixed-point unrolling and the Neumann series, respectively. Importantly, we show that our proposed instantiations satisfy the theoretical condition, and that the stochastic gradient descent (SGD) algorithm based on the phantom gradient enjoys a sound convergence property as long as the relevant hyperparameters, e.g., the damping factor, are wisely selected. Note that our method only affects, and thus accelerates, the backward formulation of the implicit models, leaving the forward pass formulation (i.e., the root-solving process) and the inference behavior unchanged so that our method is applicable to a wide range of implicit models, forward solvers, and inference strategies. We conduct an extensive set of synthetic, ablation, and large-scale experiments to both analyze the theoretical properties of the phantom gradient and validate its speedup and performances on various tasks, such as ImageNet [10] classification and Wikitext-103 [13] language modeling.
+
+Overall, our results suggest that: 1) the phantom gradient estimates an ascent direction; 2) it is applicable to large-scale tasks and is capable of achieving a strong performance which is comparable with or even better than that of the exact gradient; and 3) it significantly shortens the total training time needed for implicit models roughly by a factor of $1 . 4 \sim 1 . 7 \times$ , and even accelerates the backward passes by astonishingly $1 2 \times$ on ImageNet. We believe that our results provide strong evidence for effectively training implicit models with the lightweight phantom gradient.
+
+# 2 Method
+
+# 2.1 Inspection of Implicit Differentiation
+
+In this work, we primarily focus on the formulation of implicit models based on root-solving, represented by the DEQ models [2]. The table of notations is arranged in Appendix A. Specifically, given an equilibrium module $\mathcal { F }$ , the output of the implicit model is characterized by the solution $h ^ { * }$ to the following fixed-point equation:
+
+$$
+\begin{array} { r } { \begin{array} { r } { h ^ { * } = \mathcal { F } ( h ^ { * } , z ) , } \end{array} } \end{array}
+$$
+
+where $\boldsymbol { z } \in \mathbb { R } ^ { d _ { \boldsymbol { u } } + d _ { \boldsymbol { \theta } } }$ is the union of the module’s input $\pmb { u } \in \mathbb { R } ^ { d _ { u } }$ and parameters $\pmb { \theta } \in \mathbb { R } ^ { d _ { \pmb { \theta } } }$ , i.e., $z ^ { \top } =$ $[ \boldsymbol { u } ^ { \top } , \boldsymbol { \theta } ^ { \top } ]$ . Here, $\textbf { \em u }$ is usually a projection of the original data point $\boldsymbol { x } \in \mathbb { R } ^ { d _ { x } }$ , e.g., $\pmb { u } = \mathcal { M } ( \pmb { x } )$ . In this section, we assume $\mathcal { F }$ is a contraction mapping w.r.t. $^ { h }$ so that its Lipschitz constant $L _ { h } \ w . r . t . \ h$ is less than one, i.e., $L _ { h } < 1$ , a setting that has been analyzed in recent works $[ 1 4 , 1 5 ] ^ { 1 }$ .
+
+To differentiate through the fixed point by Eq. (1), we need to calculate the gradient of $h ^ { \ast } w . r . t .$ the input $_ z$ . By Implicit Function Theorem (IFT), we have
+
+$$
+\frac { \partial { \pmb h } ^ { * } } { \partial z } = \left. \frac { \partial \mathcal { F } } { \partial z } \right| _ { { \pmb h } ^ { * } } \left( { \pmb I } - \left. \frac { \partial \mathcal { F } } { \partial { \pmb h } } \right| _ { { \pmb h } ^ { * } } \right) ^ { - 1 } .
+$$
+
+Here, $( \partial \pmb { a } / \partial b ) _ { i j } = \partial \pmb { a } _ { j } / \partial b _ { i }$ . The equilibrium point $h ^ { * }$ of Eq. (1) is then passed to a post-processing function $\mathcal { G }$ to obtain a prediction $\hat { y } = \mathcal G ( h ^ { * } )$ . In the generic learning scenario, the training objective is the following expected loss:
+
+$$
+\begin{array} { r } { \mathcal { R } ( \pmb { \theta } ) = \mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { P } } \left[ \mathcal { L } ( \hat { \pmb { y } } ( \pmb { x } ; \pmb { \theta } ) , \pmb { y } ) \right] , } \end{array}
+$$
+
+where $\textbf {  { y } }$ is the groundtruth corresponding to the training example $_ { \textbf { \em x } }$ , and $\mathcal { P }$ is the data distribution. Here, we omit the parameters of $\mathcal { G }$ , because given the output $h ^ { * }$ of the implicit module $\mathcal { F }$ , training the post-processing part $\mathcal { G }$ is the same as training explicit neural networks. The most crucial component is the gradient of the loss function $\mathcal { L } w . r . t .$ the input vector $z ^ { \top } = [ \pmb { u } ^ { \top } , \pmb { \theta } ^ { \top } ]$ , which is used to train both the implicit module $\mathcal { F }$ and the input projection module $\mathcal { M }$ . Using Eq. (2) with the condition $h = h ^ { * }$ , we have
+
+$$
+\frac { \partial \mathcal { L } } { \partial u } = \frac { \partial \mathcal { F } } { \partial u } \left( I - \frac { \partial \mathcal { F } } { \partial h } \right) ^ { - 1 } \frac { \partial \mathcal { L } } { \partial h } , \quad \frac { \partial \mathcal { L } } { \partial \theta } = \frac { \partial \mathcal { F } } { \partial \theta } \left( I - \frac { \partial \mathcal { F } } { \partial h } \right) ^ { - 1 } \frac { \partial \mathcal { L } } { \partial h } .
+$$
+
+The gradients in Eq. (4) are in the same form w.r.t. $\textbf { \em u }$ and $\pmb \theta$ . Without loss of generality, we only discuss the gradient w.r.t. $\pmb { \theta }$ in the following sections.
+
+# 2.2 Motivation
+
+The most intriguing part lies in the Jacobian-inverse term, i.e., $\left( I - \partial \mathcal { F } / \partial h \right) ^ { - 1 }$ . Computing the inverse term by brute force is intractable due to the $\mathcal { O } ( n ^ { 3 } )$ complexity. Previous implicit models [2] approach this by solving a linear system involving a Jacobian-vector product iteratively via a gradient solver, introducing over 30 Broyden [16] iterations in the backward pass. However, the scale of the Jacobian matrix can exceed $\mathrm { 1 0 ^ { 6 } \times 1 0 ^ { 6 } }$ in the real scenarios, leading to a prohibitive cost in computing the exact gradient. For example, training a small-scale state-of-the-art implicit model on ImageNet can consume weeks using 8 GPUs while training explicit models usually takes days, demonstrating that pursuing the exact gradient severely slows down the training process of implicit models compared with explicit models.
+
+Secondly, because of the inversion operation, we cast doubt on the conditioning of the gradient and the stability of training process from the numerical aspect. The Jacobian-inverse can be numerically unstable when encountering the ill-conditioning issue. The conditioning problem might further undermine the training stability, as studied in the recent work [17].
+
+Plus, the inexact gradient [9, 18, 19, 20, 21, 22] is widely applied in the previous learning protocol, like linear propagation [23] and synthetic gradient [24]. Here, the Jacobian-inverse is used to calculate the exact gradient which is not always optimal for model training. Moreover, previous research has used a moderate gradient noise as a regularization approach [25], which has been shown to play a central role in escaping poor local minima and improving generalization ability [26, 27, 28].
+
+The concerns and observations motivate us to rethink the possibility of replacing the Jacobian-inverse term in the standard implicit differentiation with a cheaper and more stable counterpart. We believe that an exact gradient estimate is not always required, especially for a black-box layer like those in the implicit models. Hence this work designs an inexact, theoretically sound, and practically efficient gradient for training implicit models under various settings. We name the proposed gradient estimate as the phantom gradient.
+
+Suppose the Jacobian ${ { \partial } { h } ^ { * } } / { { \partial } { \theta } }$ is replaced with a matrix $\pmb { A }$ , and the corresponding phantom gradient is defined as
+
+$$
+{ \widehat { \frac { \partial { \mathcal { L } } } { \partial \theta } } } : = A \ { \frac { \partial { \mathcal { L } } } { \partial h } } .
+$$
+
+Next, we give the general condition on $\pmb { A }$ so that the phantom gradient can be guaranteed valid for optimization (Sec. 2.3), and provide two concrete instantiations of $\pmb { A }$ based on either damped fixed-point unrolling or the Neumann series (Sec. 2.4). The proofs of all our theoretical results are presented in Appendix C.
+
+# 2.3 General Condition on the Phantom Gradient
+
+Previous research on theoretical properties for the inexact gradient include several aspects, such as the gradient direction [22], the unbiasedness of the estimator [29], and the convergence theory of the stochastic algorithm [19, 30]. The following theorem formulates a sufficient condition that the phantom gradient gives an ascent direction of the loss landscape.
+
+Theorem 1. Suppose the exact gradient and the phantom gradient are given by Eq. (4) and (5), respectively. Let $\sigma _ { m a x }$ and $\sigma _ { m i n }$ be the maximal and minimal singular value of $\partial \mathcal { F } / \partial \theta$ . If
+
+$$
+\left\| A \left( I - \frac { \partial \mathcal { F } } { \partial h } \right) - \frac { \partial \mathcal { F } } { \partial \theta } \right\| < \frac { \sigma _ { m i n } ^ { 2 } } { \sigma _ { m a x } } ,
+$$
+
+then the phantom gradient provides an ascent direction of the function $\mathcal { L }$ , i.e.,
+
+$$
+\left. { \widehat { \frac { \partial { \mathcal { L } } } { \partial \theta } } } , { \frac { \partial { \mathcal { L } } } { \partial \theta } } \right. > 0 .
+$$
+
+Remark 1. Suppose only the $\left( I - \partial \mathcal { F } / \partial h \right) ^ { - 1 }$ term is replaced with a matrix $_ { D }$ , namely, $A =$ $( \partial \mathcal { F } / \partial \pmb { \theta } ) \pmb { D }$ . Then, the condition in (6) can be reduced into
+
+$$
+\left\| D \left( I - \frac { \partial \mathcal { F } } { \partial h } \right) - I \right\| < \frac { 1 } { \kappa ^ { 2 } } ,
+$$
+
+where $\kappa$ is the condition number of $\partial \mathcal { F } / \partial \theta$ . (See Appendix C.1 for the derivation.)
+
+# 2.4 Instantiations of the Phantom Gradient
+
+In this section, we present two practical instantiations of the phantom gradient. We also verify that the general condition in Theorem 1 can be satisfied if the hyperparameters in our instantiations are wisely selected.
+
+Suppose we hope to differentiate through implicit dynamics, e.g., either a root-solving process or an optimization problem. In the context of hyperparameter optimization (HO), previous solutions include differentiating through the unrolled steps of the dynamics [19] or employing the Neumann series of the Jacobian-inverse term [9]. In our case, if we solve the root of Eq. (1) via the fixed-point iteration:
+
+$$
+\pmb { h } _ { t + 1 } = \mathcal { F } ( \pmb { h } _ { t } , z ) , \quad t = 0 , 1 , \cdots , T - 1 ,
+$$
+
+then by differentiating through the unrolled steps of Eq. (9), we have
+
+$$
+\frac { \partial { \pmb h } _ { T } } { \partial { \pmb \theta } } = \sum _ { t = 0 } ^ { T - 1 } \left. \frac { \partial \mathcal { F } } { \partial { \pmb \theta } } \right| _ { { \pmb h } _ { t } ~ s = t + 1 } \left. \frac { \partial \mathcal { F } } { \partial { \pmb h } } \right| _ { { \pmb h } _ { s } } .
+$$
+
+Besides, the Neumann series of the Jacobian-inverse $\left( I - \partial \mathcal { F } / \partial h \right) ^ { - 1 }$ is
+
+$$
+\pmb { I } + \frac { \partial \mathcal { F } } { \partial \pmb { h } } + \left( \frac { \partial \mathcal { F } } { \partial \pmb { h } } \right) ^ { 2 } + \left( \frac { \partial \mathcal { F } } { \partial \pmb { h } } \right) ^ { 3 } + \cdots .
+$$
+
+Notably, computing the Jacobian $\partial \pmb { h } ^ { * } / \partial \pmb { \theta }$ using the Neumann series in (11) is equivalent to differentiating through the unrolled steps of Eq. (9) at the exact equilibrium point $h ^ { * }$ and taking the limit of infinite steps [9].
+
+Without altering the root of Eq. (1), we consider a damped variant of the fixed-point iteration:
+
+$$
+\begin{array} { r } { h _ { t + 1 } = \mathscr { F } _ { \lambda } ( h _ { t } , z ) = \lambda \mathscr { F } ( h _ { t } , z ) + ( 1 - \lambda ) h _ { t } , \quad t = 0 , 1 , \cdots , T - 1 . } \end{array}
+$$
+
+Differentiating through the unrolled steps of Eq. (12), Eq. (10) is adapted as
+
+$$
+\frac { \partial h _ { T } } { \partial \pmb { \theta } } = \lambda \sum _ { t = 0 } ^ { T - 1 }  \frac { \partial \mathcal { F } } { \partial \pmb { \theta } } | _ { h _ { t } } \prod _ { s = t + 1 } ^ { T - 1 } ( \lambda \frac { \partial \mathcal { F } } { \partial h } | _ { h _ { s } } + ( 1 - \lambda ) \pmb { I } ) .
+$$
+
+The Neumann series of $\left( I - \partial \mathcal { F } / \partial h \right) ^ { - 1 }$ is correspondingly adapted as
+
+$$
+\lambda \left( I + B + B ^ { 2 } + B ^ { 3 } + \cdots \right) , \quad \mathrm { w h e r e ~ } B = \lambda \frac { \partial \mathcal { F } } { \partial h } + ( 1 - \lambda ) I .
+$$
+
+The next theorem shows that under mild conditions, the Jacobian from the damped unrolling in Eq. (13) converges to the exact Jacobian and the Neumann series in (14) converges to the Jacobianinverse $( I - \partial \mathcal { F } / \partial h ) ^ { - 1 }$ as well.
+
+Theorem 2. Suppose the Jacobian $\partial \mathcal { F } / \partial h$ is a contraction mapping. Then,
+
+(i) the Neumann series in (14) converges to the Jacobian-inverse $\left( I - \partial \mathcal { F } / \partial h \right) ^ { - 1 }$ ; and
+
+(ii) if the function $\mathcal { F }$ is continuously differentiable w.r.t. both $^ { h }$ and $\pmb \theta$ , the sequence in Eq. (13) converges to the exact Jacobian $\partial \pmb { h } ^ { * } / \partial \pmb { \theta }$ as $T \to \infty$ , i.e.,
+
+$$
+\operatorname* { l i m } _ { T  \infty } \frac { \partial h _ { T } } { \partial \pmb { \theta } } =  \frac { \partial \mathcal { F } } { \partial \pmb { \theta } } | _ { h ^ { \ast } } ( \pmb { I } -  \frac { \partial \mathcal { F } } { \partial h } | _ { h ^ { \ast } } ) ^ { - 1 } .
+$$
+
+However, as discussed in Sec. 2.2, it is unnecessary to compute the exact gradient with infinite terms. In the following context, we introduce two instantiations of the phantom gradient based on the finite-term truncation of Eq. (13) or (14).
+
+Unrolling-based Phantom Gradient (UPG). In the unrolling form, the matrix $\pmb { A }$ is defined as
+
+$$
+A _ { k , \lambda } ^ { \mathrm { u n r } } = \lambda \sum _ { t = 0 } ^ { k - 1 }  \frac { \partial \mathcal { F } } { \partial \pmb { \theta } } | _ { h _ { t } } \prod _ { s = t + 1 } ^ { k - 1 } ( \lambda \frac { \partial \mathcal { F } } { \partial h } | _ { h _ { s } } + ( 1 - \lambda ) \pmb { I } ) .
+$$
+
+Neumann-series-based Phantom Gradient (NPG). In the Neumann form, the matrix $\pmb { A }$ is defined as
+
+$$
+\pmb { A } _ { k , \lambda } ^ { \mathrm { n e u } } = \lambda \left. \frac { \partial \mathcal { F } } { \partial \pmb { \theta } } \right| _ { h ^ { * } } \left( I + \pmb { B } + \pmb { B } ^ { 2 } + \cdots + \pmb { B } ^ { k - 1 } \right) , \quad \mathrm { w h e r e } \ \pmb { B } = \lambda \left. \frac { \partial \mathcal { F } } { \partial h } \right| _ { h ^ { * } } + ( 1 - \lambda ) I .
+$$
+
+Note that both the initial point of the fixed-point iteration (i.e., $h _ { 0 }$ in Eq. (16)) and the point at which the Neumann series is evaluated (i.e., $h ^ { * }$ in (17)) are the solution of the root-finding solver. (See Appendix B for implementation of the phantom gradient.)
+
+According to Theorem 2, the matrix $\pmb { A }$ defined by either Eq. (16) or (17) converges to the exact Jacobian ${ \bar { \partial } } h ^ { * } / \partial \theta$ as $k \to \infty$ for any $\lambda \in ( 0 , 1 ]$ . Therefore, by Theorem 2, the condition in (6) can be satisfied if a sufficiently large step $k$ is selected, since
+
+$$
+\left\| A \left( I - { \frac { \partial { \mathcal { F } } } { \partial h } } \right) - { \frac { \partial { \mathcal { F } } } { \partial \theta } } \right\| \leq \left( 1 + L _ { h } \right) \left\| A - { \frac { \partial { \mathcal { F } } } { \partial \theta } } \left( I - { \frac { \partial { \mathcal { F } } } { \partial h } } \right) ^ { - 1 } \right\| .
+$$
+
+Next, we characterize the impact of the two hyperparameters, i.e., $k$ and $\lambda$ , on the precision and conditioning of $\pmb { A }$ . Take the NPG (Eq. (17)) as an example.
+
+(i) On the precision of the phantom gradient,
+
+• a large $k$ makes the gradient estimate more accurate, as higher-order terms of the Neumann series are included, while   
+• a small $\lambda$ slows down the convergence of the Neumann series because the norm $\| B \|$ increases as $\lambda$ decreases.
+
+(ii) On the conditioning of the phantom gradient,
+
+• a large $k$ impairs the conditioning of $\pmb { A }$ since the condition number of $B ^ { k }$ grows exponentially as $k$ increases, while   
+• a small $\lambda$ helps maintain a small condition number of $\pmb { A }$ because the singular values of $\partial \mathcal { F } / \partial h$ are “smoothed” by the identity matrix.
+
+In a word, a large $k$ is preferable for a more accurate $\pmb { A }$ , while a small $\lambda$ contributes to the wellconditioning of $\pmb { A }$ . Practically, these hyperparameters should be selected in consideration of a balanced trade-off between the precision and conditioning of $\pmb { A }$ . See Sec. 3 for experimental results.
+
+# 2.5 Convergence Theory
+
+In this section, we provide the convergence guarantee of the SGD algorithm using the phantom gradient. We prove that under mild conditions, if the approximation error of the phantom gradient is sufficiently small, the SGD algorithm converges to an $\epsilon$ -approximate stationary point in the expectation sense. We will discuss the feasibility of our assumptions in Appendix C.3.
+
+Theorem 3. Suppose the loss function $\mathcal { R }$ in Eq. (3) is $\ell$ -smooth, lower-bounded, and has bounded gradient almost surely in the training process. Besides, assume the gradient in Eq. (4) is an unbiased estimator of $\nabla \mathcal { R } ( \theta )$ with a bounded covariance. If the phantom gradient in Eq. (5) is an $\epsilon$ -approximation to the gradient in Eq. (4), i.e.,
+
+$$
+\left\| { \widehat { \frac { \partial { \mathcal { L } } } { \partial \theta } } } - { \frac { \partial { \mathcal { L } } } { \partial \theta } } \right\| \leq \epsilon , \quad a l m o s t s u r e l y ,
+$$
+
+then using Eq. (5) as a stochastic first-order oracle with a step size of $\eta _ { n } = \mathcal { O } ( 1 / \sqrt { n } )$ to update $\pmb \theta$ with gradient descent, it follows after $N$ iterations that
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \frac { \sum _ { n = 1 } ^ { N } \eta _ { n } \left\| \nabla \mathcal { R } \left( \pmb { \theta } _ { n } \right) \right\| ^ { 2 } } { \sum _ { n = 1 } ^ { N } \eta _ { n } } \right] \le \mathcal { O } \left( \epsilon + \frac { \log N } { \sqrt { N } } \right) . } \end{array}
+$$
+
+Remark 2. Consider the condition in (19):
+
+$$
+\left\| \widehat { \frac { \partial \mathcal { L } } { \partial \theta } } - \frac { \partial \mathcal { L } } { \partial \theta } \right\| \leq \left\| A - \frac { \partial \mathcal { F } } { \partial \theta } \left( I - \frac { \partial \mathcal { F } } { \partial h } \right) ^ { - 1 } \right\| \left\| \frac { \partial \mathcal { L } } { \partial h } \right\| .
+$$
+
+Suppose the gradient $\partial \mathcal { L } / \partial h$ is almost surely bounded. By Theorem 2, the condition in (19) can be guaranteed as long as a sufficiently large $k$ is selected.
+
+![](images/874187b8bcf9d94135ffda4257628118a4d11dc9de88f7857d747afb5fe4f59a.jpg)  
+Figure 1: Cosine similarity between the phantom gradient and the exact gradient in the synthetic setting.
+
+# 3 Experiments
+
+In this section, we aim to answer the following questions via empirical results: (1) How precise is the phantom gradient? (2) What is the difference between the unrolling-based and the Neumannseries-based phantom gradients? (3) How is the phantom gradient influenced by the hyperparameters $k$ and $\lambda ?$ (4) How about the computational cost of the phantom gradient compared with implicit differentiation? (5) Can the phantom gradient work at large-scale settings for various tasks?
+
+We have provided some theoretical analysis and intuitions to (1), (2), and (3) in Sec. 2.4. Now we answer (1) and (2) and demonstrate the performance curves under different hyperparameters $k$ and $\lambda$ on CIFAR-10 [31]. Besides, we also study other factors that have potential influences on the training process of the state-of-the-art implicit models [2, 3] like pretraining. For (4) and (5), we conduct experiments on large-scale datasets to highlight the ultra-fast speed and competitive performances, including image classification on ImageNet [10] and language modeling on Wikitext-103 [13].
+
+We start by introducing two experiment settings. We adopt a single-layer neural network with spectral normalization [32] as the function $\mathcal { F }$ and fixed-point iterations as the equilibrium solver, which is the synthetic setting. Moreover, on the CIFAR-10 dataset, we use the MDEQ-Tiny model [3] (170K parameters) as the backbone model, denoted as the ablation setting. Additional implementation details and experimental results are presented in Appendix $\mathsf { D } ^ { 2 }$ .
+
+Precision of the Phantom Gradient. The precision of the phantom gradient is measured by its angle against the exact gradient, indicated by the cosine similarity between the two. We discuss its precision in both the synthetic setting and the ablation setting. The former is under the static and randomly generated weights, while the latter provides characterization of the training dynamics.
+
+In the synthetic setting, the function $\mathcal { F }$ is restricted to be a contraction mapping. Specifically, we directly set the Lipschitz constant of $\mathcal { F }$ as $L _ { h } = 0 . 9$ , and use 100 fixed-point iterations to solve the root $h ^ { * }$ of Eq. (1) until the relative error satisfies $\| h - \mathcal { F } ( h , z ) \| / \| h \| ^ { - } 1 0 ^ { - 5 }$ . Here, the exact gradient is estimated by backpropagation through the fixed-point iterations, and cross-validated by implicit differentiation solved with 20 iterations of the Broyden’s method [16]. In our experiment, the cosine similarity between these two gradient estimates consistently succeeds 0.9999, indicating the gradient estimate is quite accurate when the relative error of forward solver is minor. The cosine similarity between phantom gradients and exact gradients is shown in Fig. 1. It shows that the cosine similarity tends to increase as $k$ grows and that a small $\lambda$ tends to slow down the convergence of the phantom gradient, allowing it to explore in a wider range regarding the angle against the exact gradient.
+
+In the ablation setting, the precision of the phantom gradient during the training process is shown in Fig. 2. The model is trained by implicit differentiation under the official schedule3. It shows that the phantom gradient still provides an ascent direction in the real training process, as indicated by the considerable cosine similarity against the exact gradient. Interestingly, the cosine similarity slightly decays as the training progresses, which implies a possibility to construct an adaptive gradient solver for implicit models.
+
+![](images/177f666c03028cac8d033f6ef1cb3da9b3e02a09a76a80dc90d1b367f0317f4b.jpg)  
+Figure 2: Cosine similarity between the phantom gradient and the exact gradient in the real scenario. The horizontal axis corresponds to the cosine similarity, and the vertical axis to the training step.
+
+To Pretrain, or not to Pretrain? To better understand the components of the implicit models’ training schedule, we first illustrate a detailed ablation study of the baseline model in the ablation setting. The average accuracy with standard deviation is reported in Tab. 1.
+
+The MDEQ model employs a pretraining stage in which the model $\mathcal { F }$ is unrolled as a recurrent network. We study the impact of the pretraining stage, the Dropout [33] operation, and the optimizer separately. It can be seen that the unrolled pretraining stabilizes the training of the MDEQ model. Removing the pretraining stage leads to a severe performance drop and apparent training instability among different trials because the solver cannot obtain an accurate fixed point $h ^ { * }$ when the model is not adequately trained. This ablation study also suggests that the MDEQ model is a strong baseline for our method to compare with.
+
+Table 1: Ablation settings on CIFAR-10.   
+
+<table><tr><td>Method</td><td>Acc. (%)</td></tr><tr><td>Implicit Differentiation</td><td>85.0± 0.2</td></tr><tr><td>w/o Pretraining</td><td>82.3± 1.3</td></tr><tr><td>w/o Dropout Adam →SGD</td><td>83.7± 0.1</td></tr><tr><td> SGD w/o Pretraining</td><td>84.5 ± 0.3</td></tr><tr><td>UPG (A5,0.5, W/o Dropout)</td><td>82.9 ± 1.5</td></tr><tr><td>NPG (A5,0.5, w/o Dropout)</td><td>85.8±0.5</td></tr><tr><td>UPG (A9.0.5, w/ Dropout)</td><td>85.6± 0.5 86.1±0.5</td></tr></table>
+
+However, pretraining is not always indispensable for training implicit models. It introduces an extra hyperparameter, i.e., how many steps should be involved in the pretraining stage. Next, we discuss how the UPG could circumvent this issue.
+
+Trade-offs between Unrolling and Neumann. For an exact fixed point $h ^ { * }$ , i.e., $\begin{array} { r } { \begin{array} { r } { h ^ { * } = \mathcal { F } ( h ^ { * } , z ) } \end{array} } \end{array}$ , there is no difference between UPG and NPG. However, when the numerical error exists in solving $h ^ { * }$ , i.e., $\| h ^ { * } - \mathcal { F } ( h ^ { * } , z ) \| > 0$ , these two instatiations of the phantom gradient can behave differently.
+
+We note that a particular benefit of the UPG is its ability to automatically switch between the pretraining and training stages for implicit models. When the model is not sufficiently trained and the solver converges poorly (see [3]), the UPG defines a forward computation graph that is essentially equivalent to a shallow weight-tied network to refine the coarse equilibrium states. In this stage, the phantom gradient serves as a backpropagation through time (BPTT) algorithm and hence behaves as in the pretraining stage. Then, as training progresses, the solver becomes more stable and converges to the fixed point $h ^ { * }$ better. This makes the UPG behave more like the NPG. Therefore, the unrolled pretraining is gradually transited into the regular training phase based on implicit differentiation, and the hyperparameter tuning of pretraining steps can be waived. We argue that such an ability to adaptively switch training stages is benign to the implicit models’ training protocol, which is also supported by the performance gain in Tab. 1.
+
+Table 2: Complexity comparison. Mem means the memory cost, and $K$ and $k$ denote the solver’s steps and the unrolling/Neumann steps, respectively. Here, $K \gg k \approx 1$ .   
+
+<table><tr><td>Method</td><td>Time</td><td>Mem</td><td>Peak Mem</td></tr><tr><td>Implicit</td><td>O(K)</td><td>0(1)</td><td>O(k)</td></tr><tr><td>UPG</td><td>O(k）</td><td>O(k)</td><td>O(k)</td></tr><tr><td>NPG</td><td>（k）</td><td>0(1)</td><td>0(1)</td></tr></table>
+
+Although the UPG requires higher memory overhead than implicit differentiation or the NPG, it does not surpass the peak memory usage in the entire training protocol by implicit differentiation due to the pretraining stage. In the ablation setting, the MDEQ model employs a 10-layer unrolling for pretraining, which actually consumes double memory compared with a 5-step unrolling scheme, e.g., $A _ { 5 , 0 . 5 }$ in Tab. 1. In Tab. 2, we also demonstrate the time and memory complexity for implicit differentiation and the two forms of phantom gradient.
+
+![](images/92998b52ccf86b6ff21f0145c37c4cc23f3a48218c10885a6e3c395ae3d6e2ff.jpg)  
+Figure 3: Ablation studies on (a) the hyperparameters $\lambda$ and $k$ , and (b) two forms of phantom gradient.
+
+In addition, leaving the context of approximate and exact gradient aside, we also develop insights into understanding the subtle differences between UPG and NPG in terms of the state-dependent and state-free gradient. Actually, the UPG is state-dependent, which means it corresponds to the “exact” gradient of a computational sub-graph. Both the NPG and the gradient solved by implicit differentiation, however, do not exactly match the gradient of any forward computation graph unless the numerical error is entirely eliminated in both the forward and the backward passes for implicit differentiation or the one-step gradient [34, 21, 30, 22] is used for the NPG, i.e., $k =$ 1. Interestingly, we observe that models trained on the state-dependent gradient demonstrate an additional training stability regarding the Jacobian spectral radius, compared with those trained on the state-free counterpart. We empirically note that it can be seen as certain form of implicit Jacobian regularization for implicit models as a supplement to the explicit counterpart [17], indicated by the stable estimated Jacobian spectral radius during training, i.e., $\rho ( \partial \mathcal { F } _ { \lambda } / \partial h ) \approx 1$ .
+
+The experimental results also cohere with our insight. The performance curves in Fig. 3 demonstrate the influence of $\lambda$ and $k$ and further validate that the UPG is more robust to a wide range of steps $k$ than the NPG. In fact, when the Jacobian spectral radius $\rho ( \partial \mathcal { F } / \partial h )$ increases freely without proper regularization, the exploding high-order terms in the NPG could exert a negative impact on the overall direction of the phantom gradient, leading to performance degradation when a large $k$ is selected (see Fig. 3(b)). We also observe a surging of the estimated Jacobian spectral radius as well as the gradient explosion issue in the NPG experiments. On the contrary, the UPG can circumvent these problems thanks to its implicit Jacobian regularization.
+
+Table 3: Experiments using DEQ [2] and MDEQ [3] on vision and language tasks. Metrics stand for accuracy $( \% ) \uparrow$ for image classification on CIFAR-10 and ImageNet, and perplexity↓ for language modeling on Wikitext-103. JR stands for Jacobian Regularization [17]. $\dagger$ indicates additional steps in the forward equilibrium solver.   
+
+<table><tr><td>Datasets</td><td>Model</td><td>Method</td><td>Params</td><td>Metrics</td><td>Speed</td></tr><tr><td>CIFAR-10</td><td>MDEQ</td><td>Implicit</td><td>10M</td><td>93.8 ± 0.17</td><td>1.0×</td></tr><tr><td>CIFAR-10</td><td>MDEQ</td><td>UPG A5,0.5</td><td>10M</td><td>95.0 ± 0.16</td><td>1.4×</td></tr><tr><td>ImageNet</td><td>MDEQ</td><td>Implicit</td><td>18M</td><td>75.3</td><td>1.0×</td></tr><tr><td>ImageNet</td><td>MDEQ</td><td>UPG A5.0.6</td><td>18M</td><td>75.7</td><td>1.7×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PostLN)</td><td>Implicit</td><td>98M</td><td>24.0</td><td>1.0×</td></tr><tr><td>Wikitext-103</td><td>DEQ( (PostLN)</td><td>UPG A5,0.8</td><td>98M</td><td>25.7</td><td>1.7×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PreLN)</td><td>JR + Implicit</td><td>98M</td><td>24.5</td><td>1.7×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PreLN)</td><td>JR + UPG A5,0.8</td><td>98M</td><td>24.4</td><td>2.2×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PreLN)</td><td>JR + UPG A5,0.8</td><td>98M</td><td>24.0t</td><td>1.7×</td></tr></table>
+
+Phantom Gradient at Scale. We conduct large-scale experiments to verify the advantages of the phantom gradient on vision, graph, and language benchmarks. We adopt the UPG in the large-scale experiments. The results are illustrated in Tab. 3 and Tab. 4. Our method matches or surpasses the implicit differentiation training protocol on the state-of-the-art implicit models with a visible reduction on the training time. When only considering the backward pass, the acceleration for MDEQ can be remarkably $1 2 \times$ on ImageNet classification.
+
+Table 4: Experiments using IGNN [4] on graph tasks. Metrics stand for accuracy $( \% ) \uparrow$ for graph classification on COX2 and PROTEINS, Micro- $\mathrm { F l } ( \% ) \uparrow$ for node classification on PPI.   
+
+<table><tr><td>Datasets</td><td>Model</td><td>Method</td><td>Params</td><td>Metrics (%)</td></tr><tr><td>COX2</td><td>IGNN</td><td>Implicit</td><td>38K</td><td>84.1 ± 2.9</td></tr><tr><td>COX2</td><td>IGNN</td><td>UPG A5,0.5</td><td>38K</td><td>83.9 ± 3.0</td></tr><tr><td>COX2</td><td>IGNN</td><td>UPG A5,0.8</td><td>38K</td><td>83.9 ± 2.7</td></tr><tr><td>COX2</td><td>IGNN</td><td>UPG A5,1.0</td><td>38K</td><td>83.0 ± 2.9</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>Implicit</td><td>34K</td><td>78.6 ± 4.1</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>UPG A5,0.5</td><td>34K</td><td>78.4 ± 4.2</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>UPG A5,0.8</td><td>34K</td><td>78.6 ± 4.2</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>UPG A5,1.0</td><td>34K</td><td>78.8± 4.2</td></tr><tr><td>PPI</td><td>IGNN</td><td>Implicit</td><td>4.7M</td><td>97.6</td></tr><tr><td>PPI</td><td>IGNN</td><td>UPG A5,0.5</td><td>4.7M</td><td>98.2</td></tr><tr><td>PPI</td><td>IGNN</td><td>UPG A5,0.8</td><td>4.7M</td><td>97.4</td></tr><tr><td>PPI</td><td>IGNN</td><td>UPG A5,1.0</td><td>4.7M</td><td>96.2</td></tr></table>
+
+# 4 Related Work
+
+Implicit Models. Implicit models generalize the recursive forward/backward rules of neural networks and characterize their internal mechanism by some pre-specified dynamics. Based on the dynamics, the implicit mechanisms can be broadly categorized into three classes: ODE-based [1, 7], root-solving-based [2, 3, 4, 5, 11], and optimization-based [35, 36, 37] implicit models.
+
+The ODE-based implicit models [1, 7] treat the iterative update rules of residual networks as the Euler discretization of an ODE, which could be solved by any black-box ODE solver. The gradient of the differential equation is calculated using the adjoint method [38], in which the adjoint state is obtained by solving another ODE. The root-solving-based implicit models [2, 5, 3, 4, 6, 11, 22] characterize layers of neural networks by solving fixed-point equations. The equations are solved by either the black-box root-finding solver [2, 3] or the fixed-point iteration [4, 22]. The optimization-based implicit models [35, 36, 37, 39, 34, 21, 40, 41] leverage the optimization programs as layers of neural networks. Previous works have studied differentiable layers of quadratic programming [35], submodular optimization [36], maximum satisfiability (MAXSAT) problems [37], and structured decomposition [21]. As for the backward passes, implicit differentiation is applied to the problemdefining equations of the root-solving-based models [2, 3] or the KKT conditions of the optimizationbased models [35]. As such, the gradient can be obtained from solving the backward linear system.
+
+In this work, we focus on the root-solving-based implicit models. Theoretical works towards root-solving-based implicit models include the well-posedness [5, 4], monotone operators [11], global convergence [42, 41], and Lipschitz analysis [15]. We look into the theoretical aspect of the gradient-based algorithm in training implicit models and the efficient practice guidance. With these considerations, we show that implicit models of the same architecture could enjoy faster training speed and strong generalization in practical applications by using the phantom gradient.
+
+Non-End-to-End Optimization in Deep Learning. Non-end-to-end optimization aims to replace the standard gradient-based training of deep architectures with modular or weakly modular training without the entire forward and backward passes. Currently, there are mainly three research directions in this field, namely, the auxiliary variable methods [43, 44, 45, 46, 47, 48, 49], target propagation [50, 51, 52], and synthetic gradient [24, 53, 54]. The auxiliary variable methods [43, 44, 45, 46, 47, 48, 49] formulate the optimization of neural networks as constrained optimization problems, in which the layer-wise activations are considered as trainable auxiliary variables. Then, the equality constraints are relaxed as penalty terms added to the objectives so that the parameters and auxiliary variables can be divided into blocks and thus optimized in parallel. The target propagation method [50, 51, 52] trains each module by having its activations regress to the pre-assigned targets, which are propagated backwards from the downstream modules. Specifically, the auto-encoder architecture is used to reconstruct targets at each layer. Finally, the synthetic gradient method [24, 53, 54] estimates the local gradient of neural networks using auxiliary models, and employ the synthetic gradient in place of the exact gradient to perform parameter update. In this way, the forward and backward passes are decoupled and can be executed in an asynchronous manner.
+
+Our work is in line with the non-end-to-end optimization research since we also aims to decouple the forward and backward passes of neural networks. However, we show that finding a reasonable “target” or a precise gradient estimate is not always necessary in training deep architectures. Our paper paves a path that an inexact but well-conditioned gradient estimate can contribute to both fast training and competitive generalization of implicit models.
+
+Differentiation through Implicit Dynamics. Differentiation through certain implicit dynamics is an important aspect in a wide range of research fields, including bilevel optimization [19, 9], metalearning [18, 55, 30], and sensitivity analysis [56]. Since the gradient usually cannot be computed analytically, researchers have to implicitly differentiate the dynamics at the converged point. The formula of the gradient typically contains a term of Jacobian-inverse (or Hessian-inverse), which is computationally prohibitive for large-scale models. (See Eq. (2) in our case.) Herein, several techniques have been developed to approximate the matrix inverse in the previous literature.
+
+An intuitive solution is to differentiate through the unrolled steps of a numerical solver of the dynamics [57, 58, 8]. In particular, if a single step is unrolled, it reduces to the well-known one-step gradient [59, 18, 60, 34, 30, 21, 22], in which the inverse of Jacobian/Hessian is simply approximated by an identity matrix. On the contrary, unrolling a small number of steps may induce a bias [9], while the memory and computational cost grows linearly as the number of unrolled steps increases. Towards this issue, Shaban et al. [19] propose to truncate the long-term dependencies and differentiate through only the last $L$ steps. In fact, if the dynamics have converged to a stationary point, the finite-term truncation in Shaban et al. [19] is exactly the Neumann approximation of the Jacobian-inverse with the first $L$ terms. Based on this, Lorraine et al. [9] directly use the truncated Neumann series as an approximation of the Jacobian-inverse. Besides the unrolling-based methods, optimization-based approaches [61, 55] have been studied in this field as well. Since the Jacobian-inverse-vector product can be viewed as solution of a linear system, algorithms like the conjugate gradient method can be used to solve it.
+
+# 5 Limitation and Future Work
+
+The main limitation of this work lies in the hyperparameter tuning of the phantom gradient, especially for the damping factor $\lambda$ , which directly controls the gradient’s precision and conditioning, the implicit Jacobian regularization for UPG, the stability for NPG, and the final generalization behaviors. However, it has not been a hindrance to the application of phantom gradients in training implicit models as one can tune the hyperparameter according to the validation loss in the early training stage.
+
+Regarding future works, we would like to highlight the following aspects: (1) eliminating the bias of the current phantom gradient, (2) constructing an adaptive gradient solver for implicit models, (3) analyzing the damping factor to provide practical guidance, (4) investigating the implicit Jacobian regularization, and (5) understanding how different noises in the gradients can impact the training of implicit models under different loss landscapes.
+
+# 6 Conclusion
+
+In this work, we explore the possibility of training implicit models via the efficient approximate phantom gradient. We systematically analyze the general condition of a gradient estimate so that the implicit model can be guaranteed to converge to an approximate stationary point of the loss function. Specifically, we give a sufficient condition under which a first-order oracle could always find an ascent direction of the loss landscape in the training process. Moreover, we introduce two instantiations of the proposed phantom gradient, based on either the damped fixed-point unrolling or the Neumann series. The proposed method shows a $1 . 4 \sim 1 . 7 \times$ acceleration with comparable or better performances on large-scale benchmarks. Overall, this paper provides a practical perspective on training implicit models with theoretical guarantees.
+
+# Acknowledgments
+
+Zhouchen Lin was supported by the NSF China (No.s 61625301 and 61731018), NSFC Tianyuan Fund for Mathematics (No. 12026606) and Project 2020BD006 supported by PKU-Baidu Fund. Yisen Wang was partially supported by the National Natural Science Foundation of China under Grant 62006153, and Project 2020BD006 supported by PKU-Baidu Fund. Shaojie Bai was sponsored by a grant from the Bosch Center for Artificial Intelligence.
+
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+[59] Jelena Luketina, Mathias Berglund, Klaus Greff, and Tapani Raiko. Scalable Gradient-Based Tuning of Continuous Regularization Hyperparameters. In International Conference on Machine Learning (ICML), pages 2952–2960, 2016. 10   
+[60] Hanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: Differentiable Architecture Search. International Conference on Learning Representations (ICLR), 2018. 10   
+[61] Fabian Pedregosa. Hyperparameter Optimization with Approximate Gradient. In International Conference on Machine Learning (ICML), pages 737–746, 2016. 10

parse/train/3EwcMzmUbNd/3EwcMzmUbNd_content_list.json

@@ -0,0 +1,1499 @@
+[
+    {
+        "type": "text",
+        "text": "On Training Implicit Models ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zhengyang Geng1,2∗ Xin-Yu Zhang2∗ Shaojie Bai4 Yisen Wang2,3 Zhouchen Lin2,3,5† ",
+        "bbox": [
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+            199,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Zhejiang Lab, China 2Key Lab. of Machine Perception, School of AI, Peking University 3Institute for Artificial Intelligence, Peking University 4Carnegie Mellon University 5Pazhou Lab, China ",
+        "bbox": [
+            200,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            292,
+            535,
+            310
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper focuses on training implicit models of infinite layers. Specifically, previous works employ implicit differentiation and solve the exact gradient for the backward propagation. However, is it necessary to compute such an exact but expensive gradient for training? In this work, we propose a novel gradient estimate for implicit models, named phantom gradient, that 1) forgoes the costly computation of the exact gradient; and 2) provides an update direction empirically preferable to the implicit model training. We theoretically analyze the condition under which an ascent direction of the loss landscape could be found, and provide two specific instantiations of the phantom gradient based on the damped unrolling and Neumann series. Experiments on large-scale tasks demonstrate that these lightweight phantom gradients significantly accelerate the backward passes in training implicit models by roughly $1 . 7 \\times$ , and even boost the performance over approaches based on the exact gradient on ImageNet. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            535,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Conventional neural networks are typically constructed by explicitly stacking multiple linear and non-linear operators in a feed-forward manner. Recently, the implicitly-defined models [1, 2, 3, 4, 5] have attracted increasing attentions and are able to match the state-of-the-art results by explicit models on several vision [3, 6], language [2] and graph [4] tasks. These works treat the evolution of the intermediate hidden states as a certain form of dynamics, such as fixed-point equations [2, 3] or ordinary differential equations (ODEs) [1, 7], which represents infinite latent states. The forward passes of implicit models are therefore formulated as solving the underlying dynamics, by either black-box ODE solvers [1, 7] or root-finding algorithms [2, 3]. As for the backward passes, however, directly differentiating through the forward pass trajectories could induce a heavy memory overhead [8, 9]. To this end, researchers have developed memory-efficient backpropagation via implicit differentiation, such as solving a Jacobian-based linear fixed-point equation for the backward pass of deep equilibrium models (DEQs) [2], which eventually makes the backpropagation trajectories independent of the forward passes. This technique allows one to train implicit models with essentially constant memory consumption, as we only need to store the final output and the layer itself without saving any intermediate states. However, in order to estimate the exact gradient by implicit differentiation, implicit models have to rely on expensive black-box solvers for backward passes, e.g., ODE solvers or root-solving algorithms. These black-box solver usually makes the gradient computation very costly in practice, even taking weeks to train state-of-the-art implicit models on ImageNet [10] with 8 GPUs. ",
+        "bbox": [
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+            566,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This work investigates fast approximate gradients for training implicit models. We found that a firstorder oracle that produces good gradient estimates is enough to efficiently and effectively train implicit models, circumventing laboriously computing the exact gradient as in prior arts [2, 3, 4, 11, 12]. We develop a framework in which a balanced trade-off is made between the precision and conditioning of the gradient estimate. Specifically, we provide the general condition under which the phantom gradient can provide an ascent direction of the loss landscape. We further propose two instantiations of phantom gradients in the context of DEQ models, which are based on the the damped fixed-point unrolling and the Neumann series, respectively. Importantly, we show that our proposed instantiations satisfy the theoretical condition, and that the stochastic gradient descent (SGD) algorithm based on the phantom gradient enjoys a sound convergence property as long as the relevant hyperparameters, e.g., the damping factor, are wisely selected. Note that our method only affects, and thus accelerates, the backward formulation of the implicit models, leaving the forward pass formulation (i.e., the root-solving process) and the inference behavior unchanged so that our method is applicable to a wide range of implicit models, forward solvers, and inference strategies. We conduct an extensive set of synthetic, ablation, and large-scale experiments to both analyze the theoretical properties of the phantom gradient and validate its speedup and performances on various tasks, such as ImageNet [10] classification and Wikitext-103 [13] language modeling. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overall, our results suggest that: 1) the phantom gradient estimates an ascent direction; 2) it is applicable to large-scale tasks and is capable of achieving a strong performance which is comparable with or even better than that of the exact gradient; and 3) it significantly shortens the total training time needed for implicit models roughly by a factor of $1 . 4 \\sim 1 . 7 \\times$ , and even accelerates the backward passes by astonishingly $1 2 \\times$ on ImageNet. We believe that our results provide strong evidence for effectively training implicit models with the lightweight phantom gradient. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Method ",
+        "text_level": 1,
+        "bbox": [
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+            406,
+            271,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Inspection of Implicit Differentiation ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            435,
+            468,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we primarily focus on the formulation of implicit models based on root-solving, represented by the DEQ models [2]. The table of notations is arranged in Appendix A. Specifically, given an equilibrium module $\\mathcal { F }$ , the output of the implicit model is characterized by the solution $h ^ { * }$ to the following fixed-point equation: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/425749c5016357bdf5f75d23a7d909e0058318d3182339152cb5209d7cd34917.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { r } { h ^ { * } = \\mathcal { F } ( h ^ { * } , z ) , } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
+            518,
+            552,
+            535
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { z } \\in \\mathbb { R } ^ { d _ { \\boldsymbol { u } } + d _ { \\boldsymbol { \\theta } } }$ is the union of the module’s input $\\pmb { u } \\in \\mathbb { R } ^ { d _ { u } }$ and parameters $\\pmb { \\theta } \\in \\mathbb { R } ^ { d _ { \\pmb { \\theta } } }$ , i.e., $z ^ { \\top } =$ $[ \\boldsymbol { u } ^ { \\top } , \\boldsymbol { \\theta } ^ { \\top } ]$ . Here, $\\textbf { \\em u }$ is usually a projection of the original data point $\\boldsymbol { x } \\in \\mathbb { R } ^ { d _ { x } }$ , e.g., $\\pmb { u } = \\mathcal { M } ( \\pmb { x } )$ . In this section, we assume $\\mathcal { F }$ is a contraction mapping w.r.t. $^ { h }$ so that its Lipschitz constant $L _ { h } \\ w . r . t . \\ h$ is less than one, i.e., $L _ { h } < 1$ , a setting that has been analyzed in recent works $[ 1 4 , 1 5 ] ^ { 1 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To differentiate through the fixed point by Eq. (1), we need to calculate the gradient of $h ^ { \\ast } w . r . t .$ the input $_ z$ . By Implicit Function Theorem (IFT), we have ",
+        "bbox": [
+            174,
+            598,
+            823,
+            627
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/76a7210ffa7019ebd5a4c7ec70165760d755d89ccf8c64a1cedeb855b8980161.jpg",
+        "text": "$$\n\\frac { \\partial { \\pmb h } ^ { * } } { \\partial z } = \\left. \\frac { \\partial \\mathcal { F } } { \\partial z } \\right| _ { { \\pmb h } ^ { * } } \\left( { \\pmb I } - \\left. \\frac { \\partial \\mathcal { F } } { \\partial { \\pmb h } } \\right| _ { { \\pmb h } ^ { * } } \\right) ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            627,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Here, $( \\partial \\pmb { a } / \\partial b ) _ { i j } = \\partial \\pmb { a } _ { j } / \\partial b _ { i }$ . The equilibrium point $h ^ { * }$ of Eq. (1) is then passed to a post-processing function $\\mathcal { G }$ to obtain a prediction $\\hat { y } = \\mathcal G ( h ^ { * } )$ . In the generic learning scenario, the training objective is the following expected loss: ",
+        "bbox": [
+            173,
+            666,
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+            710
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49adf10d57baedc5a463588007cbb2e798ad28aa353480dba72d79044107fc6f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { R } ( \\pmb { \\theta } ) = \\mathbb { E } _ { ( \\pmb { x } , \\pmb { y } ) \\sim \\mathcal { P } } \\left[ \\mathcal { L } ( \\hat { \\pmb { y } } ( \\pmb { x } ; \\pmb { \\theta } ) , \\pmb { y } ) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            712,
+            616,
+            729
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\textbf {  { y } }$ is the groundtruth corresponding to the training example $_ { \\textbf { \\em x } }$ , and $\\mathcal { P }$ is the data distribution. Here, we omit the parameters of $\\mathcal { G }$ , because given the output $h ^ { * }$ of the implicit module $\\mathcal { F }$ , training the post-processing part $\\mathcal { G }$ is the same as training explicit neural networks. The most crucial component is the gradient of the loss function $\\mathcal { L } w . r . t .$ the input vector $z ^ { \\top } = [ \\pmb { u } ^ { \\top } , \\pmb { \\theta } ^ { \\top } ]$ , which is used to train both the implicit module $\\mathcal { F }$ and the input projection module $\\mathcal { M }$ . Using Eq. (2) with the condition $h = h ^ { * }$ , we have ",
+        "bbox": [
+            173,
+            729,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f32292a35280caa1d1f2ca64d670ca1a38600152fef9c8242b83ecfc634de43d.jpg",
+        "text": "$$\n\\frac { \\partial \\mathcal { L } } { \\partial u } = \\frac { \\partial \\mathcal { F } } { \\partial u } \\left( I - \\frac { \\partial \\mathcal { F } } { \\partial h } \\right) ^ { - 1 } \\frac { \\partial \\mathcal { L } } { \\partial h } , \\quad \\frac { \\partial \\mathcal { L } } { \\partial \\theta } = \\frac { \\partial \\mathcal { F } } { \\partial \\theta } \\left( I - \\frac { \\partial \\mathcal { F } } { \\partial h } \\right) ^ { - 1 } \\frac { \\partial \\mathcal { L } } { \\partial h } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            811,
+            715,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The gradients in Eq. (4) are in the same form w.r.t. $\\textbf { \\em u }$ and $\\pmb \\theta$ . Without loss of generality, we only discuss the gradient w.r.t. $\\pmb { \\theta }$ in the following sections. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 Motivation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            289,
+            106
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The most intriguing part lies in the Jacobian-inverse term, i.e., $\\left( I - \\partial \\mathcal { F } / \\partial h \\right) ^ { - 1 }$ . Computing the inverse term by brute force is intractable due to the $\\mathcal { O } ( n ^ { 3 } )$ complexity. Previous implicit models [2] approach this by solving a linear system involving a Jacobian-vector product iteratively via a gradient solver, introducing over 30 Broyden [16] iterations in the backward pass. However, the scale of the Jacobian matrix can exceed $\\mathrm { 1 0 ^ { 6 } \\times 1 0 ^ { 6 } }$ in the real scenarios, leading to a prohibitive cost in computing the exact gradient. For example, training a small-scale state-of-the-art implicit model on ImageNet can consume weeks using 8 GPUs while training explicit models usually takes days, demonstrating that pursuing the exact gradient severely slows down the training process of implicit models compared with explicit models. ",
+        "bbox": [
+            173,
+            114,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Secondly, because of the inversion operation, we cast doubt on the conditioning of the gradient and the stability of training process from the numerical aspect. The Jacobian-inverse can be numerically unstable when encountering the ill-conditioning issue. The conditioning problem might further undermine the training stability, as studied in the recent work [17]. ",
+        "bbox": [
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+            247,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Plus, the inexact gradient [9, 18, 19, 20, 21, 22] is widely applied in the previous learning protocol, like linear propagation [23] and synthetic gradient [24]. Here, the Jacobian-inverse is used to calculate the exact gradient which is not always optimal for model training. Moreover, previous research has used a moderate gradient noise as a regularization approach [25], which has been shown to play a central role in escaping poor local minima and improving generalization ability [26, 27, 28]. ",
+        "bbox": [
+            174,
+            309,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The concerns and observations motivate us to rethink the possibility of replacing the Jacobian-inverse term in the standard implicit differentiation with a cheaper and more stable counterpart. We believe that an exact gradient estimate is not always required, especially for a black-box layer like those in the implicit models. Hence this work designs an inexact, theoretically sound, and practically efficient gradient for training implicit models under various settings. We name the proposed gradient estimate as the phantom gradient. ",
+        "bbox": [
+            173,
+            385,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Suppose the Jacobian ${ { \\partial } { h } ^ { * } } / { { \\partial } { \\theta } }$ is replaced with a matrix $\\pmb { A }$ , and the corresponding phantom gradient is defined as ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/630ccc056a66737f5a33e3f37edbee44dfae02dabd2da9bdf7ff27de9208d2ef.jpg",
+        "text": "$$\n{ \\widehat { \\frac { \\partial { \\mathcal { L } } } { \\partial \\theta } } } : = A \\ { \\frac { \\partial { \\mathcal { L } } } { \\partial h } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            446,
+            497,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next, we give the general condition on $\\pmb { A }$ so that the phantom gradient can be guaranteed valid for optimization (Sec. 2.3), and provide two concrete instantiations of $\\pmb { A }$ based on either damped fixed-point unrolling or the Neumann series (Sec. 2.4). The proofs of all our theoretical results are presented in Appendix C. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 General Condition on the Phantom Gradient ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            603,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Previous research on theoretical properties for the inexact gradient include several aspects, such as the gradient direction [22], the unbiasedness of the estimator [29], and the convergence theory of the stochastic algorithm [19, 30]. The following theorem formulates a sufficient condition that the phantom gradient gives an ascent direction of the loss landscape. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Suppose the exact gradient and the phantom gradient are given by Eq. (4) and (5), respectively. Let $\\sigma _ { m a x }$ and $\\sigma _ { m i n }$ be the maximal and minimal singular value of $\\partial \\mathcal { F } / \\partial \\theta$ . If ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e07a6d40eb5dd14c9ca8603412d7f6fc45f526478e91efa3f7d0f40385bda3f3.jpg",
+        "text": "$$\n\\left\\| A \\left( I - \\frac { \\partial \\mathcal { F } } { \\partial h } \\right) - \\frac { \\partial \\mathcal { F } } { \\partial \\theta } \\right\\| < \\frac { \\sigma _ { m i n } ^ { 2 } } { \\sigma _ { m a x } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            719,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "then the phantom gradient provides an ascent direction of the function $\\mathcal { L }$ , i.e., ",
+        "bbox": [
+            173,
+            758,
+            679,
+            772
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39bf6ec3423283ae3f420d6e901a61f8748e9746644a5b5a61acf4751605029c.jpg",
+        "text": "$$\n\\left. { \\widehat { \\frac { \\partial { \\mathcal { L } } } { \\partial \\theta } } } , { \\frac { \\partial { \\mathcal { L } } } { \\partial \\theta } } \\right. > 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            775,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. Suppose only the $\\left( I - \\partial \\mathcal { F } / \\partial h \\right) ^ { - 1 }$ term is replaced with a matrix $_ { D }$ , namely, $A =$ $( \\partial \\mathcal { F } / \\partial \\pmb { \\theta } ) \\pmb { D }$ . Then, the condition in (6) can be reduced into ",
+        "bbox": [
+            173,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56b965fc9489eca867c53690fb166d74618c3ddab5bdb45d1aca5e7239f73e2e.jpg",
+        "text": "$$\n\\left\\| D \\left( I - \\frac { \\partial \\mathcal { F } } { \\partial h } \\right) - I \\right\\| < \\frac { 1 } { \\kappa ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            858,
+            598,
+            893
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\kappa$ is the condition number of $\\partial \\mathcal { F } / \\partial \\theta$ . (See Appendix C.1 for the derivation.) ",
+        "bbox": [
+            171,
+            896,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.4 Instantiations of the Phantom Gradient ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            488,
+            106
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present two practical instantiations of the phantom gradient. We also verify that the general condition in Theorem 1 can be satisfied if the hyperparameters in our instantiations are wisely selected. ",
+        "bbox": [
+            176,
+            116,
+            825,
+            159
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Suppose we hope to differentiate through implicit dynamics, e.g., either a root-solving process or an optimization problem. In the context of hyperparameter optimization (HO), previous solutions include differentiating through the unrolled steps of the dynamics [19] or employing the Neumann series of the Jacobian-inverse term [9]. In our case, if we solve the root of Eq. (1) via the fixed-point iteration: ",
+        "bbox": [
+            173,
+            164,
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+            233
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f916ed51b79a67d1484a0888da1120b668af7a3eeb368542b8fa439c657a0ec6.jpg",
+        "text": "$$\n\\pmb { h } _ { t + 1 } = \\mathcal { F } ( \\pmb { h } _ { t } , z ) , \\quad t = 0 , 1 , \\cdots , T - 1 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "then by differentiating through the unrolled steps of Eq. (9), we have ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/448758686636112d8e5f0360602d9c61dc31a71e2efef613f3bca2d7bf6cdf84.jpg",
+        "text": "$$\n\\frac { \\partial { \\pmb h } _ { T } } { \\partial { \\pmb \\theta } } = \\sum _ { t = 0 } ^ { T - 1 } \\left. \\frac { \\partial \\mathcal { F } } { \\partial { \\pmb \\theta } } \\right| _ { { \\pmb h } _ { t } ~ s = t + 1 } \\left. \\frac { \\partial \\mathcal { F } } { \\partial { \\pmb h } } \\right| _ { { \\pmb h } _ { s } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            271,
+            617,
+            315
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Besides, the Neumann series of the Jacobian-inverse $\\left( I - \\partial \\mathcal { F } / \\partial h \\right) ^ { - 1 }$ is ",
+        "bbox": [
+            173,
+            324,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1c720ad997f88041ef6e7d8073c298d1e696d1a5ac259c5110c6b46a0d595da.jpg",
+        "text": "$$\n\\pmb { I } + \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { h } } + \\left( \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { h } } \\right) ^ { 2 } + \\left( \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { h } } \\right) ^ { 3 } + \\cdots .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            344,
+            632,
+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Notably, computing the Jacobian $\\partial \\pmb { h } ^ { * } / \\partial \\pmb { \\theta }$ using the Neumann series in (11) is equivalent to differentiating through the unrolled steps of Eq. (9) at the exact equilibrium point $h ^ { * }$ and taking the limit of infinite steps [9]. ",
+        "bbox": [
+            173,
+            387,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Without altering the root of Eq. (1), we consider a damped variant of the fixed-point iteration: ",
+        "bbox": [
+            171,
+            435,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a4780383c39aacd152d55f9ed333b5e38b8c92f20b56a754fccdce21a2e6018.jpg",
+        "text": "$$\n\\begin{array} { r } { h _ { t + 1 } = \\mathscr { F } _ { \\lambda } ( h _ { t } , z ) = \\lambda \\mathscr { F } ( h _ { t } , z ) + ( 1 - \\lambda ) h _ { t } , \\quad t = 0 , 1 , \\cdots , T - 1 . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            455,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Differentiating through the unrolled steps of Eq. (12), Eq. (10) is adapted as ",
+        "bbox": [
+            171,
+            478,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bfa26beeadb2736b084f5e67cb75922c6a1dbb04ca3d078e723775c33438501f.jpg",
+        "text": "$$\n\\frac { \\partial h _ { T } } { \\partial \\pmb { \\theta } } = \\lambda \\sum _ { t = 0 } ^ { T - 1 }  \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { \\theta } } | _ { h _ { t } } \\prod _ { s = t + 1 } ^ { T - 1 } ( \\lambda \\frac { \\partial \\mathcal { F } } { \\partial h } | _ { h _ { s } } + ( 1 - \\lambda ) \\pmb { I } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            500,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Neumann series of $\\left( I - \\partial \\mathcal { F } / \\partial h \\right) ^ { - 1 }$ is correspondingly adapted as ",
+        "bbox": [
+            174,
+            549,
+            642,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5ae297511071cbe0a1265188dca455530e9c4bc3513b36723157da17492d37c5.jpg",
+        "text": "$$\n\\lambda \\left( I + B + B ^ { 2 } + B ^ { 3 } + \\cdots \\right) , \\quad \\mathrm { w h e r e ~ } B = \\lambda \\frac { \\partial \\mathcal { F } } { \\partial h } + ( 1 - \\lambda ) I .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            571,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The next theorem shows that under mild conditions, the Jacobian from the damped unrolling in Eq. (13) converges to the exact Jacobian and the Neumann series in (14) converges to the Jacobianinverse $( I - \\partial \\mathcal { F } / \\partial h ) ^ { - 1 }$ as well. ",
+        "bbox": [
+            176,
+            607,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Suppose the Jacobian $\\partial \\mathcal { F } / \\partial h$ is a contraction mapping. Then, ",
+        "bbox": [
+            174,
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+            671
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(i) the Neumann series in (14) converges to the Jacobian-inverse $\\left( I - \\partial \\mathcal { F } / \\partial h \\right) ^ { - 1 }$ ; and ",
+        "bbox": [
+            202,
+            681,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(ii) if the function $\\mathcal { F }$ is continuously differentiable w.r.t. both $^ { h }$ and $\\pmb \\theta$ , the sequence in Eq. (13) converges to the exact Jacobian $\\partial \\pmb { h } ^ { * } / \\partial \\pmb { \\theta }$ as $T \\to \\infty$ , i.e., ",
+        "bbox": [
+            200,
+            705,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/855adce49aa72adefa0a2f1e997c1bfc9e5838060e61ffce21e66e43779a4a8f.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { T  \\infty } \\frac { \\partial h _ { T } } { \\partial \\pmb { \\theta } } =  \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { \\theta } } | _ { h ^ { \\ast } } ( \\pmb { I } -  \\frac { \\partial \\mathcal { F } } { \\partial h } | _ { h ^ { \\ast } } ) ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            739,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, as discussed in Sec. 2.2, it is unnecessary to compute the exact gradient with infinite terms. In the following context, we introduce two instantiations of the phantom gradient based on the finite-term truncation of Eq. (13) or (14). ",
+        "bbox": [
+            174,
+            794,
+            825,
+            838
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Unrolling-based Phantom Gradient (UPG). In the unrolling form, the matrix $\\pmb { A }$ is defined as ",
+        "bbox": [
+            169,
+            849,
+            805,
+            867
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edd1c2ab677ab6cb86b02aff1554db02adf0a8ffefad8a2ba1bb1cb2c0c3a1e0.jpg",
+        "text": "$$\nA _ { k , \\lambda } ^ { \\mathrm { u n r } } = \\lambda \\sum _ { t = 0 } ^ { k - 1 }  \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { \\theta } } | _ { h _ { t } } \\prod _ { s = t + 1 } ^ { k - 1 } ( \\lambda \\frac { \\partial \\mathcal { F } } { \\partial h } | _ { h _ { s } } + ( 1 - \\lambda ) \\pmb { I } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            871,
+            681,
+            915
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Neumann-series-based Phantom Gradient (NPG). In the Neumann form, the matrix $\\pmb { A }$ is defined as ",
+        "bbox": [
+            169,
+            90,
+            826,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5961b279433958061a4dc84dc6856e268c15a62789efc5d6fc90f959dda0a469.jpg",
+        "text": "$$\n\\pmb { A } _ { k , \\lambda } ^ { \\mathrm { n e u } } = \\lambda \\left. \\frac { \\partial \\mathcal { F } } { \\partial \\pmb { \\theta } } \\right| _ { h ^ { * } } \\left( I + \\pmb { B } + \\pmb { B } ^ { 2 } + \\cdots + \\pmb { B } ^ { k - 1 } \\right) , \\quad \\mathrm { w h e r e } \\ \\pmb { B } = \\lambda \\left. \\frac { \\partial \\mathcal { F } } { \\partial h } \\right| _ { h ^ { * } } + ( 1 - \\lambda ) I .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            118,
+            777,
+            154
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that both the initial point of the fixed-point iteration (i.e., $h _ { 0 }$ in Eq. (16)) and the point at which the Neumann series is evaluated (i.e., $h ^ { * }$ in (17)) are the solution of the root-finding solver. (See Appendix B for implementation of the phantom gradient.) ",
+        "bbox": [
+            174,
+            161,
+            825,
+            204
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "According to Theorem 2, the matrix $\\pmb { A }$ defined by either Eq. (16) or (17) converges to the exact Jacobian ${ \\bar { \\partial } } h ^ { * } / \\partial \\theta$ as $k \\to \\infty$ for any $\\lambda \\in ( 0 , 1 ]$ . Therefore, by Theorem 2, the condition in (6) can be satisfied if a sufficiently large step $k$ is selected, since ",
+        "bbox": [
+            173,
+            209,
+            825,
+            252
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5919a9482fa1eb01e65fbd599dfd64211a798e9e7bb864f1eb0313f40dd158d7.jpg",
+        "text": "$$\n\\left\\| A \\left( I - { \\frac { \\partial { \\mathcal { F } } } { \\partial h } } \\right) - { \\frac { \\partial { \\mathcal { F } } } { \\partial \\theta } } \\right\\| \\leq \\left( 1 + L _ { h } \\right) \\left\\| A - { \\frac { \\partial { \\mathcal { F } } } { \\partial \\theta } } \\left( I - { \\frac { \\partial { \\mathcal { F } } } { \\partial h } } \\right) ^ { - 1 } \\right\\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            253,
+            720,
+            296
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we characterize the impact of the two hyperparameters, i.e., $k$ and $\\lambda$ , on the precision and conditioning of $\\pmb { A }$ . Take the NPG (Eq. (17)) as an example. ",
+        "bbox": [
+            173,
+            304,
+            823,
+            333
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(i) On the precision of the phantom gradient, ",
+        "bbox": [
+            209,
+            340,
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+            356
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• a large $k$ makes the gradient estimate more accurate, as higher-order terms of the Neumann series are included, while   \n• a small $\\lambda$ slows down the convergence of the Neumann series because the norm $\\| B \\|$ increases as $\\lambda$ decreases. ",
+        "bbox": [
+            251,
+            358,
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+            416
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(ii) On the conditioning of the phantom gradient, ",
+        "bbox": [
+            205,
+            420,
+            529,
+            434
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• a large $k$ impairs the conditioning of $\\pmb { A }$ since the condition number of $B ^ { k }$ grows exponentially as $k$ increases, while   \n• a small $\\lambda$ helps maintain a small condition number of $\\pmb { A }$ because the singular values of $\\partial \\mathcal { F } / \\partial h$ are “smoothed” by the identity matrix. ",
+        "bbox": [
+            251,
+            438,
+            825,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In a word, a large $k$ is preferable for a more accurate $\\pmb { A }$ , while a small $\\lambda$ contributes to the wellconditioning of $\\pmb { A }$ . Practically, these hyperparameters should be selected in consideration of a balanced trade-off between the precision and conditioning of $\\pmb { A }$ . See Sec. 3 for experimental results. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.5 Convergence Theory ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            560,
+            356,
+            577
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide the convergence guarantee of the SGD algorithm using the phantom gradient. We prove that under mild conditions, if the approximation error of the phantom gradient is sufficiently small, the SGD algorithm converges to an $\\epsilon$ -approximate stationary point in the expectation sense. We will discuss the feasibility of our assumptions in Appendix C.3. ",
+        "bbox": [
+            173,
+            585,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3. Suppose the loss function $\\mathcal { R }$ in Eq. (3) is $\\ell$ -smooth, lower-bounded, and has bounded gradient almost surely in the training process. Besides, assume the gradient in Eq. (4) is an unbiased estimator of $\\nabla \\mathcal { R } ( \\theta )$ with a bounded covariance. If the phantom gradient in Eq. (5) is an $\\epsilon$ -approximation to the gradient in Eq. (4), i.e., ",
+        "bbox": [
+            173,
+            643,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/59b98e544d1610a66b617741428c1f2a0667a099cc1079c75bc6da6efd9b091d.jpg",
+        "text": "$$\n\\left\\| { \\widehat { \\frac { \\partial { \\mathcal { L } } } { \\partial \\theta } } } - { \\frac { \\partial { \\mathcal { L } } } { \\partial \\theta } } \\right\\| \\leq \\epsilon , \\quad a l m o s t s u r e l y ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            702,
+            617,
+            744
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "then using Eq. (5) as a stochastic first-order oracle with a step size of $\\eta _ { n } = \\mathcal { O } ( 1 / \\sqrt { n } )$ to update $\\pmb \\theta$ with gradient descent, it follows after $N$ iterations that ",
+        "bbox": [
+            173,
+            746,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5fcbca60bc1c5895239a9f2e4326c998979c39c279ea5c25e15a66aa842529ee.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\frac { \\sum _ { n = 1 } ^ { N } \\eta _ { n } \\left\\| \\nabla \\mathcal { R } \\left( \\pmb { \\theta } _ { n } \\right) \\right\\| ^ { 2 } } { \\sum _ { n = 1 } ^ { N } \\eta _ { n } } \\right] \\le \\mathcal { O } \\left( \\epsilon + \\frac { \\log N } { \\sqrt { N } } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            776,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 2. Consider the condition in (19): ",
+        "bbox": [
+            173,
+            824,
+            465,
+            838
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1dbd2b026d38b112048b63918f5477143b353f5a10406f7c96f6cb69b856598d.jpg",
+        "text": "$$\n\\left\\| \\widehat { \\frac { \\partial \\mathcal { L } } { \\partial \\theta } } - \\frac { \\partial \\mathcal { L } } { \\partial \\theta } \\right\\| \\leq \\left\\| A - \\frac { \\partial \\mathcal { F } } { \\partial \\theta } \\left( I - \\frac { \\partial \\mathcal { F } } { \\partial h } \\right) ^ { - 1 } \\right\\| \\left\\| \\frac { \\partial \\mathcal { L } } { \\partial h } \\right\\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            839,
+            671,
+            881
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Suppose the gradient $\\partial \\mathcal { L } / \\partial h$ is almost surely bounded. By Theorem 2, the condition in (19) can be guaranteed as long as a sufficiently large $k$ is selected. ",
+        "bbox": [
+            173,
+            883,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/874187b8bcf9d94135ffda4257628118a4d11dc9de88f7857d747afb5fe4f59a.jpg",
+        "image_caption": [
+            "Figure 1: Cosine similarity between the phantom gradient and the exact gradient in the synthetic setting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            265,
+            312,
+            282
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we aim to answer the following questions via empirical results: (1) How precise is the phantom gradient? (2) What is the difference between the unrolling-based and the Neumannseries-based phantom gradients? (3) How is the phantom gradient influenced by the hyperparameters $k$ and $\\lambda ?$ (4) How about the computational cost of the phantom gradient compared with implicit differentiation? (5) Can the phantom gradient work at large-scale settings for various tasks? ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We have provided some theoretical analysis and intuitions to (1), (2), and (3) in Sec. 2.4. Now we answer (1) and (2) and demonstrate the performance curves under different hyperparameters $k$ and $\\lambda$ on CIFAR-10 [31]. Besides, we also study other factors that have potential influences on the training process of the state-of-the-art implicit models [2, 3] like pretraining. For (4) and (5), we conduct experiments on large-scale datasets to highlight the ultra-fast speed and competitive performances, including image classification on ImageNet [10] and language modeling on Wikitext-103 [13]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We start by introducing two experiment settings. We adopt a single-layer neural network with spectral normalization [32] as the function $\\mathcal { F }$ and fixed-point iterations as the equilibrium solver, which is the synthetic setting. Moreover, on the CIFAR-10 dataset, we use the MDEQ-Tiny model [3] (170K parameters) as the backbone model, denoted as the ablation setting. Additional implementation details and experimental results are presented in Appendix $\\mathsf { D } ^ { 2 }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Precision of the Phantom Gradient. The precision of the phantom gradient is measured by its angle against the exact gradient, indicated by the cosine similarity between the two. We discuss its precision in both the synthetic setting and the ablation setting. The former is under the static and randomly generated weights, while the latter provides characterization of the training dynamics. ",
+        "bbox": [
+            174,
+            555,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the synthetic setting, the function $\\mathcal { F }$ is restricted to be a contraction mapping. Specifically, we directly set the Lipschitz constant of $\\mathcal { F }$ as $L _ { h } = 0 . 9$ , and use 100 fixed-point iterations to solve the root $h ^ { * }$ of Eq. (1) until the relative error satisfies $\\| h - \\mathcal { F } ( h , z ) \\| / \\| h \\| ^ { - } 1 0 ^ { - 5 }$ . Here, the exact gradient is estimated by backpropagation through the fixed-point iterations, and cross-validated by implicit differentiation solved with 20 iterations of the Broyden’s method [16]. In our experiment, the cosine similarity between these two gradient estimates consistently succeeds 0.9999, indicating the gradient estimate is quite accurate when the relative error of forward solver is minor. The cosine similarity between phantom gradients and exact gradients is shown in Fig. 1. It shows that the cosine similarity tends to increase as $k$ grows and that a small $\\lambda$ tends to slow down the convergence of the phantom gradient, allowing it to explore in a wider range regarding the angle against the exact gradient. ",
+        "bbox": [
+            174,
+            617,
+            825,
+            768
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the ablation setting, the precision of the phantom gradient during the training process is shown in Fig. 2. The model is trained by implicit differentiation under the official schedule3. It shows that the phantom gradient still provides an ascent direction in the real training process, as indicated by the considerable cosine similarity against the exact gradient. Interestingly, the cosine similarity slightly decays as the training progresses, which implies a possibility to construct an adaptive gradient solver for implicit models. ",
+        "bbox": [
+            174,
+            776,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/177f666c03028cac8d033f6ef1cb3da9b3e02a09a76a80dc90d1b367f0317f4b.jpg",
+        "image_caption": [
+            "Figure 2: Cosine similarity between the phantom gradient and the exact gradient in the real scenario. The horizontal axis corresponds to the cosine similarity, and the vertical axis to the training step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To Pretrain, or not to Pretrain? To better understand the components of the implicit models’ training schedule, we first illustrate a detailed ablation study of the baseline model in the ablation setting. The average accuracy with standard deviation is reported in Tab. 1. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The MDEQ model employs a pretraining stage in which the model $\\mathcal { F }$ is unrolled as a recurrent network. We study the impact of the pretraining stage, the Dropout [33] operation, and the optimizer separately. It can be seen that the unrolled pretraining stabilizes the training of the MDEQ model. Removing the pretraining stage leads to a severe performance drop and apparent training instability among different trials because the solver cannot obtain an accurate fixed point $h ^ { * }$ when the model is not adequately trained. This ablation study also suggests that the MDEQ model is a strong baseline for our method to compare with. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f6d627af009a7afe0da8424603054ad7199efa8e66b5f46b2f5c08402bc782ce.jpg",
+        "table_caption": [
+            "Table 1: Ablation settings on CIFAR-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Acc. (%)</td></tr><tr><td>Implicit Differentiation</td><td>85.0± 0.2</td></tr><tr><td>w/o Pretraining</td><td>82.3± 1.3</td></tr><tr><td>w/o Dropout Adam →SGD</td><td>83.7± 0.1</td></tr><tr><td> SGD w/o Pretraining</td><td>84.5 ± 0.3</td></tr><tr><td>UPG (A5,0.5, W/o Dropout)</td><td>82.9 ± 1.5</td></tr><tr><td>NPG (A5,0.5, w/o Dropout)</td><td>85.8±0.5</td></tr><tr><td>UPG (A9.0.5, w/ Dropout)</td><td>85.6± 0.5 86.1±0.5</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, pretraining is not always indispensable for training implicit models. It introduces an extra hyperparameter, i.e., how many steps should be involved in the pretraining stage. Next, we discuss how the UPG could circumvent this issue. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Trade-offs between Unrolling and Neumann. For an exact fixed point $h ^ { * }$ , i.e., $\\begin{array} { r } { \\begin{array} { r } { h ^ { * } = \\mathcal { F } ( h ^ { * } , z ) } \\end{array} } \\end{array}$ , there is no difference between UPG and NPG. However, when the numerical error exists in solving $h ^ { * }$ , i.e., $\\| h ^ { * } - \\mathcal { F } ( h ^ { * } , z ) \\| > 0$ , these two instatiations of the phantom gradient can behave differently. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We note that a particular benefit of the UPG is its ability to automatically switch between the pretraining and training stages for implicit models. When the model is not sufficiently trained and the solver converges poorly (see [3]), the UPG defines a forward computation graph that is essentially equivalent to a shallow weight-tied network to refine the coarse equilibrium states. In this stage, the phantom gradient serves as a backpropagation through time (BPTT) algorithm and hence behaves as in the pretraining stage. Then, as training progresses, the solver becomes more stable and converges to the fixed point $h ^ { * }$ better. This makes the UPG behave more like the NPG. Therefore, the unrolled pretraining is gradually transited into the regular training phase based on implicit differentiation, and the hyperparameter tuning of pretraining steps can be waived. We argue that such an ability to adaptively switch training stages is benign to the implicit models’ training protocol, which is also supported by the performance gain in Tab. 1. ",
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+        "type": "table",
+        "img_path": "images/4e6fc18bf18adb307efde9de6360ac5604792f0232435ee108b62e99c3d815d2.jpg",
+        "table_caption": [
+            "Table 2: Complexity comparison. Mem means the memory cost, and $K$ and $k$ denote the solver’s steps and the unrolling/Neumann steps, respectively. Here, $K \\gg k \\approx 1$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Time</td><td>Mem</td><td>Peak Mem</td></tr><tr><td>Implicit</td><td>O(K)</td><td>0(1)</td><td>O(k)</td></tr><tr><td>UPG</td><td>O(k）</td><td>O(k)</td><td>O(k)</td></tr><tr><td>NPG</td><td>（k）</td><td>0(1)</td><td>0(1)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although the UPG requires higher memory overhead than implicit differentiation or the NPG, it does not surpass the peak memory usage in the entire training protocol by implicit differentiation due to the pretraining stage. In the ablation setting, the MDEQ model employs a 10-layer unrolling for pretraining, which actually consumes double memory compared with a 5-step unrolling scheme, e.g., $A _ { 5 , 0 . 5 }$ in Tab. 1. In Tab. 2, we also demonstrate the time and memory complexity for implicit differentiation and the two forms of phantom gradient. ",
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+    {
+        "type": "image",
+        "img_path": "images/92998b52ccf86b6ff21f0145c37c4cc23f3a48218c10885a6e3c395ae3d6e2ff.jpg",
+        "image_caption": [
+            "Figure 3: Ablation studies on (a) the hyperparameters $\\lambda$ and $k$ , and (b) two forms of phantom gradient. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In addition, leaving the context of approximate and exact gradient aside, we also develop insights into understanding the subtle differences between UPG and NPG in terms of the state-dependent and state-free gradient. Actually, the UPG is state-dependent, which means it corresponds to the “exact” gradient of a computational sub-graph. Both the NPG and the gradient solved by implicit differentiation, however, do not exactly match the gradient of any forward computation graph unless the numerical error is entirely eliminated in both the forward and the backward passes for implicit differentiation or the one-step gradient [34, 21, 30, 22] is used for the NPG, i.e., $k =$ 1. Interestingly, we observe that models trained on the state-dependent gradient demonstrate an additional training stability regarding the Jacobian spectral radius, compared with those trained on the state-free counterpart. We empirically note that it can be seen as certain form of implicit Jacobian regularization for implicit models as a supplement to the explicit counterpart [17], indicated by the stable estimated Jacobian spectral radius during training, i.e., $\\rho ( \\partial \\mathcal { F } _ { \\lambda } / \\partial h ) \\approx 1$ . ",
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+    {
+        "type": "text",
+        "text": "The experimental results also cohere with our insight. The performance curves in Fig. 3 demonstrate the influence of $\\lambda$ and $k$ and further validate that the UPG is more robust to a wide range of steps $k$ than the NPG. In fact, when the Jacobian spectral radius $\\rho ( \\partial \\mathcal { F } / \\partial h )$ increases freely without proper regularization, the exploding high-order terms in the NPG could exert a negative impact on the overall direction of the phantom gradient, leading to performance degradation when a large $k$ is selected (see Fig. 3(b)). We also observe a surging of the estimated Jacobian spectral radius as well as the gradient explosion issue in the NPG experiments. On the contrary, the UPG can circumvent these problems thanks to its implicit Jacobian regularization. ",
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+    {
+        "type": "table",
+        "img_path": "images/be600d4a6950ec6a82fe1381e8efbef5e921c9e9dee6e8e940657b314b3c0b06.jpg",
+        "table_caption": [
+            "Table 3: Experiments using DEQ [2] and MDEQ [3] on vision and language tasks. Metrics stand for accuracy $( \\% ) \\uparrow$ for image classification on CIFAR-10 and ImageNet, and perplexity↓ for language modeling on Wikitext-103. JR stands for Jacobian Regularization [17]. $\\dagger$ indicates additional steps in the forward equilibrium solver. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Model</td><td>Method</td><td>Params</td><td>Metrics</td><td>Speed</td></tr><tr><td>CIFAR-10</td><td>MDEQ</td><td>Implicit</td><td>10M</td><td>93.8 ± 0.17</td><td>1.0×</td></tr><tr><td>CIFAR-10</td><td>MDEQ</td><td>UPG A5,0.5</td><td>10M</td><td>95.0 ± 0.16</td><td>1.4×</td></tr><tr><td>ImageNet</td><td>MDEQ</td><td>Implicit</td><td>18M</td><td>75.3</td><td>1.0×</td></tr><tr><td>ImageNet</td><td>MDEQ</td><td>UPG A5.0.6</td><td>18M</td><td>75.7</td><td>1.7×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PostLN)</td><td>Implicit</td><td>98M</td><td>24.0</td><td>1.0×</td></tr><tr><td>Wikitext-103</td><td>DEQ( (PostLN)</td><td>UPG A5,0.8</td><td>98M</td><td>25.7</td><td>1.7×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PreLN)</td><td>JR + Implicit</td><td>98M</td><td>24.5</td><td>1.7×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PreLN)</td><td>JR + UPG A5,0.8</td><td>98M</td><td>24.4</td><td>2.2×</td></tr><tr><td>Wikitext-103</td><td>DEQ (PreLN)</td><td>JR + UPG A5,0.8</td><td>98M</td><td>24.0t</td><td>1.7×</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Phantom Gradient at Scale. We conduct large-scale experiments to verify the advantages of the phantom gradient on vision, graph, and language benchmarks. We adopt the UPG in the large-scale experiments. The results are illustrated in Tab. 3 and Tab. 4. Our method matches or surpasses the implicit differentiation training protocol on the state-of-the-art implicit models with a visible reduction on the training time. When only considering the backward pass, the acceleration for MDEQ can be remarkably $1 2 \\times$ on ImageNet classification. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/56703f5ee215b28ffe865ab6202ac66bcfb437af3733667d16f72a56afd80c0b.jpg",
+        "table_caption": [
+            "Table 4: Experiments using IGNN [4] on graph tasks. Metrics stand for accuracy $( \\% ) \\uparrow$ for graph classification on COX2 and PROTEINS, Micro- $\\mathrm { F l } ( \\% ) \\uparrow$ for node classification on PPI. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Model</td><td>Method</td><td>Params</td><td>Metrics (%)</td></tr><tr><td>COX2</td><td>IGNN</td><td>Implicit</td><td>38K</td><td>84.1 ± 2.9</td></tr><tr><td>COX2</td><td>IGNN</td><td>UPG A5,0.5</td><td>38K</td><td>83.9 ± 3.0</td></tr><tr><td>COX2</td><td>IGNN</td><td>UPG A5,0.8</td><td>38K</td><td>83.9 ± 2.7</td></tr><tr><td>COX2</td><td>IGNN</td><td>UPG A5,1.0</td><td>38K</td><td>83.0 ± 2.9</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>Implicit</td><td>34K</td><td>78.6 ± 4.1</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>UPG A5,0.5</td><td>34K</td><td>78.4 ± 4.2</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>UPG A5,0.8</td><td>34K</td><td>78.6 ± 4.2</td></tr><tr><td>PROTEINS</td><td>IGNN</td><td>UPG A5,1.0</td><td>34K</td><td>78.8± 4.2</td></tr><tr><td>PPI</td><td>IGNN</td><td>Implicit</td><td>4.7M</td><td>97.6</td></tr><tr><td>PPI</td><td>IGNN</td><td>UPG A5,0.5</td><td>4.7M</td><td>98.2</td></tr><tr><td>PPI</td><td>IGNN</td><td>UPG A5,0.8</td><td>4.7M</td><td>97.4</td></tr><tr><td>PPI</td><td>IGNN</td><td>UPG A5,1.0</td><td>4.7M</td><td>96.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+            420,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Implicit Models. Implicit models generalize the recursive forward/backward rules of neural networks and characterize their internal mechanism by some pre-specified dynamics. Based on the dynamics, the implicit mechanisms can be broadly categorized into three classes: ODE-based [1, 7], root-solving-based [2, 3, 4, 5, 11], and optimization-based [35, 36, 37] implicit models. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The ODE-based implicit models [1, 7] treat the iterative update rules of residual networks as the Euler discretization of an ODE, which could be solved by any black-box ODE solver. The gradient of the differential equation is calculated using the adjoint method [38], in which the adjoint state is obtained by solving another ODE. The root-solving-based implicit models [2, 5, 3, 4, 6, 11, 22] characterize layers of neural networks by solving fixed-point equations. The equations are solved by either the black-box root-finding solver [2, 3] or the fixed-point iteration [4, 22]. The optimization-based implicit models [35, 36, 37, 39, 34, 21, 40, 41] leverage the optimization programs as layers of neural networks. Previous works have studied differentiable layers of quadratic programming [35], submodular optimization [36], maximum satisfiability (MAXSAT) problems [37], and structured decomposition [21]. As for the backward passes, implicit differentiation is applied to the problemdefining equations of the root-solving-based models [2, 3] or the KKT conditions of the optimizationbased models [35]. As such, the gradient can be obtained from solving the backward linear system. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we focus on the root-solving-based implicit models. Theoretical works towards root-solving-based implicit models include the well-posedness [5, 4], monotone operators [11], global convergence [42, 41], and Lipschitz analysis [15]. We look into the theoretical aspect of the gradient-based algorithm in training implicit models and the efficient practice guidance. With these considerations, we show that implicit models of the same architecture could enjoy faster training speed and strong generalization in practical applications by using the phantom gradient. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Non-End-to-End Optimization in Deep Learning. Non-end-to-end optimization aims to replace the standard gradient-based training of deep architectures with modular or weakly modular training without the entire forward and backward passes. Currently, there are mainly three research directions in this field, namely, the auxiliary variable methods [43, 44, 45, 46, 47, 48, 49], target propagation [50, 51, 52], and synthetic gradient [24, 53, 54]. The auxiliary variable methods [43, 44, 45, 46, 47, 48, 49] formulate the optimization of neural networks as constrained optimization problems, in which the layer-wise activations are considered as trainable auxiliary variables. Then, the equality constraints are relaxed as penalty terms added to the objectives so that the parameters and auxiliary variables can be divided into blocks and thus optimized in parallel. The target propagation method [50, 51, 52] trains each module by having its activations regress to the pre-assigned targets, which are propagated backwards from the downstream modules. Specifically, the auto-encoder architecture is used to reconstruct targets at each layer. Finally, the synthetic gradient method [24, 53, 54] estimates the local gradient of neural networks using auxiliary models, and employ the synthetic gradient in place of the exact gradient to perform parameter update. In this way, the forward and backward passes are decoupled and can be executed in an asynchronous manner. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our work is in line with the non-end-to-end optimization research since we also aims to decouple the forward and backward passes of neural networks. However, we show that finding a reasonable “target” or a precise gradient estimate is not always necessary in training deep architectures. Our paper paves a path that an inexact but well-conditioned gradient estimate can contribute to both fast training and competitive generalization of implicit models. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Differentiation through Implicit Dynamics. Differentiation through certain implicit dynamics is an important aspect in a wide range of research fields, including bilevel optimization [19, 9], metalearning [18, 55, 30], and sensitivity analysis [56]. Since the gradient usually cannot be computed analytically, researchers have to implicitly differentiate the dynamics at the converged point. The formula of the gradient typically contains a term of Jacobian-inverse (or Hessian-inverse), which is computationally prohibitive for large-scale models. (See Eq. (2) in our case.) Herein, several techniques have been developed to approximate the matrix inverse in the previous literature. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "An intuitive solution is to differentiate through the unrolled steps of a numerical solver of the dynamics [57, 58, 8]. In particular, if a single step is unrolled, it reduces to the well-known one-step gradient [59, 18, 60, 34, 30, 21, 22], in which the inverse of Jacobian/Hessian is simply approximated by an identity matrix. On the contrary, unrolling a small number of steps may induce a bias [9], while the memory and computational cost grows linearly as the number of unrolled steps increases. Towards this issue, Shaban et al. [19] propose to truncate the long-term dependencies and differentiate through only the last $L$ steps. In fact, if the dynamics have converged to a stationary point, the finite-term truncation in Shaban et al. [19] is exactly the Neumann approximation of the Jacobian-inverse with the first $L$ terms. Based on this, Lorraine et al. [9] directly use the truncated Neumann series as an approximation of the Jacobian-inverse. Besides the unrolling-based methods, optimization-based approaches [61, 55] have been studied in this field as well. Since the Jacobian-inverse-vector product can be viewed as solution of a linear system, algorithms like the conjugate gradient method can be used to solve it. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Limitation and Future Work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
+            444,
+            575
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The main limitation of this work lies in the hyperparameter tuning of the phantom gradient, especially for the damping factor $\\lambda$ , which directly controls the gradient’s precision and conditioning, the implicit Jacobian regularization for UPG, the stability for NPG, and the final generalization behaviors. However, it has not been a hindrance to the application of phantom gradients in training implicit models as one can tune the hyperparameter according to the validation loss in the early training stage. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Regarding future works, we would like to highlight the following aspects: (1) eliminating the bias of the current phantom gradient, (2) constructing an adaptive gradient solver for implicit models, (3) analyzing the damping factor to provide practical guidance, (4) investigating the implicit Jacobian regularization, and (5) understanding how different noises in the gradients can impact the training of implicit models under different loss landscapes. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            756,
+            299,
+            773
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we explore the possibility of training implicit models via the efficient approximate phantom gradient. We systematically analyze the general condition of a gradient estimate so that the implicit model can be guaranteed to converge to an approximate stationary point of the loss function. Specifically, we give a sufficient condition under which a first-order oracle could always find an ascent direction of the loss landscape in the training process. Moreover, we introduce two instantiations of the proposed phantom gradient, based on either the damped fixed-point unrolling or the Neumann series. The proposed method shows a $1 . 4 \\sim 1 . 7 \\times$ acceleration with comparable or better performances on large-scale benchmarks. Overall, this paper provides a practical perspective on training implicit models with theoretical guarantees. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            92,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhouchen Lin was supported by the NSF China (No.s 61625301 and 61731018), NSFC Tianyuan Fund for Mathematics (No. 12026606) and Project 2020BD006 supported by PKU-Baidu Fund. Yisen Wang was partially supported by the National Natural Science Foundation of China under Grant 62006153, and Project 2020BD006 supported by PKU-Baidu Fund. Shaojie Bai was sponsored by a grant from the Bosch Center for Artificial Intelligence. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            204,
+            266,
+            220
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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parse/train/BJInEZsTb/BJInEZsTb.md

@@ -0,0 +1,396 @@
+# LEARNING REPRESENTATIONS AND GENERATIVE MODELS FOR 3D POINT CLOUDS
+
+Panos Achlioptas ，\* Department of Computer Science Stanford University, USA
+
+Olga Diamanti Department of Computer Science Stanford University, USA
+
+Ioannis Mitliagkas   
+Department of Computer Science and Operations Research   
+University of Montréal, Canada
+
+Leonidas Guibas Department of Computer Science Stanford University, USA
+
+# ABSTRACT
+
+Three-dimensional geometric data offer an excellent domain for studying representation learning and generative modeling. In this paper, we look at geometric data represented as point clouds. We introduce a deep autoencoder (AE) network with state-of-the-art reconstruction quality and generalization ability. The learned representations outperform existing methods on 3D recognition tasks and enable basic shape editing via simple algebraic manipulations, such as semantic part editing, shape analogies and shape interpolation. We perform a thorough study of different generative models including: GANs operating on the raw point clouds, significantly improved GANs trained in the fixed latent space of our AEs and, Gaussian mixture models (GMM). For our quantitative evaluation we propose measures of sample fidelity and diversity based on matchings between sets of point clouds. Interestingly, our careful evaluation of generalization, fidelity and diversity reveals that GMMs trained in the latent space of our AEs produce the best results.
+
+# 1 INTRODUCTION
+
+Three-dimensional (3D) representations of real-life objects are a core tool for vision, robotics, medicine, augmented and virtual reality applications. Recent encodings like view-based projections, volumetric grids and graphs, complement more traditional shape representations such as 3D meshes, level set functions, curve-based CAD models and constructive solid geometry (Botsch et al., 2010). These encodings, while effective in their respective domains (e.g. acquisition or rendering), are often poor in semantics. For example, naïvely interpolating between two different cars in a view-based representation does not yield a representation of an “intermediate” car. Furthermore, these raw, high-dimensional representations are typically not well suited for the design of generative models via classic statistical methods. As such, editing and designing new objects with such representations frequently involves the construction and manipulation of complex, object-specific parametric models that link the semantics to the representation. This may require significant expertise and effort.
+
+Recent advances in deep learning bring the promise of a data-driven approach. In domains where data is plentiful, deep learning tools have eliminated the need for hand-crafting features and models. Deep learning architectures like autoencoders (AEs) (Rumelhart et al., 1988; Kingma & Welling, 2013) and Generative Adversarial Networks (GANs) (Goodfellow et al., 2014; Radford et al., 2015; Denton et al., 2015; Che et al., 2016) are successful at learning complex data representations and generating realistic samples from complex underlying distributions. Recently, deep learning architectures for view-based projections (Su et al., 2015; Wei et al., 2016; Kalogerakis et al., 2016), volumetric grids (Qi et al., 2016b; Wu et al., 2015; Hegde & Zadeh, 2016) and graphs (Bruna et al., 2013; Henaff et al., 2015; Defferrard et al., 2016; Yi et al., 2016b) have appeared in the 3D machine learning literature.
+
+In this paper we focus on point clouds, a relatively unexplored 3D modality. Point clouds provide a homogeneous, expressive and compact representation of surface geometry, easily amenable to geometric operations. These properties make them attractive from a learning point of view. In addition, they come up as the output of common range-scanning acquisition pipelines used in devices like the Kinect and iPhone’s recent face identification feature. Only a handful of deep architectures for 3D point clouds exist in the literature: PointNet (Qi et al., 2016a; 2017) successfully tackled classification and segmentation tasks; Kalogerakis et al. (2016) used point-clouds as an intermediate step in their pipeline; Fan et al. (2016) used pointclouds as the underlying representation to extract 3D information from 2D images. We provide the first results that use deep architectures with the focus of learning representations and generative models for point clouds.
+
+Generative models have garnered increased attention recently in the deep learning community with the introduction of GANs (Goodfellow et al., 2014). An issue with GAN-based generative pipelines is that training them is notoriously hard and unstable (Salimans et al., 2016). More importantly, there is no universally accepted way to evaluate generative models. In evaluating generative models one is interested in both fidelity, i.e. how much the generated points resemble the actual data, and coverage, i.e. what fraction of the data distribution a generated sample represents. The latter is especially important given the tendency of certain GANs to exhibit mode collapse. We provide simple methods to deal with both issues (training and evaluation) in our target domain. Our specific contributions are:
+
+• We design a new AE architecture—inspired by recent architectures used for classification (Qi et al., 2016a)—that is capable of learning compact representations of point clouds with excellent reconstruction quality even on unseen samples. The learned representations are (i) good for classification via simple methods (SVM), improving on the state of the art (Wu et al., 2016); (ii) suitable for meaningful interpolations and semantic operations.   
+We create the first set of generative models which (i) can generate point clouds measurably similar to the training data and held-out test data; (ii) provide good coverage of the training and test dataset. We argue that jointly learning the representation and training the GAN is unnecessary for our modality. We propose a workflow that first learns a representation by training an AE with a compact bottleneck layer, then trains a plain GAN in that fixed latent representation. Intuitively, training a GAN inside a compact, low-dimensional representation is easier. We point to theory (Arjovsky & Bottou, 2017) that supports this idea, and verify it empirically. Latent GANs are much easier to train than monolithic (raw) GANs and achieve superior reconstruction with much better coverage. Somewhat surprisingly, GMMs trained in the latent space of fixed AEs achieve the best performance across the board.   
+We show that multi-class GANs work almost on par with dedicated GANs trained per-objectcategory, as long as they are trained in the latent space.   
+To support our qualitative evaluation, we perform a careful study of various old and new metrics, in terms of their applicability (i) as objectives for learning good representations; (ii) for the evaluation of generated samples. We find that a commonly used point cloud metric, Chamfer distance, fails to discriminate certain pathological cases from good examples. We also propose fidelity and coverage metrics for our generative models, based on an optimal matching between two different samples, e.g. a set of point clouds generated by the model and a held-out test set.
+
+The rest of this paper is organized as follows: Section 2 outlines the necessary background and building blocks for our work and introduces our evaluation metrics. Section 3 introduces our models for latent representations and generation of point clouds. In Section 4, we evaluate all of our models both quantitatively and qualitatively, and analyze their behaviour. Further results and evaluation can be found in the appendix. The code for all our models is publicly available 1.
+
+# 2 BACKGROUND
+
+Autoencoders. Autoencoders (AE - inset) are deep architectures that aim to reproduce their input. They are especially useful, when they contain a narrow bottleneck layer between input and output. Upon successful training, the bottleneck layer corresponds to a lowdimensional representation, a code for the dataset. The Encoder (E) learns to compress a data point $_ { \textbf { \em x } }$ into its latent representation, $_ z$ . The Decoder (D) can then reproduce $_ { \textbf { \em x } }$ from its encoded version $_ { z }$ .
+
+![](images/b002868012e8863fc8fa0a00d10a2933b2e6bec54c7cb367d6d0fa8b9e750211.jpg)
+
+Generative Adversarial Networks. GANs are state-of-the-art generative models. The basic architecture (inset) is based on a adversarial game between a generator (G) and a discriminator (D). The generator aims to synthesize samples that look indistinguishable from real data (drawn from $\begin{array} { r } { \mathbf { \boldsymbol { x } } \sim p _ { \mathrm { d a t a } } , } \end{array}$ ) by passing a randomly drawn sample $z \sim p _ { z }$ through the generator function $G$ . The discriminator tries to tell synthesized from real samples. The most commonly used losses for the discriminator and generator networks are:
+
+![](images/689ac7d9f5a66cbc8a0f2f3daa77cf4aa5a8aa0b8ad2f646ec4a4986d8e8e593.jpg)
+
+$$
+\begin{array} { r c l } { { J ^ { ( D ) } ( { \pmb \theta } ^ { ( D ) } , { \pmb \theta } ^ { ( G ) } ) } } & { { = } } & { { - \mathbb { E } _ { { \pmb x } \sim p _ { \mathrm { d a t a } } } \log D ( { \pmb x } ) - \mathbb { E } _ { { \pmb z } \sim p _ { z } } \log \left( 1 - D \left( G ( { \pmb z } ) \right) \right) , } } \\ { { J ^ { ( G ) } ( { \pmb \theta } ^ { ( D ) } , { \pmb \theta } ^ { ( G ) } ) } } & { { = } } & { { - \mathbb { E } _ { { \pmb z } \sim p _ { z } } \log D ( G ( { \pmb z } ) ) , } } \end{array}
+$$
+
+where $\pmb \theta ^ { ( D ) } , \pmb \theta ^ { ( G ) }$ are the parameters for the discriminator and the generator network respectively. In addition to the classical GAN formulation, we also use the improved Wasserstein GAN (Gulrajani et al., 2017), which has shown improved stability during training.
+
+Challenges specific to point cloud geometry. Point clouds as an input modality present a unique set of challenges when building a network architecture. As an example, the convolution operator – now ubiquitous in image-processing pipelines – requires the signal (in our case, geometry) to be defined on top of an underlying grid-like structure. Such a structure is not available in raw point clouds, which renders them significantly more difficult to encode than e.g. images or voxel grids. Recent classification work on point clouds (PointNet – Qi et al. (2016a)) bypasses this issue by circumventing 2D convolutions. Another issue with point clouds as a representation is that they are unordered - any permutation of a point set still describes the same shape. This complicates comparisons between two point sets, typically needed to define a loss function. This unorderedness of point clouds also creates the need for making the encoded feature permutation invariant.
+
+Point-set distances. Two permutation-invariant metrics for comparing unordered point sets have been proposed in the literature (Fan et al., 2016). On the one hand, the Earth Mover’s distance (EMD) (Rubner et al., 2000) is the solution of a transportation problem which attempts to transform one set to the other. For two equally sized subsets $S _ { 1 } \subseteq \bar { R ^ { 3 } } , S _ { 2 } \subseteq R ^ { 3 }$ , their EMD is defined by $d _ { E M D } ( S _ { 1 } , S _ { 2 } ) \ = \ \operatorname* { m i n } _ { \phi : S _ { 1 } \to S _ { 2 } } \sum _ { x \in S _ { 1 } } \| x - \phi ( x ) \| _ { 2 }$ where $\phi$ is a bijection. Interpreted as a loss, EMD is differentiable almost everywhere. On the other hand, the Chamfer (pseudo)-distance (CD) measures the squared distance between each point in one set to its nearest neighbor in the other set: $d _ { C H } ( S _ { 1 } , S _ { 2 } ) = \sum _ { x \in S _ { 1 } } \operatorname* { m i n } _ { y \in S _ { 2 } } \| x - y \| _ { 2 } ^ { 2 } + \sum _ { y \in S _ { 2 } } \operatorname* { m i n } _ { x \in S _ { 1 } } \| x - y \| _ { 2 } ^ { 2 } .$ . It is still differentiable but more computationally efficient.
+
+Evaluation Metrics for representations and generative models. In the remainder of the paper, we will frequently need to compare a given set (distribution) of points clouds, whether reconstructed or synthesized, to its ground truth counterpart. For example, one might want to assess the quality of a representation model, in terms of how well it matches the training set or a held-out test set. Such a comparison might be done to evaluate the faithfulness and/or diversity of a generative model, and measure potential mode-collapse. To measure how well a point-cloud distribution $A$ matches a ground truth distribution $G$ , we use the following metrics:
+
+Coverage. For each point-cloud in $A$ we find its closest neighbor in $G$ ; closeness can be computed using either CD or EMD, thus yielding two different metrics, COV-CD and COV-EMD. Coverage is measured as the fraction of the point-clouds in $G$ that were matched to point-clouds in $A$ . A high coverage score typically indicates that most of $G$ is roughly represented within $A$ .
+
+Minimum Matching Distance (MMD). Coverage is not representative of the fidelity of $A$ with respect to $G$ as matched elements need not be close. To capture fidelity, we match every point cloud of $G$ to the one in $A$ with the minimum distance (MMD) and report the average of distances in the matching. Either of the structural distances can be used, yielding MMD-CD and MMD-EMD. MMD measures the distances in the pairwise matchings, so it correlates with how realistic the elements of $A$ are.
+
+Jensen-Shannon Divergence (JSD). The Jensen-Shannon divergence between marginal distributions defined over the euclidean 3D space. Assuming point cloud data that are axis-aligned and a canonical voxel grid in the ambient space; one can measure the degree to which point clouds of $A$ tend to occupy similar locations as those of $B$ . To that end, we count the number of points lying within each voxel across all point clouds of $A$ , and correspondingly for $B$ and report the JSD between the obtained empirical distributions.
+
+# 3 REPRESENTATION AND GENERATIVE MODELS
+
+In this section we describe the architectures of our representation and generative models for point clouds, starting from our autoencoder design. Later, we introduce a GAN architecture tailored to point-cloud data, followed by a more efficient pipeline that first learns an AE and the trains a much smaller GAN in the learned latent space, and a simpler generative model based on Gaussian Mixtures.
+
+# 3.1 LEARNING REPRESENTATIONS OF 3D POINT CLOUDS
+
+The input to our AE network is a point cloud with 2048 points $( 2 0 4 8 \times 3$ matrix), representing a 3D shape. The encoder architecture follows the principle of Qi et al. (2016a): 1-D convolutional layers with kernel size 1 and increasing number of features, ending with a "symmetric" function. This approach encodes every point independently and uses a permutation-invariant (symmetric) function to make a joint representation. In our implementation we use 5 1-D conv layers, each followed by a ReLU and a batch-norm layer. The output of the last 1-D conv layer is passed to a feature-wise maximum to produce a $k$ -dimensional vector which is the basis for our latent space. The decoder transforms the latent vector with 3 fully connected layers, the first two having ReLUs, to produce a $2 0 4 8 \times 3$ output. For a permutation invariant objective, we explore both the efficient EMD-distance approximation (Fan et al., 2016) and the Chamfer-Distance as our structural losses; this yields two distinct AE models, referred to as AE-EMD and AE-CD (detailed architecture parameters can be found in Appendix A). To determine an appropriate size for the latent-space, we constructed 8 (otherwise architecturally identical) AEs with bottleneck sizes $k \in \{ 4 , 8 \ldots , 5 1 2 \}$ and trained them with point-clouds of a single object class, under the two losses. We repeated this procedure with pseudo-random weight initializations three times (see appendix, Fig. 15) and found that $k = 1 2 8$ had the best generalization error on the test data, while achieving minimal reconstruction error on the train split.
+
+# 3.2 GENERATIVE MODELS FOR POINT CLOUDS
+
+Raw point cloud GAN (r-GAN). The first version of our generative model operates directly on the raw $2 0 4 8 \times 3$ point set input – to the best of our knowledge this work is the first to present a GAN for point clouds. The architecture of the discriminator is identical to the AE (modulo the filter-sizes and the number of neurons), without any batch-norm and with leaky ReLUs (Maas et al., 2013) instead or ReLUs. The output of the last fully connected layer is fed into a sigmoid neuron. The generator takes as input a 128-dimensional noise vector and maps it to a $2 0 4 8 \times 3$ output by 5 FC-ReLU layers.
+
+Latent-space GAN (l-GAN). In our l-GAN, instead of operating on the raw point cloud input, we pass the data through our pre-trained autoencoder, trained separately for each object class with the EMD (or Chamfer) loss function. Both the generator and the discriminator of the GAN then operate on the 128- dimensional bottleneck variable of the AE. Finally, once the GAN training is over, the output of the generator is decoded to a point cloud via the AE decoder. The architecture for the l-GAN is significantly simpler than the one of the r-GAN. We found that very shallow designs for both the generator and discriminator (in our case, one hidden FC layer for the generator and two FC for the discriminator) are sufficient to produce realistic results.
+
+![](images/6ae4cb0bcbbb997ba63f322b6f59a604049e849237cbb5838072aada2abd3ec5.jpg)
+
+Gaussian Mixture Model. In addition to the l-GANs, we also train a family of Gaussian Mixture Models (GMMs) on the latent spaces learned by our AEs. We fitted GMMs with varying numbers of Gaussian components, and experimented with both diagonal and full covariance matrices for the Gaussians. The GMMs can be turned into point-cloud generators by first sampling the latent-space from the GMM distribution and accordingly using the AE’s decoder, similarly to the l-GANs.
+
+![](images/0e7ad4d9ebaa94fae2c3e060380294246ffb3e3580ff1a66b63702df844247b2.jpg)  
+Figure 1: Reconstructions of unseen shapes from the test split of the input data. The leftmost image of each pair shows the ground truth shape, the rightmost the shape produced after encoding and decoding using our class-specific AEs.
+
+# 4 EVALUATION AND RESULTS
+
+Our source for shapes is the ShapeNet repository (Chang et al., 2015); we pre-center all shapes into a sphere of diameter 1. Unless otherwise stated, we train specific per-class models, and split the models in each class into training/testing/validation set using a $8 5 \% - 5 \% - 1 0 \%$ split.
+
+# 4.1 EVALUATING THE LATENT REPRESENTATION
+
+Classification. A common technique for evaluating the quality of unsupervised representation learning algorithms is to apply them as feature extractors on supervised datasets and evaluate the performance of linear models fitted on top of these features. We use this technique to evaluate the performance of the latent “features” computed by our AE. For this experiment to be meaningful, the AE was trained across all different shape categories: we used 57,000 models from ShapeNet from 55 categories of man-made objects. Exclusively for this experiment, we used a bigger bottleneck of 512, increased the number of neurons and applied batch-norm to the decoder as well. To obtain features for an input 3D shape, we feed forward to the network its point-cloud and extract the 512-dimensional bottleneck layer vector. This feature is then processed by a linear classification SVM trained on the de-facto 3D classification benchmark of ModelNet (Wu et al., 2015). Table 1 shows comparative results. Note that previous state of the art (Wu et al., 2016) uses several layers of a GAN to derive a 7168-long feature; our 512-dimensional feature is more intuitive and parsimonious.
+
+<table><tr><td>Dataset</td><td>SPH[1]</td><td>LFD[2]</td><td>T-L-Net[3]</td><td>VConv-DAE[4]</td><td>3D-GAN[5]</td><td>ours - EMD</td><td>ours - CD</td></tr><tr><td>MN10</td><td>79.8%</td><td>79.9%</td><td>-</td><td>80.5%</td><td>91.0%</td><td>95.4%</td><td>95.4%</td></tr><tr><td>MN40</td><td>68.2%</td><td>75.5%</td><td>74.4%</td><td>75.5%</td><td>83.3%</td><td>84.0%</td><td>84.5%</td></tr></table>
+
+Table 1: Classification performance on ModelNet40 and ModelNet10. All methods train a linear SVM with features derived in an unsupervised manner. Comparing to [1] Kazhdan et al. (2003), [2] Chen et al. (2003), [3] Girdhar et al. (2016a), [4] Sharma et al. (2016), [5] Wu et al. (2016).
+
+The decoupling of latent representation from generation allows flexibly choosing the AE loss, which can effect the learned feature. On ModelNet10, which includes primarily larger objects and fewer categories than ModelNet40, the EMD and CD losses perform equivalently. On the other hand, when the variation within the collection increases, CD produces better results. This is perhaps due to its more local and less smooth nature, which allows it to understand rough edges and some high frequency geometric details. Finally, note that since our AEs were not trained on ModelNet, this experiment also demonstrates the domain-robustness of our learned features.
+
+Qualitative Evaluation. To visually assess the quality of the learned representation, we show some reconstruction results in Fig. 1. Here, we use our AEs to encode samples from the test split of the ground truth dataset (the leftmost of each pair of images) and then decode them and compare them visually to the input (the rightmost image). These results show the ability of our learned representation to generalize to unseen shapes. In addition to reconstruction, our learned latent representation enables a number of interesting shape editing applications, including shape interpolations (Fig. 2), part editing and shape analogies. More results are showcased in Appendix F.
+
+Generalization Ability. Our AEs are able to reconstruct unseen shapes; this is highlighted not only in the results of Figure 1, but also in quantitative measurements of the fidelity and coverage of the reconstructed ground truth datasets (see appendix, Table 6) and by the comparable reconstruction quality on the training vs. test splits (see appendix, Figure 15).
+
+![](images/e34f0a4b5b8e877a2f37be134ea7c95ce5ae9860ec8d5f5669f011c838b5bb6d.jpg)  
+Figure 2: Interpolating between different point clouds, using our latent space representation.
+
+# 4.2 EVALUATING THE GENERATIVE MODELS
+
+We train and compare a total of five generative models on the data distribution of point-clouds of the chair category. We begin by establishing the two AEs with the 128-dimensional bottleneck, trained with the CD or EMD loss respectively – referred to as AE-CD and AE-EMD. Both AEs were stopped at the epoch at which the average reconstruction error with respect to our validation dataset was minimized. We train an l-GAN in each of the AE-CD and AE-EMD latent spaces. In the space associated only with the AE-EMD we train a further two models: an identical (architecture-wise) l-GAN that utilizes the Wasserstein objective with gradient-penalty (Gulrajani et al., 2017), and a family of GMMs. Lastly, we also train an r-GAN directly on the point cloud data.
+
+Model Selection. All GANs are trained for maximally 2000 epochs; for each GAN, we select one of its training epochs to obtain the “final” model, based on how well the synthetic results match the ground-truth distribution. Specifically, at a given epoch, we use the GAN to generate a set of synthetic point clouds, and measure the distance between this set and the validation set (Section 2). We avoid measuring this distance using MMD-EMD, given the high computational cost of EMD. Instead, we use either the JSD or MMD-CD metrics to compare the synthetic dataset to the validation dataset. To further reduce the computational cost of model selection, we only check every 100 epochs (50 for r-GAN). The epochs at which the various models were selected using the JSD criterion are shown in Table 3. Using the same criterion, we also select the number and covariance type of Gaussian components for the GMM, and obtain the optimal value of 32 components. GMMs performed much better with full (as opposed to diagonal) covariance matrices, suggesting strong correlations between the latent dimensions (see appendix Fig. 17). When using MMD-CD as the selection criterion, we obtain models of similar quality and at similar stopping epochs (see appendix, Table 13); the optimal number of Gaussians in this case was 40.
+
+Table 2: Evaluating 5 generators on train-split of chair dataset on epochs/models selected via minimal JSD on the validation-split. We also compare against the volumetric approach of Wu et al. (2016). Note that the average classification score attained by the ground-truth point clouds was $8 4 . 7 \%$ .   
+
+<table><tr><td>Metric</td><td>r-GAN</td><td>Wu et al. (2016)</td><td>1-GAN (AE-CD)</td><td>1-GAN (AE-EMD)</td><td>1-WGAN (AE-EMD)</td><td>GMM (AE-EMD)</td></tr><tr><td>JSD</td><td>0.1660</td><td>0.1705</td><td>0.0372</td><td>0.0188</td><td>0.0077</td><td>0.0048</td></tr><tr><td>Classification</td><td>84.10</td><td>87.00</td><td>96.10</td><td>94.53</td><td>89.35</td><td>87.40</td></tr><tr><td>MMD-CD</td><td>0.0017</td><td>0.0042</td><td>0.0015</td><td>0.0018</td><td>0.0015</td><td>0.0014</td></tr></table>
+
+Quantitative Evaluation. Upon selection of the models, we compare them with respect to their capacity to generate synthetic samples. In two different sets of experiments, we measure how well the distribution of the generated samples resembles both the train and test splits of the ground truth distribution, by using our models to generate a set of synthetic point clouds and employing the metrics from Section 2 to compare against the train or test set distributions respectively. The train split results are reported in Table 2. We also show the average classification probability for those samples being recognized as a chair using the PointNet classifier (Qi et al., 2016a), which is state of the art for classifying point clouds. A similar experiment is ran to measure how well the synthetic samples match the test split dataset; here, we repeat the experiment with three pseudo-random seeds and report the average measurements in Table 3, for various comparison metrics. Perhaps surprisingly, training a simple Gaussian mixture model in the latent space of the EMD-based AE yields the best results in terms of both fidelity and coverage. Furthermore, GMMs are particularly easy to train. Additionally, the achieved fidelity and coverage are very close to the reconstruction baseline, namely, the lower bounds for the JSD and MMD achieved by the AE on which the generative models operate (see appendix, Table 6). For example, the AE-EMD achieved an MMD-EMD of 0.05 with respect to the ground truth training data , which is comparable with the MMD-EMD value of 0.06 achieved by the GMMs with respect to the test data. Finally, by comparing Table 2 and Table 3 we can again establish the generalization ability of our models, since their performance for the training vs. testing splits is comparable. This is highlighted in more detail in Fig. 16 in the appendix.
+
+Table 3: Evaluating 5 generators on test-split of chair dataset on epochs/models selected via minimal JSD on the validation-split. The reported scores are averages of 3 pseudo-random repetitions. GMM32-F stands for a GMM with 32 Gaussian components with full covariances.   
+
+<table><tr><td>Method</td><td>Epoch</td><td>JSD</td><td>MMD-CD</td><td>MMD-EMD</td><td>COV-EMD</td><td>COV-CD</td></tr><tr><td>r-GAN</td><td>1700</td><td>0.1764</td><td>0.0020</td><td>0.1230</td><td>19.0</td><td>52.3</td></tr><tr><td>1-GAN (AE-CD)</td><td>300</td><td>0.0486</td><td>0.0020</td><td>0.0796</td><td>32.2</td><td>59.4</td></tr><tr><td>1-GAN (AE-EMD)</td><td>100</td><td>0.0308</td><td>0.0023</td><td>0.0697</td><td>57.1</td><td>59.3</td></tr><tr><td>1-WGAN (AE-EMD)</td><td>1800</td><td>0.0227</td><td>0.0019</td><td>0.0660</td><td>66.9</td><td>67.6</td></tr><tr><td>GMM-32-F (AE-EMD)</td><td>-</td><td>0.0202</td><td>0.0018</td><td>0.0651</td><td>67.4</td><td>68.9</td></tr></table>
+
+Note: The number of synthetic point clouds we generate for the train split experiment is equal to the size of the train dataset. For the test split experiment, as well as for the validation split comparisons done for model selection, we generate synthetic datasets that are three times bigger than the ground truth dataset (the test resp. validation set); this is possible due to the relatively small size of the test resp. validation sets, and helps reduce sampling bias. This is only necessary when measuring MMD or Coverage statistics.
+
+Fidelity of metrics. In Table 3 we note that the MMD-CD distance to the test set appears to be relatively small for the r-GANs. This seeming advantage of the r-GANs is counter to what a qualitative inspection of the results yields. We attribute this effect to the inadequacy of the chamfer distance to distinguish pathological cases. Some examples of such behaviour are showcased in Fig. 3. We show two triplets of images: in each triplet, an r-GAN and an l-GAN is used to generate a synthetic set of point clouds; the left triplet shows an l-GAN on the AE-CD and the right an l-GAN on the AE-EMD. For a given ground truth point cloud from the test set (leftmost image of each triplet), we find its nearest neighbor in each synthetic set under the chamfer distance - the middle image in each triplet shows the nearest neighbor in the synthetic results of the r-GAN and the right most image the nearest neighbor in the l-GAN set. We report the distances between these nearest neighbors and the ground truth using both CD and EMD (in-image numbers). Note that the CD values miss the fact that the r-GAN results are visibly of lesser quality. The underlying reason appears to be that r-GANs tend to generate clouds with many points concentrated in the areas that are most likely to be occupied in the underlying shape class (e.g. the seat of chairs in the figure). This implies that one of the two terms in the CD –namely, the one going from the synthetic point cloud to the ground truth– is likely to be very small for r-GAN results. The “blindness” of the CD metric to only partial matches between shapes has the additional interesting side-effect that the CD-based coverage metric is consistently bigger than that reported by EMD, as noted in Table 3. Instead, the EMD distance promotes a one-to-one mapping and thus correlates more strongly to visual quality; this means that it heavily penalizes the r-GAN result both in terms of MMD and coverage.
+
+Training trends. We performed extensive measurements during training of our models, to understand their behavior during training, as shown in Fig. 4. On the left, we plot the JSD distance between the ground truth test set and synthetic datasets generated by the various models at various epochs of training. On the right, we also plot the EMD-based MMD and Coverage between the same two sets, where larger marker symbols denote a higher epoch. In general, r-GAN struggles to provide good coverage of the test set no matter the metric used; which alludes to the well-established fact that end-to-end GANs are generally difficult to train. The l-GAN (AE-CD) performs better in terms of fidelity with much fewer epochs as measured by JSD/MMD-EMD, but its coverage remains low. We attribute this to the CD promoting unnatural topologies – cf. Fig. 3 that visually shows this phenomenon. Switching to an EMD-based AE for the representation and otherwise using the same latent GAN architecture (l-GAN, AE-EMD), yields a dramatic improvement in coverage and fidelity. Both l-GANs though suffer from the known issue of mode collapse: Half-way through training, first coverage starts dropping with fidelity still at good levels, which implies that they are overfitting a small subset of the data. Later on, this is followed by a more catastrophic collapse, with coverage dropping as low as $0 . 5 \%$ . Switching to a latent WGAN largely eliminates this collapse, as expected.
+
+![](images/ebdfaae66ff2c99e16e8826a2b8d5c3c0524b5c979b6c34868df8bef94e6de76.jpg)  
+Figure 3: The CD distance is less faithful than EMD to visual quality of synthetic results; in this case it favors r-GAN results, due to the presence of high-density areas in the synthesized point sets.
+
+![](images/ab7c0c780945b58d5ac15ebbe3cbe4bdf1288528a3b9e2579efbe474d112da05.jpg)  
+Figure 4: Training trends for the various generative models, in terms of coverage / fidelity to the ground truth test dataset. On the right, the curve markers indicate epochs 1, 10, 100, 200, 400, 1000, 1500, 2000, with larger symbols denoting higher epochs. See text for more details.
+
+Comparisons to voxel-based methods. To the best of our knowledge we are the first to propose GANs on point-cloud data. To find out how our models fare against other 3D generative methods, in Table 2 and Fig. 4 we compare to a recent voxel-grid based approach (Wu et al., 2016) in terms of the JSD on the training set of the chair category - other shape categories can be found in the appendix (Table 10). We convert their voxel grid output into a point-set with 2048 points by performing farthest-point-sampling on the isosurface of the grid values. Per the authors’ suggestion, we used an isovalue parameter of 0.1 and isolated the largest connected component from the isosurface. Since Wu et al. (2016) do not use any train/test split, we perform 5 rounds of sampling 1k synthetic results from their models and report the best values of the respective evaluation metrics. The r-GAN mildly outperforms Wu et al. (2016) in terms of its diversity (as measured by JSD/MMD), while also creating realistic-looking results, as shown by the classification score. The l-GANs perform even better, both in terms of classification and diversity, with less training epochs. Note also that the training time for one epoch of the l-GAN is more than an order of magnitude smaller than for the r-GAN, due to its much smaller architecture and dimensionality. For fairness, we acknowledge that since Wu et al. (2016) operates on voxel grids, it is not necessarily on equal standing when it comes to generating point clouds.
+
+Qualitative evaluation. In Fig. 5, we show some synthetic results produced by our l-GANs (top row) and the 32-component GMM, both trained on the AE-EMD latent space. We notice high quality results from either model - this highlights the strength of our learned representation, which makes it possible for the simple GMM model to perform well. The shapes (after decoding) corresponding to the 32 means of the Gaussian components can be found in the appendix (Fig. 18), as well as results using the r-GAN (see appendix, Fig. 14). The l-GAN produces crisper and less noisy results than the r-GAN, demonstrating an advantage of using a good structural loss on the decoupled, pre-trained AE.
+
+![](images/77cbb375fb987c359c79a95aed19ec012462d9ef4f949763242a2d289d59bc09.jpg)  
+Figure 5: Synthetic point clouds generated by samples produced with l-GAN (top) and 32-component GMM (bottom), both trained on the latent space of an AE using the EMD loss.
+
+Extensions to multiple classes We have performed experiments with an AE-EMD trained on a mixed set containing point clouds from 5 categories (chair, airplane, car, table, sofa). The training and testing datasets for this AE were constructed by randomly picking and adding models from each class; 2K models per class for the training set, 200 models for testing and 100 for validation. The multi-class AE has the same bottleneck size of 128 and was trained for 1000 epochs. We compare against the class-specific AEs with the 85-5-10 train-val-test-split, which we trained for 500 epochs. The precise AE model in all cases was selected based on the minimal reconstruction loss on the the respective validation set. On top of all six AEs, we train six l-WGANs for 2K epochs, and evaluate their fidelity/coverage using the MMD-CD between the respective testing sets and a synthesized dataset of $3 \mathbf { x }$ the size, as above. It turns out that the l-WGANs based on the multi-class AE perform similarly to the dedicated class-specifically trained ones (Table 4). A qualitative comparison (Fig. 6) also reveals that by using a multi-class AE-EMD we do not sacrifice much in terms of visual quality compared to the dedicated AEs.
+
+![](images/e873322fc85dcc8430e4f0ff1c6855153aceab89ecfb1f1fe6acdc2310f534e7.jpg)  
+Figure 6: Synthetic point clouds generated by samples produced with l-WGANs trained in the latent space of an AE-EMD trained on a multi-class dataset.
+
+<table><tr><td></td><td>airplane</td><td>car</td><td>chair</td><td>sofa</td><td>table</td><td>average</td><td>multi-class</td></tr><tr><td>train</td><td>0.0004</td><td>0.0006</td><td>0.0015</td><td>0.0011</td><td>0.0013</td><td>0.0010</td><td>0.0011</td></tr><tr><td>test</td><td>0.0006</td><td>0.0007</td><td>0.0019</td><td>0.0014</td><td>0.0017</td><td>0.0013</td><td>0.0014</td></tr></table>
+
+Table 4: MMD-CD measurements for l-WGANs stopped at the two-thousand epoch and trained on the latent spaces of dedicated (left 5 columns) and multi-class EMD-AEs (right column). The “average” measurement is computed as the weighted average of the per-class values, using the number of train resp. test examples for each class as weights.
+
+Limitations. Fig. 7 shows some failure cases of our models. Chairs with rare geometries (left two images) are sometimes not faithfully decoded. Additionally, the AEs may miss high-frequency geometric details, e.g. a hole in the back of a chair (middle), thus altering the style of the input shape. Finally, the r-GAN often struggles to create realistic-looking shapes (right) for some shape classes – while the r-GAN chairs that are easily visually recognizable, it has a harder time on cars. Designing more robust raw-GANs for point clouds remain an interesting avenue for future work.
+
+# 5 RELATED WORK
+
+A number of recent works (Wu et al. (2016), Wang et al. (2016), Girdhar et al. (2016b), Brock et al. (2016), Maimaitimin et al. (2017), Zhu et al. (2016)) have explored generative and discriminative representations for geometry. They operate on different modalities, typically voxel grids or view-based image projections. To the best of our knowledge, our work is the first to study such representations for point clouds.
+
+![](images/346aaaf4542eb907ea84050f1fe747140891ac6595116e6a983b5142db54bd2f.jpg)  
+Figure 7: Limitations: The AEs might fail to reconstruct shapes of uncommon/overly detailed geometry (left four images). The r-GAN may synthesize noisy/unrealistic results, cf. a car (right).
+
+Training Gaussian mixture models (GMM) in the latent space of an autoencoder is closely related to VAEs (Kingma & Welling, 2013). One documented issue with VAEs is over-regularization: the regularization term associated with the prior, is often so strong that reconstruction quality suffers (Bowman et al., 2015; Sønderby et al., 2016; Kingma et al., 2016; Dilokthanakul et al., 2016). The literature contains methods that start only with a reconstruction penalty and slowly increase the weight of the regularizer. In our case, we find that fixing the AE before we train our generative models yields good results.
+
+# 6 CONCLUSION
+
+We presented a novel set of architectures for 3D point-cloud representation learning and generation. Our results show good generalization to unseen data and our representations encode meaningful semantics. In particular our generative models are able to produce faithful samples and cover most of the ground truth distribution without memorizing a few examples. Interestingly, we see that the best-performing generative model in our experiments is a GMM trained in the fixed latent space of an AE. While, this might not be a universal result, it suggests that simple classic tools should not be dismissed. A thorough investigation on the conditions under which simple latent GMMs are as powerful as adversarially trained models would be of significant interest.
+
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+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pp. 2226–2234, 2016.
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+Abhishek Sharma, Oliver Grau, and Mario Fritz. Vconv-dae: Deep volumetric shape learning without object labels. In Geometry Meets Deep Learning Workshop at European Conference on Computer Vision (ECCV-W), 2016.
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+Casper Kaae Sønderby, Tapani Raiko, Lars Maaløe, Søren Kaae Sønderby, and Ole Winther. How to train deep variational autoencoders and probabilistic ladder networks. arXiv preprint arXiv:1602.02282, 2016.
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+Hang Su, Subhransu Maji, Evangelos Kalogerakis, and Erik G. Learned-Miller. Multi-view convolutional neural networks for 3d shape recognition. In 2015 IEEE International Conference on Computer Vision, ICCV 2015, Santiago, Chile, December 7-13, 2015, pp. 945–953, 2015. doi: 10.1109/ICCV.2015.114.
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+Maxim Tatarchenko, Alexey Dosovitskiy, and Thomas Brox. Octree generating networks: Efficient convolutional architectures for high-resolution 3d outputs. arXiv preprint arXiv:1703.09438, 2017.
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+Jiajun Wu, Chengkai Zhang, Tianfan Xue, Bill Freeman, and Josh Tenenbaum. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems 29, pp. 82–90. Curran Associates, Inc., 2016.
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+Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2015, Boston, MA, USA, June 7-12, 2015, pp. 1912–1920, 2015. doi: 10.1109/CVPR.2015.7298801.
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+Li Yi, Vladimir G. Kim, Duygu Ceylan, I-Chao Shen, Mengyan Yan, Hao Su, Cewu Lu, Qixing Huang, Alla Sheffer, and Leonidas J. Guibas. A scalable active framework for region annotation in 3d shape collections. ACM Trans. Graph., (6):210:1–210:12, 2016a.
+
+Li Yi, Hao Su, Xingwen Guo, and Leonidas J. Guibas. Syncspeccnn: Synchronized spectral CNN for 3d shape segmentation. CoRR, 2016b.
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+Zhuotun Zhu, Xinggang Wang, Song Bai, Cong Yao, and Xiang Bai. Deep learning representation using autoencoder for 3d shape retrieval. 2016.
+
+# A AE DETAILS
+
+The encoding layers of our AEs were implemented as 1D-convolutions with ReLUs, with kernel size of 1 and stride of 1, i.e. treating each 3D point independently. Their decoding layers, were MLPs built with FC-ReLUs. We used Adam (Kingma & Ba, 2014) with initial learning rate of 0.0005, $\beta _ { 1 }$ of 0.9 and a batch size of 50 to train all AEs.
+
+# A.1 AE USED FOR SVM-BASED EXPERIMENTS
+
+For the AE mentioned in the beginning of Section 4.1 and which was used for the SVM-related experiments, we used an encoder with 128, 128, 256 and 512 filters in each of its layers and a decoder with 1024, 2048, $2 0 4 8 \times 3$ neurons, respectively. Batch normalization was used between every layer. We also used online data augmentation by applying random rotations along the gravity- $\mathbf { \tau } ( \mathbf { Z } )$ -axis to the input point-clouds of each batch. We trained this AE for 1000 epochs with the CD loss and for 1100 with the EMD.
+
+# A.2 ALL OTHER AES
+
+For all other AEs, the encoder had 64, 128, 128, 256 and $k$ filters at each layer, with $k$ being the bottle-neck size. The decoder was comprised by 3 FC-ReLU layers with 256, 256, $2 0 4 8 \times 3$ neurons each. We trained these AEs for a maximum of 500 epochs when using single class data and 1000 epochs for the sole experiment involving 5 shape classes (end of Section 4.2)
+
+Remark. Different AE setups (denoising/regularised) brought no noticeable advantage over our “vanilla” architecture. Adding drop-out layers resulted in worse reconstructions and using batch-norm on the encoder only, sped up training and gave us slightly better generalization error when the AE was trained with single-class data.
+
+# B R-GAN DETAILS
+
+The discriminator’s first 5 layers are 1D-convolutions with stride/kernel of size 1 and $\{ 6 4 , 1 2 8 , 2 5 6 , 2 5 6 , 5 1 2 \}$ filters each; interleaved with leaky-ReLU. They are followed by a featurewise max-pool. The last 2 FC-leaky-ReLU layers have $\{ 1 2 8 , 6 4 \}$ , neurons each and they lead to single sigmoid neuron. We used 0.2 units of leak.
+
+The generator consists of 5 FC-ReLU layers with $\{ 6 4 , 1 2 8 , 5 1 2 , 1 0 2 4 , 2 0 4 8 \times 3 \}$ neurons each. We trained r-GAN with Adam with an initial learning rate of 0.0001, and $b e t a _ { 1 }$ of 0.5 in batches of size 50. The noise vector was drawn by a spherical Gaussian of 128 dimensions with zero mean and 0.2 units of standard deviation.
+
+# C L-GAN DETAILS
+
+The discriminator consists of 2 FC-ReLU layers with $\{ 2 5 6 , 5 1 2 \}$ neurons each and a final FC layer with a single sigmoid neuron. The generator consists of 2 FC-ReLUs with $\{ 1 2 8 , k = 1 2 8 \}$ neurons each. When used the l-Wasserstein-GAN, we used a gradient penalty regularizer $\lambda = 1 0$ and trained the critic for 5 iterations per one iteration of the generator. The training parameters (learning rate, batch size) and the generator’s noise distribution were the same as those used for the r-GAN.
+
+# D SVM PARAMETERS FOR AUTOENCODER EVALUATION
+
+For the classification experiments of Section 4.1 we used a one-versus-rest linear SVM classifier with an $l _ { 2 }$ norm penalty and balanced class weights. The exact optimization parameters can be found in Table 5.
+
+<table><tr><td rowspan=2 colspan=1>Structural Loss</td><td rowspan=1 colspan=3>ModelNet40</td><td rowspan=1 colspan=3>ModelNet10</td></tr><tr><td rowspan=1 colspan=1>C-penalty</td><td rowspan=1 colspan=1>intercept</td><td rowspan=1 colspan=1>loss</td><td rowspan=1 colspan=1>C-penalty</td><td rowspan=1 colspan=1>intercept</td><td rowspan=1 colspan=1>loss</td></tr><tr><td rowspan=1 colspan=1>EMD</td><td rowspan=1 colspan=1>0.09</td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>hinge</td><td rowspan=1 colspan=1>0.02</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>squared-hinge</td></tr><tr><td rowspan=1 colspan=1>CD</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>0.4</td><td rowspan=1 colspan=1>squared-hinge</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.2</td><td rowspan=1 colspan=1>squared-hinge</td></tr></table>
+
+Table 5: Training parameters of SVMs used in each dataset with each structural loss of the AE. $C$ -penalty: term controlling the trade-off between the size of the learned margin and the misclassification’s rate; intercept: extra dimension appended on the input features to center them; loss: svm’s optimization loss function.
+
+# E AE RECONSTRUCTION QUALITY
+
+Table 6 shows the reconstruction quality of the two AEs (CD- and EMD-based), in terms of the JSD of the reconstructed datasets with respect to their ground truth counterparts. Note that, in general, the quality of reconstruction is comparable between the training and test datasets, indicating that the AEs are indeed able to generalize.
+
+<table><tr><td>Method</td><td>JSD (Tr)</td><td>JSD (Te)</td><td>MMD-CD (Tr)</td><td>MMD-EMD(Tr)</td></tr><tr><td>AE-CD</td><td>0.0216</td><td>0.0243</td><td>0.0004</td><td>0.0753</td></tr><tr><td>AE-EMD</td><td>0.0028</td><td>0.0067</td><td>0.0005</td><td>0.0527</td></tr></table>
+
+Table 6: Effect of loss-type for AE reconstructions. The EMD loss gives rise to reconstructions with significantly better JSD compared to Chamfer. MMD-measurements favor the AE that was trained with the same loss under which the MMD measurement is computed. (Tr: Train split, Te: Test split)
+
+# F APPLICATIONS OF THE LATENT SPACE REPRESENTATION
+
+For shape editing applications, we use the embedding we learned with the AE-EMD trained across all 55 object classes, not separately per-category. This showcases its ability to encode features for different shapes, and enables interesting applications involving different kinds of shapes.
+
+![](images/09f6741119e0afd67c068bdc4f104b57f03cced0ca171a499ef6717360095fc4.jpg)  
+Figure 8: Editing parts in point clouds using vector arithmetic on the AE latent space. Left to right: tuning the appearance of cars towards the shape of convertibles, adding armrests to chairs, removing handle from mug.
+
+Editing shape parts. We use the shape annotations of $\mathrm { Y i }$ et al.Yi et al. (2016a) as guidance to modify shapes. As an example, assume that a given object category (e.g. chairs) can be further subdivided into two sub-categories $\mathcal { A }$ and $\boldsymbol { B }$ : every object $A \in { \mathcal { A } }$ possesses a certain structural property (e.g. has armrests, is four-legged, etc.) and objects $B \in B$ do not. Using our latent representation we can model this structural difference between the two sub-categories by the difference between their average latent representations $\mathbf { x } _ { B } - \mathbf { x } _ { A }$ , where $\mathbf { x } _ { \mathcal { A } } = \sum _ { A \in \mathcal { A } } \mathbf { x } _ { A } , \mathbf { x } _ { \mathcal { B } } = \sum _ { B \in \mathcal { B } } \mathbf { x } _ { B }$ . Then, given an object $A \ \in \ A$ , we can change its property by transforming its latent representation: $x _ { A ^ { \prime } } = x _ { A } + \mathbf { x } _ { B } - \mathbf { x } _ { A }$ , and decode $\mathbf { x } _ { \ r { A ^ { \prime } } }$ to obtain $A ^ { \prime } \in B$ . This process is shown in Figure 8. Note that the height of chairs with armrests is on average $13 \%$ smaller than the chairs without, which is reflected in the output of this process.
+
+![](images/25b3143932a765f892815d25dfc3a02f7968aca79b9ff43dd2af53ab75b07e23.jpg)  
+Figure 9: Interpolating between different point clouds, using our latent space representation. Note the interpolation between structurally and topologically different shapes.
+
+Interpolating shapes. By linearly interpolating between the latent representations of two shapes and decoding the result we obtain intermediate variants between the two shapes. This produces a “morph-like” sequence with the two shapes at its end points (Fig. 2, 9). Our latent representation is powerful enough to support removing and merging shape parts, which enables morphing between shapes of significantly different appearance. Our cross-category latent representation enables morphing between shapes of different classes, cfg. the second row for an interpolation between a bench and a sofa.
+
+Shape analogies. Another demonstration of the euclidean nature of the latent space is demonstrated by finding “analogous” shapes by a combination of linear manipulations and euclidean nearestneighbor searching. Concretely, we find the difference vector between $A$ and $A ^ { \prime }$ , we add it to shape $B$ and search in the latent space for the nearest-neighbor of that result, which yields shape $B ^ { \prime }$ . We demonstrate the finding in Fig. 10 with images taken from the meshes used to derive the underlying point-clouds to help the visualization. Finding shape analogies has been of interest recently in the geometry processing community Rustamov et al. (2013).
+
+# G MORE COMPARISONS WITH VOXEL-BASED METHODS
+
+In this section we include preliminary results of point-cloud generators that work in conjunction with voxel-based AEs. We followed the same strategy as we did with the l-GAN but instead of using a point-cloud autoencoder we learned the latent space by an AE that works with occupancy grids of 3D shapes. For generation we used a full-GMM model with 32 centers, which was established as our best model in our previous experiments. We tried two different grid resolutions: $3 2 ^ { 3 }$ and $6 4 ^ { 3 }$ on ShapeNet’s chair class. To compare with our established “pure” point-cloud generators we converted the generated voxel-grids into 2048 points by first extracting a mesh from the grids using isosurfacing and then sampling points on the mesh using uniform area-wise sampling. We also compare against Wu et al.’s Wu et al. (2016) voxel-based GANs, which represent the “raw” GAN architecture for the case of voxel grids. For quantitative results and more details see Table 7.
+
+Discussion. First, we see that the latent AE-based GMM models outperform Wu et al.’s “raw” GAN architecture by a big margin. In terms of coverage, using the latent representation (voxel GMM)
+
+![](images/4a44e0949f48894409ac1ad4c4fb1f35b0afd0fc0822836a4dab68cf39e95a1f.jpg)  
+Figure 10: Shape Analogies using our learned representation. Shape $B ^ { \prime }$ relates to $B$ in the same way that shape $A ^ { \prime }$ relates to $A$ .
+
+provides a vast improvement over the “raw” voxel GAN architecture (Wu et al.). This indicates an advantage of using latent representations for generation in the voxel modality as well. Second, we note that the performance of the $6 4 ^ { 3 }$ voxel-based GMM is comparable to the one operating at $3 2 ^ { 3 }$ resolution. This suggests that the main factor affecting fidelity of the results is not the lack of high-frequency details in the ground-truth data. Third, our point-cloud-based models outperform voxel-based models in terms of the fidelity between their output and the ground truth, as measured by the MMD.
+
+The bigger coverage boost of the voxel-based latent-space models compared to the MMD, is likely due to the way the coverage metric is computed: one matches all generated shapes against the ground truth, regardless of the quality of the generated shape. Voxel-based models frequently produce shapes with missing components (see Fig. 11); even extremely partial instances (“outliers”) will be matched (however poorly) to an arbitrary ground truth model. This effect likely artificially increases the coverage. The histogram in Figure 12 shows distances between the GMM-generated samples and their closest matches in the ground truth. The heavier “tail” for the voxel-based method indicates the presence of such poor quality matchings. Qualitative inspection of the ground truth models that were covered by the voxel-based output but not by the point-cloud output confirmed that the covering came mostly from very poor quality partial shapes.
+
+<table><tr><td>Metric</td><td>“raw” 64³-voxel GAN Wu et al. (2016)</td><td>latent 323-voxel GMM-32</td><td>latent 643-voxel GMM-32</td><td>latent 1283-octtree GMM-32</td><td>latent point-cloud GMM-32</td></tr><tr><td>MMD-CD</td><td>0.0046</td><td>0.0025</td><td>0.0025</td><td>0.0024</td><td>0.0018</td></tr><tr><td>MMD-EMD</td><td>0.0915</td><td>0.0742</td><td>0.0729</td><td>0.0750</td><td>0.0651</td></tr><tr><td>COV-CD</td><td>19.6</td><td>63.5</td><td>60.3</td><td>60.9</td><td>68.9</td></tr><tr><td>COV-EMD</td><td>22.4</td><td>66.6</td><td>64.8</td><td>64.7</td><td>67.4</td></tr></table>
+
+Table 7: MMD and Coverage metrics evaluated on the output of voxel-based methods at resolutions $3 2 ^ { 3 }$ and $6 4 ^ { 3 }$ and (oct-tree based) $1 2 8 ^ { 3 }$ , matched against the chair test set, using the same protocol as in Table 3 of the main paper. Our volumetric models use GMMs with full covariances and 32 centers and 64 or 256-dimensional latent codes (for the $3 2 ^ { 3 }$ , $6 4 ^ { 3 }$ and $1 2 8 ^ { 3 }$ respectively). For the mesh conversion we used the marching cubes algorithm ((Lewiner et al., 2003)) with an iso-surface value of 0.5. The rightmost column shows the results with our point-cloud based GMM.
+
+# G.1 VOXEL AE DETAILS
+
+Our voxel-based AEs are fully-convolutional with the encoders consisting of 3D-Conv-ReLU layers and the decoders of 3D-Conv-Relu-transpose layers. Below, we list the parameters of consecutive layers, listed left-to-right. The layer parameters are denoted in the following manner: (number of filters, filter size). Each conv/conv-tranpose has a stride of 2 except the last layer of the $3 2 ^ { 3 }$ decoder which has 4. In the last layer of the decoders we do not use a non-linearity. The abbreviation "bn" stands for batch-normalization.
+
+![](images/95b3fbfa79af7fd90c9c2ef8fab2a46099dcbdf77f1ed8637e158fd311c6fd98.jpg)  
+Figure 11: Point clouds extracted from synthetic voxel-based results, after isosurfacing and point sampling. Note the missing components and appearance of noise.
+
+![](images/be854cf60d68ba1b187407236a79824207a6b6845a8a257d6f388db62aafda80.jpg)  
+Figure 12: Histograms of MMD-distances: EMD (left) and Chamfer (right), for a purely point-cloudbased generative model (GMM with 32 full-covariance components, in orange) and a voxel-based model (a latent-GAN trained on a voxel-based AE of resolution $6 4 ^ { 3 }$ , in blue). Note the larger MMD values for the voxel based approach, indicating results of lower fidelity.
+
+• $3 2 ^ { 3 }$ - model Encoder: Input $ ( 3 2 , 6 )  ( 3 2 , 6 )  \mathrm { b n }  ( 6 4 , 4 )  ( 6 4 , 2 )  \mathrm { b n }  ( 6 4 , 2 )$ Decoder: (6 $4 , 2 )  ( 3 2 , 4 )  \mathrm { b n }  ( 3 2 , 6 )  ( 1 , 8 )  \mathrm { C }$ utput   
+• $6 4 ^ { 3 }$ - model Encoder: $\mathrm { I n p u t } \to ( 3 2 , 6 ) \to ( 3 2 , 6 ) \to \mathrm { b n } \to ( 6 4 , 4 ) \to ( 6 4 , 4 ) \to \mathrm { b n } \to ( 6 4 , 2 )$ $ ( 6 4 , 2 )$ Decoder: $( 6 4 , 2 )  ( 3 2 , 4 )  \mathrm { b n }  ( 3 2 , 6 )  ( 3 2 , 6 )  \mathrm { b n }  ( 3 2 , 8 )  ( 1 , 8 )$ Output
+
+We train each AE for 100 epochs with Adam under the binary cross-entropy loss. The learning rate was 0.001, the $\beta _ { 1 } 0 . 9$ and the batch size 64. To validate our voxel AE architectures, we compared them in terms of reconstruction quality to the state-of-the-art method of Tatarchenko et al. (2017) and obtained comparable results, as demonstrated in Table 8.
+
+<table><tr><td>Voxel Resolution</td><td>32</td><td>64</td></tr><tr><td>Ours</td><td>92.7</td><td>88.4</td></tr><tr><td>(Tatarchenko et al., 2017)</td><td>93.9</td><td>90.4</td></tr></table>
+
+Table 8: Reconstruction quality statistics for our dense voxel-based AE and the one of Tatarchenko et al. (2017) for the ShapeNetCars dataset. Both approaches use a 0.5 occupancy threshold and the train-test split of Tatarchenko et al. (2017). Reconstruction quality is measured by measuring the intersection-over-union between the input and synthesized voxel grids, namely the ratio between the volume in the voxel grid that is 1 in both grids divided by the volume that is 1 in at least one grid.
+
+# H MEMORIZATION BASELINE
+
+Here we compare our GMM-generator against a model that memorizes the training data of the chair class. To do this, we either consider the entire training set or randomly sub-sample it, to create sets of different sizes. We then evaluate our metrics between these ”memorized” sets and the point-clouds of test split (see Table 9). The coverage/fidelity obtained by our generative models (last row) is slightly lower than the equivalent in size case (third row) as expected: memorizing the training set produces good coverage/fidelity with respect to the test set when they are both drawn from the same population. This speaks for the validity of our metrics. Naturally, the advantage of using a learned representation lies in learning the structure of the underlying space instead of individual samples, which enables compactly representing the data and generating novel shapes as demonstrated by our interpolations. In particular, note that while some mode collapse is present in our generative results, as indicated by the $\sim 1 0 \%$ drop in coverage, the achieved MMD of our generative models is almost identical to that of the memorization case, indicating excellent fidelity.
+
+<table><tr><td>Sample Set Size</td><td>COV-CD</td><td>MMD-CD</td><td>COV-EMD</td><td>MMD-EMD</td></tr><tr><td>Entire |Trainl</td><td>97.3</td><td>0.0013</td><td>98.2</td><td>0.0545</td></tr><tr><td>1 × ITestl</td><td>54.0</td><td>0.0023</td><td>51.9</td><td>0.0699</td></tr><tr><td>3 × ITestl</td><td>79.4</td><td>0.0018</td><td>78.6</td><td>0.0633</td></tr><tr><td>Full-GMM/32 (3 × ITestl)</td><td>68.9</td><td>0.0018</td><td>67.4</td><td>0.0651</td></tr></table>
+
+Table 9: Quantitative results of a baseline sampling/memorizing model, for different sizes of sets sampled from the training set and evaluated against the test split. The first three rows show results of a memorizing model, while the third row corresponds to our generative model. The first row shows the results of memorizing the entire training chair dataset. The second and third rows show the averages of three repetitions of the sub-sampling procedure with different random seeds.
+
+# I MORE COMPARISONS WITH WU ET AL.
+
+In Tables 10, 11, 12 we provide more comparisons with Wu et al. (2015) for the major ShapeNet classes for which the authors have made publicly available their models. In Table 10 we provide JSD-based comparisons for two of our models (see details in main paper 4.2.) In Tables 11, 12 we provide MMD/Coverage comparisons on the test split following the same protocol as in Table 3.   
+
+<table><tr><td rowspan="2">Class</td><td rowspan="2">Wu et al. (2016) train+test</td><td colspan="2">L-GAN (AE-EMD)</td><td colspan="2">Full GMM/32 (AE-EMD)</td></tr><tr><td>train</td><td>test</td><td>train</td><td>test</td></tr><tr><td>airplane</td><td>1</td><td>0.0149</td><td>0.0268</td><td>0.0065</td><td>0.0191</td></tr><tr><td>car</td><td>0.1890</td><td>0.0081</td><td>0.0109</td><td>0.0063</td><td>0.0108</td></tr><tr><td>rifle</td><td>0.2012</td><td>0.0212</td><td>0.0364</td><td>0.0092</td><td>0.0214</td></tr><tr><td>sofa</td><td>0.1812</td><td>0.0102</td><td>0.0102</td><td>0.0102</td><td>0.0101</td></tr><tr><td>table</td><td>0.2472</td><td>0.0058</td><td>0.0177</td><td>0.0035</td><td>0.0143</td></tr></table>
+
+Table 10: JSD-based comparison between Wu et al. (2016) and our generative models. Full GMM/32 stands for a GM model trained on the latent space of our AE with the EMD structural loss. Note that the l-GAN here uses the same “vanilla” adversarial objective as Wu et al. (2016).
+
+Table 11: EMD based MMD and Coverage comparison between Wu et al. (2016) and our generative model on the test split of each class. Full GMM/32 stands for a GM model trained on the latent space of our AE with the EMD structural loss. Note that Wu et al. used all models of each class for training.   
+
+<table><tr><td rowspan="2">Class</td><td colspan="2">MMD-EMD</td><td colspan="2">COV-EMD</td></tr><tr><td>Wu et al. (2016)</td><td>Full GMM/32</td><td>Wu et al. (2016)</td><td>Full GMM/32</td></tr><tr><td>airplane</td><td>■</td><td>0.0387</td><td>=</td><td>69.6</td></tr><tr><td>car</td><td>0.0591</td><td>0.0418</td><td>28.6</td><td>65.3</td></tr><tr><td>rifle</td><td>0.0512</td><td>0.0459</td><td>69.0</td><td>74.8</td></tr><tr><td>sofa</td><td>0.0773</td><td>0.0554</td><td>52.53</td><td>66.6</td></tr><tr><td>table</td><td>0.1038</td><td>0.0615</td><td>18.35</td><td>71.1</td></tr></table>
+
+Table 12: CD based MMD and Coverage comparison between Wu et al. (2016) and our generative model on the test split of each class. Full GMM/32 stands for a GM model trained on the latent space of our AE with the EMD structural loss. Note that Wu et al. used all models of each class for training.   
+
+<table><tr><td rowspan="2">Class</td><td colspan="2">MMD-CD</td><td colspan="2">COV-CD</td></tr><tr><td>Wu et al. (2016)</td><td>Full GMM/32</td><td>Wu et al. (2016)</td><td>Full GMM/32</td></tr><tr><td>airplane</td><td>1</td><td>0.0005</td><td>1</td><td>71.1</td></tr><tr><td>car</td><td>0.0015</td><td>0.0007</td><td>22.9</td><td>63.0</td></tr><tr><td>rifle</td><td>0.0008</td><td>0.0005</td><td>56.7</td><td>71.7</td></tr><tr><td>sofa</td><td>0.0027</td><td>0.0013</td><td>42.40</td><td>75.5</td></tr><tr><td>table</td><td>0.0058</td><td>0.0016</td><td>16.7</td><td>71.7</td></tr></table>
+
+# J FURTHER EVALUATION AND RESULTS
+
+![](images/67cb6b86c09ecd5df290f84bc32e9c20c4a826322434d559f04ea872549c2061.jpg)  
+Figure 13: Confusion matrix for the SVM-based classification of Section 4.1, for the Chamfer loss on ModelNet40. The class pairs most confused by the classifier are dresser/nightstand, flower pot/plant. Better viewed in the electronic version.
+
+![](images/597ef7d0c709eb6ff6c80993c012c3f24ed5fd05587caf455a3c613804d3ec88.jpg)  
+Figure 14: Synthetic results produced by the r-GAN. From left to right: airplanes, car, chairs, sofas.
+
+![](images/78799565b473123ceb8603033317a67ecfae2b3cfb6ab584d34ec0417fa3ed25.jpg)  
+Figure 15: The optimal bottleneck size was fixed at 128 by observing the reconstruction loss of the AEs, shown here for various bottleneck sizes.
+
+![](images/52ff709d6eb480bc909ac2cb5797f51c9595ba8ad7f85a369c48ef10bec50579.jpg)  
+Figure 16: Generalization error of the various GAN models, at various training epochs. Generalization is estimated using the JSD (left) and MMD-CD (right) metrics, which measure closeness of the synthetic results to the training resp. test ground truth distributions. The plots show the measurements of various GANs.
+
+![](images/6236445f3696e900457cd11b12eb3a199bcb143e8eb2bcb5ac0dee15b7d2b4c2.jpg)  
+Figure 17: GMM model selection. GMMs with a varying number of Gaussians and covariance type are trained on the latent space learned by and AE trained with EMD and a bottleneck of 128. Models with a full covariance matrix achieve significantly smaller JSD than models trained with diagonal covariance. For those with full covariance, 30 or more clusters seem sufficient to achieve minimal JSD. On the right, the values in a typical covariance matrix of a Gaussian component are shown in pseudocolor - note the strong off-diagonal components.
+
+![](images/1e5ea7a99ac4290d2a2ef8f4b9041ce73b2f22215a5c310b0acc0cd7ab32d167.jpg)  
+Figure 18: The 32 centers of the GMM fitted to the latent codes, and decoded using the decoder of the AE-EMD.
+
+Table 13: Evaluation of five generators on test-split of chair data on epochs/models that were selected via minimal MMD-CD on the validation-split. The reported scores are averages of three pseudo-random repetitions. Compare this with Table 3. Note that the overall quality of the selected models remains the same, irrespective of the metric used for the selection. GMM-40-F stands for a GMM with 40 Gaussian components with full covariances.   
+
+<table><tr><td>Method</td><td>Epoch</td><td>JSD</td><td>MMD-CD</td><td>MMD-EMD</td><td>COV-EMD</td><td>COV-CD</td></tr><tr><td>r-GAN</td><td>1350</td><td>0.1893</td><td>0.0020</td><td>0.1265</td><td>19.4</td><td>54.7</td></tr><tr><td>1-GAN (AE-CD)</td><td>300</td><td>0.0463</td><td>0.0020</td><td>0.0800</td><td>32.6</td><td>58.2</td></tr><tr><td>1-GAN (AE-EMD)</td><td>200</td><td>0.0319</td><td>0.0022</td><td>0.0684</td><td>57.6</td><td>58.7</td></tr><tr><td>1-WGAN (AE-EMD)</td><td>1700</td><td>0.0240</td><td>0.0020</td><td>0.0664</td><td>64.2</td><td>64.7</td></tr><tr><td>GMM-40-F (AE-EMD)</td><td>-</td><td>0.0182</td><td>0.0018</td><td>0.0646</td><td>68.6</td><td>69.3</td></tr></table>
+
+![](images/43ffae34acf8bc3c9ab916a6616c6b201050c2501a0fd784772b6b5903ea30e5.jpg)  
+Figure 19: Training trends in terms of the MMD-CD metric for the various GANs. Here, we sample a set of synthetic point-clouds for each model, of size 3x the size of the ground truth test dataset, and measure how well this synthetic dataset matches the ground truth in terms of MMD-CD. This plot complements Fig. 4 (left), where a different evaluation measure was used - note the similar behavior.

parse/train/Fo6S5-3Dx_/Fo6S5-3Dx_.md

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+# DEEP EVOLUTIONARY LEARNING FOR MOLECULAR DESIGN
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this paper, we propose a deep evolutionary learning (DEL) process that integrates fragment-based deep generative model and multi-objective evolutionary computation for molecular design. Our approach enables (1) evolutionary operations in the latent space of the generative model, rather than the structural space, to generate novel promising molecular structures for the next evolutionary generation, and (2) generative model fine-tuning using newly generated highquality samples. Thus, DEL implements a data-model co-evolution concept which improves both sample population and generative model learning. Experiments on two public datasets indicate that sample population obtained by DEL exhibits improved property distributions, and dominates samples generated by multiobjective Bayesian optimization algorithms.
+
+# 1 INTRODUCTION
+
+A drug is a molecule that binds to a target (e.g. protein) to inhibit or activate specific pathways in pathogens or host cells that cause abnormal phenotype. Drug discovery and development is a costly and time-consuming process, which is compounded by personalized medicines development for cancer or other complex and rare diseases. Computational drug discovery has been shown to accelerate the whole discovery process using simulations and machine intelligence. However, challenge remains in this field by demands for a robust and unbiased feature representation theories for molecules and their corresponding receptors, and efficient search algorithms. The rise of AI and data science provides us with a unique opportunity to reevaluate the problem and develop fast intelligent search or design approaches (Gromski et al., 2019; Chen et al., 2018). These new technologies claim differences from the traditional ones in two aspects: (1) features can be automatically learned using embedding techniques on a large number of training samples, and (2) high-level relationships (in supervised case) and complex distributions (in unsupervised case) can be captured using appropriate deep architectures. Recently, new representation theories and architectures have been proposed in the domain of molecular generation. There exist two new major methods to present a molecule for a machine learning algorithm. The first method converts a molecule structure to a string, such as the simplified molecular-input line-entry system (SMILES) string (Weininger, 1988), and adopt natural language processing (NLP) methods for supervised or unsupervised learning. The second uses an undirected graph to present a molecular structure and applies graph convolutional neural networks (Duvenaud et al., 2015). Three major families of AI algorithms have been developed for novel drug discovery: deep generative models (DGMs), reinforcement learning, and the combination of both.
+
+As one family of major neural probabilistic models for data modelling, generative autoencoders (e.g. variational autoencoder (VAE) (Kingma & Welling, 2014)) have been adopted to learn on either SMILES strings (Romez-Bombarelli et al., 2018) or molecular graphs (Simonovsky & Komodakis, 2018) with corresponding physical and biochemical properties for molecular generation. By integrating the generative adversarial nets and autoencoders, adversarial autoencoders (AAEs) have also been well applied to molecular design (Kadurin et al., 2017). The advantage of using generative autoencoders is that, molecules, as discrete objects in our world, are mapped to the continuous latent space, whose landscape can be organized by their properties, which helps generate new structures with preferred property values. However, critical problems remain due to imperfect representation methods. When SMILES strings are used in VAE, the model suffers from imbalance of tokens in embedding, generation of invalid structures, and the problem where two almost identical molecules have markedly different canonical SMILES strings. When using graph as molecular representation in VAE, a technical difficulty is to design an effective graph decoder. In addition to other heuristic methods, a SMILES decoder can be used to pair with a graph encoder. While DGMs offer convenience of searching in latent space, reinforcement learning algorithms can directly search in the molecules’ structural space by adding or deleting bounds and atoms (Zhou et al., 2019). In the Markov decision process (MDP) for drug design, the agent is a molecular generator, the molecular structure indicates the state, the actions are modifications to the current structure, and a simulator (e.g. surrogate model) is often used as the environment to provide reward. Furthermore, generative and predictive models can be integrated in MDP to form deep reinforcement learning (DRL) methods, where the generative model is trained as a policy approximation and the predictive model can be used as a value function approximation (You et al., 2018; Popova et al., 2018). Search in the discrete input space and inefficient learning are arguable concerns to be addressed when applying reinforcement-learning-based solutions for compound design.
+
+Interestingly, as an old peer of reinforcement learning for black-box optimizations, evolutionary computation (EC) methods (Eiben & Smith, 2015) have been catching up with promising performances in modern optimization, design and modelling problems. Besnard et al. (2012) present a strategy for evolution of ligands along multiple properties in the structural space, where a library of knowledge-based chemical structural transformation is used as the mutation operator. Interactions between EC and neural networks have mainly focused on network evolution and neural surrogate models for fitness functions. For examples, EC has been used at a large scale for neuroevolution that leads to evolution of neural network architectures (Stanley et al., 2019); feedforward neural network is commonly used as fitness function in EC (Mandal et al., 2019). Furthermore, it has been recently discovered that evolutionary strategy (ES) can perform competitively with reinforcement learning in game AI (Salimans et al., 2017). ES (Wierstra et al., 2014) and estimation of distribution algorithms (EDA) (Hauschild & Pelikan, 2011) build parameterized search distributions over promising points and either employ gradient information or sample from such a probabilistic model to find better points. Both probabilistic strategies from EC can potentially be used as alternatives to Bayesian optimization (BO) in (continuous) black-box optimizations. A single-objective BO has been recently applied to molecule optimization in the latent space of VAE (Romez-Bombarelli et al., 2018).
+
+Since model learning is essentially parameter estimation from the statistical modelling perspective, the quality of data in deep learning is crucial for model performance. Data augmentation is becoming a new strategy in deep learning to improve the training of a model. For example, in computer vision, transformations (such as rotation and flipping) of images are used to increase the sample size when the original data set is insufficient (Perez & Wang, 2017; Cubuk et al., 2019; Shorten & Khoshgoftaar, 2019). In NLP, a text dataset can be augmented using tricks such as replacing words or phrases with their synonyms (Wei & Zou, 2019), and resorting aids from other language models (e.g. word embedding and neural machine translations) (Sennrich et al., 2016; Wei & Zou, 2019). Basically, these methods either increase the data by transforming existing information which can only alleviate the limit of certain techniques (e.g. convolution), or indirectly borrow new information from other sources (e.g. methods in NLP).
+
+In summary, even though modern machine learning, evolutionary computation, and data science methods have been applied to molecular design and achieved promising results, we are still challenged by three chief issues. (1) An effective representation method for compound structures, that is encoding-decoding friendly and invariant to multimorphic forms, is still missing. (2) New ideas are expected for effective representation and coding of discrete structures in EC. And, (3) quality of data can be further improved as current data augmentation tricks only increase the number of samples but does not specifically address data quality. In this paper, we propose a novel deep evolutionary learning (DEL) process that combines the merits of deep generative model and multi-objective evolutionary computation together for molecular design. Specifically, our work has three major contributions. (1) In our approach, latent representations of phenotypic samples in a population serve as genotypic codes for evolutionary operations.This approach differs from traditional evolutionary algorithms that search in the original space of a problem. Specifically, our framework’s DGM encoder projects molecular structures in a population from discrete space to continuous latent space where evolutionary operations are applied to help explore the latent representation space. Subsequently, the DGM decoder maps the genotypic representation to the phenotypic space for generating new molecules with desired property values. (2) In each evolutionary generation, the newly formed population containing novel competitive molecules can be used to further fine-tune the DGM. This approach is an innovative data augmentation strategy that enriches training data with novel highquality samples. The whole DEL process implements a new learning paradigm that co-evolves data and model alternatingly through multiple evolutionary generations. (3) Our comprehensive experiments demonstrate that DEL is able to produce populations of novel samples with improved values of properties and outperforms state-of-the-art multi-objective BO algorithms (MOBO).
+
+# 2 METHOD
+
+The proposed deep evolutionary learning (DEL) process combines deep learning and multi-objective EC through the latent representation space of molecules. One of the theoretical innovations of our approach is that it demonstrates that EC methods are extendable to corresponding deep versions. The main idea is illustrated in Figure 1a and formally presented in Algorithm 1 (see Appendix A.1). This algorithm consists of the following steps. (a) A VAE (as molecule modeller) and a multilayer perceptron neural network (MLP, property predictor as regularizer) are pretrained using all the original training data to start the first evolutionary generation, or, if not in the first generation, using samples from the previous population. (b) Training samples (if first generation) or population samples (otherwise) are projected to the latent space using the encoder of the VAE. (c) Based on non-dominated ranking and crowding distances of samples with respect to multiple properties, evolutionary operations (selection, recombination/crossover and mutation) are conducted on the latent representations of the samples. (d) Given these new latent codes after evolutionary operations, new molecule samples are generated by the decoder of the VAE. (e) Properties of these generated samples are obtained using a simulator (e.g. RDKit (Landrum, 2006) in our experiment). (f) New samples with good desired properties and good samples from the previous generation form the new population. (g) Steps (b-f) are iterated for multiple generations. (h) The final population is returned.
+
+![](images/796b1548fe0f99eabe321f8a59bb0e6f56541bec3a979e6fcc810d7a262c645b.jpg)  
+Figure 1: Deep evolutionary learning process and deep generative model integrated in DEL.
+
+The major advantages of our DEL algorithm over existing interactions between EC and neural networks can be explained as follows. (a) We directly evolve a collection of data rather than many neural network structures and parameters. Data evolution tends to be more efficient than direct evolution of model structures and parameters. (b) The single neural network model (i.e. DGM in DEL) can be indirectly improved through learning on the evolved data with modern gradient-based variational learning and inference algorithms. Thus, the improvement of populations along evolutionary generations can be viewed as an effective data augmentation strategy that includes novel and highquality samples for further training of the neural network. (c) The continuous latent representation space established by the encoder of DGM can be naturally used as encoding (genotypic) space for evolutionary computation. Thus, evolutionary operations are carried out in the latent space instead of the discrete structural input space, allowing more efficient and smooth exploration, because the latent space is often multimodal and can be organized by properties (regularized by the property predictor) and evolutionary operations in this space can help the search escape from local regions and explore new regions of interest. (d) The multi-objective operations - non-dominated sorting and crowding distance, can help identify competitive and diverse parent samples to breed offspring. In summary, DEL takes advantages of both multi-objective EC and probabilistic neural model learning. The DGM, multi-objective components (non-dominated sorting and crowding distance), evolutionary operations, formation of new populations are discussed in details as below.
+
+# 2.1 FRAGVAE FOR FRAGMENT-BASED MOLECULAR MODELLING
+
+In our DEL process, we adopted a VAE model originally for fragment-based molecular generation (Podda et al., 2020). The concept of fragment-based drug design (FBDD) was introduced in (Shuker et al., 1996). In FBDD-based approaches, small organic molecules that bind to proximal subsites of a protein are identified, optimized, and linked together to produce high-affinity ligands. Wet-lab approaches for FBDD include $\mathrm { X }$ -ray crystallography and NMR spectroscopy. Compared to atom-based drug design, FBDD has the following advantages (Erlanson, 2011). (1) The search space in FBDD is much smaller $1 0 ^ { 7 }$ versus $1 0 ^ { 6 0 }$ ). (2) Identifying a fragment with certain affinity to the target may mean finding a pharmacophore. (3) Fragment-based synthesis could be more efficient than high-throughput screening. Majority fragmentation methods, which break a molecule into parts, are based on synthetic accessibility. For examples, RECAP (retrosynthetic combinatorial analysis procedure) is a method that breaks bonds formed by chemical reactions (Lewell et al., 1998); BRICS (breaking of retrosynthetically interesting chemical substructures) generates a more elaborated set of fragmentation rules along synthetically accessible bonds and generates more fragments than RECAP (Degen et al., 2008). BRICS is used in (Podda et al., 2020) to chop a SMILES string into several fragments. Then, fragment embeddings are produced using Word2Vec (Mikolov et al., 2013). Next, the sequences of fragments are modelled by a GRU-based VAE. In our work, a multi-head feedforward neural network component for predicting values of properties is added to the original model such that the latent representations can be regularized by properties of interest. Additionally, we normalize the three loss terms using batch size, and allow tuning of the weights among the loss terms. A crucial implementation bug in the original VAE model was also corrected (see Appendix A.2). Hereafter, we name this modified VAE model for fragments as FragVAE whose architecture is displayed in Figure 1b.
+
+We denote the encoder parameter by $\phi$ , the decoder parameter by $\pmb { \theta }$ , and the property predictor network by $f _ { \psi } ( z )$ parameterized by $\psi$ . The objective (to be minimized) of this DGM employed in DEL is a weighted combination of three terms:
+
+$$
+\begin{array} { r } { l _ { \phi , \theta , \psi } = - \mathrm { E } _ { q _ { \phi } ( z | x ) } [ \log p _ { \theta } ( x | h ) ] + \beta \mathrm { K L } \big ( q _ { \phi } ( z | x ) | | p _ { \theta } ( z ) \big ) + \alpha \mathrm { E } _ { q _ { \phi } ( z | x ) } \big [ \mathrm { M S E } ( f _ { \psi } ( z ) , y ) \big ] , } \end{array}
+$$
+
+where the first term is to reduce the reconstruction error, the second term is to regularize the posterior latent distribution with a simple prior, and the third term uses mean squared error of property prediction to further regularize the posterior distribution of latent codes. Previous studies unveil that VAE can easily fail on modelling text data because of the training imbalance between the reconstruction error (difficult to reduce once the KL divergence becomes very small) and the KL divergence (easy to diminish to zero). Thus, proper trade-off between the reconstruction error and KL divergence through $\beta$ -VAE (Higgins et al., 2017) is vital in text generation and molecular generation (Yan et al., 2020; Bowman et al., 2016). In practice, the value of $\beta$ should be smaller than 1. To look for a suitable value of $\beta$ , we design a versatile function, called $\beta$ -function, as formulated below,
+
+$$
+\beta ( t ) = \operatorname* { m i n } \Big \{ \operatorname* { m a x } \big \{ a e ^ { k ( 1 - \frac { T } { t } ) } , l \big \} , u \Big \} ,
+$$
+
+where $T$ represents the total number of epochs, $t \in \{ 1 , 2 , \cdots , T \}$ indicates the current index of epoch, $k$ controls the incremental speed, $a$ defines the amplitude, $l$ and $u$ serve as lower and upper bounds respectively for the value of $\beta$ . With different settings, a variety of curves of this function are shown in Figure 6 (see Appendix A.7). The value of $\alpha$ can be set similarly in FragVAE.
+
+# 2.2 NON-DOMINATION RANK AND CROWDING DISTANCE
+
+To get non-domination rank and crowding distance of a feasible solution for guiding the sample selection (Section 2.3) and the population merging (Section 2.4), the fast non-dominated sort and crowding comparison methods are adopted from the classic NSGA-II algorithm for multi-objective optimization (Deb et al., 2002). The properties in molecular design are treated as objectives. In an optimization problem with $K$ objectives $\bar { f } ( z ) = \{ f _ { 1 } ( z ) , \cdot \cdot \cdot , f _ { K } ( \bar { z } ) \}$ , feasible solution $z _ { 1 }$ is said to dominate $z _ { 2 }$ (denoted by $z _ { 1 } \prec z _ { 2 } ,$ ), if $\forall k \in \{ 1 , \cdots , K \}$ : $f _ { k } ( z _ { 1 } ) \leq f _ { k } ( z _ { 2 } )$ and $\exists k \in \{ 1 , \cdots , K \}$ : $f _ { k } ( z _ { 1 } ) < f _ { k } ( z _ { 2 } )$ . Using this concept of domination, all feasible solutions in a collection can be sorted to form Pareto frontiers (or fronts, ranks) $\mathcal { F } = \{ \mathcal { F } _ { 1 } , \mathcal { F } _ { 2 } , \cdot \cdot \cdot \}$ . Samples in the same frontier do not dominate each other. Frontier ${ \mathcal { F } } _ { i }$ dominates ${ \mathcal { F } } _ { j }$ for $j > i$ . Thus, we define function $F ( z )$ to retrieve the rank (i.e. frontier index) of any feasible solution $_ { z }$ in the population.
+
+The crowding distance of a feasible solution is computed as the normalized perimeter of the cuboid formed by its immediate neighbours along all objective axes. To compute the crowding distance of $z _ { i }$ , the normalized distance between its nearest neighbours above (denoted by $z _ { a }$ ) and below (denoted by $z _ { b }$ ) it w.r.t. the $k$ -th objective axis is calculated using $\begin{array} { r } { d _ { k } ( z _ { i } ) = \frac { f _ { k } ( z _ { a } ) - f _ { k } ( z _ { b } ) } { f _ { k } ^ { \operatorname* { m a x } } - f _ { k } ^ { \operatorname* { m i n } } } } \end{array}$ , where $f _ { k } ^ { \mathrm { m a x } }$ and idual $f _ { k } ^ { \mathrm { m i n } }$ are respectively the maximal and minimal values ots are summed up to form the crowding distance of h: $k$ these. The $z _ { i }$ $\begin{array} { r } { d ( z _ { i } ) = \sum _ { k = 1 } ^ { K } d _ { k } ( z _ { i } ) } \end{array}$
+
+Using the two concepts, partial order can be defined. We say $z _ { 1 } \prec _ { n } z _ { 2 }$ if either (1) $z _ { \mathrm { 1 } } \prec z _ { \mathrm { 2 } }$ (that is $F ( z _ { 1 } ) < F ( z _ { 2 } ) )$ , or (2) $F ( z _ { 1 } ) = F ( z _ { 2 } )$ and $d ( z _ { 1 } ) > d ( z _ { 2 } )$ . When two solutions have same rank, the one with larger crowding distance is preferred because it helps maintain a diverse population.
+
+# 2.3 EVOLUTIONARY OPERATIONS
+
+The evolutionary operations include parent selection, recombination and mutation to produce new offspring in the evolutionary process. Binary tournament selection is applied to select one out of two randomly drawn samples from the current population. In such a selection process, let us suppose $z _ { 1 }$ and $z _ { 2 }$ are randomly taken from the population and $z _ { 1 } \prec _ { n } z _ { 2 }$ . Sample $z _ { 1 }$ will be selected with selection probability $p _ { s }$ which is close to one, and $z _ { 2 }$ will be selected with a small chance $1 - p _ { s }$ . This selection process is repeated $M$ times to thus find $M$ parents where $M$ is the fixed population size. A pair of such parents will produce two children through recombination and mutation operations. Given two parents’ latent representation $z _ { p 1 }$ and $z _ { p 2 }$ , there are two recombination options - linear and discrete methods, to produce their new children $\hat { z } _ { 1 }$ and $\hat { z } _ { 2 }$ . For linear recombination, $\hat { z } _ { 1 } =$ $z _ { p 1 } + r _ { 1 } \big ( z _ { p 2 } - z _ { p 1 } \big )$ and $\hat { z } _ { 2 } = z _ { p 1 } + r _ { 2 } ( z _ { p 2 } - z _ { p 1 } )$ where $r _ { 1 } = - d \substack { + ( 1 + 2 d ) \alpha _ { 1 } }$ , $r _ { 2 } = - d \ t ( 1 \tplus d ) \alpha _ { 2 }$ , $d = 0 . 2 5$ , and $\alpha _ { 1 } , \alpha _ { 2 } \sim \mathrm { U n i f o r m } ( 0 , 1 )$ . For discrete method, supposing a latent representation vector is of length $L$ , an integer $l$ is randomly drawn from $\{ 1 , \cdots , L - 1 \}$ such that $\begin{array} { r } { \hat { z } _ { 1 } ^ { \ast } = \left[ z _ { p 1 } [ 1 : \right. } \end{array}$ $l ] , z _ { p 2 } [ l + 1 : L ] ]$ and $\hat { z } _ { 2 } ~ = ~ \big [ z _ { p 2 } [ 1 ~ : ~ l ] , z _ { p 1 } [ l ~ + ~ 1 ~ : ~ L ] \big ]$ . After crossover, a new sample $\hat { z } _ { m }$ $( m \in \{ 1 , \cdots , M \} )$ will have a small mutation probability $p _ { m }$ (say 0.01) of getting mutation. For $\hat { z } _ { m }$ , a random value $r$ is drawn from $\mathrm { U n i f o r m } ( 0 , 1 )$ . If $r < p _ { m }$ , then a random integer $l$ is randomly selected from $\{ 1 , \cdots , L \}$ such that the $l$ -th position of $\hat { z } _ { m }$ is replaced with a value drawn from standard Gaussian distribution: $\hat { z } _ { m , l } \sim \mathcal { N } ( 0 , 1 )$ .
+
+# 2.4 FORMING NEW POPULATION
+
+Ideally we need to maintain excellent and diverse populations. After possible mutation operations, all $M$ genotypic coding vectors will pass through the decoder of the DGM to produce phenotypic samples. All valid samples (supposed in set $\hat { \mathcal { P } } _ { t + 1 } \mathrm { ~ . ~ }$ ) will be kept to merge with the previous population (denoted by $\mathcal { P } _ { t }$ ) to produce a new generation (denoted by $\mathcal { P } _ { t + 1 }$ ). To implement it, all samples in $\hat { \mathcal { P } } _ { t + 1 } + \mathcal { P } _ { t }$ are sorted based on their non-domination ranks first and then on their crowding distances. Finally, only the top $M$ samples are taken from them to form the new population.
+
+# 3 EXPERIMENTS
+
+The performance of DEL was investigated on the ZINC (Irwin & Shoichet, 2005) and PCBA (Wang et al., 2016) datasets. These data were processed in the work of Podda et al. (2020). ZINC and PCBA are respectively composed of 227,945 and 383,790 molecules with two or more fragments. More statistics of both data can be found in (Podda et al., 2020). We comprehensively investigated the empirical performance of FragVAE and DEL. Three properties (QED: quantitative estimation of drug-likeness, SAS: synthetic accessibility score, and logP: water-octanol partition coefficient) are selected as objectives in DEL. Molecules with large QED, low SAS, and small logP values are prioritized. Incorporation of other properties (e.g. binding affinity, structure-property relationship, and ADME) will be considered in future work. QED, a scalarization of eight molecular properties (including logP) (Bickerton et al., 2012), is an adequate initial screening step for drug candidates. As we dive into more specific applications, selective properties can be tailored for subsequent screening. Thus, explicit usage of logP as one objective in DEL can help better assess lipophilicity, a key factor in drug design for some diseases, e.g. kidney and heart problems.
+
+# 3.1 EVALUATION OF FRAGVAE
+
+As FragVAE is a significant modification of the original model used in (Podda et al., 2020), we investigated the impact of $\beta$ value to the performance of FragVAE in terms of loss function values through Figure 7 (see Appendix A.7). Other hyperparameter values can be found in Appendix A.3. One can see that a large $\beta$ value can quickly reduce the KL loss to near zero which leads to stagnant reductions of reconstruction error and property regression error – the notorious posterior collapse problem (Goyal et al., 2017), because it is much easier to reduce the KL divergence than the reconstruction error in complex sequence modelling. Using a suitable small value of $\beta$ would allow the continuous decrease of the reconstruction error and property regression error. This observation is consistent with discoveries in language generative models (Yan et al., 2020; Bowman et al., 2016).
+
+Table 2 (see Appendix A.6) shows the validity, novelty and diversity of 20,000 samples generated from trained FragVAEs using standard Normal prior to sample $_ { z }$ followed by the decoder. Results of previous language-model-based and graph-based methods are also given for comparison. In general, the validity is defined as the ratio of number of valid generated samples to total number of generated samples. To clarify, the perfect validity reported in Podda et al. (2020) is actually calculated as the ratio of valid generated SMILES strings after discarding invalid fragment sequences versus total number of valid fragment sequences, i.e. Validity (SMILES) in Table 2. We found that this ratio is always 1 in fragment-based models. To have a better understanding about the model, we hence computed the validity of fragment sequences as the percentage of number of valid fragment sequences to total number of generated fragment sequences, i.e. Validity (Fragments) in Table 2. The novelty is defined as the ratio of number of generated novel valid molecules that do not exist in the training data versus total number of generated valid samples. The diversity is calculated as the percentage of generated unique valid samples among total number of generated valid samples. When the value of $\beta$ is very small (0.01), the posterior $p ( \boldsymbol { z } | \boldsymbol { x } )$ is highly different from the simple standard Normal prior $p ( z )$ . Thus, it is reasonable to see relatively low diversity in samples derived using standard Normal distribution. However, it does not imply that FragVAE with a very small value of $\beta$ is poor at learning latent representation. In fact, previous work in $\beta$ -VAE shows that small values of $\beta$ tend to encourage disentangled representations and form latent clusters (Li et al., 2020).
+
+The property distributions of generated samples can be important indicators to the proximity of generated samples with actual samples. Figure 2 shows the distributions of QED, SAS and logP in samples generated using standard Normal prior. Large $\beta$ value can lead to the sample SAS distribution appearing at the right side of the actual SAS distribution. Interestingly, the property distributions when using $\beta = 0 . 0 1$ do not resemble actual data, implying that using a very small value of $\beta$ would lead to latent representations that deviate from standard normal distributions. To summarize, $\beta \leq 0 . 1$ can prevent the model training from posterior collapse and can form structured latent representation space which is useful for latent space exploration using optimization techniques.
+
+![](images/c28b66a8e93fb7f363c8f8219d17ab5408b96cf6e4f623a49672dd261a99c8ee.jpg)  
+Figure 2: Property and structural feature distributions over 20K randomly sampled molecules.
+
+# 3.2 DEL PERFORMANCE
+
+We executed DEL processes using fixed or annealed loss trade-off weights: (1) $\beta = 0 . 1$ and $\alpha = 1$ , (2) $\beta = 0 . 0 1$ and $\alpha = 1$ , (3) $\beta = 0 . 1$ and $\alpha = 1$ in the initial training of FragVAE, then annealing $\beta$ to 0.4 and $\alpha$ to 4 in the second generation of DEL (denoted by $\beta = 0 . 1  0 . 4$ , $\alpha = 1  4$ ), and similarly (4) $\beta = 0 . 0 1  0 . 4$ , $\alpha = 1  4$ . Other hyperparameter values can be found in Appendix A.4. The change of losses is shown in Figure 8. We observe that (1) all settings lead to similar reconstruction error convergence, (2) smaller values of $\beta$ tend to obtain smaller property prediction error, (3) annealing can help reduce the KL divergence compared to the corresponding fixed settings.
+
+Figure 3 shows population validity, novelty and diversity in different DEL processes. In this chart, validity (SMILES) is the validity of SMILES strings in the population; validity (fragments) is the validity of fragment sequences sampled using FragVAE after evolutionary operations; novelty is the ratio of population samples that are not in the training data against the population size; and diversity is the ratio of unique population samples against the population size. We observe that almost all samples in the populations are novel; and the population samples in the last generation are quite diverse, ranging from 0.798 to 0.988. Table 9 lists the increasing numbers of high-quality novel molecules discovered along the DEL processes. In contrast to the training molecules visualized in Figures 9 and 10 (Appendix), DEL is able to discover novel and diverse high-quality molecules (see Figures 11-18 in Appendix). Furthermore, the distributions of properties and structural features of samples along the evolutionary process are compared in Figure 4 (and Figures 19-21 in Appendix). It can be seen that the process can be quite different from the actual data distribution and is able to gradually improve the distributions of QED, SAS and logP in population samples towards the preferred goals. Interestingly, samples randomly generated using standard Normal prior do not clearly show this trend (Figures 22-25 in Appendix).
+
+![](images/f7ce6fb50daa0521e2eddad7dc75088017a6c25e0118c1bcf1a9f5634ae09bee.jpg)  
+Figure 3: Population validity, novelty and diversity during DEL processes.
+
+![](images/d9aa22a62d1418d4a4359486de4528487c54afb7b0ae77954c8692660de64b6f.jpg)  
+Figure 4: Property & structural feature distributions of DEL populations $\mathrm { \beta } = 0 . 0 1 \to 0 . 4$ , $\alpha { = } 1  4$ ).
+
+# 3.3 COMPARISON WITH MULTI-OBJECTIVE BAYESIAN OPTIMIZATION
+
+DEL was compared with two MOBO methods: q-Pareto Efficient Global Optimization (qParEGO) and $\mathbf { q }$ -Expected Hypervolume Improvement (qEHVI) (Daulton et al., 2020). These MOBO methods were run in the latent space of FragVAE trained in the first generation of DEL using all training samples. Hyperparameter settings of these algorithms are listed in Appendix A.5. Figure 26 shows the hypervolumes of both algorithms and the quasi-random baseline which selects candidates from a scrambled Sobol sequence along batches. Hypervolumes obtained using these models are plotted in Figure 26. It shows that qParEGO and qEHVI work better with $\beta = 0 . 0 1$ than that with $\beta = 0 . 1$ .
+
+To qualitatively compare DEL with the MOBO algorithms, the first five Pareto fronts obtained using DEL and last five batches obtained using qParEGO and qEHVI are visualized in Figure 5 and Figure 27 in Appendix. We can see that the solutions from qParEGO and qEHVI are behind the Pareto fronts from DEL. To quantitatively compare DEL with qParEGO and qEHVI, we conducted nondominated sorting on the combination of the first Pareto front of DEL and the last six batches of qParEGO and qEHVI and report the results in Table 1. It shows that all solutions from the DEL Pareto front stay in the new integrated Pareto front, while almost all solutions from qParEGO and qEHVI are behind the integrated Pareto front. Furthermore, as indicated in Table 3 in Appendix,
+
+DEL runs more efficiently than these MOBO algorithms, even though the population size (20,000) of DEL is much larger than the batch sizes (8) of MOBO algorithms. It has been a well-known challenge to scale up BO algorithms.
+
+![](images/9cdbdd01d164450844dd31d4254992ba47f57e33cc28a14b3f835759372985c2.jpg)  
+Figure 5: Pareto fronts of DEL and MOBO algorithms $\beta = 0 . 0 1  0 . 4$ ). In the legend, the last batch of qParEGO or qEHVI is written as Front 1.
+
+Table 1: Non-dominated sorting of combination of DEL, qParEGO and qEHVI Pareto fronts.   
+
+<table><tr><td colspan="2"></td><td colspan="3">Pareto Front Size for Comparison</td><td colspan="3">In New Pareto Front</td></tr><tr><td>Data</td><td>DELHyperparameter</td><td>DEL</td><td>qParEGO</td><td>qEHVI</td><td>DEL</td><td>qParEGO</td><td>qEHVI</td></tr><tr><td>ZINC</td><td>β=0.1→0.4</td><td>243</td><td>46</td><td>45</td><td>243 (100%)</td><td>0(0%)</td><td>0(0%)</td></tr><tr><td>ZINC</td><td>β= 0.01→ 0.4</td><td>200</td><td>46</td><td>40</td><td>200 (100%)</td><td>0(0%)</td><td>1(2.5%)</td></tr><tr><td>PCBA</td><td>β=0.1→0.4</td><td>228</td><td>46</td><td>36</td><td>228 (100%)</td><td>0 (0%)</td><td>0(0%)</td></tr><tr><td>PCBA</td><td>β= 0.01→ 0.4</td><td>183</td><td>46</td><td>43</td><td>183 (100%)</td><td>1(2.17%)</td><td>2 (4.65%)</td></tr></table>
+
+# 3.4 ABLATION STUDIES
+
+By default, DEL uses the property predictor for latent representation regularization, fine-tunes FragVAE using new population data in each generation, forms new child latent codes using the linear crossover method, and fixes the population size to 20K. Variants were created by (1) disabling the property predictor, (2) disabling finetuning, (3) using the discrete crossover method, and (4) allowing a much larger population size (100K). While it is difficult to compare these variants in terms of validity, novelty and diversity of population samples (see Figure 28 in Appendix), it turns out that non-dominated sorting is an informative method for comparison. Table 4, in Appendix A.6, indicates that DEL with the property predictor outperforms the variant without it. Table 5 shows that FragVAE finetuning in DEL can help obtain better Pareto fronts. Table 6 implies that both linear and discrete crossover operations behave well in DEL. Also, DEL with larger population size can form better Pareto fronts (see Table 7). Additionally, we applied non-dominated sorting to compare the quality of Pareto fronts obtained using different values of $\beta$ on DEL, and found that the integrated Pareto front consists of samples relatively evenly from all settings (Table 8).
+
+# 4 CONCLUSION
+
+In this paper, we presented our DEL framework where a fragment-based VAE is integrated such that evolutionary exploration is conducted in the continuous latent representation space rather than the discrete structural space. Our intensive experiments show that DEL is able to generate novel populations of molecules with improved properties, and outperforms state-of-the-art multi-objective Bayesian optimization algorithms. Applications of DEL are certainly not restricted to design of small molecules. As future work, DEL will be tested on different datasets, other design problems, and more specific applications. Other types of DGMs and search strategies will be explored to further enhance DEL. New MOBO algorithms for latent-space based optimization need to be studied further to address issues such as scalability, unknown invalid domains in latent space, and the curse of dimensionality. Github link of our PyTorch and BoTorch implementation will be available.
+
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+
+# A APPENDIX
+
+# A.1 DEL ALGORITHM
+
+# Algorithm 1: DEL Algorithm
+
+Inputs: $X _ { 0 }$ , $Y _ { 0 }$ : training samples and corresponding properties; Result: population of designed molecules; learned DGM; 1 for each evolutionary generation do 2 if first generation then 3 $X = X _ { 0 }$ ; $Y = Y _ { 0 }$ ; 4 Train DGM using $\{ X , Y \}$ ; // use all training data to learn DGM 5 $X , Y = \operatorname { s u b s e t } ( X , Y , M )$ ; // a subset of $M$ examples from the original training data 6 else 7 Train DGM using $\{ X , Y \}$ ; // use population to further train DGM 8 end 9 end 10 Z = Encoder $( X )$ ; // obtain latent representations of samples in $\boldsymbol { x }$ 11 $r , \mathcal { F } =$ nondominated sort $( Y )$ ; // $\pmb { r }$ : non-domination ranks, $\mathcal { F }$ : Pareto frontiers 12 $d =$ compute crowding distance $( Y , { \mathcal { F } } )$ ; 13 $Z ^ { \prime } = \operatorname { E v { \bar { O p } } } ( Z , r , d )$ ; // evolutionary operations: selection, recombination and mutation 14 X0 = Decoder $( Z ^ { \prime } )$ ; // generate novel samples 15 $\mathbf { Y } ^ { \prime } =$ get property $( X ^ { \prime } )$ ; // obtain properties of the generated samples using a simulator, e.g. RDKit 16 $\boldsymbol { X }$ , $\mathbf { Y } =$ form new population $( { \pmb X } , { \pmb Y } , { \pmb X } ^ { \prime } , { \pmb Y } ^ { \prime } , { \pmb M } )$ ; // keep the top $M$ good samples from the combination of $\{ X , Y \}$ and $\{ X ^ { \prime } , Y ^ { \prime } \}$ 17 end 18 Return X , Y , DGM
+
+# A.2 BUG CORRECTION
+
+In the VAE implementation by Podda et al. (2020), the incorrect use of view function makes the forward flow of information messed up among different samples in a batch.
+
+Listing 1: Wrong use of view function in the original VAE model in (Podda et al., 2020). , s t a t e $=$ s e l f . r n n ( p a c k e d , s t a t e ) # n u m l a y e r s b y b a t c h b y h i d d e n s i z e −> b a t c h b y h i d d e n l a y e r \* h i d d e n s i z e s t a t e $=$ s t a t e . v i e w ( b a t c h s i z e , s e l f . h i d d e n s i z e \* s e l f . h i d d e n l a y e r s ) mean $=$ s e l f . r n n 2 m e a n ( s t a t e ) # mean : b a t c h b y l a t e n t s i z e l o g v a r $=$ s e l f . r n n 2 l o g v ( s t a t e ) # l o g v : b a t c h b y l a t e n t s i z e
+
+Listing 2: Our correction.
+
+s t a t e $=$ s e l f . r n n ( p a c k e d , s t a t e ) # n u m l a y e r s b y b a t c h b y h i d d e n s i z e −> b a t c h b y n u m l a y e r s b y h i d d e n s i z e s t a t e $=$ s t a t e . t r a n s p o s e $^ { ( 1 , 0 ) }$ s t a t e $=$ s t a t e . f l a t t e n ( s t a r t d i m $= 1$ ) # b a t c h b y h i d d e n l a y e r \* h i d d e n s i z e mean $=$ s e l f . r n n 2 m e a n ( s t a t e ) # mean : b a t c h b y l a t e n t s i z e l o g v a r $=$ s e l f . r n n 2 l o g v ( s t a t e ) # l o g v : b a t c h b y l a t e n t s i z e
+
+# A.3 HYPERPARAMETER SETTING FOR FRAGVAE
+
+When not specified in the main text, the following values of hyperparameters are used in experiments.
+
+• size of the embedding layer: 128
+
+• mask low-frequency fragments: Yes   
+• masking frequency: 2   
+• number of low frequency clusters: 5   
+• window for word2vec embedding: 3   
+• number of recurrent neurons per layer: 128   
+• number of recurrent layers: 2   
+• size of the VAE latent space: 64   
+• maximum length of the sampled sequence: 10   
+• batch size: 128   
+• shuffle batches: Yes   
+• number of epochs to train: 50   
+• learning rate: 0.0005   
+• dropout for the recurrent layers: 0.3   
+• annealing step size for the scheduler: 4   
+• annealing rate for the scheduler: 0.8   
+• threshold to clip the gradient norm: 5   
+• number of layers in MLP for properties: 2   
+• number of hidden units in each hidden layer of MLP: 64   
+• weight on property regression loss $( \alpha )$ : 1   
+• weight on KL divergence $( \beta )$ : {0.01, 0.1, 1}
+
+A.4 HYPERPARAMETER SETTING FOR DEL
+
+When not specified in the main text, the following values of hyperparameters are used in experiments.
+
+• number of evolutionary generations: 10   
+• popularization size: {20000 (default), 100000}   
+• number of epochs in the initial training of DGM: 50   
+• annealing step size for the scheduler in initial training of DGM: 4   
+• annealing rate for the scheduler: 0.8   
+• number of epochs in a subsequent training of DGM: 30   
+• learning rate in the initial training of DGM: 0.0005   
+• annealing step size for the scheduler in initial training: 2   
+• probability in tournament selection: 0.95   
+• crossover method: {linear (default), discrete}   
+• mutation rate: 0.01
+
+A.5 HYPERPARAMETER SETTING FOR MOBO
+
+When not specified in the main text, the following values of hyperparameters are used in experiments.
+
+• number of initial data points: 1000   
+• number of batches: 30   
+• batch size: 8   
+• number of MC samples: 128
+
+A.6 ADDITIONAL TABLES
+
+Table 2: Performance of FragVAE in comparison with existing methods.   
+
+<table><tr><td>Model</td><td>Data</td><td>Validity (SMILES)</td><td>Validity (Fragments)</td><td>Novelty</td><td>Diversity</td></tr><tr><td>ChemVAE GrammarVAE SDVAE GraphVAE CGVAE</td><td>ZINC ZINC ZINC ZINC ZINC</td><td>0.170 0.310 0.435 0.140 1.000</td><td></td><td>0.980 1.000 = 1.000 1.000</td><td>0.310 0.108 1 0.316 0.998</td></tr><tr><td>NeVAE FragVAE (β=1.00)</td><td>ZINC ZINC</td><td>1.000 1.000</td><td>■ 0.922</td><td>0.999 1.000</td><td>1.000 0.961</td></tr><tr><td>FragVAE (β=0.10)</td><td>ZINC</td><td>1.000</td><td>0.953</td><td>0.999</td><td>0.985</td></tr><tr><td>FragVAE(β=0.01)</td><td>ZINC</td><td>1.000</td><td>0.655</td><td>0.997</td><td>0.809</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FragVAE (β=1.00)</td><td>PCBA</td><td>1.000</td><td>0.443</td><td>1.000</td><td>0.925</td></tr><tr><td>FragVAE (β=0.10)</td><td>PCBA</td><td>1.000</td><td>0.481</td><td>0.983</td><td>0.886</td></tr><tr><td></td><td>PCBA</td><td>1.000</td><td>0.777</td><td>0.997</td><td></td></tr><tr><td>FragVAE (β=0.01)</td><td></td><td></td><td></td><td></td><td>0.632</td></tr></table>
+
+Table 3: Running time of DEL and MOBO. Since both methods involve training FragVAE, the FragVAE training time was not counted in this table. Format: Hours:Minutes:Seconds. Note: when $\beta$ is annealed to 0.4 in DEL, $\alpha$ is annealed from 1 to 4. All experiments were carried out on a Dell Precision 5820 Workstation equipped with an Intel Xeon W-2255 CPU (10C), RAM of 128GB, and a Nvidia Quadro RTX 6000 (24GB) GPU.   
+
+<table><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Hyperparameter</td><td rowspan=1 colspan=1>DEL</td><td rowspan=1 colspan=1>MOBO</td></tr><tr><td rowspan=1 colspan=1>ZINCZINC</td><td rowspan=1 colspan=1>β=0.1→0.4β = 0.01 → 0.4</td><td rowspan=1 colspan=1>3:03:493:08:09</td><td rowspan=1 colspan=1>25:23:01141:49:37</td></tr><tr><td rowspan=1 colspan=1>PCBAPCBA</td><td rowspan=1 colspan=1>β=0.1→0.4β = 0.01 → 0.4</td><td rowspan=1 colspan=1>3:01:321:50:20</td><td rowspan=1 colspan=1>31:16:1845:31:38</td></tr></table>
+
+Table 4: Non-dominated sorting of combination of Pareto fronts from DEL with and without property prediction (PP) component as latent space regularization.   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>InNewPareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DELParameter</td><td rowspan=1 colspan=1>With PP</td><td rowspan=1 colspan=1>Without PP</td><td rowspan=1 colspan=1>With PP</td><td rowspan=1 colspan=1>Without PP</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.1β=0.1</td><td rowspan=1 colspan=1>235224</td><td rowspan=1 colspan=1>234108</td><td rowspan=1 colspan=1>206 (87.66%)216 (96.43%)</td><td rowspan=1 colspan=1>0(0%)0(0%)</td></tr></table>
+
+Table 5: Non-dominated sorting of combination of Pareto fronts from DEL with and without DGM finetuning (FT) phrase.   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>In New Pareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Parameter</td><td rowspan=1 colspan=1>With FT</td><td rowspan=1 colspan=1>Without FT</td><td rowspan=1 colspan=1>With FT</td><td rowspan=1 colspan=1>Without FT</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.1β=0.1</td><td rowspan=1 colspan=1>235224</td><td rowspan=1 colspan=1>193214</td><td rowspan=1 colspan=1>235 (100%)199 (88.84%)</td><td rowspan=1 colspan=1>0 (0%)0(0%)</td></tr></table>
+
+Table 6: Non-dominated sorting of combination of Pareto fronts from DEL with linear or discrete crossover operation. Note: when $\beta$ is annealed to 0.4, $\alpha$ is annealed from 1 to 4.   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>InNewPareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Parameter</td><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>Discrete</td><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>Discrete</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.01→0.4β = 0.01 → 0.4</td><td rowspan=1 colspan=1>200183</td><td rowspan=1 colspan=1>181195</td><td rowspan=1 colspan=1>196 (98.00%)146 (79.78%)</td><td rowspan=1 colspan=1>173 (95.58%)184 (94.36%)</td></tr></table>
+
+A.7 ADDITIONAL FIGURES
+
+Table 7: Non-dominated sorting of combination of Pareto fronts from DEL with population sizes of 20K and 100K. Note: when $\beta$ is annealed to 0.4, $\alpha$ is annealed from 1 to 4.   
+
+<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>In New Pareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Parameter</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>100K</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>100K</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.1→0.4β= 0.1 → 0.4</td><td rowspan=1 colspan=1>243228</td><td rowspan=1 colspan=1>316354</td><td rowspan=1 colspan=1>78 (32.10%)88 (38.60%)</td><td rowspan=1 colspan=1>304 (96.20%)334 (94.35%)</td></tr></table>
+
+Table 8: Non-dominated sorting of combination Pareto fronts from DEL with different values of $\beta$ Note: when $\beta$ is annealed to 0.4, $\alpha$ is annealed from 1 to 4. Population size: 20K.   
+
+<table><tr><td></td><td colspan="4">Pareto Front Size for Comparison</td><td colspan="4">In New Pareto Front</td></tr><tr><td>Data</td><td></td><td></td><td></td><td>β=0.1β=0.1→0.4β=0.01β=0.01→0.4</td><td>β=0.1</td><td>β=0.1→0.4</td><td>β=0.01</td><td>β=0.01→0.4</td></tr><tr><td>ZINC</td><td>235</td><td>243</td><td>228</td><td>200</td><td>195(82.98%)</td><td>186(76.54%)</td><td>170(74.56%)</td><td>143(71.50%)</td></tr><tr><td>PCBA</td><td>224</td><td>228</td><td>189</td><td>183</td><td>169(75.45%)</td><td>183(80.26%)</td><td>122(64.55%)</td><td>111(60.66%)</td></tr></table>
+
+Table 9: Numbers of novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the 1st, 5th and final (10th) generations of DEL. Numbers of training molecules satisfying same conditions are also given. Note: when $\beta$ is annealed to 0.4, $\alpha$ is annealed from 1 to 4.   
+
+<table><tr><td>Data</td><td>Hyperparameter</td><td>Train</td><td>DEL(1)</td><td>DEL(5)</td><td>DEL(F)</td></tr><tr><td>ZINC</td><td>β=0.1</td><td>406</td><td>226</td><td>14</td><td>40</td></tr><tr><td>ZINC ZINC</td><td>β= 0.1 → 0.4</td><td>-</td><td></td><td>22</td><td>32</td></tr><tr><td></td><td>β =0.01</td><td>-</td><td></td><td>54</td><td>115</td></tr><tr><td>ZINC</td><td>β = 0.01 →0.4</td><td>1</td><td>3</td><td>9</td><td>20</td></tr><tr><td>PCBA</td><td>β 三 ：0.1</td><td>196</td><td>4</td><td>13</td><td>27</td></tr><tr><td>PCBA</td><td>β 三 0.1→ 0.4</td><td>1</td><td>10</td><td>33</td><td>52</td></tr><tr><td>PCBA</td><td>β =0.01</td><td></td><td>0</td><td>8</td><td>16</td></tr><tr><td>PCBA</td><td>β= 0.01 → 0.4</td><td>=</td><td>4</td><td>9</td><td>10</td></tr></table>
+
+![](images/328b6d651080a5518fe92dc230d1ef865a134b0c74e8e19c2f2076d0fa8e1e06.jpg)  
+Figure 6: Different of forms of the $\beta$ -function. The total number of epochs is $T = 1 0 0$ .
+
+![](images/cf6acad7e815d28bcf6de1c5e6a817e86f3ee5884ec5732eb3accd5c73ae61e8.jpg)  
+Figure 7: Loss of FragVAE in initial training on ZINC and PCBA respectively.
+
+![](images/7dc3634a8cc5465345a0ddfa487e586e7e8d497188199065a872c6919ede4776.jpg)  
+Figure 8: Loss of of FragVAE in DEL on ZINC and PCBA respectively.
+
+![](images/1fd02795e1ba4d96b7253b15692d60e1a07210a48f43d6d027bf0e09523a9979.jpg)  
+Figure 9: ZINC training molecules that satisfy properties QED≥ 0.88, $\mathrm { S A S } { \le } \ 3$ , and $\mathrm { l o g P \le 1 }$ . Note: 196 molecules satisfy these conditions, but 64 are visualized.
+
+![](images/dc8483339e12e7ce17feb421a56e94d3857dac87b13fb316462716a16e2cb6a4.jpg)  
+Figure 10: PCBA training molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ . Note: 406 molecules satisfy these conditions, but 64 are visualized.
+
+![](images/ef35348a43423ccd0019a79cb6e48a7c8be10d1d1efe34ab90c1e60683f3dacd.jpg)  
+Figure 11: Novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\beta = 0 . 1$ , $\alpha = 1$ .
+
+![](images/7ea716c19ba88a986d55d50d66b4c69dcd77a6214a9b6cb51862c22d46759d97.jpg)  
+Figure 12: Novel molecules that satisfy properties QE $\mathrm { D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\beta = 0 . 1  0 . 4$ , $\alpha = 1  4$ .
+
+![](images/172ac080ef55216a9030c88707c52ba819ffb87c66f6cb1b897817235f70d0c7.jpg)  
+Figure 13: Novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\beta = 0 . 0 1$ , $\alpha = 1$ .
+
+![](images/806ff87c94e38c19ef495c75301659f701bd1b5aae14d172da0db77cd5e8cccd.jpg)  
+Figure 14: Novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8 , \mathrm { S A S } \leq 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\beta = 0 . 0 1  0 . 4$ , $\alpha = 1  4$ .
+
+![](images/e8a6adf652a4f14dd67d7916b6d690ab5aae4c7f0d88d519378988000de179ec.jpg)  
+Figure 15: Novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8 , \mathrm { S A S } \leq 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\beta = 0 . 1$ , $\alpha = 1$ .
+
+![](images/35a0a618896d192b72930ef12d4822e13dfa99d64a4b6f753f34b28166e817c2.jpg)  
+Figure 16: Novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\beta = 0 . 1 \to 0 . 4$ , $\alpha = 1  4$ .
+
+![](images/7a98a373e39b584dc4538fe81c05c380bdb18d92cd9a7fb2311bd64e10c2d683.jpg)  
+Figure 17: Novel molecules that satisfy properties QE $\mathrm { D \geq 0 . 8 8 }$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\beta = 0 . 0 1$ , $\alpha = 1$ .
+
+![](images/dd0f2739d6ed68f68e37c700db304d114aca75025c20b7ab80931fc7afdca46e.jpg)  
+Figure 18: Novel molecules that satisfy properties $\mathrm { Q E D } \geq 0 . 8 8$ , $\mathrm { S A S } \le 3$ , and $\mathrm { l o g P \le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\beta = 0 . 0 1  0 . 4$ , $\alpha = 1  4$ .
+
+![](images/6a0a15d3f0d6bf47f1866b819db5e2e1fca329240ee33b0166456c39c30395c2.jpg)  
+Figure 19: Property and structural feature distributions of population samples during DEL $\beta = 0 . 1$ ).
+
+![](images/67bc0b4c81048eb4cfdc98e413d01fb25f64752d0db278b75060246553f20f4d.jpg)  
+Figure 20: Property and structural feature distributions of population samples during DEL $\beta =$ $0 . 1  0 . 4$ ).
+
+![](images/98a8c8b650d38981128378951283efc70338aaf76b2167c7759a1030ba0d87a0.jpg)  
+Figure 21: Property and structural feature distributions of population samples during DEL $\beta =$ 0.01).
+
+![](images/e5f677e00c8bd8817dbe030353fe6a77b089aec371fba7d29eee0397a4f6290d.jpg)  
+Figure 22: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL $\beta = 0 . 1 $ ).
+
+![](images/54f7aa24e3ee1a9789d0e90cfea59f5623e96e01bc0d83ce4335eb0d0a4630aa.jpg)  
+Figure 23: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL $\beta = 0 . 1  0 . 4$ ).
+
+![](images/11db239c64935ba9a85ee39f3e8155ab627ec1d9eba86bdec28c8204ec1932a6.jpg)  
+Figure 24: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL ( $\beta = 0 . 0 1$ ).
+
+![](images/1de784e2340f7d61a2743f0c807a2ee7c1a7cc8a69b830f8fe3447d8bd117bfc.jpg)  
+Figure 25: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL $\beta = 0 . 0 1  0 . 4$ ).
+
+![](images/2f5ae9516423a62c293d0e50ce5b806f01a7c1a1cc0d2cc73a5dab1ef0f80fff.jpg)  
+Figure 26: Hypervolume along batches when running Sobol random search, qParEGO, and qEHVI.
+
+![](images/ba4e8b0d755b63ebd955c60e6d827a81d4974cc4eefedb4c9713f2a740582baa.jpg)  
+Figure 27: Pareto fronts of DEL and MOBO algorithms $\beta = 0 . 1 $ . In the legend, the last batch of qParEGO or qEHVI is written as Front 1.
+
+![](images/63b287ba7dd5a16b8d67e19055298098726a60a3692270784ef5c7618042b008.jpg)  
+Figure 28: Validity, novelty and diversity of population samples (after evolutionary operations and before merging with previous population) in different variants of DEL.

parse/train/Fo6S5-3Dx_/Fo6S5-3Dx__content_list.json

@@ -0,0 +1,1414 @@
+[
+    {
+        "type": "text",
+        "text": "DEEP EVOLUTIONARY LEARNING FOR MOLECULAR DESIGN ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
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+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a deep evolutionary learning (DEL) process that integrates fragment-based deep generative model and multi-objective evolutionary computation for molecular design. Our approach enables (1) evolutionary operations in the latent space of the generative model, rather than the structural space, to generate novel promising molecular structures for the next evolutionary generation, and (2) generative model fine-tuning using newly generated highquality samples. Thus, DEL implements a data-model co-evolution concept which improves both sample population and generative model learning. Experiments on two public datasets indicate that sample population obtained by DEL exhibits improved property distributions, and dominates samples generated by multiobjective Bayesian optimization algorithms. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            443,
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+            458
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A drug is a molecule that binds to a target (e.g. protein) to inhibit or activate specific pathways in pathogens or host cells that cause abnormal phenotype. Drug discovery and development is a costly and time-consuming process, which is compounded by personalized medicines development for cancer or other complex and rare diseases. Computational drug discovery has been shown to accelerate the whole discovery process using simulations and machine intelligence. However, challenge remains in this field by demands for a robust and unbiased feature representation theories for molecules and their corresponding receptors, and efficient search algorithms. The rise of AI and data science provides us with a unique opportunity to reevaluate the problem and develop fast intelligent search or design approaches (Gromski et al., 2019; Chen et al., 2018). These new technologies claim differences from the traditional ones in two aspects: (1) features can be automatically learned using embedding techniques on a large number of training samples, and (2) high-level relationships (in supervised case) and complex distributions (in unsupervised case) can be captured using appropriate deep architectures. Recently, new representation theories and architectures have been proposed in the domain of molecular generation. There exist two new major methods to present a molecule for a machine learning algorithm. The first method converts a molecule structure to a string, such as the simplified molecular-input line-entry system (SMILES) string (Weininger, 1988), and adopt natural language processing (NLP) methods for supervised or unsupervised learning. The second uses an undirected graph to present a molecular structure and applies graph convolutional neural networks (Duvenaud et al., 2015). Three major families of AI algorithms have been developed for novel drug discovery: deep generative models (DGMs), reinforcement learning, and the combination of both. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As one family of major neural probabilistic models for data modelling, generative autoencoders (e.g. variational autoencoder (VAE) (Kingma & Welling, 2014)) have been adopted to learn on either SMILES strings (Romez-Bombarelli et al., 2018) or molecular graphs (Simonovsky & Komodakis, 2018) with corresponding physical and biochemical properties for molecular generation. By integrating the generative adversarial nets and autoencoders, adversarial autoencoders (AAEs) have also been well applied to molecular design (Kadurin et al., 2017). The advantage of using generative autoencoders is that, molecules, as discrete objects in our world, are mapped to the continuous latent space, whose landscape can be organized by their properties, which helps generate new structures with preferred property values. However, critical problems remain due to imperfect representation methods. When SMILES strings are used in VAE, the model suffers from imbalance of tokens in embedding, generation of invalid structures, and the problem where two almost identical molecules have markedly different canonical SMILES strings. When using graph as molecular representation in VAE, a technical difficulty is to design an effective graph decoder. In addition to other heuristic methods, a SMILES decoder can be used to pair with a graph encoder. While DGMs offer convenience of searching in latent space, reinforcement learning algorithms can directly search in the molecules’ structural space by adding or deleting bounds and atoms (Zhou et al., 2019). In the Markov decision process (MDP) for drug design, the agent is a molecular generator, the molecular structure indicates the state, the actions are modifications to the current structure, and a simulator (e.g. surrogate model) is often used as the environment to provide reward. Furthermore, generative and predictive models can be integrated in MDP to form deep reinforcement learning (DRL) methods, where the generative model is trained as a policy approximation and the predictive model can be used as a value function approximation (You et al., 2018; Popova et al., 2018). Search in the discrete input space and inefficient learning are arguable concerns to be addressed when applying reinforcement-learning-based solutions for compound design. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, as an old peer of reinforcement learning for black-box optimizations, evolutionary computation (EC) methods (Eiben & Smith, 2015) have been catching up with promising performances in modern optimization, design and modelling problems. Besnard et al. (2012) present a strategy for evolution of ligands along multiple properties in the structural space, where a library of knowledge-based chemical structural transformation is used as the mutation operator. Interactions between EC and neural networks have mainly focused on network evolution and neural surrogate models for fitness functions. For examples, EC has been used at a large scale for neuroevolution that leads to evolution of neural network architectures (Stanley et al., 2019); feedforward neural network is commonly used as fitness function in EC (Mandal et al., 2019). Furthermore, it has been recently discovered that evolutionary strategy (ES) can perform competitively with reinforcement learning in game AI (Salimans et al., 2017). ES (Wierstra et al., 2014) and estimation of distribution algorithms (EDA) (Hauschild & Pelikan, 2011) build parameterized search distributions over promising points and either employ gradient information or sample from such a probabilistic model to find better points. Both probabilistic strategies from EC can potentially be used as alternatives to Bayesian optimization (BO) in (continuous) black-box optimizations. A single-objective BO has been recently applied to molecule optimization in the latent space of VAE (Romez-Bombarelli et al., 2018). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Since model learning is essentially parameter estimation from the statistical modelling perspective, the quality of data in deep learning is crucial for model performance. Data augmentation is becoming a new strategy in deep learning to improve the training of a model. For example, in computer vision, transformations (such as rotation and flipping) of images are used to increase the sample size when the original data set is insufficient (Perez & Wang, 2017; Cubuk et al., 2019; Shorten & Khoshgoftaar, 2019). In NLP, a text dataset can be augmented using tricks such as replacing words or phrases with their synonyms (Wei & Zou, 2019), and resorting aids from other language models (e.g. word embedding and neural machine translations) (Sennrich et al., 2016; Wei & Zou, 2019). Basically, these methods either increase the data by transforming existing information which can only alleviate the limit of certain techniques (e.g. convolution), or indirectly borrow new information from other sources (e.g. methods in NLP). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, even though modern machine learning, evolutionary computation, and data science methods have been applied to molecular design and achieved promising results, we are still challenged by three chief issues. (1) An effective representation method for compound structures, that is encoding-decoding friendly and invariant to multimorphic forms, is still missing. (2) New ideas are expected for effective representation and coding of discrete structures in EC. And, (3) quality of data can be further improved as current data augmentation tricks only increase the number of samples but does not specifically address data quality. In this paper, we propose a novel deep evolutionary learning (DEL) process that combines the merits of deep generative model and multi-objective evolutionary computation together for molecular design. Specifically, our work has three major contributions. (1) In our approach, latent representations of phenotypic samples in a population serve as genotypic codes for evolutionary operations.This approach differs from traditional evolutionary algorithms that search in the original space of a problem. Specifically, our framework’s DGM encoder projects molecular structures in a population from discrete space to continuous latent space where evolutionary operations are applied to help explore the latent representation space. Subsequently, the DGM decoder maps the genotypic representation to the phenotypic space for generating new molecules with desired property values. (2) In each evolutionary generation, the newly formed population containing novel competitive molecules can be used to further fine-tune the DGM. This approach is an innovative data augmentation strategy that enriches training data with novel highquality samples. The whole DEL process implements a new learning paradigm that co-evolves data and model alternatingly through multiple evolutionary generations. (3) Our comprehensive experiments demonstrate that DEL is able to produce populations of novel samples with improved values of properties and outperforms state-of-the-art multi-objective BO algorithms (MOBO). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The proposed deep evolutionary learning (DEL) process combines deep learning and multi-objective EC through the latent representation space of molecules. One of the theoretical innovations of our approach is that it demonstrates that EC methods are extendable to corresponding deep versions. The main idea is illustrated in Figure 1a and formally presented in Algorithm 1 (see Appendix A.1). This algorithm consists of the following steps. (a) A VAE (as molecule modeller) and a multilayer perceptron neural network (MLP, property predictor as regularizer) are pretrained using all the original training data to start the first evolutionary generation, or, if not in the first generation, using samples from the previous population. (b) Training samples (if first generation) or population samples (otherwise) are projected to the latent space using the encoder of the VAE. (c) Based on non-dominated ranking and crowding distances of samples with respect to multiple properties, evolutionary operations (selection, recombination/crossover and mutation) are conducted on the latent representations of the samples. (d) Given these new latent codes after evolutionary operations, new molecule samples are generated by the decoder of the VAE. (e) Properties of these generated samples are obtained using a simulator (e.g. RDKit (Landrum, 2006) in our experiment). (f) New samples with good desired properties and good samples from the previous generation form the new population. (g) Steps (b-f) are iterated for multiple generations. (h) The final population is returned. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/796b1548fe0f99eabe321f8a59bb0e6f56541bec3a979e6fcc810d7a262c645b.jpg",
+        "image_caption": [
+            "Figure 1: Deep evolutionary learning process and deep generative model integrated in DEL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            438,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The major advantages of our DEL algorithm over existing interactions between EC and neural networks can be explained as follows. (a) We directly evolve a collection of data rather than many neural network structures and parameters. Data evolution tends to be more efficient than direct evolution of model structures and parameters. (b) The single neural network model (i.e. DGM in DEL) can be indirectly improved through learning on the evolved data with modern gradient-based variational learning and inference algorithms. Thus, the improvement of populations along evolutionary generations can be viewed as an effective data augmentation strategy that includes novel and highquality samples for further training of the neural network. (c) The continuous latent representation space established by the encoder of DGM can be naturally used as encoding (genotypic) space for evolutionary computation. Thus, evolutionary operations are carried out in the latent space instead of the discrete structural input space, allowing more efficient and smooth exploration, because the latent space is often multimodal and can be organized by properties (regularized by the property predictor) and evolutionary operations in this space can help the search escape from local regions and explore new regions of interest. (d) The multi-objective operations - non-dominated sorting and crowding distance, can help identify competitive and diverse parent samples to breed offspring. In summary, DEL takes advantages of both multi-objective EC and probabilistic neural model learning. The DGM, multi-objective components (non-dominated sorting and crowding distance), evolutionary operations, formation of new populations are discussed in details as below. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.1 FRAGVAE FOR FRAGMENT-BASED MOLECULAR MODELLING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In our DEL process, we adopted a VAE model originally for fragment-based molecular generation (Podda et al., 2020). The concept of fragment-based drug design (FBDD) was introduced in (Shuker et al., 1996). In FBDD-based approaches, small organic molecules that bind to proximal subsites of a protein are identified, optimized, and linked together to produce high-affinity ligands. Wet-lab approaches for FBDD include $\\mathrm { X }$ -ray crystallography and NMR spectroscopy. Compared to atom-based drug design, FBDD has the following advantages (Erlanson, 2011). (1) The search space in FBDD is much smaller $1 0 ^ { 7 }$ versus $1 0 ^ { 6 0 }$ ). (2) Identifying a fragment with certain affinity to the target may mean finding a pharmacophore. (3) Fragment-based synthesis could be more efficient than high-throughput screening. Majority fragmentation methods, which break a molecule into parts, are based on synthetic accessibility. For examples, RECAP (retrosynthetic combinatorial analysis procedure) is a method that breaks bonds formed by chemical reactions (Lewell et al., 1998); BRICS (breaking of retrosynthetically interesting chemical substructures) generates a more elaborated set of fragmentation rules along synthetically accessible bonds and generates more fragments than RECAP (Degen et al., 2008). BRICS is used in (Podda et al., 2020) to chop a SMILES string into several fragments. Then, fragment embeddings are produced using Word2Vec (Mikolov et al., 2013). Next, the sequences of fragments are modelled by a GRU-based VAE. In our work, a multi-head feedforward neural network component for predicting values of properties is added to the original model such that the latent representations can be regularized by properties of interest. Additionally, we normalize the three loss terms using batch size, and allow tuning of the weights among the loss terms. A crucial implementation bug in the original VAE model was also corrected (see Appendix A.2). Hereafter, we name this modified VAE model for fragments as FragVAE whose architecture is displayed in Figure 1b. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "We denote the encoder parameter by $\\phi$ , the decoder parameter by $\\pmb { \\theta }$ , and the property predictor network by $f _ { \\psi } ( z )$ parameterized by $\\psi$ . The objective (to be minimized) of this DGM employed in DEL is a weighted combination of three terms: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/52a6317c8b12df71a1a6d0da27d3910c3884dff03c590162bbc5dc24c4806fff.jpg",
+        "text": "$$\n\\begin{array} { r } { l _ { \\phi , \\theta , \\psi } = - \\mathrm { E } _ { q _ { \\phi } ( z | x ) } [ \\log p _ { \\theta } ( x | h ) ] + \\beta \\mathrm { K L } \\big ( q _ { \\phi } ( z | x ) | | p _ { \\theta } ( z ) \\big ) + \\alpha \\mathrm { E } _ { q _ { \\phi } ( z | x ) } \\big [ \\mathrm { M S E } ( f _ { \\psi } ( z ) , y ) \\big ] , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the first term is to reduce the reconstruction error, the second term is to regularize the posterior latent distribution with a simple prior, and the third term uses mean squared error of property prediction to further regularize the posterior distribution of latent codes. Previous studies unveil that VAE can easily fail on modelling text data because of the training imbalance between the reconstruction error (difficult to reduce once the KL divergence becomes very small) and the KL divergence (easy to diminish to zero). Thus, proper trade-off between the reconstruction error and KL divergence through $\\beta$ -VAE (Higgins et al., 2017) is vital in text generation and molecular generation (Yan et al., 2020; Bowman et al., 2016). In practice, the value of $\\beta$ should be smaller than 1. To look for a suitable value of $\\beta$ , we design a versatile function, called $\\beta$ -function, as formulated below, ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5847fb99884959e58a88ee62c645c688c5e63c01fbefc6bb971153e56e0b1e05.jpg",
+        "text": "$$\n\\beta ( t ) = \\operatorname* { m i n } \\Big \\{ \\operatorname* { m a x } \\big \\{ a e ^ { k ( 1 - \\frac { T } { t } ) } , l \\big \\} , u \\Big \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $T$ represents the total number of epochs, $t \\in \\{ 1 , 2 , \\cdots , T \\}$ indicates the current index of epoch, $k$ controls the incremental speed, $a$ defines the amplitude, $l$ and $u$ serve as lower and upper bounds respectively for the value of $\\beta$ . With different settings, a variety of curves of this function are shown in Figure 6 (see Appendix A.7). The value of $\\alpha$ can be set similarly in FragVAE. ",
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+    {
+        "type": "text",
+        "text": "2.2 NON-DOMINATION RANK AND CROWDING DISTANCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To get non-domination rank and crowding distance of a feasible solution for guiding the sample selection (Section 2.3) and the population merging (Section 2.4), the fast non-dominated sort and crowding comparison methods are adopted from the classic NSGA-II algorithm for multi-objective optimization (Deb et al., 2002). The properties in molecular design are treated as objectives. In an optimization problem with $K$ objectives $\\bar { f } ( z ) = \\{ f _ { 1 } ( z ) , \\cdot \\cdot \\cdot , f _ { K } ( \\bar { z } ) \\}$ , feasible solution $z _ { 1 }$ is said to dominate $z _ { 2 }$ (denoted by $z _ { 1 } \\prec z _ { 2 } ,$ ), if $\\forall k \\in \\{ 1 , \\cdots , K \\}$ : $f _ { k } ( z _ { 1 } ) \\leq f _ { k } ( z _ { 2 } )$ and $\\exists k \\in \\{ 1 , \\cdots , K \\}$ : $f _ { k } ( z _ { 1 } ) < f _ { k } ( z _ { 2 } )$ . Using this concept of domination, all feasible solutions in a collection can be sorted to form Pareto frontiers (or fronts, ranks) $\\mathcal { F } = \\{ \\mathcal { F } _ { 1 } , \\mathcal { F } _ { 2 } , \\cdot \\cdot \\cdot \\}$ . Samples in the same frontier do not dominate each other. Frontier ${ \\mathcal { F } } _ { i }$ dominates ${ \\mathcal { F } } _ { j }$ for $j > i$ . Thus, we define function $F ( z )$ to retrieve the rank (i.e. frontier index) of any feasible solution $_ { z }$ in the population. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The crowding distance of a feasible solution is computed as the normalized perimeter of the cuboid formed by its immediate neighbours along all objective axes. To compute the crowding distance of $z _ { i }$ , the normalized distance between its nearest neighbours above (denoted by $z _ { a }$ ) and below (denoted by $z _ { b }$ ) it w.r.t. the $k$ -th objective axis is calculated using $\\begin{array} { r } { d _ { k } ( z _ { i } ) = \\frac { f _ { k } ( z _ { a } ) - f _ { k } ( z _ { b } ) } { f _ { k } ^ { \\operatorname* { m a x } } - f _ { k } ^ { \\operatorname* { m i n } } } } \\end{array}$ , where $f _ { k } ^ { \\mathrm { m a x } }$ and idual $f _ { k } ^ { \\mathrm { m i n } }$ are respectively the maximal and minimal values ots are summed up to form the crowding distance of h: $k$ these. The $z _ { i }$ $\\begin{array} { r } { d ( z _ { i } ) = \\sum _ { k = 1 } ^ { K } d _ { k } ( z _ { i } ) } \\end{array}$ ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Using the two concepts, partial order can be defined. We say $z _ { 1 } \\prec _ { n } z _ { 2 }$ if either (1) $z _ { \\mathrm { 1 } } \\prec z _ { \\mathrm { 2 } }$ (that is $F ( z _ { 1 } ) < F ( z _ { 2 } ) )$ , or (2) $F ( z _ { 1 } ) = F ( z _ { 2 } )$ and $d ( z _ { 1 } ) > d ( z _ { 2 } )$ . When two solutions have same rank, the one with larger crowding distance is preferred because it helps maintain a diverse population. ",
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+    {
+        "type": "text",
+        "text": "2.3 EVOLUTIONARY OPERATIONS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The evolutionary operations include parent selection, recombination and mutation to produce new offspring in the evolutionary process. Binary tournament selection is applied to select one out of two randomly drawn samples from the current population. In such a selection process, let us suppose $z _ { 1 }$ and $z _ { 2 }$ are randomly taken from the population and $z _ { 1 } \\prec _ { n } z _ { 2 }$ . Sample $z _ { 1 }$ will be selected with selection probability $p _ { s }$ which is close to one, and $z _ { 2 }$ will be selected with a small chance $1 - p _ { s }$ . This selection process is repeated $M$ times to thus find $M$ parents where $M$ is the fixed population size. A pair of such parents will produce two children through recombination and mutation operations. Given two parents’ latent representation $z _ { p 1 }$ and $z _ { p 2 }$ , there are two recombination options - linear and discrete methods, to produce their new children $\\hat { z } _ { 1 }$ and $\\hat { z } _ { 2 }$ . For linear recombination, $\\hat { z } _ { 1 } =$ $z _ { p 1 } + r _ { 1 } \\big ( z _ { p 2 } - z _ { p 1 } \\big )$ and $\\hat { z } _ { 2 } = z _ { p 1 } + r _ { 2 } ( z _ { p 2 } - z _ { p 1 } )$ where $r _ { 1 } = - d \\substack { + ( 1 + 2 d ) \\alpha _ { 1 } }$ , $r _ { 2 } = - d \\ t ( 1 \\tplus d ) \\alpha _ { 2 }$ , $d = 0 . 2 5$ , and $\\alpha _ { 1 } , \\alpha _ { 2 } \\sim \\mathrm { U n i f o r m } ( 0 , 1 )$ . For discrete method, supposing a latent representation vector is of length $L$ , an integer $l$ is randomly drawn from $\\{ 1 , \\cdots , L - 1 \\}$ such that $\\begin{array} { r } { \\hat { z } _ { 1 } ^ { \\ast } = \\left[ z _ { p 1 } [ 1 : \\right. } \\end{array}$ $l ] , z _ { p 2 } [ l + 1 : L ] ]$ and $\\hat { z } _ { 2 } ~ = ~ \\big [ z _ { p 2 } [ 1 ~ : ~ l ] , z _ { p 1 } [ l ~ + ~ 1 ~ : ~ L ] \\big ]$ . After crossover, a new sample $\\hat { z } _ { m }$ $( m \\in \\{ 1 , \\cdots , M \\} )$ will have a small mutation probability $p _ { m }$ (say 0.01) of getting mutation. For $\\hat { z } _ { m }$ , a random value $r$ is drawn from $\\mathrm { U n i f o r m } ( 0 , 1 )$ . If $r < p _ { m }$ , then a random integer $l$ is randomly selected from $\\{ 1 , \\cdots , L \\}$ such that the $l$ -th position of $\\hat { z } _ { m }$ is replaced with a value drawn from standard Gaussian distribution: $\\hat { z } _ { m , l } \\sim \\mathcal { N } ( 0 , 1 )$ . ",
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+    {
+        "type": "text",
+        "text": "2.4 FORMING NEW POPULATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Ideally we need to maintain excellent and diverse populations. After possible mutation operations, all $M$ genotypic coding vectors will pass through the decoder of the DGM to produce phenotypic samples. All valid samples (supposed in set $\\hat { \\mathcal { P } } _ { t + 1 } \\mathrm { ~ . ~ }$ ) will be kept to merge with the previous population (denoted by $\\mathcal { P } _ { t }$ ) to produce a new generation (denoted by $\\mathcal { P } _ { t + 1 }$ ). To implement it, all samples in $\\hat { \\mathcal { P } } _ { t + 1 } + \\mathcal { P } _ { t }$ are sorted based on their non-domination ranks first and then on their crowding distances. Finally, only the top $M$ samples are taken from them to form the new population. ",
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+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The performance of DEL was investigated on the ZINC (Irwin & Shoichet, 2005) and PCBA (Wang et al., 2016) datasets. These data were processed in the work of Podda et al. (2020). ZINC and PCBA are respectively composed of 227,945 and 383,790 molecules with two or more fragments. More statistics of both data can be found in (Podda et al., 2020). We comprehensively investigated the empirical performance of FragVAE and DEL. Three properties (QED: quantitative estimation of drug-likeness, SAS: synthetic accessibility score, and logP: water-octanol partition coefficient) are selected as objectives in DEL. Molecules with large QED, low SAS, and small logP values are prioritized. Incorporation of other properties (e.g. binding affinity, structure-property relationship, and ADME) will be considered in future work. QED, a scalarization of eight molecular properties (including logP) (Bickerton et al., 2012), is an adequate initial screening step for drug candidates. As we dive into more specific applications, selective properties can be tailored for subsequent screening. Thus, explicit usage of logP as one objective in DEL can help better assess lipophilicity, a key factor in drug design for some diseases, e.g. kidney and heart problems. ",
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+    {
+        "type": "text",
+        "text": "3.1 EVALUATION OF FRAGVAE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As FragVAE is a significant modification of the original model used in (Podda et al., 2020), we investigated the impact of $\\beta$ value to the performance of FragVAE in terms of loss function values through Figure 7 (see Appendix A.7). Other hyperparameter values can be found in Appendix A.3. One can see that a large $\\beta$ value can quickly reduce the KL loss to near zero which leads to stagnant reductions of reconstruction error and property regression error – the notorious posterior collapse problem (Goyal et al., 2017), because it is much easier to reduce the KL divergence than the reconstruction error in complex sequence modelling. Using a suitable small value of $\\beta$ would allow the continuous decrease of the reconstruction error and property regression error. This observation is consistent with discoveries in language generative models (Yan et al., 2020; Bowman et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "Table 2 (see Appendix A.6) shows the validity, novelty and diversity of 20,000 samples generated from trained FragVAEs using standard Normal prior to sample $_ { z }$ followed by the decoder. Results of previous language-model-based and graph-based methods are also given for comparison. In general, the validity is defined as the ratio of number of valid generated samples to total number of generated samples. To clarify, the perfect validity reported in Podda et al. (2020) is actually calculated as the ratio of valid generated SMILES strings after discarding invalid fragment sequences versus total number of valid fragment sequences, i.e. Validity (SMILES) in Table 2. We found that this ratio is always 1 in fragment-based models. To have a better understanding about the model, we hence computed the validity of fragment sequences as the percentage of number of valid fragment sequences to total number of generated fragment sequences, i.e. Validity (Fragments) in Table 2. The novelty is defined as the ratio of number of generated novel valid molecules that do not exist in the training data versus total number of generated valid samples. The diversity is calculated as the percentage of generated unique valid samples among total number of generated valid samples. When the value of $\\beta$ is very small (0.01), the posterior $p ( \\boldsymbol { z } | \\boldsymbol { x } )$ is highly different from the simple standard Normal prior $p ( z )$ . Thus, it is reasonable to see relatively low diversity in samples derived using standard Normal distribution. However, it does not imply that FragVAE with a very small value of $\\beta$ is poor at learning latent representation. In fact, previous work in $\\beta$ -VAE shows that small values of $\\beta$ tend to encourage disentangled representations and form latent clusters (Li et al., 2020). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The property distributions of generated samples can be important indicators to the proximity of generated samples with actual samples. Figure 2 shows the distributions of QED, SAS and logP in samples generated using standard Normal prior. Large $\\beta$ value can lead to the sample SAS distribution appearing at the right side of the actual SAS distribution. Interestingly, the property distributions when using $\\beta = 0 . 0 1$ do not resemble actual data, implying that using a very small value of $\\beta$ would lead to latent representations that deviate from standard normal distributions. To summarize, $\\beta \\leq 0 . 1$ can prevent the model training from posterior collapse and can form structured latent representation space which is useful for latent space exploration using optimization techniques. ",
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+        "img_path": "images/c28b66a8e93fb7f363c8f8219d17ab5408b96cf6e4f623a49672dd261a99c8ee.jpg",
+        "image_caption": [
+            "Figure 2: Property and structural feature distributions over 20K randomly sampled molecules. "
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+        "type": "text",
+        "text": "3.2 DEL PERFORMANCE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We executed DEL processes using fixed or annealed loss trade-off weights: (1) $\\beta = 0 . 1$ and $\\alpha = 1$ , (2) $\\beta = 0 . 0 1$ and $\\alpha = 1$ , (3) $\\beta = 0 . 1$ and $\\alpha = 1$ in the initial training of FragVAE, then annealing $\\beta$ to 0.4 and $\\alpha$ to 4 in the second generation of DEL (denoted by $\\beta = 0 . 1  0 . 4$ , $\\alpha = 1  4$ ), and similarly (4) $\\beta = 0 . 0 1  0 . 4$ , $\\alpha = 1  4$ . Other hyperparameter values can be found in Appendix A.4. The change of losses is shown in Figure 8. We observe that (1) all settings lead to similar reconstruction error convergence, (2) smaller values of $\\beta$ tend to obtain smaller property prediction error, (3) annealing can help reduce the KL divergence compared to the corresponding fixed settings. ",
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+    {
+        "type": "text",
+        "text": "Figure 3 shows population validity, novelty and diversity in different DEL processes. In this chart, validity (SMILES) is the validity of SMILES strings in the population; validity (fragments) is the validity of fragment sequences sampled using FragVAE after evolutionary operations; novelty is the ratio of population samples that are not in the training data against the population size; and diversity is the ratio of unique population samples against the population size. We observe that almost all samples in the populations are novel; and the population samples in the last generation are quite diverse, ranging from 0.798 to 0.988. Table 9 lists the increasing numbers of high-quality novel molecules discovered along the DEL processes. In contrast to the training molecules visualized in Figures 9 and 10 (Appendix), DEL is able to discover novel and diverse high-quality molecules (see Figures 11-18 in Appendix). Furthermore, the distributions of properties and structural features of samples along the evolutionary process are compared in Figure 4 (and Figures 19-21 in Appendix). It can be seen that the process can be quite different from the actual data distribution and is able to gradually improve the distributions of QED, SAS and logP in population samples towards the preferred goals. Interestingly, samples randomly generated using standard Normal prior do not clearly show this trend (Figures 22-25 in Appendix). ",
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+        "image_caption": [
+            "Figure 3: Population validity, novelty and diversity during DEL processes. "
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+        "img_path": "images/d9aa22a62d1418d4a4359486de4528487c54afb7b0ae77954c8692660de64b6f.jpg",
+        "image_caption": [
+            "Figure 4: Property & structural feature distributions of DEL populations $\\mathrm { \\beta } = 0 . 0 1 \\to 0 . 4$ , $\\alpha { = } 1  4$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 COMPARISON WITH MULTI-OBJECTIVE BAYESIAN OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "DEL was compared with two MOBO methods: q-Pareto Efficient Global Optimization (qParEGO) and $\\mathbf { q }$ -Expected Hypervolume Improvement (qEHVI) (Daulton et al., 2020). These MOBO methods were run in the latent space of FragVAE trained in the first generation of DEL using all training samples. Hyperparameter settings of these algorithms are listed in Appendix A.5. Figure 26 shows the hypervolumes of both algorithms and the quasi-random baseline which selects candidates from a scrambled Sobol sequence along batches. Hypervolumes obtained using these models are plotted in Figure 26. It shows that qParEGO and qEHVI work better with $\\beta = 0 . 0 1$ than that with $\\beta = 0 . 1$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To qualitatively compare DEL with the MOBO algorithms, the first five Pareto fronts obtained using DEL and last five batches obtained using qParEGO and qEHVI are visualized in Figure 5 and Figure 27 in Appendix. We can see that the solutions from qParEGO and qEHVI are behind the Pareto fronts from DEL. To quantitatively compare DEL with qParEGO and qEHVI, we conducted nondominated sorting on the combination of the first Pareto front of DEL and the last six batches of qParEGO and qEHVI and report the results in Table 1. It shows that all solutions from the DEL Pareto front stay in the new integrated Pareto front, while almost all solutions from qParEGO and qEHVI are behind the integrated Pareto front. Furthermore, as indicated in Table 3 in Appendix, ",
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+    {
+        "type": "text",
+        "text": "DEL runs more efficiently than these MOBO algorithms, even though the population size (20,000) of DEL is much larger than the batch sizes (8) of MOBO algorithms. It has been a well-known challenge to scale up BO algorithms. ",
+        "bbox": [
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+        "image_caption": [
+            "Figure 5: Pareto fronts of DEL and MOBO algorithms $\\beta = 0 . 0 1  0 . 4$ ). In the legend, the last batch of qParEGO or qEHVI is written as Front 1. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "table",
+        "img_path": "images/b79dd815bc0732dbc40f5a088a92ae1cab445c2e436b3bfe0072bc1dc8929e26.jpg",
+        "table_caption": [
+            "Table 1: Non-dominated sorting of combination of DEL, qParEGO and qEHVI Pareto fronts. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"3\">Pareto Front Size for Comparison</td><td colspan=\"3\">In New Pareto Front</td></tr><tr><td>Data</td><td>DELHyperparameter</td><td>DEL</td><td>qParEGO</td><td>qEHVI</td><td>DEL</td><td>qParEGO</td><td>qEHVI</td></tr><tr><td>ZINC</td><td>β=0.1→0.4</td><td>243</td><td>46</td><td>45</td><td>243 (100%)</td><td>0(0%)</td><td>0(0%)</td></tr><tr><td>ZINC</td><td>β= 0.01→ 0.4</td><td>200</td><td>46</td><td>40</td><td>200 (100%)</td><td>0(0%)</td><td>1(2.5%)</td></tr><tr><td>PCBA</td><td>β=0.1→0.4</td><td>228</td><td>46</td><td>36</td><td>228 (100%)</td><td>0 (0%)</td><td>0(0%)</td></tr><tr><td>PCBA</td><td>β= 0.01→ 0.4</td><td>183</td><td>46</td><td>43</td><td>183 (100%)</td><td>1(2.17%)</td><td>2 (4.65%)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.4 ABLATION STUDIES ",
+        "text_level": 1,
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+        "text": "By default, DEL uses the property predictor for latent representation regularization, fine-tunes FragVAE using new population data in each generation, forms new child latent codes using the linear crossover method, and fixes the population size to 20K. Variants were created by (1) disabling the property predictor, (2) disabling finetuning, (3) using the discrete crossover method, and (4) allowing a much larger population size (100K). While it is difficult to compare these variants in terms of validity, novelty and diversity of population samples (see Figure 28 in Appendix), it turns out that non-dominated sorting is an informative method for comparison. Table 4, in Appendix A.6, indicates that DEL with the property predictor outperforms the variant without it. Table 5 shows that FragVAE finetuning in DEL can help obtain better Pareto fronts. Table 6 implies that both linear and discrete crossover operations behave well in DEL. Also, DEL with larger population size can form better Pareto fronts (see Table 7). Additionally, we applied non-dominated sorting to compare the quality of Pareto fronts obtained using different values of $\\beta$ on DEL, and found that the integrated Pareto front consists of samples relatively evenly from all settings (Table 8). ",
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+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we presented our DEL framework where a fragment-based VAE is integrated such that evolutionary exploration is conducted in the continuous latent representation space rather than the discrete structural space. Our intensive experiments show that DEL is able to generate novel populations of molecules with improved properties, and outperforms state-of-the-art multi-objective Bayesian optimization algorithms. Applications of DEL are certainly not restricted to design of small molecules. As future work, DEL will be tested on different datasets, other design problems, and more specific applications. Other types of DGMs and search strategies will be explored to further enhance DEL. New MOBO algorithms for latent-space based optimization need to be studied further to address issues such as scalability, unknown invalid domains in latent space, and the curse of dimensionality. Github link of our PyTorch and BoTorch implementation will be available. ",
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+        "text": "REFERENCES   \nTJ. Besnard, G.F. Ruda, V. Setola, K. Abecassis, R.M. Rodriguiz, X. Huang, S. Norval, M.F. Sassano, A.I. Shin, L.A. Webster, F.R.C. Simeons, L. Stojanovski, A. Prat, N.G. Seidah, D.B. Constam, G.R. Bickerton, K.D. Read, W.C. Wetsel, I.H. Gilbert, B.L. Roth, and A.L. Hopkins. Automated design of ligands to polypharmacological profiles. Nature, 412:215–220, 2012.   \nG.R. Bickerton, G.V. Paolini, J. Besnard, S. Muresan, and A.L. Hopkins. Quantifying the chemical beauty of drugs. Nature Chemistry, 4:90–98, 2012.   \nS.R. Bowman, L. Vilnis, O. Vinyals, A.M. Dai, R. Jozefowicz, and S. Bengio. Generating sentences from a continuous space. In SIGNLL Conference on Computational Natural Language Learning (CoNLL), pp. 10–21, 2016.   \nH. Chen, O. Engkvist, Y. Wang, M. Olivecrona, and T. Blaschke. The rise of deep learning in drug discovery. Drug Discovery Today, 23(6):1241–1259, 2018.   \nE.D. Cubuk, B. Zoph, D. Mane, V. Vasudevan, and Q.V. Le. AutoAugment: Learning augmentation strategies from data. In CVPR, 2019.   \nS. Daulton, M. Balandat, and E. Bakshy. Differentiable expected hypervolume improvement for parallel multi-objective Bayesian optimization. arXiv Preprint, pp. arXiv:2006.05078, 2020.   \nK. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182–197, 2002.   \nJ. Degen, C. Wegscheid-Gerlach, A. Zaliani, and M. Rarey. On the art of compiling and using ‘drug-like’ chemical fragment spaces. ChemMedChem, 3(10):1503–1507, 2008.   \nD. Duvenaud, D. Maclaurin, J. Aguilera-Iparraguirre, R. Bombarell, T. Hirzel, A. Aspuru-Guzik, and R.P. Adams. Convolutional networks on graphs for learning molecular fingerprints. In NIPS, 2015.   \nA.E. Eiben and J. Smith. From evolutionary computation to the evolution of things. Nature, 521 (2014):476–482, 2015.   \nD.A. Erlanson. Introduction to fragment-based drug discovery. Topics in Current Chemistry, 317: 1–32, 2011.   \nA. Goyal, A. Sordoni, M. Cote, N.R. Ke, and Y. Bengio. Z-Forcing: Training stochastic recurrent networks. In NIPS, 2017.   \nP.S. Gromski, A.B. Henson, J.M. Granda, and L. Cronin. How to explore chemical space using algorithms and automation. Nature Review Chemistry, 3:119–128, 2019.   \nM. Hauschild and M. Pelikan. An introduction and survey of estimation of distribution algorithms. Swarm and Evolutionary Computation, 1(3):111–128, 2011.   \nI. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner. beta-VAE: Learning basic visual concepts with a constrained variational framework. In ICLR, 2017.   \nJ.J. Irwin and B.K. Shoichet. ZINC - a free database of commercially available compounds for virtual screening. Journal of Chemical Information and Modeling, 45(1):177–182, 2005.   \nA. Kadurin, S. Nikolenko, K. Khrabrov, A. Aliper, and A. Zhavoronkov. druGAN: An advanced generative adversarial autoencoder model for de novo generation of new molecules with desired molecular properties in silico. Molecular Pharmaceutics, 14:3098–3104, 2017.   \nD.P. Kingma and M. Welling. Auto-encoding variational Bayes. In International Conference on Learning Representations, 2014.   \nG. Landrum. RDKit: Open-source cheminformatics, 2006. URL http://www.rdkit.org.   \nX.Q. Lewell, D.B. Judd, S.P. Watson, and M.M. Hann. RECAP - retrosynthetic combinatorial analysis procedure: a powerful new technique for identifying privileged molecular fragments with useful applications in combinatorial chemistry. Journal of Chemical Information and Computer Sciences, 38(3):511–522, 1998.   \nX. Li, I. Kiringa, T. Yeap, X. Zhu, and Y. Li. Anomaly detection based on unsupervised disentangled representation learning in combination with manifold learning. In International Joint Conference on Neural Networks, 2020.   \nS. Mandal, T.A. Anderson, J. Gottschlich, S. Zhou, and A. Muzahid. Learning fitness functions for genetic algorithms. arXiv Preprint, pp. arXiv:1908.08783, 2019.   \nT. Mikolov, K. Chen, G.S. Corrado, and J. Dean. Efficient estimation of word representations in vector space. In International Conference on Learning Representations, 2013.   \nL. Perez and J. Wang. The effectiveness of data augmentation in image classification using deep learning. arXiv Preprint, pp. arXiv:1712.04621, 2017.   \nM. Podda, D. Bacciu, and A. Micheli. A deep generative model for fragment-based molecule generation. In International Conference on Artificial Intelligence and Statistics, pp. 2240–2250, 2020.   \nM. Popova, O. Isayev, and A. Tropsha. Deep reinforcement learning for de novo drug design. Science Advances, 4:eaap7885, 2018.   \nR. Romez-Bombarelli, J.N. Wei, D. Duvenaud, J.M. Hernarndez-Lobato, B. Sanchez-Lengeling, D. Sheberla, J. Aguilera-Iparraguirre, T.D. Hirzel, R.P. Adams, and A. Aspuru-Guzik. Automatic chemical design using a data-driven continuous representation of molecules. ACS Central Science, 4:268–276, 2018.   \nT. Salimans, J. Ho, X. Chen, S. Sidor, and I. Sutskever. Evolution strategies as a scalable alternative to reinforcement learning. arXiv Reprint, pp. arXiv:1703.03864, 2017.   \nR. Sennrich, B. Haddow, and A. Birch. Improving neural machine translation models with monolingual data. In ACL, 2016.   \nC. Shorten and T.M. Khoshgoftaar. A survey on image data augmentation for deep learning. Journal of Big Data, 6:60, 2019.   \nS.B. Shuker, P.J. Hajduk, R.P. Meadows, and S.W. Fesik. Discovering high-affinity ligands for proteins: SAR by NMR. Science, 274(5292):11531–1534, 1996.   \nM. Simonovsky and N. Komodakis. GraphVAE: Towards generation of small graphs using variational autoencoders. In ICANN, pp. 412–422, 2018.   \nK.O. Stanley, J. Clune, J. Lehman, and R. Miikkulainen. Designing neural networks through neuroevolution. Nature Machine Intelligence, 1:24–30, 2019.   \nY. Wang, S.H. Bryant, T. Cheng, J. Wang, A. Gindulyte, B.A. Shoemaker, P.A. Thiessen, S. He, and J. Zhang. PubChem BioAssay: 2017 update. Nucleic Acids Research, 45(D1):D955–D963, 2016.   \nJ. Wei and K. Zou. EDA: Easy data augmentation techniques for boosting performance on text classification tasks. In EMNLP-IJCNLP, 2019.   \nD. Weininger. SMILES, a chemical language and information system. 1. introduction to methodology and encoding rules. Journal of Chemical Information and Computer Sciences, 28(1):31–36, 1988.   \nD. Wierstra, T. Schaul, T. Glasmachers, Y. Sun, J. Peters, and J. Schmidhuber. Natural evolution strategies. Journal of Machine Learning Research, 15(2014):949–980, 2014.   \nC. Yan, S. Wang, J. Yang, T. Xu, and J. Huang. Re-balancing variational autoencoder loss for molecule sequence generation. arXiv Preprint, pp. arXiv:1910.00698, 2020.   \nJ. You, B. Liu, R. Ying, V. Pande, and J. Leskovec. Graph convolutional policy network for goaldirected molecular graph generation. In NeurIPS, 2018.   \nZ. Zhou, S. Kearnes, L. Li, R.N. Zare, and P. Riley. Optimization of molecules vis deep reinforcement learning. Scientific Reports, 9:10752, 2019. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A.1 DEL ALGORITHM ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Algorithm 1: DEL Algorithm ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Inputs: $X _ { 0 }$ , $Y _ { 0 }$ : training samples and corresponding properties; Result: population of designed molecules; learned DGM; 1 for each evolutionary generation do 2 if first generation then 3 $X = X _ { 0 }$ ; $Y = Y _ { 0 }$ ; 4 Train DGM using $\\{ X , Y \\}$ ; // use all training data to learn DGM 5 $X , Y = \\operatorname { s u b s e t } ( X , Y , M )$ ; // a subset of $M$ examples from the original training data 6 else 7 Train DGM using $\\{ X , Y \\}$ ; // use population to further train DGM 8 end 9 end 10 Z = Encoder $( X )$ ; // obtain latent representations of samples in $\\boldsymbol { x }$ 11 $r , \\mathcal { F } =$ nondominated sort $( Y )$ ; // $\\pmb { r }$ : non-domination ranks, $\\mathcal { F }$ : Pareto frontiers 12 $d =$ compute crowding distance $( Y , { \\mathcal { F } } )$ ; 13 $Z ^ { \\prime } = \\operatorname { E v { \\bar { O p } } } ( Z , r , d )$ ; // evolutionary operations: selection, recombination and mutation 14 X0 = Decoder $( Z ^ { \\prime } )$ ; // generate novel samples 15 $\\mathbf { Y } ^ { \\prime } =$ get property $( X ^ { \\prime } )$ ; // obtain properties of the generated samples using a simulator, e.g. RDKit 16 $\\boldsymbol { X }$ , $\\mathbf { Y } =$ form new population $( { \\pmb X } , { \\pmb Y } , { \\pmb X } ^ { \\prime } , { \\pmb Y } ^ { \\prime } , { \\pmb M } )$ ; // keep the top $M$ good samples from the combination of $\\{ X , Y \\}$ and $\\{ X ^ { \\prime } , Y ^ { \\prime } \\}$ 17 end 18 Return X , Y , DGM ",
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+    {
+        "type": "text",
+        "text": "A.2 BUG CORRECTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the VAE implementation by Podda et al. (2020), the incorrect use of view function makes the forward flow of information messed up among different samples in a batch. ",
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+    {
+        "type": "text",
+        "text": "Listing 1: Wrong use of view function in the original VAE model in (Podda et al., 2020). , s t a t e $=$ s e l f . r n n ( p a c k e d , s t a t e ) # n u m l a y e r s b y b a t c h b y h i d d e n s i z e −> b a t c h b y h i d d e n l a y e r \\* h i d d e n s i z e s t a t e $=$ s t a t e . v i e w ( b a t c h s i z e , s e l f . h i d d e n s i z e \\* s e l f . h i d d e n l a y e r s ) mean $=$ s e l f . r n n 2 m e a n ( s t a t e ) # mean : b a t c h b y l a t e n t s i z e l o g v a r $=$ s e l f . r n n 2 l o g v ( s t a t e ) # l o g v : b a t c h b y l a t e n t s i z e ",
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+    {
+        "type": "text",
+        "text": "Listing 2: Our correction. ",
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+    },
+    {
+        "type": "text",
+        "text": "s t a t e $=$ s e l f . r n n ( p a c k e d , s t a t e ) # n u m l a y e r s b y b a t c h b y h i d d e n s i z e −> b a t c h b y n u m l a y e r s b y h i d d e n s i z e s t a t e $=$ s t a t e . t r a n s p o s e $^ { ( 1 , 0 ) }$ s t a t e $=$ s t a t e . f l a t t e n ( s t a r t d i m $= 1$ ) # b a t c h b y h i d d e n l a y e r \\* h i d d e n s i z e mean $=$ s e l f . r n n 2 m e a n ( s t a t e ) # mean : b a t c h b y l a t e n t s i z e l o g v a r $=$ s e l f . r n n 2 l o g v ( s t a t e ) # l o g v : b a t c h b y l a t e n t s i z e ",
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+    {
+        "type": "text",
+        "text": "A.3 HYPERPARAMETER SETTING FOR FRAGVAE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "When not specified in the main text, the following values of hyperparameters are used in experiments. ",
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+    {
+        "type": "text",
+        "text": "• size of the embedding layer: 128 ",
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+    {
+        "type": "text",
+        "text": "• mask low-frequency fragments: Yes   \n• masking frequency: 2   \n• number of low frequency clusters: 5   \n• window for word2vec embedding: 3   \n• number of recurrent neurons per layer: 128   \n• number of recurrent layers: 2   \n• size of the VAE latent space: 64   \n• maximum length of the sampled sequence: 10   \n• batch size: 128   \n• shuffle batches: Yes   \n• number of epochs to train: 50   \n• learning rate: 0.0005   \n• dropout for the recurrent layers: 0.3   \n• annealing step size for the scheduler: 4   \n• annealing rate for the scheduler: 0.8   \n• threshold to clip the gradient norm: 5   \n• number of layers in MLP for properties: 2   \n• number of hidden units in each hidden layer of MLP: 64   \n• weight on property regression loss $( \\alpha )$ : 1   \n• weight on KL divergence $( \\beta )$ : {0.01, 0.1, 1} ",
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+    {
+        "type": "text",
+        "text": "A.4 HYPERPARAMETER SETTING FOR DEL ",
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+    {
+        "type": "text",
+        "text": "When not specified in the main text, the following values of hyperparameters are used in experiments. ",
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+    {
+        "type": "text",
+        "text": "• number of evolutionary generations: 10   \n• popularization size: {20000 (default), 100000}   \n• number of epochs in the initial training of DGM: 50   \n• annealing step size for the scheduler in initial training of DGM: 4   \n• annealing rate for the scheduler: 0.8   \n• number of epochs in a subsequent training of DGM: 30   \n• learning rate in the initial training of DGM: 0.0005   \n• annealing step size for the scheduler in initial training: 2   \n• probability in tournament selection: 0.95   \n• crossover method: {linear (default), discrete}   \n• mutation rate: 0.01 ",
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+    {
+        "type": "text",
+        "text": "A.5 HYPERPARAMETER SETTING FOR MOBO ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When not specified in the main text, the following values of hyperparameters are used in experiments. ",
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+    {
+        "type": "text",
+        "text": "• number of initial data points: 1000   \n• number of batches: 30   \n• batch size: 8   \n• number of MC samples: 128 ",
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+    {
+        "type": "text",
+        "text": "A.6 ADDITIONAL TABLES ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7b588959462bcbe017c4e8e4f28e66c5069dce8c76c08ef370acb0ec66a4a590.jpg",
+        "table_caption": [
+            "Table 2: Performance of FragVAE in comparison with existing methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Data</td><td>Validity (SMILES)</td><td>Validity (Fragments)</td><td>Novelty</td><td>Diversity</td></tr><tr><td>ChemVAE GrammarVAE SDVAE GraphVAE CGVAE</td><td>ZINC ZINC ZINC ZINC ZINC</td><td>0.170 0.310 0.435 0.140 1.000</td><td></td><td>0.980 1.000 = 1.000 1.000</td><td>0.310 0.108 1 0.316 0.998</td></tr><tr><td>NeVAE FragVAE (β=1.00)</td><td>ZINC ZINC</td><td>1.000 1.000</td><td>■ 0.922</td><td>0.999 1.000</td><td>1.000 0.961</td></tr><tr><td>FragVAE (β=0.10)</td><td>ZINC</td><td>1.000</td><td>0.953</td><td>0.999</td><td>0.985</td></tr><tr><td>FragVAE(β=0.01)</td><td>ZINC</td><td>1.000</td><td>0.655</td><td>0.997</td><td>0.809</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FragVAE (β=1.00)</td><td>PCBA</td><td>1.000</td><td>0.443</td><td>1.000</td><td>0.925</td></tr><tr><td>FragVAE (β=0.10)</td><td>PCBA</td><td>1.000</td><td>0.481</td><td>0.983</td><td>0.886</td></tr><tr><td></td><td>PCBA</td><td>1.000</td><td>0.777</td><td>0.997</td><td></td></tr><tr><td>FragVAE (β=0.01)</td><td></td><td></td><td></td><td></td><td>0.632</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Running time of DEL and MOBO. Since both methods involve training FragVAE, the FragVAE training time was not counted in this table. Format: Hours:Minutes:Seconds. Note: when $\\beta$ is annealed to 0.4 in DEL, $\\alpha$ is annealed from 1 to 4. All experiments were carried out on a Dell Precision 5820 Workstation equipped with an Intel Xeon W-2255 CPU (10C), RAM of 128GB, and a Nvidia Quadro RTX 6000 (24GB) GPU. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Hyperparameter</td><td rowspan=1 colspan=1>DEL</td><td rowspan=1 colspan=1>MOBO</td></tr><tr><td rowspan=1 colspan=1>ZINCZINC</td><td rowspan=1 colspan=1>β=0.1→0.4β = 0.01 → 0.4</td><td rowspan=1 colspan=1>3:03:493:08:09</td><td rowspan=1 colspan=1>25:23:01141:49:37</td></tr><tr><td rowspan=1 colspan=1>PCBAPCBA</td><td rowspan=1 colspan=1>β=0.1→0.4β = 0.01 → 0.4</td><td rowspan=1 colspan=1>3:01:321:50:20</td><td rowspan=1 colspan=1>31:16:1845:31:38</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/df557786b22cafc6cdb4931414555eb675feb4518c606ba6d5d47f2ab31701cd.jpg",
+        "table_caption": [
+            "Table 4: Non-dominated sorting of combination of Pareto fronts from DEL with and without property prediction (PP) component as latent space regularization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>InNewPareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DELParameter</td><td rowspan=1 colspan=1>With PP</td><td rowspan=1 colspan=1>Without PP</td><td rowspan=1 colspan=1>With PP</td><td rowspan=1 colspan=1>Without PP</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.1β=0.1</td><td rowspan=1 colspan=1>235224</td><td rowspan=1 colspan=1>234108</td><td rowspan=1 colspan=1>206 (87.66%)216 (96.43%)</td><td rowspan=1 colspan=1>0(0%)0(0%)</td></tr></table>",
+        "bbox": [
+            174,
+            517,
+            802,
+            577
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/30d4858dd6d5c5ba63d11fe0b3e70edd8121f8b9d17e1ae4855e226b721a4ab1.jpg",
+        "table_caption": [
+            "Table 5: Non-dominated sorting of combination of Pareto fronts from DEL with and without DGM finetuning (FT) phrase. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>In New Pareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Parameter</td><td rowspan=1 colspan=1>With FT</td><td rowspan=1 colspan=1>Without FT</td><td rowspan=1 colspan=1>With FT</td><td rowspan=1 colspan=1>Without FT</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.1β=0.1</td><td rowspan=1 colspan=1>235224</td><td rowspan=1 colspan=1>193214</td><td rowspan=1 colspan=1>235 (100%)199 (88.84%)</td><td rowspan=1 colspan=1>0 (0%)0(0%)</td></tr></table>",
+        "bbox": [
+            174,
+            628,
+            803,
+            689
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/ca6180a73faaed9eda822faff50220519f3d166febca0f0c8c9e3fac14306c61.jpg",
+        "table_caption": [
+            "Table 6: Non-dominated sorting of combination of Pareto fronts from DEL with linear or discrete crossover operation. Note: when $\\beta$ is annealed to 0.4, $\\alpha$ is annealed from 1 to 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>InNewPareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Parameter</td><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>Discrete</td><td rowspan=1 colspan=1>Linear</td><td rowspan=1 colspan=1>Discrete</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.01→0.4β = 0.01 → 0.4</td><td rowspan=1 colspan=1>200183</td><td rowspan=1 colspan=1>181195</td><td rowspan=1 colspan=1>196 (98.00%)146 (79.78%)</td><td rowspan=1 colspan=1>173 (95.58%)184 (94.36%)</td></tr></table>",
+        "bbox": [
+            173,
+            742,
+            823,
+            803
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.7 ADDITIONAL FIGURES ",
+        "bbox": [
+            174,
+            827,
+            377,
+            842
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2693ed14b3fa0110d9f0677c262c2ee3929778f20a61706df214cad89eb6c202.jpg",
+        "table_caption": [
+            "Table 7: Non-dominated sorting of combination of Pareto fronts from DEL with population sizes of 20K and 100K. Note: when $\\beta$ is annealed to 0.4, $\\alpha$ is annealed from 1 to 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>Pareto Front Size for Comparison</td><td rowspan=1 colspan=2>In New Pareto Front</td></tr><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>DEL Parameter</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>100K</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>100K</td></tr><tr><td rowspan=1 colspan=1>ZINCPCBA</td><td rowspan=1 colspan=1>β=0.1→0.4β= 0.1 → 0.4</td><td rowspan=1 colspan=1>243228</td><td rowspan=1 colspan=1>316354</td><td rowspan=1 colspan=1>78 (32.10%)88 (38.60%)</td><td rowspan=1 colspan=1>304 (96.20%)334 (94.35%)</td></tr></table>",
+        "bbox": [
+            174,
+            166,
+            810,
+            227
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b725b5eba8ac556d8278efae4ac5cad2967b91eca2627019e6750fcc9cd745eb.jpg",
+        "table_caption": [
+            "Table 8: Non-dominated sorting of combination Pareto fronts from DEL with different values of $\\beta$ Note: when $\\beta$ is annealed to 0.4, $\\alpha$ is annealed from 1 to 4. Population size: 20K. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Pareto Front Size for Comparison</td><td colspan=\"4\">In New Pareto Front</td></tr><tr><td>Data</td><td></td><td></td><td></td><td>β=0.1β=0.1→0.4β=0.01β=0.01→0.4</td><td>β=0.1</td><td>β=0.1→0.4</td><td>β=0.01</td><td>β=0.01→0.4</td></tr><tr><td>ZINC</td><td>235</td><td>243</td><td>228</td><td>200</td><td>195(82.98%)</td><td>186(76.54%)</td><td>170(74.56%)</td><td>143(71.50%)</td></tr><tr><td>PCBA</td><td>224</td><td>228</td><td>189</td><td>183</td><td>169(75.45%)</td><td>183(80.26%)</td><td>122(64.55%)</td><td>111(60.66%)</td></tr></table>",
+        "bbox": [
+            173,
+            325,
+            857,
+            380
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/23f0f3b2d83c0ed5f0c0d0eed730882d0a82a87daf6744472979fc8754539116.jpg",
+        "table_caption": [
+            "Table 9: Numbers of novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the 1st, 5th and final (10th) generations of DEL. Numbers of training molecules satisfying same conditions are also given. Note: when $\\beta$ is annealed to 0.4, $\\alpha$ is annealed from 1 to 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Data</td><td>Hyperparameter</td><td>Train</td><td>DEL(1)</td><td>DEL(5)</td><td>DEL(F)</td></tr><tr><td>ZINC</td><td>β=0.1</td><td>406</td><td>226</td><td>14</td><td>40</td></tr><tr><td>ZINC ZINC</td><td>β= 0.1 → 0.4</td><td>-</td><td></td><td>22</td><td>32</td></tr><tr><td></td><td>β =0.01</td><td>-</td><td></td><td>54</td><td>115</td></tr><tr><td>ZINC</td><td>β = 0.01 →0.4</td><td>1</td><td>3</td><td>9</td><td>20</td></tr><tr><td>PCBA</td><td>β 三 ：0.1</td><td>196</td><td>4</td><td>13</td><td>27</td></tr><tr><td>PCBA</td><td>β 三 0.1→ 0.4</td><td>1</td><td>10</td><td>33</td><td>52</td></tr><tr><td>PCBA</td><td>β =0.01</td><td></td><td>0</td><td>8</td><td>16</td></tr><tr><td>PCBA</td><td>β= 0.01 → 0.4</td><td>=</td><td>4</td><td>9</td><td>10</td></tr></table>",
+        "bbox": [
+            269,
+            493,
+            728,
+            622
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/328b6d651080a5518fe92dc230d1ef865a134b0c74e8e19c2f2076d0fa8e1e06.jpg",
+        "image_caption": [
+            "Figure 6: Different of forms of the $\\beta$ -function. The total number of epochs is $T = 1 0 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            267,
+            684,
+            732,
+            859
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf6acad7e815d28bcf6de1c5e6a817e86f3ee5884ec5732eb3accd5c73ae61e8.jpg",
+        "image_caption": [
+            "Figure 7: Loss of FragVAE in initial training on ZINC and PCBA respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            238,
+            787,
+            761
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7dc3634a8cc5465345a0ddfa487e586e7e8d497188199065a872c6919ede4776.jpg",
+        "image_caption": [
+            "Figure 8: Loss of of FragVAE in DEL on ZINC and PCBA respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            119,
+            790,
+            869
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fd02795e1ba4d96b7253b15692d60e1a07210a48f43d6d027bf0e09523a9979.jpg",
+        "image_caption": [
+            "Figure 9: ZINC training molecules that satisfy properties QED≥ 0.88, $\\mathrm { S A S } { \\le } \\ 3$ , and $\\mathrm { l o g P \\le 1 }$ . Note: 196 molecules satisfy these conditions, but 64 are visualized. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            228,
+            877,
+            770
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/dc8483339e12e7ce17feb421a56e94d3857dac87b13fb316462716a16e2cb6a4.jpg",
+        "image_caption": [
+            "Figure 10: PCBA training molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ . Note: 406 molecules satisfy these conditions, but 64 are visualized. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            228,
+            880,
+            770
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef35348a43423ccd0019a79cb6e48a7c8be10d1d1efe34ab90c1e60683f3dacd.jpg",
+        "image_caption": [
+            "Figure 11: Novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\\beta = 0 . 1$ , $\\alpha = 1$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            143,
+            872,
+            474
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ea716c19ba88a986d55d50d66b4c69dcd77a6214a9b6cb51862c22d46759d97.jpg",
+        "image_caption": [
+            "Figure 12: Novel molecules that satisfy properties QE $\\mathrm { D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\\beta = 0 . 1  0 . 4$ , $\\alpha = 1  4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            546,
+            874,
+            854
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/172ac080ef55216a9030c88707c52ba819ffb87c66f6cb1b897817235f70d0c7.jpg",
+        "image_caption": [
+            "Figure 13: Novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\\beta = 0 . 0 1$ , $\\alpha = 1$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            228,
+            879,
+            770
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/806ff87c94e38c19ef495c75301659f701bd1b5aae14d172da0db77cd5e8cccd.jpg",
+        "image_caption": [
+            "Figure 14: Novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8 , \\mathrm { S A S } \\leq 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on ZINC with hyperparameter: $\\beta = 0 . 0 1  0 . 4$ , $\\alpha = 1  4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            68,
+            875,
+            371
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/e8a6adf652a4f14dd67d7916b6d690ab5aae4c7f0d88d519378988000de179ec.jpg",
+        "image_caption": [
+            "Figure 15: Novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8 , \\mathrm { S A S } \\leq 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\\beta = 0 . 1$ , $\\alpha = 1$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            558,
+            877,
+            819
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "image",
+        "img_path": "images/35a0a618896d192b72930ef12d4822e13dfa99d64a4b6f753f34b28166e817c2.jpg",
+        "image_caption": [
+            "Figure 16: Novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\\beta = 0 . 1 \\to 0 . 4$ , $\\alpha = 1  4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            137,
+            879,
+            612
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a98a373e39b584dc4538fe81c05c380bdb18d92cd9a7fb2311bd64e10c2d683.jpg",
+        "image_caption": [
+            "Figure 17: Novel molecules that satisfy properties QE $\\mathrm { D \\geq 0 . 8 8 }$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\\beta = 0 . 0 1$ , $\\alpha = 1$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            671,
+            874,
+            853
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/dd0f2739d6ed68f68e37c700db304d114aca75025c20b7ab80931fc7afdca46e.jpg",
+        "image_caption": [
+            "Figure 18: Novel molecules that satisfy properties $\\mathrm { Q E D } \\geq 0 . 8 8$ , $\\mathrm { S A S } \\le 3$ , and $\\mathrm { l o g P \\le 1 }$ in the final generation of DEL trained on PCBA with hyperparameter: $\\beta = 0 . 0 1  0 . 4$ , $\\alpha = 1  4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            64,
+            875,
+            267
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a0a15d3f0d6bf47f1866b819db5e2e1fca329240ee33b0166456c39c30395c2.jpg",
+        "image_caption": [
+            "Figure 19: Property and structural feature distributions of population samples during DEL $\\beta = 0 . 1$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            372,
+            815,
+            568
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/67bc0b4c81048eb4cfdc98e413d01fb25f64752d0db278b75060246553f20f4d.jpg",
+        "image_caption": [
+            "Figure 20: Property and structural feature distributions of population samples during DEL $\\beta =$ $0 . 1  0 . 4$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            660,
+            815,
+            854
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/98a8c8b650d38981128378951283efc70338aaf76b2167c7759a1030ba0d87a0.jpg",
+        "image_caption": [
+            "Figure 21: Property and structural feature distributions of population samples during DEL $\\beta =$ 0.01). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            121,
+            815,
+            314
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "image",
+        "img_path": "images/e5f677e00c8bd8817dbe030353fe6a77b089aec371fba7d29eee0397a4f6290d.jpg",
+        "image_caption": [
+            "Figure 22: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL $\\beta = 0 . 1 $ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            396,
+            815,
+            590
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "image",
+        "img_path": "images/54f7aa24e3ee1a9789d0e90cfea59f5623e96e01bc0d83ce4335eb0d0a4630aa.jpg",
+        "image_caption": [
+            "Figure 23: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL $\\beta = 0 . 1  0 . 4$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            672,
+            815,
+            867
+        ],
+        "page_idx": 23
+    },
+    {
+        "type": "image",
+        "img_path": "images/11db239c64935ba9a85ee39f3e8155ab627ec1d9eba86bdec28c8204ec1932a6.jpg",
+        "image_caption": [
+            "Figure 24: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL ( $\\beta = 0 . 0 1$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            189,
+            815,
+            383
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "image",
+        "img_path": "images/1de784e2340f7d61a2743f0c807a2ee7c1a7cc8a69b830f8fe3447d8bd117bfc.jpg",
+        "image_caption": [
+            "Figure 25: Property and structural feature distributions of randomly sampled molecules using FragVAE during DEL $\\beta = 0 . 0 1  0 . 4$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            603,
+            815,
+            797
+        ],
+        "page_idx": 24
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f5ae9516423a62c293d0e50ce5b806f01a7c1a1cc0d2cc73a5dab1ef0f80fff.jpg",
+        "image_caption": [
+            "Figure 26: Hypervolume along batches when running Sobol random search, qParEGO, and qEHVI. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            137,
+            815,
+            541
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba4e8b0d755b63ebd955c60e6d827a81d4974cc4eefedb4c9713f2a740582baa.jpg",
+        "image_caption": [
+            "Figure 27: Pareto fronts of DEL and MOBO algorithms $\\beta = 0 . 1 $ . In the legend, the last batch of qParEGO or qEHVI is written as Front 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            636,
+            820,
+            849
+        ],
+        "page_idx": 25
+    },
+    {
+        "type": "image",
+        "img_path": "images/63b287ba7dd5a16b8d67e19055298098726a60a3692270784ef5c7618042b008.jpg",
+        "image_caption": [
+            "Figure 28: Validity, novelty and diversity of population samples (after evolutionary operations and before merging with previous population) in different variants of DEL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            198,
+            877,
+            784
+        ],
+        "page_idx": 26
+    }
+]

parse/train/S1lF8xHYwS/S1lF8xHYwS.md

@@ -0,0 +1,360 @@
+# UNSUPERVISED DOMAIN ADAPTATION THROUGH SELF-SUPERVISION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper addresses unsupervised domain adaptation, the setting where labeled training data is available on a source domain, but the goal is to have good performance on a target domain with only unlabeled data. Like much of previous work, we seek to align the learned representations of the source and target domains while preserving discriminability. The way we accomplish alignment is by learning to perform auxiliary self-supervised task(s) on both domains simultaneously. Each self-supervised task brings the two domains closer together along the direction relevant to that task. Training this jointly with the main task classifier on the source domain is shown to successfully generalize to the unlabeled target domain. The presented objective is straightforward to implement and easy to optimize. We achieve state-of-the-art results on four out of seven standard benchmarks, and competitive results on segmentation adaptation. We also demonstrate that our method composes well with another popular pixel-level adaptation method.
+
+# 1 INTRODUCTION
+
+Visual distribution shifts are fundamental to our constantly evolving world. We humans face them all the time, e.g. when we navigate a foreign city, read text in a new font, or recognize objects in an environment we have never encountered before. These real-world challenges to the human visual perception have direct parallels in computer vision. Formally, a distribution shift happens when a model is trained on data from one distribution (source), but the goal is to make good predictions on some other distribution (target) that shares the label space with the source. Often computational models struggle even for pairs of distributions that humans find intuitively similar.
+
+Our paper studies the setting of unsupervised domain adaptation, with labeled data in the source domain, but only unlabeled data in the target domain. The general philosophy of the field is to induce alignment of the source and target domains through some transformation. In the context of deep learning, a convolutional neural network maps images to learned representations in some feature space, so inducing alignment is done by making the distribution shifts small between the source and target in this shared feature space (Csurka, 2017; Wang & Deng, 2018; Gopalan et al., 2011). If, in addition, such representations preserve discriminability on the source domain, then we can learn a good classifier on the source, which now generalizes to the target under the reasonable assumption that the representations of the two domains have the same ground truth.
+
+Most existing approaches implement this philosophy of alignment by minimizing a measurement of distributional discrepancy in the feature space (details in section 2), often some form of maximum mean discrepancy (MMD) e.g. Long et al. (2017), or a learned discriminator of the source and target as an approximation to the total variation distance e.g. Ganin et al. (2016). Both measurements lead to the formulation of the training objective as a minimax optimization problem a.k.a. adversarial learning, which is known to be very difficult to solve. Unless carefully balanced, the push and pull in opposite directions can often cause wild fluctuations in the discrepancy loss and lead to sudden divergence (details in section 2). Therefore, we propose to avoid minimax optimization altogether through a very different approach.
+
+Our main idea is to achieve alignment between the source and target domains by training a model on the same task in both domains simultaneously. Indeed, if we had labels in both domains, we could simply use our original classification task for this. However, since we lack labels in the target domain, we propose to use a self-supervised auxiliary task, which creates its own labels directly from the data (see section 3). In fact, we can use multiple self-supervised tasks, each one aligning the two domains along a direction of variation relevant to that task. Jointly training all the self-supervised tasks on both domains together with the original task on the source domain produces well-aligned representations as shown in Figure 1.
+
+![](images/795172d944f106b788fa84d04f7378b8ec489da53f9412ba67b01771d23a1719.jpg)  
+Figure 1: We propose a method for unsupervised domain adaptation that uses self-supervision to align the learned representations of two domains in a shared feature space. Here we visualize how these representations might be aligned in the feature space. a) Without our method, the source domain is far away from the target domain, and a source classifier cannot generalize to the target. b) Training a shared representation to support one self-supervised task on both domains can align the source and target along one direction. c) Using multiple self-supervised tasks can further align the domains along multiple directions. Now the source and target are close in this shared feature space, and the source classifier can hope to generalize to the target.
+
+Like all of deep learning, we can only empirically verify that at least in our experiments, the model does not overfit by internally creating a different decision boundary for each domain along different dimensions, which would then yield bad results. Recent research suggests that stochastic gradient descent is indeed unlikely to find such overfitting solutions with a costly decision boundary of high complexity (implicit regularization), even though the models have enough capacity (Zhang et al., 2016a; Neyshabur et al., 2017b; Arora et al., 2018).
+
+The key contribution of our work is to draw a connection between unsupervised domain adaptation and self-supervised learning. While we do not propose any fundamentally new self-supervised tasks, we offer insights in section 3 on how to select the right ones for adaptation, and propose in section 4 a novel training algorithm on those tasks, using batches of samples from both domains. Additionally, we demonstrate that domain alignment could be achieved with a simple and stable algorithm, without the need for adversarial learning. In section 5, we report state-of-the-art results on several standard benchmarks.
+
+# 2 RELATED WORK
+
+In this section we provide a brief overview of the two fields that our work would like to bridge.
+
+# 2.1 UNSUPERVISED DOMAIN ADAPTATION
+
+Methods for unsupervised domain adaptation in computer vision can be divided into three broad classes. The dominant class, which our work belongs to, aims to induce alignment between the source and the target domains in some feature space. This has been done by optimizing for some measurement of distributional discrepancy. One popular measurement is the maximum mean discrepancy (MMD) – the distance between the mean of the two domains in some reproducing kernel Hilbert space, where the kernel is chosen to maximize the distance (Bousmalis et al., 2016; Long et al., 2015; 2017). Another way to obtain a measurement of discrepancy is to train an adversarial discriminator that distinguishes between the two domains (Ganin & Lempitsky, 2014; Ganin et al., 2016; Tzeng et al., 2017). However, both MMD and adversarial training are formulated as minimax optimization problems, which are widely known, both in theory and practice, to be very difficult (Fedus et al., 2017; Duchi et al., 2016; Liang, 2017; Jin et al., 2019). Since the optimization landscape is much more complex than in standard supervised learning, training often does not converge or converges to a bad local minimum (Goodfellow, 2016; Nagarajan & Kolter, 2017; Li et al., 2017), and requires carefully balancing the two sets of parameters (for minimization and maximization) so one does not dominate the other (Salimans et al., 2016; Neyshabur et al., 2017a).
+
+To make minimax optimization easier, researchers have proposed numerous modifications to the loss function, network design, and training procedure (Arjovsky et al., 2017; Gulrajani et al., 2017; Karras et al., 2017; Courty et al., 2017; Sun & Saenko, 2016; Shu et al., 2018; Sener et al., 2016). Over the years, these modifications have yielded practical improvements on many standard benchmarks, but have also made the state-of-the-art algorithms very complicated. Often practitioners are not sure which tricks are necessary for which applications, and implementing these tricks can be bug-prone and frustrating. To make matters worse, since there is no labeled target data available for a validation set, practitioners have no way to perform hyper-parameter tuning or early stopping.
+
+The second class of methods directly transforms the source images to resemble the target images with generative models (Taigman et al., 2016; Hoffman et al., 2017; Bousmalis et al., 2017). While similar to the first class in the philosophy of alignment, these methods operate on image pixels directly instead of an intermediate representation space, and therefore can benefit from an additional round of adaptation in some representation space. In subsection 5.2 we demonstrate that composing our method with a popular pixel-level method yields stronger performance than either alone.
+
+The third class of methods uses a model trained on the labeled source data to estimate labels on the target data, then trains on some of those estimated pseudo-labels (e.g. the most confident ones), therefore bootstrapping through the unlabeled target data. Sometimes called self-ensembling (French et al., 2017), this technique is borrowed from semi-supervised learning, where it is called co-training (Saito et al., 2017; Zou et al., 2018; Chen et al., 2018; 2011). In contrast, our method uses joint training (of the main and self-supervised tasks), different from co-training in every aspect except the name.
+
+# 2.2 SELF-SUPERVISED FEATURE LEARNING
+
+The emerging field of self-supervised learning uses the machinery of supervised learning on problems where external supervision is not available. The idea is to use data itself as supervision for auxiliary (also called “pretext”) tasks that learn deep feature representations which will hopefully be informative for downstream “real” tasks. Many such auxiliary tasks have been proposed in the literature, including colorization (predicting the chrominance channels of an image given its luminance) (Zhang et al., 2016b; Larsson et al., 2017; Zhang et al., 2017), image inpainting Pathak et al. (2016), spatial context prediction (Doersch et al., 2015), solving jigsaw puzzles (Noroozi & Favaro, 2016), image rotation prediction (Gidaris et al., 2018), predicting audio from video (Owens et al., 2016), contrastive predictive coding (Oord et al., 2018), etc. Researchers have also experimented with self-supervision on videos (Wang & Gupta, 2015; Wang et al., 2019), and combining multiple self-supervised tasks together (Doersch & Zisserman, 2017).
+
+Typically, self-supervision is used as a pre-training step on unlabeled data (e.g. the ImageNet training set without labels) to initialize a deep learning model, followed by fine-tuning on a labeled training set (e.g. PASCAL VOC) and evaluating on the corresponding test set. Instead, in this paper, we train the self-supervised tasks together with the main supervised task, encouraging a consistent representation that both aligns the two domains and does well on the main task1.
+
+Recently, self-supervision has also been used for other problem settings, such as improving robustness (Hendrycks et al., 2019), domain generalization (Carlucci et al., 2019) and few-short learning (Su et al., 2019). The most relevant paper to us is Ghifary et al. (2016), which uses self-supervision for unsupervised domain adaptation, but not through alignment. Their algorithm trains a denoising autoencoder Vincent et al. (2008) only on the target data, together with the main classifier only on the labeled source data. They argue theoretically that this is better than training the autoencoder on both domains together. However, their theory is based on the critical assumption that the domains are already aligned, which is rarely true in practice. Consequently, their empirical results are much weaker than ours, as discussed in section 5. Please see Appendix A for more detailed comparisons with these works.
+
+The design of auxiliary self-supervised tasks is an exciting area of research in itself, with many successful examples listed in section 2. However, not all of them are suitable for unsupervised domain adaptation. In order to induce alignment between the source and target, the labels created by selfsupervision should not require capturing information on the very factors where the domains are meaninglessly different, i.e. the factors of variation that we are trying to eliminate through adaptation.
+
+A particularly unsuitable tasks are these that try to predict pixels of the original image, as image inpainting (Pathak et al., 2016), colorization (Zhang et al., 2016b; Larsson et al., 2017; Zhang et al., 2017) or denoising autoencoder(Vincent et al., 2008). The success of pixel-wise reconstruction depends strongly on brightness information, or other factors of variation in overall appearance (e.g. sunny vs. coudy) that are typically irrelevant to high-level visual concepts. Thus, instead of inducing alignment, learning a pixel reconstruction task would instead serve to further separate the domains. We have experimented with using the colorization task and the denoising autoencoder for our training algorithm, and found their performance little better than the source only baseline, sometimes even worse! 2
+
+Table 1: Transformations and labels on a sample input image created by two of the self-supervised tasks used in our work.   
+
+<table><tr><td>Task</td><td colspan="3">Images and self-supervised labels</td></tr><tr><td>Rotation</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">Location</td><td></td><td></td><td>90°180°270°</td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>(0,0）(1,0）(0,1）(1,1)</td><td></td></tr></table>
+
+In general, classification tasks that predict structural labels seems better suited for our purpose than reconstruction tasks that predict pixels. Therefore, we have settled on three classification-based self-supervised tasks that combine simplicity with high performance:
+
+Rotation Prediction: (Gidaris et al., 2018) An input image is rotated in 90-degree increments i.e. $0 ^ { \circ }$ , $9 0 ^ { \circ }$ , $1 8 0 ^ { \circ }$ , and $2 7 0 ^ { \circ }$ ; the task is to predict the angle of rotation as a four-way classification problem.
+
+Flip Prediction: An input image is randomly flipped vertically; the task is to predict whether the image is flipped or not 3.
+
+Patch Location Prediction: Patches are randomly cropped out of an input image; the task is to predict where the patches come from 4.
+
+Consider a trivial illustrative example where the source and target are exactly the same except that the target pixels are all scaled down by a constant factor (e.g. daylight to dusk transition). All three of the aforementioned forms of self-supervision are suitable for this example, because pixel scaling i.e. brightness is “orthogonal” to the prediction of rotation, flip and location.
+
+# 4 METHOD
+
+Our training algorithm is simple once a set of $K$ self-supervised tasks are selected. We already have the loss function on the main prediction task, denoted $L _ { 0 }$ , that we do not have target labels for. Each self-supervised task corresponds to a loss function $\mathcal { L } _ { k }$ for $k = 1 . . . K$ . So altogether, our optimization problem is set up as a combination of these $K + 1$ loss functions, as is done for standard multi-task learning (see Figure 2). Implementation details are included in Appendix B.
+
+![](images/9c367ab6075ed42c17750fd5c6ed7ab393c659e26edb8df14bdae382b1f775b7.jpg)  
+Figure 2: Our method jointly trains a supervised head on labeled source data and self-supervised heads on unbaled data from both domains. The heads use high-level features from a shared encoder, which learns to align the feature distributions.
+
+Each loss function corresponds to a different “head” $h _ { k }$ for $k = 0 . . . K$ , which produces predictions in the respective label space. All the task-specific heads (including $h _ { 0 }$ for the actual prediction task) share a common feature extractor $\phi$ . Altogether, the parameters of $\phi$ and $h _ { k } , k = 0 , . . k$ are the learned variables i.e. the free parameters of our optimization problem.
+
+In our paper, $\phi$ is a deep convolutional neural network, and every $h _ { k }$ is simply a linear layer i.e. multiplication by a matrix of size output space dimension $\times$ feature space dimension. If $k \mathrm { t h }$ task is classification in nature, then $h _ { k }$ also has a softmax or sigmoid (for multi-class or binary) following the linear layer. The output space dimension is only four for rotation and location classification, and two for flip and location regression. Depending on the network architecture used, the feature space dimension ranges between 64 and 512. The point is to make every $h _ { k }$ low capacity, so the heads are forced to share high-level features, as is desirable for inducing alignment. A linear map from the highest-level features to the output is the smallest possible head and performs well empirically.
+
+Let ${ \cal { S } } = \{ ( x _ { i } , y _ { i } ) , i = 1 . . . m \}$ contain the labeled source data, and $T = \{ ( x _ { i } ) , i = 1 . . . n \}$ contain the unlabeled target data. $L _ { 0 }$ for the main prediction task takes in the labeled source data, and produces the following term in our objective:
+
+$$
+\mathcal { L } _ { 0 } \big ( S ; \phi , h _ { 0 } \big ) = \sum _ { ( x , y ) \in S } L _ { 0 } \big ( h _ { 0 } \big ( \phi ( x ) \big ) , y \big ) .
+$$
+
+Each self-supervised task $F _ { k }$ for $k = 1 . . . K$ modifies the input samples with some transformation $f _ { k }$ and creates labels $\tilde { y }$ . Denote $F _ { k } ( S ) = \{ ( f _ { k } ( x _ { i } ) , \tilde { y } _ { i } ) , i = 1 . . . m \}$ as the self-supervised samples generated from the source samples (with the original labels discarded), and $F _ { k } ( \bar { T ( \cdot ) } = \{ ( f _ { k } ( x _ { i } ) \bar { , } \tilde { y } _ { i } ) , \bar { \cdot } =$ $\left. 1 . . n \right\}$ from the target. Then the loss $L _ { k }$ of each task $k = 1 . . . K$ produces the following term:
+
+$$
+\mathcal { L } _ { k } ( S , T ; \phi , h _ { k } ) = \sum _ { ( f _ { k } ( x ) , \tilde { y } ) \in F ( S ) } L _ { k } ( h _ { k } ( \phi ( f _ { k } ( x ) ) ) , \tilde { y } ) + \sum _ { ( f _ { k } ( x ) , \tilde { y } ) \in F ( T ) } L _ { k } ( h _ { k } ( \phi ( f _ { k } ( x ) ) ) , \tilde { y } ) .
+$$
+
+Note that $\mathcal { L } _ { k } ( S , T ; \phi , h _ { k } )$ for $k = 1 . . . K$ , unlike $\mathcal { L } _ { 0 } ( S ; \phi , h _ { 0 } )$ , take in both the source and target data; as we emphasize for many times throughout the paper, this is critical for inducing alignment. Altogether, our optimization problem can be formalized as in multi-task learning 5:
+
+$$
+\operatorname* { m i n } _ { \phi , h _ { k } , k = 1 \ldots K } \quad { \mathcal { L } } _ { 0 } ( S ; \phi , h _ { 0 } ) + \sum _ { k = 1 } ^ { K } { \mathcal { L } } _ { k } ( S , T ; \phi , h _ { k } ) .
+$$
+
+![](images/f8fec6c4048a733744578d415f2110d66a542b8d6f4a9a9be3d77830439c6ed6.jpg)  
+Figure 3: Results on MNIST MNIST-M (left) and CIFAR- $\mathrm { \cdot 1 0 {  } S T L \mathrm { - } 1 0 }$ (right). Test error converges smoothly on the source and target domains for the main task as well as the self-supervised task. This kind of smooth convergence is often seen in supervised learning, but rarely in adversarial learning. The centroid distance (linear MMD) between the feature distributions of the two domains converges alongside, even though it is never explicitly optimized for.
+
+At test-time, we discard the self-supervised heads and use $h _ { 0 } ( \phi ( x ) )$ .
+
+# 4.1 A HEURISTIC FOR HYPER-PARAMETER TUNING AND EARLY STOPPING
+
+As previously mentioned, in the unsupervised domain adaptation setting, there is no target label available and therefore no target validation set, so typical strategies for hyper-parameter tuning and early stopping that require a validation set cannot be applied. This problem remains underappreciated; in fact, it is often unclear how previous works select their hyper-parameters or detemine when training is finished, both of which can be important factors that impact performance, especially for complex algorithms using adversarial learning. In this subsection we describe a simple heuristic 6, merely as rule-of-thumb to make things work instead of a technical innovation. Like almost all of deep learning, there is no statistical guarantee on this heuristic, but it is shown to be practically effective in our experiments (see Figure 3). We only hope that it serves as the first guess of a solution towards this underappreciated problem.
+
+The main idea is that, because our method never explicitly optimizes for any measurement of distributional discrepancy, these measurements can instead be used for hyper-parameter tuning and early stopping. Since it would be counterproductive to introduce additional parameters in order to perform hyper-parameter tuning, we simply use the distance between the mean of the source and target samples in the learned representation space, as produced by $\phi$ . Formally, this can be expressed as
+
+$$
+D ( S ^ { \prime } , T ^ { \prime } ; \phi ) = \biggl \| \frac { 1 } { m } \sum _ { x \in S ^ { \prime } } \phi ( x ) - \frac { 1 } { n } \sum _ { x \in T ^ { \prime } } \phi ( x ) \biggl \| _ { 2 } ,
+$$
+
+where $S ^ { \prime }$ and $T ^ { \prime }$ are unlabeled source and target validation sets 7.
+
+Our heuristic combines $D ( S ^ { \prime } , T ^ { \prime } ; \phi )$ and the main task error on the (labeled) source validation set. Denote $\mathbf { v } = ( v _ { 1 } , . . . , v _ { T } )$ and $\mathbf { w } = ( w _ { 1 } , . . . , w _ { T } )$ the measurement vectors of those two quantities respectively over $T$ epochs. The final measurement vector is $\mathbf { u } = \mathbf { v } / \operatorname* { m i n } ( \mathbf { v } ) + \mathbf { w } / \operatorname* { m i n } ( \mathbf { \bar { w } } )$ i.e. a normalized sum of the two vectors; the epoch at which we perform early stopping is then simply arg $\operatorname* { m i n } _ { t \in \{ 1 . . . T \} } \mathbf { u } _ { t }$ . Intuitively, this heuristic roughly corresponds to our goal of inducing alignment while preserving discriminability.
+
+<table><tr><td>Source Target</td><td>MNIST MNIST-M</td><td>MNIST SVHN</td><td>SVHN MNIST</td><td>MNIST USPS</td><td>USPS MNIST</td><td>CIFAR-10 STL-10</td><td>STL-10 CIFAR-10</td></tr><tr><td>DANN (Ganin et al.,2016)</td><td>81.5</td><td>35.7</td><td>73.6</td><td>-</td><td>-</td><td>-</td><td>=</td></tr><tr><td>DRCN (Ghifary et al.,2016)</td><td>-</td><td>40.1</td><td>82.0</td><td>=</td><td></td><td>66.4</td><td>58.7</td></tr><tr><td>DSN (Bousmalis et al., 2016)</td><td>83.2</td><td>=</td><td>82.7</td><td></td><td>=</td><td>1</td><td>1</td></tr><tr><td>kNN-Ad (Sener et al., 2016)</td><td>86.7</td><td>40.3</td><td>78.8</td><td>=</td><td></td><td>=</td><td>1</td></tr><tr><td>PixelDA (Bousmalis et al., 2017)</td><td>98.2</td><td></td><td></td><td></td><td></td><td>=</td><td>=</td></tr><tr><td>ATT (Saito et al., 2017)</td><td>94.2</td><td>52.8</td><td>86.2</td><td></td><td></td><td>-</td><td>1</td></tr><tr><td>II-Model (French et al., 2017)</td><td>-</td><td>71.4</td><td>92.0</td><td>=</td><td></td><td>76.3</td><td>64.2</td></tr><tr><td>ADDA (Tzeng et al., 2017)</td><td>=</td><td>1</td><td>76.0</td><td>89.4</td><td>90.1</td><td>-</td><td>1</td></tr><tr><td>CyCADA (Hoffman et al., 2017)</td><td>=</td><td>-</td><td>90.4</td><td>95.6</td><td>96.5</td><td>1</td><td>1</td></tr><tr><td>VADA (Shu et al.,2018)</td><td>97.7</td><td>47.5</td><td>97.9</td><td>1</td><td>1</td><td>80.0</td><td>73.5</td></tr><tr><td>DIRT-T (Shu et al., 2018)</td><td>98.9</td><td>54.5</td><td>99.4</td><td>=</td><td>1</td><td>-</td><td>75.3</td></tr><tr><td>VADA (IN) (Shu et al., 2018)</td><td>95.7</td><td>73.3</td><td>94.5</td><td>=</td><td></td><td>78.3</td><td>71.4</td></tr><tr><td>DIRT-T (IN) (Shu et al., 2018)</td><td>98.7</td><td>76.5</td><td>99.4</td><td>-</td><td>-</td><td>1</td><td>73.3</td></tr><tr><td>Source only VADA &amp; DIRT-T</td><td>58.5</td><td>27.9</td><td>77.0</td><td>-</td><td></td><td>76.3</td><td>63.6</td></tr><tr><td>Source only VADA &amp; DIRT-T (IN)</td><td>59.9</td><td>40.9</td><td>82.4</td><td>-</td><td>-</td><td>77.0</td><td>62.6</td></tr><tr><td>Source only our method</td><td>44.9</td><td>30.5</td><td>92.2</td><td>94.7</td><td>81.4</td><td>75.6</td><td>58.8</td></tr><tr><td>R</td><td>98.9</td><td>61.3</td><td>85.88</td><td>96.5</td><td>90.2</td><td>81.2</td><td>66.9</td></tr><tr><td>R+L+F</td><td>-</td><td>1</td><td>=</td><td>-</td><td>1</td><td>82.1</td><td>75.5</td></tr></table>
+
+Table 2: Test accuracy $( \% )$ on standard domain adaptation benchmarks. Our results are organized according to the self-supervised task(s) used: R for rotation, L for location, and F for flip. We achieve state-of-the-art accuracy on four out of the seven benchmarks.
+
+# 5 EXPERIMENTS
+
+# 5.1 SEVEN BENCHMARKS FOR OBJECT RECOGNITION
+
+The seven benchmarks are based on the six datasets described in Appendix D, each with a predefined training set / test set split, and labels are available on both splits. Previous works (cited in Table 2) have created those seven benchmarks by picking pairs of datasets with the same label space, treating one as the source and the other as the target, and training on the training set of the source with labels revealed and of the target with labels hidden. Following the standard setup of the field, labels on the target test set should only be used for evaluation, not for hyper-parameter tuning or early stopping; therefore we apply the heuristic described in subsection 4.1.
+
+For the two natural scene benchmarks, we use all three tasks described in section 3: rotation, location and flip prediction. For the five benchmarks on digits we do not use location because it yields trivial solutions that do not encourage the learning of semantic concepts. Given the image of a digit cropped into the four quadrants, location prediction can be trivially solved by looking at the four corners where a white stroke determines the category. Adding flip to the digits does not hurt performance, but does not improve significantly either, so we do not report those results separately.
+
+As shown in Table 2, despite the simplicity of our method, we achieve state-of-the-art accuracy on four out of the seven benchmarks. In addition, we show the source only results from our closest competitor (VADA and DIRT-T), and note that our source only results are in fact lower than theirs on those very benchmarks that we perform the best on; this indicates that our success is indeed due to effectiveness in adaptation instead of the base architecture.
+
+Our method fails on the pair of benchmarks with SVHN, on which rotation yields trivial solutions. Because SVHN digits are cropped from house numbers with multiple digits, majority of the images have parts of the adjacent digits on the side. The main task head needs to look at the center, but the rotation head learns to look at the periphery and cheat. This failure case shows that the success of our method is tied to how well the self-supervised task fits the application. Practioners should use their domain knowledge evaluate how well the task fits, instead of blithely apply it to everything.
+
+Also seen in Table 2 is that our method excels at object recognition in natural scenes, especially with all three tasks together. For STL- $1 0 {  }$ CIFAR-10, our base model is considerably worse than that of VADA and DIRT-T, but still beats all the baselines. Adding location and flip gives an improvement of $8 \%$ , on top of the $8 \%$ already over source only. This is not surprising since those tasks were originally developed for ImageNet pre-training i.e. object recognition in natural scenes – our method is very successful when the task fits the application well.
+
+Table 3: Test accuracy $( \% )$ on $\mathrm { G T A } 5 $ Cityscapes. Our method significantly improves over source only, and also over CyCADA when combined. This indicates that additional self-supervision using our training algorithm further aligns the domains.   
+
+<table><tr><td></td><td colspan="13">GTA5→ Cityscapes</td><td colspan="7"></td></tr><tr><td></td><td></td><td>seeapra 12.7</td><td>Buiping 39.6</td><td></td><td></td><td></td><td></td><td>trggee</td><td>usrs ogen</td><td>enee</td><td>Eirrin</td><td>3</td><td>uos.iad</td><td></td><td>u</td><td></td><td>1</td><td></td><td></td><td>mooiritte</td><td>aeleir</td><td>mIoU</td></tr><tr><td>Source only Ours</td><td>28.8 69.9</td><td></td><td>22.7</td><td>69.7</td><td>9.4 18.1</td><td>3.5 9.9</td><td>18.1 13.5</td><td>22.7 18.7</td><td>9.4 8.9</td><td>80.9 80.3</td><td>12.4 19.4</td><td>45.8 58.4</td><td></td><td>53.9 53.8</td><td>9.6 2.6</td><td>74.7 75.1</td><td>20.9 13.6</td><td>15.0 5.2</td><td>0.0 0.3</td><td>19.4 8.1</td><td>3.9 1.2</td><td>25.3 28.9</td></tr><tr><td>CyCADA</td><td></td><td>79.1 33.1</td><td>77.923.4</td><td></td><td></td><td>17.332.1</td><td></td><td></td><td>33.3 31.8 81.5</td><td></td><td>26.7</td><td>69.0</td><td>62.8</td><td></td><td></td><td>14.774.520.925.6</td><td></td><td></td><td>6.9</td><td></td><td>18.820.4</td><td>39.5</td></tr><tr><td>Ours + CyCADA</td><td>86.6</td><td>37.8</td><td>80.8</td><td>29.7</td><td></td><td>16.4</td><td>28.9</td><td>30.9</td><td>22.2</td><td>83.8</td><td>37.1</td><td>76.9</td><td>60.1</td><td>7.8</td><td>84.1</td><td></td><td>30.8</td><td>32.1</td><td>1.2</td><td>23.2</td><td>13.3</td><td>41.2</td></tr><tr><td>Oracle</td><td>97.3</td><td>79.8</td><td>88.6</td><td>32.5</td><td></td><td>48.2</td><td>56.3</td><td>63.6</td><td>73.3</td><td>89.0</td><td>58.9</td><td>93.0</td><td>78.2</td><td>55.2</td><td>92.2</td><td></td><td>45.0</td><td>67.3</td><td>39.6</td><td>49.9</td><td>73.6</td><td>67.4</td></tr></table>
+
+# 5.2 BENCHMARK FOR SEMANTIC SEGMENTATION
+
+To experiment with our method in more diverse applications, we also evaluate on a challenging simulation-to-real benchmark for semantic segmentation – $\mathbf { \partial } \cdot \mathbf { G T A } 5  \mathbf { \langle }$ Cityscapes. GTA5 (Richter et al., 2016) contains 24,966 video frames taken from the computer game, where dense segmentation labels are automatically given by the game engine. Cityscapes (Cordts et al., 2016) contains 5,000 video frames taken from real-world dash-cams. The main task is to classify every pixel in an image as one of the 19 classes shared across both datasets, and accuracy is measured by intersection over union (IoU). The best possible results are given as the oracle, when the labels on Cityscapes are available for training, so typical supervised learning methods are applicable.
+
+Our results are shown in Table 3, and implementation details in Appendix F. Our self-supervised tasks were designed for classification, where the label on an image depends on its global content, while in segmentation the labels tend to be highly local. Nevertheless, with very little modification, we see significant improvements over the source only baseline. In Appendix G we provide visualizations of the segmentation results, and make qualitative comparisons between those produced by the baseline and our method.
+
+We also experiment with combining our method with another popular unsupervised domain adaptation method – CyCADA (Hoffman et al., 2017), designed specifically for segmentation. Surprisingly, when operating on top of images produced by this already very strong baseline, our method further improves performance. This demonstrates that pixel-level adaptation methods might still benefit from an additional round of adaptation by inducing alignment through self-supervision. We emphasize that these results are obtained with a very simple instantiation of our method, as a start towards the development of self-supervised tasks more suitable for semantic segmentation.
+
+# 6 DISCUSSION
+
+We hope that this work encourages future researchers in unsupervised domain adaptation to consider the study of self-supervision as an alternative to adversarial learning, and researchers in selfsupervision to consider designing tasks and evaluating them in our problem setting. Most selfsupervised tasks today were originally designed for pre-training and evaluated in terms of accuracy gains on a downstream recognition, localization or detection tasks. It will be interesting to see if new self-supervised tasks can arise from the motivation of adaptation, for which alignment is the key objective. Moreover, domain experts could perhaps incorporate their dataset specific knowledge into the design of a self-supervised task specifically for their application.
+
+One additional advantage of our method, not considered in this paper, is that it might be particularly amenable to very small target sample size, when those other methods based on adversarial learning cannot accurately estimate the target distribution. We leave this topic for the future work.
+
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+
+# APPENDIX
+
+# A ADDITIONAL DISCUSSION ON SPECIFIC PAPERS RELATED TO OURS
+
+In this section we discuss the four papers mentioned in section 2 that are related to ours, but different in training algorithm, philosophy or problem setting.
+
+Deep Reconstruction-Classification Networks (Ghifary et al., 2016). This method works in unsupervised domain adaptation, the same problem setting as ours. It learns a denoising autoencoder on the target, together with the main classifier on the source. However, they use only the target domain for reconstruction, claiming both empirically and theoretically that it is better than using both domains together. In contrast, we use both the source and target for self-supervision. These two algorithmic differences really reflect our fundamental difference in philosophy. They take the target data as analogous to the unlabeled (source) data in semi-supervised learning, so any form of self-supervision suitable for semi-supervised learning is good enough; their theoretical analysis is also directly borrowed from that of semi-supervised learning, which concludes that it is necessary and sufficient to only use the target data for self-supervision. We take data from both domains for self-supervision in order to align their feature distributions; also, both conceptually and empirically, we cannot use reconstruction tasks because they are unsuitable for inducing alignment (see section 3).
+
+Empirical experiments further support our arguments in three ways: 1) If we were to only use the target for our self-supervision tasks, we would perform barely better than the source only baseline, even on MNIST MNIST-M, the easiest benchmark where we would otherwise observe a huge improvement. 2) If we were to use the reconstruction task i.e. a denoising autoencoder, we would again perform barely better than the source only baseline, and sometimes even worse, as described in section 3. 3) As shown in Table 2, our results are much better than theirs on all of the benchmarks by a large margin. This shows that implementing the philosophy of alignment, developed for unsupervised domain adaptation, is much superior in our problem setting to blithely borrowing from semi-supervised learning.
+
+In addition to the three classes of methods covered in section 2, unsupervised domain adaptation has also been studied using sample-to-sample correspondence through graph matching (Das & George Lee, 2018; Das & Lee, 2018); these methods are less relevant to ours and not discussed in detail.
+
+Using Self-Supervised Learning Can Improve Model Robustness and Uncertainty (Hendrycks et al., 2019). This paper studies the setting of robustness, where there is no sample provided, even unlabeled, from the target domain. Their method jointly trains for the main task and the self-supervised task only on the source domain, really because there is no target data of any kind. Because their problem setting is very challenging (with so little information provided on the target), their primary evaluation is on a dataset of CIFAR-10 images with the addition of 15 types of common corruptions (e.g. impulse noise, JPEG compression, snow weather, motion blur and pixelation), simulated by an algorithm. No idea on unsupervised domain adaptation is mentioned.
+
+Domain Generalization by Solving Jigsaw Puzzles (Carlucci et al., 2019). This paper studies two setting. The first is the robustness setting, exactly the same as in Hendrycks et al. (2019), except that evaluation is done on MNIST MNIST-M and MNIST ${ } ^ { \prime } {  } S \mathbf { V } \mathbf { H } \mathbf { N }$ . Their baseline is also a method from the robustness community that trains on adversarial examples and uses no target data. Again, because their problem setting is very challenging, the accuracy is low for both their proposed method and the baseline. The second setting is called domain generalization, which is very similar to meta-learning. The goal is to perform well simultaenously on multiple distributions, all labeled. Evaluation is done using the mean accuracy on all the domains. Beside the name, there is little similarity between the setting of unsupervised domain adaptation and domain generalization, which has no unsupervised component.
+
+Boosting Supervision with Self-Supervision for Few-shot Learning (Su et al., 2019). As evident from the title, this paper studies the setting of few-shot learning, where the goal is to perform well on a very small dataset. Again, there is no unsupervised component and little connection to our setting.
+
+Most of our results have already been produced when Carlucci et al. (2019) and Su et al. (2019) were presented at a conference. Hendrycks et al. (2019) was submitted to NeurIPS 2019 and should be considered concurrent work.
+
+# B ADDITIONAL ALGORITHMIC DETAILS
+
+In practice and for our experiments, the source and target datasets are often imbalanced in size. If we were to blithely solve for the objective in Equation 2, the domain with a larger dataset would carry more weight for every $\mathcal { L } _ { k } ( S , T ; \phi , h _ { k } ) , k = 1 . . . K$ because we are summing over all samples in both datasets. We would like to keep the two sums inside $\mathcal { L } _ { k }$ roughly balanced such that in terms of features produced by $\phi$ , neither of the two domains would dominate the other. This is easy to achieve in our implementation through balanced batches. When we need to sample a batch for task $k$ , we simply sample half the batch from the source, and another half from the target, then put them together. In the end, our implementation requires little change on top of an existing supervised learning codebase. Each self-supervised task is defined as a module, and adding a new one only amounts to defining the structural modifications and the loss function.
+
+We optimize the objective in Equation 2 with stochastic gradient descent (SGD). One can simply take a joint step on Equation 2, which covers all the losses for $k = 0$ and $k = 1 . . . K$ . However, the implementation would then have to store the gradients with respect to all $K + 1$ losses together. For memory efficiency, our implementation loops over $k = 1 . . . K$ ; for each self-supervised task $k$ , it samples a batch of combined source and target data (without their original labels), structurally modifies the images by $f _ { k }$ , creates new labels according to the modification, and obtains a loss for $h _ { k }$ and $\phi$ on this batch. So a gradient step is taken for each $k = 1 . . . K$ , before finally a batch of original source images and labels are sampled for a gradient step on $h _ { 0 }$ and $\phi$ . In terms of test accuracy, these two implementations make little difference (usually less than $1 \%$ ), and the choice simply comes down to a time-memory trade-off.
+
+Because our source only baseline on $\mathrm { S T L } { \cdot } 1 0  \mathrm { C I F A R } { \cdot } 1 0$ is considerably worse than the closest competitor DIRT-T, we borrow from their implementation and use dropout regularization (Srivastava et al., 2014) with $p = 0 . 5$ , only on this benchmark. This makes our source only baseline closer to theirs, so to allow for a fair comparison of the adaptation methods. Without this modification, our accuracy for source only, R, and $\mathrm { R + L + F }$ are respectively 56.1, 65.6 and $7 4 . 0 \%$ .
+
+# C ADDITIONAL DISCUSSION ON THE MEAN DISTANCE
+
+In this section we answer some potential questions about our selection rule, which uses the mean distance, from the perspective of a possibly confused reader.
+
+$Q$ : The motivation of the paper is that methods based on minimax optimization, such as those using kernel-MMD, are difficult to optimize. Since you are using the mean distance (MMD under the linear kernel) for hyper-parameter tuning and early stopping, which is a form of model selection, how are you different from those other methods and why is optimization easy for you?
+
+$A$ : Our method is fundamentally different from those other methods based on MMD, and therefore more amenable to optimization in the following two ways: 1) Even though we use the mean distance, which is a form of MMD, we never pose a minimax optimization problem. The model parameters minimize the loss functions of the tasks, which we hope makes the mean distance small (see Figure 3). Model selection using subsection 4.1 also minimizes the mean distance. In our method, all the loss functions, for model parameters and hyper-parameters, work towards the same goal of inducing alignment, so optimization is easier. 2) Even though hyper-parameter tuning and early stopping are a forms of model selection just like training, there is a qualitative difference in the degrees of freedom involved. For subsection 4.1, when performing early stopping for example, optimization amounts to a grid search over the epochs after training is finished, and there is only one degree of freedom.
+
+$Q$ : Why is the mean distance suitable for model selection when it comes to hyper-parameter tuning and early stopping, but not regular training?
+
+$A$ : Again, we agree that hyper-parameter tuning and early stopping are a forms of model selection. However, unlike model parameters ranging in the hundred of thousands, there are at most a few hyper-parameters and only one parameter for early stopping. Model parameters can easily overfit to the mean distance while those few degrees of freedom we use cannot. This is precisely also the reason why previous works using MMD resort to minimax optimization.
+
+# D DETAILS OF THE SIX DATASETS USED FOR OBJECT RECOGNITION
+
+1) MNIST (LeCun et al., 1998): greyscale images of handwritten digits 0 to 9; 60,000 samples in the training set and 10,000 in the test set.
+
+2) MNIST-M (Ganin et al., 2016): constructed by blending MNIST digits with random color patches from the BSDS500 dataset Arbelaez et al. (2011); same training / test set size as MNIST.
+
+3) SVHN (Netzer et al., 2011): colored images of cropped out house numbers from Google Street View; the task is to classify the digit at the center; 73,257 samples in the training set, 26,032 in the test set and 531,131 easier samples for additional training.
+
+4) USPS: greyscale images of handwritten digits only slightly different from MNIST; 7291 samples in the training set and 2007 in the test set.
+
+5) CIFAR-10 (Krizhevsky & Hinton, 2009): colored images of 10 classes of centered natural scene objects; 50,000 samples in the training set and 10,000 in the test set.
+
+6) STL-10 (Coates et al., 2011): colored images of objects only slightly different from CIFAR-10;   
+5000 samples in the training set and 8000 in the test set.
+
+Because CIFAR-10 and STL-10 differ in one class category, we follow common practice (Shu et al., 2018; French et al., 2017; Ghifary et al., 2016) and delete the offending categories, so each of these two datasets actually only has 9 classes.
+
+# E IMPLEMENTATION DETAILS ON THE OBJECT RECOGNITION BENCHMARKS
+
+We use a 26-layer pre-activation ResNet (He et al., 2016) as our test-time model $h _ { 0 } ( \phi ( x ) )$ , where $h _ { 0 }$ is the last linear layer that makes the predictions, and $\phi$ is everything before that. For unsupervised domain adaptation, there is no consensus on what base architecture to use among the previous works. Our choice is simply base on the widespread adoption of the ResNet architecture and the ease of implementation. In Table 2 we provide the source only results using our base architecture, and the ones from Shu et al. (2018), our closest competitor. Our source only results are in fact worse than theirs, indicating that our improvements are indeed made through adaptation.
+
+At training time, the self-supervised heads $h _ { k } , k = 1 . . . K$ are simply linear layers connected to the end of $\phi$ as discussed in section 4. There is no other modification on the standard ResNet. For all experiments on the object recognition benchmarks, we optimize our model with SGD using weight decay 5e-4 and momentum 0.9, with a batch size of 128. We use an initial learning rate of 0.1 and a two milestone schedule, where the learning rate drops by a factor of 10 at each milestone. All these optimization hyper-parameters are taken directly from the standard literature (He et al., 2016; Huang et al., 2016; Guo et al., 2017) without any modification for our problem setting. We select the total number of epochs and the two milestones based on convergence of the source classifier $h _ { 0 }$ and unsupervised classifiers $h _ { 1 } , . . . , h _ { K }$ . Early stopping is done using the selection heuristic discussed in subsection 4.1. For fair comparison with our baselines, we do not perform data augmentation, following previous works (Hoffman et al., 2017; Sener et al., 2016; Ghifary et al., 2016).
+
+# F IMPLEMENTATION DETAILS ON GTA5 CITYSCAPES
+
+For our experiments, we initialize our model from the DeepLab-v3 architecture (Chen et al., 2017), pre-trained on ImageNet, as commonly done in the field. Each self-supervised head consists of a global average pooling layer on the pre-logit layer, followed by a single linear layer. To take advantage of the large size of the natural scene images, we use the continuous i.e. regression version of location prediction. The self-supervised head is trained on the square loss, to regress the coordinates (in two dimensions) that the patch is cropped from. Natural for the regression version, instead of cropping from the quadrants like for the classification version on the small datasets, we instead crop out $4 0 0 \times 4 0 0$ patches taken at random from the segmentation scenes. We optimize our model with SGD using a learning rate of 0.007 for 15,000 iterations, with a batch size of 48. Once again, we use the selection heuristic in subsection 4.1 for early-stopping.
+
+![](images/3b95329aeb1141d373e0aac7d1097346c51996a0c1192aeba9a892e068e9bbed.jpg)  
+Table 4: Qualitative comparison of segmentation results using our method, the baseline, and the ground truth label.
+
+# G QUALITATIVE COMPARISON ON GTA5 CITYSCAPES
+
+We thank the anonymous Reviewer 3 for suggeting us to add these visualizations. This section shows segmentation results produced before and after adaptation(as shown in Table 4), alongside the original input image and the ground truth label.
+
+Qualitatively, from visual inspection of these images, we can see that the quality of produced segmentations improves drastically after adaptation with our method. In particular, we note that location prediction as a self-supervised task helps to prevent nonsensical label configurations, which are often produced by the baseline. For example, the baseline model sometimes predicts “building” or “terrain” for pixels directly in front of a car. These errors are largely corrected for by our proposed adaptation method, leading to more accurate “road” predictions.

parse/train/S1lF8xHYwS/S1lF8xHYwS_content_list.json

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+    {
+        "type": "text",
+        "text": "UNSUPERVISED DOMAIN ADAPTATION THROUGH SELF-SUPERVISION ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper addresses unsupervised domain adaptation, the setting where labeled training data is available on a source domain, but the goal is to have good performance on a target domain with only unlabeled data. Like much of previous work, we seek to align the learned representations of the source and target domains while preserving discriminability. The way we accomplish alignment is by learning to perform auxiliary self-supervised task(s) on both domains simultaneously. Each self-supervised task brings the two domains closer together along the direction relevant to that task. Training this jointly with the main task classifier on the source domain is shown to successfully generalize to the unlabeled target domain. The presented objective is straightforward to implement and easy to optimize. We achieve state-of-the-art results on four out of seven standard benchmarks, and competitive results on segmentation adaptation. We also demonstrate that our method composes well with another popular pixel-level adaptation method. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Visual distribution shifts are fundamental to our constantly evolving world. We humans face them all the time, e.g. when we navigate a foreign city, read text in a new font, or recognize objects in an environment we have never encountered before. These real-world challenges to the human visual perception have direct parallels in computer vision. Formally, a distribution shift happens when a model is trained on data from one distribution (source), but the goal is to make good predictions on some other distribution (target) that shares the label space with the source. Often computational models struggle even for pairs of distributions that humans find intuitively similar. ",
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+    {
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+        "text": "Our paper studies the setting of unsupervised domain adaptation, with labeled data in the source domain, but only unlabeled data in the target domain. The general philosophy of the field is to induce alignment of the source and target domains through some transformation. In the context of deep learning, a convolutional neural network maps images to learned representations in some feature space, so inducing alignment is done by making the distribution shifts small between the source and target in this shared feature space (Csurka, 2017; Wang & Deng, 2018; Gopalan et al., 2011). If, in addition, such representations preserve discriminability on the source domain, then we can learn a good classifier on the source, which now generalizes to the target under the reasonable assumption that the representations of the two domains have the same ground truth. ",
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+    {
+        "type": "text",
+        "text": "Most existing approaches implement this philosophy of alignment by minimizing a measurement of distributional discrepancy in the feature space (details in section 2), often some form of maximum mean discrepancy (MMD) e.g. Long et al. (2017), or a learned discriminator of the source and target as an approximation to the total variation distance e.g. Ganin et al. (2016). Both measurements lead to the formulation of the training objective as a minimax optimization problem a.k.a. adversarial learning, which is known to be very difficult to solve. Unless carefully balanced, the push and pull in opposite directions can often cause wild fluctuations in the discrepancy loss and lead to sudden divergence (details in section 2). Therefore, we propose to avoid minimax optimization altogether through a very different approach. ",
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+    {
+        "type": "text",
+        "text": "Our main idea is to achieve alignment between the source and target domains by training a model on the same task in both domains simultaneously. Indeed, if we had labels in both domains, we could simply use our original classification task for this. However, since we lack labels in the target domain, we propose to use a self-supervised auxiliary task, which creates its own labels directly from the data (see section 3). In fact, we can use multiple self-supervised tasks, each one aligning the two domains along a direction of variation relevant to that task. Jointly training all the self-supervised tasks on both domains together with the original task on the source domain produces well-aligned representations as shown in Figure 1. ",
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+        "image_caption": [
+            "Figure 1: We propose a method for unsupervised domain adaptation that uses self-supervision to align the learned representations of two domains in a shared feature space. Here we visualize how these representations might be aligned in the feature space. a) Without our method, the source domain is far away from the target domain, and a source classifier cannot generalize to the target. b) Training a shared representation to support one self-supervised task on both domains can align the source and target along one direction. c) Using multiple self-supervised tasks can further align the domains along multiple directions. Now the source and target are close in this shared feature space, and the source classifier can hope to generalize to the target. "
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+    {
+        "type": "text",
+        "text": "Like all of deep learning, we can only empirically verify that at least in our experiments, the model does not overfit by internally creating a different decision boundary for each domain along different dimensions, which would then yield bad results. Recent research suggests that stochastic gradient descent is indeed unlikely to find such overfitting solutions with a costly decision boundary of high complexity (implicit regularization), even though the models have enough capacity (Zhang et al., 2016a; Neyshabur et al., 2017b; Arora et al., 2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The key contribution of our work is to draw a connection between unsupervised domain adaptation and self-supervised learning. While we do not propose any fundamentally new self-supervised tasks, we offer insights in section 3 on how to select the right ones for adaptation, and propose in section 4 a novel training algorithm on those tasks, using batches of samples from both domains. Additionally, we demonstrate that domain alignment could be achieved with a simple and stable algorithm, without the need for adversarial learning. In section 5, we report state-of-the-art results on several standard benchmarks. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
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+        "text": "In this section we provide a brief overview of the two fields that our work would like to bridge. ",
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+    {
+        "type": "text",
+        "text": "2.1 UNSUPERVISED DOMAIN ADAPTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Methods for unsupervised domain adaptation in computer vision can be divided into three broad classes. The dominant class, which our work belongs to, aims to induce alignment between the source and the target domains in some feature space. This has been done by optimizing for some measurement of distributional discrepancy. One popular measurement is the maximum mean discrepancy (MMD) – the distance between the mean of the two domains in some reproducing kernel Hilbert space, where the kernel is chosen to maximize the distance (Bousmalis et al., 2016; Long et al., 2015; 2017). Another way to obtain a measurement of discrepancy is to train an adversarial discriminator that distinguishes between the two domains (Ganin & Lempitsky, 2014; Ganin et al., 2016; Tzeng et al., 2017). However, both MMD and adversarial training are formulated as minimax optimization problems, which are widely known, both in theory and practice, to be very difficult (Fedus et al., 2017; Duchi et al., 2016; Liang, 2017; Jin et al., 2019). Since the optimization landscape is much more complex than in standard supervised learning, training often does not converge or converges to a bad local minimum (Goodfellow, 2016; Nagarajan & Kolter, 2017; Li et al., 2017), and requires carefully balancing the two sets of parameters (for minimization and maximization) so one does not dominate the other (Salimans et al., 2016; Neyshabur et al., 2017a). ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To make minimax optimization easier, researchers have proposed numerous modifications to the loss function, network design, and training procedure (Arjovsky et al., 2017; Gulrajani et al., 2017; Karras et al., 2017; Courty et al., 2017; Sun & Saenko, 2016; Shu et al., 2018; Sener et al., 2016). Over the years, these modifications have yielded practical improvements on many standard benchmarks, but have also made the state-of-the-art algorithms very complicated. Often practitioners are not sure which tricks are necessary for which applications, and implementing these tricks can be bug-prone and frustrating. To make matters worse, since there is no labeled target data available for a validation set, practitioners have no way to perform hyper-parameter tuning or early stopping. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The second class of methods directly transforms the source images to resemble the target images with generative models (Taigman et al., 2016; Hoffman et al., 2017; Bousmalis et al., 2017). While similar to the first class in the philosophy of alignment, these methods operate on image pixels directly instead of an intermediate representation space, and therefore can benefit from an additional round of adaptation in some representation space. In subsection 5.2 we demonstrate that composing our method with a popular pixel-level method yields stronger performance than either alone. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The third class of methods uses a model trained on the labeled source data to estimate labels on the target data, then trains on some of those estimated pseudo-labels (e.g. the most confident ones), therefore bootstrapping through the unlabeled target data. Sometimes called self-ensembling (French et al., 2017), this technique is borrowed from semi-supervised learning, where it is called co-training (Saito et al., 2017; Zou et al., 2018; Chen et al., 2018; 2011). In contrast, our method uses joint training (of the main and self-supervised tasks), different from co-training in every aspect except the name. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 SELF-SUPERVISED FEATURE LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The emerging field of self-supervised learning uses the machinery of supervised learning on problems where external supervision is not available. The idea is to use data itself as supervision for auxiliary (also called “pretext”) tasks that learn deep feature representations which will hopefully be informative for downstream “real” tasks. Many such auxiliary tasks have been proposed in the literature, including colorization (predicting the chrominance channels of an image given its luminance) (Zhang et al., 2016b; Larsson et al., 2017; Zhang et al., 2017), image inpainting Pathak et al. (2016), spatial context prediction (Doersch et al., 2015), solving jigsaw puzzles (Noroozi & Favaro, 2016), image rotation prediction (Gidaris et al., 2018), predicting audio from video (Owens et al., 2016), contrastive predictive coding (Oord et al., 2018), etc. Researchers have also experimented with self-supervision on videos (Wang & Gupta, 2015; Wang et al., 2019), and combining multiple self-supervised tasks together (Doersch & Zisserman, 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Typically, self-supervision is used as a pre-training step on unlabeled data (e.g. the ImageNet training set without labels) to initialize a deep learning model, followed by fine-tuning on a labeled training set (e.g. PASCAL VOC) and evaluating on the corresponding test set. Instead, in this paper, we train the self-supervised tasks together with the main supervised task, encouraging a consistent representation that both aligns the two domains and does well on the main task1. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Recently, self-supervision has also been used for other problem settings, such as improving robustness (Hendrycks et al., 2019), domain generalization (Carlucci et al., 2019) and few-short learning (Su et al., 2019). The most relevant paper to us is Ghifary et al. (2016), which uses self-supervision for unsupervised domain adaptation, but not through alignment. Their algorithm trains a denoising autoencoder Vincent et al. (2008) only on the target data, together with the main classifier only on the labeled source data. They argue theoretically that this is better than training the autoencoder on both domains together. However, their theory is based on the critical assumption that the domains are already aligned, which is rarely true in practice. Consequently, their empirical results are much weaker than ours, as discussed in section 5. Please see Appendix A for more detailed comparisons with these works. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The design of auxiliary self-supervised tasks is an exciting area of research in itself, with many successful examples listed in section 2. However, not all of them are suitable for unsupervised domain adaptation. In order to induce alignment between the source and target, the labels created by selfsupervision should not require capturing information on the very factors where the domains are meaninglessly different, i.e. the factors of variation that we are trying to eliminate through adaptation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A particularly unsuitable tasks are these that try to predict pixels of the original image, as image inpainting (Pathak et al., 2016), colorization (Zhang et al., 2016b; Larsson et al., 2017; Zhang et al., 2017) or denoising autoencoder(Vincent et al., 2008). The success of pixel-wise reconstruction depends strongly on brightness information, or other factors of variation in overall appearance (e.g. sunny vs. coudy) that are typically irrelevant to high-level visual concepts. Thus, instead of inducing alignment, learning a pixel reconstruction task would instead serve to further separate the domains. We have experimented with using the colorization task and the denoising autoencoder for our training algorithm, and found their performance little better than the source only baseline, sometimes even worse! 2 ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/576ffc63038ed4f1092e63dcc9f10394e9a007c0e6b4b50b73951296aad7687b.jpg",
+        "table_caption": [
+            "Table 1: Transformations and labels on a sample input image created by two of the self-supervised tasks used in our work. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td colspan=\"3\">Images and self-supervised labels</td></tr><tr><td>Rotation</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"3\">Location</td><td></td><td></td><td>90°180°270°</td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>(0,0）(1,0）(0,1）(1,1)</td><td></td></tr></table>",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "In general, classification tasks that predict structural labels seems better suited for our purpose than reconstruction tasks that predict pixels. Therefore, we have settled on three classification-based self-supervised tasks that combine simplicity with high performance: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Rotation Prediction: (Gidaris et al., 2018) An input image is rotated in 90-degree increments i.e. $0 ^ { \\circ }$ , $9 0 ^ { \\circ }$ , $1 8 0 ^ { \\circ }$ , and $2 7 0 ^ { \\circ }$ ; the task is to predict the angle of rotation as a four-way classification problem. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Flip Prediction: An input image is randomly flipped vertically; the task is to predict whether the image is flipped or not 3. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Patch Location Prediction: Patches are randomly cropped out of an input image; the task is to predict where the patches come from 4. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Consider a trivial illustrative example where the source and target are exactly the same except that the target pixels are all scaled down by a constant factor (e.g. daylight to dusk transition). All three of the aforementioned forms of self-supervision are suitable for this example, because pixel scaling i.e. brightness is “orthogonal” to the prediction of rotation, flip and location. ",
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+    {
+        "type": "text",
+        "text": "4 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our training algorithm is simple once a set of $K$ self-supervised tasks are selected. We already have the loss function on the main prediction task, denoted $L _ { 0 }$ , that we do not have target labels for. Each self-supervised task corresponds to a loss function $\\mathcal { L } _ { k }$ for $k = 1 . . . K$ . So altogether, our optimization problem is set up as a combination of these $K + 1$ loss functions, as is done for standard multi-task learning (see Figure 2). Implementation details are included in Appendix B. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c367ab6075ed42c17750fd5c6ed7ab393c659e26edb8df14bdae382b1f775b7.jpg",
+        "image_caption": [
+            "Figure 2: Our method jointly trains a supervised head on labeled source data and self-supervised heads on unbaled data from both domains. The heads use high-level features from a shared encoder, which learns to align the feature distributions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Each loss function corresponds to a different “head” $h _ { k }$ for $k = 0 . . . K$ , which produces predictions in the respective label space. All the task-specific heads (including $h _ { 0 }$ for the actual prediction task) share a common feature extractor $\\phi$ . Altogether, the parameters of $\\phi$ and $h _ { k } , k = 0 , . . k$ are the learned variables i.e. the free parameters of our optimization problem. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In our paper, $\\phi$ is a deep convolutional neural network, and every $h _ { k }$ is simply a linear layer i.e. multiplication by a matrix of size output space dimension $\\times$ feature space dimension. If $k \\mathrm { t h }$ task is classification in nature, then $h _ { k }$ also has a softmax or sigmoid (for multi-class or binary) following the linear layer. The output space dimension is only four for rotation and location classification, and two for flip and location regression. Depending on the network architecture used, the feature space dimension ranges between 64 and 512. The point is to make every $h _ { k }$ low capacity, so the heads are forced to share high-level features, as is desirable for inducing alignment. A linear map from the highest-level features to the output is the smallest possible head and performs well empirically. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let ${ \\cal { S } } = \\{ ( x _ { i } , y _ { i } ) , i = 1 . . . m \\}$ contain the labeled source data, and $T = \\{ ( x _ { i } ) , i = 1 . . . n \\}$ contain the unlabeled target data. $L _ { 0 }$ for the main prediction task takes in the labeled source data, and produces the following term in our objective: ",
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+    {
+        "type": "equation",
+        "img_path": "images/01f2f1e2c6e4ae7896e84e051643bf4535236136032e7303b062d145726248c9.jpg",
+        "text": "$$\n\\mathcal { L } _ { 0 } \\big ( S ; \\phi , h _ { 0 } \\big ) = \\sum _ { ( x , y ) \\in S } L _ { 0 } \\big ( h _ { 0 } \\big ( \\phi ( x ) \\big ) , y \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Each self-supervised task $F _ { k }$ for $k = 1 . . . K$ modifies the input samples with some transformation $f _ { k }$ and creates labels $\\tilde { y }$ . Denote $F _ { k } ( S ) = \\{ ( f _ { k } ( x _ { i } ) , \\tilde { y } _ { i } ) , i = 1 . . . m \\}$ as the self-supervised samples generated from the source samples (with the original labels discarded), and $F _ { k } ( \\bar { T ( \\cdot ) } = \\{ ( f _ { k } ( x _ { i } ) \\bar { , } \\tilde { y } _ { i } ) , \\bar { \\cdot } =$ $\\left. 1 . . n \\right\\}$ from the target. Then the loss $L _ { k }$ of each task $k = 1 . . . K$ produces the following term: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aa2b915220b97275fae4a366e5ef33879b5074f8c13d18d56c2e8d8fc39f4138.jpg",
+        "text": "$$\n\\mathcal { L } _ { k } ( S , T ; \\phi , h _ { k } ) = \\sum _ { ( f _ { k } ( x ) , \\tilde { y } ) \\in F ( S ) } L _ { k } ( h _ { k } ( \\phi ( f _ { k } ( x ) ) ) , \\tilde { y } ) + \\sum _ { ( f _ { k } ( x ) , \\tilde { y } ) \\in F ( T ) } L _ { k } ( h _ { k } ( \\phi ( f _ { k } ( x ) ) ) , \\tilde { y } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that $\\mathcal { L } _ { k } ( S , T ; \\phi , h _ { k } )$ for $k = 1 . . . K$ , unlike $\\mathcal { L } _ { 0 } ( S ; \\phi , h _ { 0 } )$ , take in both the source and target data; as we emphasize for many times throughout the paper, this is critical for inducing alignment. Altogether, our optimization problem can be formalized as in multi-task learning 5: ",
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+    {
+        "type": "equation",
+        "img_path": "images/520f4b9853a8249e7dd6006d18bf6e2a37f29c40d3ae45281ec33c89ef65cdec.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\phi , h _ { k } , k = 1 \\ldots K } \\quad { \\mathcal { L } } _ { 0 } ( S ; \\phi , h _ { 0 } ) + \\sum _ { k = 1 } ^ { K } { \\mathcal { L } } _ { k } ( S , T ; \\phi , h _ { k } ) .\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/f8fec6c4048a733744578d415f2110d66a542b8d6f4a9a9be3d77830439c6ed6.jpg",
+        "image_caption": [
+            "Figure 3: Results on MNIST MNIST-M (left) and CIFAR- $\\mathrm { \\cdot 1 0 {  } S T L \\mathrm { - } 1 0 }$ (right). Test error converges smoothly on the source and target domains for the main task as well as the self-supervised task. This kind of smooth convergence is often seen in supervised learning, but rarely in adversarial learning. The centroid distance (linear MMD) between the feature distributions of the two domains converges alongside, even though it is never explicitly optimized for. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "At test-time, we discard the self-supervised heads and use $h _ { 0 } ( \\phi ( x ) )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 A HEURISTIC FOR HYPER-PARAMETER TUNING AND EARLY STOPPING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As previously mentioned, in the unsupervised domain adaptation setting, there is no target label available and therefore no target validation set, so typical strategies for hyper-parameter tuning and early stopping that require a validation set cannot be applied. This problem remains underappreciated; in fact, it is often unclear how previous works select their hyper-parameters or detemine when training is finished, both of which can be important factors that impact performance, especially for complex algorithms using adversarial learning. In this subsection we describe a simple heuristic 6, merely as rule-of-thumb to make things work instead of a technical innovation. Like almost all of deep learning, there is no statistical guarantee on this heuristic, but it is shown to be practically effective in our experiments (see Figure 3). We only hope that it serves as the first guess of a solution towards this underappreciated problem. ",
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+    {
+        "type": "text",
+        "text": "The main idea is that, because our method never explicitly optimizes for any measurement of distributional discrepancy, these measurements can instead be used for hyper-parameter tuning and early stopping. Since it would be counterproductive to introduce additional parameters in order to perform hyper-parameter tuning, we simply use the distance between the mean of the source and target samples in the learned representation space, as produced by $\\phi$ . Formally, this can be expressed as ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b8d0f35be1b7cd79af42def4d2df6d5b39f185a7604eaa7639014cc6a20464f.jpg",
+        "text": "$$\nD ( S ^ { \\prime } , T ^ { \\prime } ; \\phi ) = \\biggl \\| \\frac { 1 } { m } \\sum _ { x \\in S ^ { \\prime } } \\phi ( x ) - \\frac { 1 } { n } \\sum _ { x \\in T ^ { \\prime } } \\phi ( x ) \\biggl \\| _ { 2 } ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $S ^ { \\prime }$ and $T ^ { \\prime }$ are unlabeled source and target validation sets 7. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our heuristic combines $D ( S ^ { \\prime } , T ^ { \\prime } ; \\phi )$ and the main task error on the (labeled) source validation set. Denote $\\mathbf { v } = ( v _ { 1 } , . . . , v _ { T } )$ and $\\mathbf { w } = ( w _ { 1 } , . . . , w _ { T } )$ the measurement vectors of those two quantities respectively over $T$ epochs. The final measurement vector is $\\mathbf { u } = \\mathbf { v } / \\operatorname* { m i n } ( \\mathbf { v } ) + \\mathbf { w } / \\operatorname* { m i n } ( \\mathbf { \\bar { w } } )$ i.e. a normalized sum of the two vectors; the epoch at which we perform early stopping is then simply arg $\\operatorname* { m i n } _ { t \\in \\{ 1 . . . T \\} } \\mathbf { u } _ { t }$ . Intuitively, this heuristic roughly corresponds to our goal of inducing alignment while preserving discriminability. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/45c4c2babe9ab675c7dd09cb531f3357966def9c8029acd337e7c84a8a1335c6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source Target</td><td>MNIST MNIST-M</td><td>MNIST SVHN</td><td>SVHN MNIST</td><td>MNIST USPS</td><td>USPS MNIST</td><td>CIFAR-10 STL-10</td><td>STL-10 CIFAR-10</td></tr><tr><td>DANN (Ganin et al.,2016)</td><td>81.5</td><td>35.7</td><td>73.6</td><td>-</td><td>-</td><td>-</td><td>=</td></tr><tr><td>DRCN (Ghifary et al.,2016)</td><td>-</td><td>40.1</td><td>82.0</td><td>=</td><td></td><td>66.4</td><td>58.7</td></tr><tr><td>DSN (Bousmalis et al., 2016)</td><td>83.2</td><td>=</td><td>82.7</td><td></td><td>=</td><td>1</td><td>1</td></tr><tr><td>kNN-Ad (Sener et al., 2016)</td><td>86.7</td><td>40.3</td><td>78.8</td><td>=</td><td></td><td>=</td><td>1</td></tr><tr><td>PixelDA (Bousmalis et al., 2017)</td><td>98.2</td><td></td><td></td><td></td><td></td><td>=</td><td>=</td></tr><tr><td>ATT (Saito et al., 2017)</td><td>94.2</td><td>52.8</td><td>86.2</td><td></td><td></td><td>-</td><td>1</td></tr><tr><td>II-Model (French et al., 2017)</td><td>-</td><td>71.4</td><td>92.0</td><td>=</td><td></td><td>76.3</td><td>64.2</td></tr><tr><td>ADDA (Tzeng et al., 2017)</td><td>=</td><td>1</td><td>76.0</td><td>89.4</td><td>90.1</td><td>-</td><td>1</td></tr><tr><td>CyCADA (Hoffman et al., 2017)</td><td>=</td><td>-</td><td>90.4</td><td>95.6</td><td>96.5</td><td>1</td><td>1</td></tr><tr><td>VADA (Shu et al.,2018)</td><td>97.7</td><td>47.5</td><td>97.9</td><td>1</td><td>1</td><td>80.0</td><td>73.5</td></tr><tr><td>DIRT-T (Shu et al., 2018)</td><td>98.9</td><td>54.5</td><td>99.4</td><td>=</td><td>1</td><td>-</td><td>75.3</td></tr><tr><td>VADA (IN) (Shu et al., 2018)</td><td>95.7</td><td>73.3</td><td>94.5</td><td>=</td><td></td><td>78.3</td><td>71.4</td></tr><tr><td>DIRT-T (IN) (Shu et al., 2018)</td><td>98.7</td><td>76.5</td><td>99.4</td><td>-</td><td>-</td><td>1</td><td>73.3</td></tr><tr><td>Source only VADA &amp; DIRT-T</td><td>58.5</td><td>27.9</td><td>77.0</td><td>-</td><td></td><td>76.3</td><td>63.6</td></tr><tr><td>Source only VADA &amp; DIRT-T (IN)</td><td>59.9</td><td>40.9</td><td>82.4</td><td>-</td><td>-</td><td>77.0</td><td>62.6</td></tr><tr><td>Source only our method</td><td>44.9</td><td>30.5</td><td>92.2</td><td>94.7</td><td>81.4</td><td>75.6</td><td>58.8</td></tr><tr><td>R</td><td>98.9</td><td>61.3</td><td>85.88</td><td>96.5</td><td>90.2</td><td>81.2</td><td>66.9</td></tr><tr><td>R+L+F</td><td>-</td><td>1</td><td>=</td><td>-</td><td>1</td><td>82.1</td><td>75.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Test accuracy $( \\% )$ on standard domain adaptation benchmarks. Our results are organized according to the self-supervised task(s) used: R for rotation, L for location, and F for flip. We achieve state-of-the-art accuracy on four out of the seven benchmarks. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 SEVEN BENCHMARKS FOR OBJECT RECOGNITION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The seven benchmarks are based on the six datasets described in Appendix D, each with a predefined training set / test set split, and labels are available on both splits. Previous works (cited in Table 2) have created those seven benchmarks by picking pairs of datasets with the same label space, treating one as the source and the other as the target, and training on the training set of the source with labels revealed and of the target with labels hidden. Following the standard setup of the field, labels on the target test set should only be used for evaluation, not for hyper-parameter tuning or early stopping; therefore we apply the heuristic described in subsection 4.1. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the two natural scene benchmarks, we use all three tasks described in section 3: rotation, location and flip prediction. For the five benchmarks on digits we do not use location because it yields trivial solutions that do not encourage the learning of semantic concepts. Given the image of a digit cropped into the four quadrants, location prediction can be trivially solved by looking at the four corners where a white stroke determines the category. Adding flip to the digits does not hurt performance, but does not improve significantly either, so we do not report those results separately. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As shown in Table 2, despite the simplicity of our method, we achieve state-of-the-art accuracy on four out of the seven benchmarks. In addition, we show the source only results from our closest competitor (VADA and DIRT-T), and note that our source only results are in fact lower than theirs on those very benchmarks that we perform the best on; this indicates that our success is indeed due to effectiveness in adaptation instead of the base architecture. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our method fails on the pair of benchmarks with SVHN, on which rotation yields trivial solutions. Because SVHN digits are cropped from house numbers with multiple digits, majority of the images have parts of the adjacent digits on the side. The main task head needs to look at the center, but the rotation head learns to look at the periphery and cheat. This failure case shows that the success of our method is tied to how well the self-supervised task fits the application. Practioners should use their domain knowledge evaluate how well the task fits, instead of blithely apply it to everything. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Also seen in Table 2 is that our method excels at object recognition in natural scenes, especially with all three tasks together. For STL- $1 0 {  }$ CIFAR-10, our base model is considerably worse than that of VADA and DIRT-T, but still beats all the baselines. Adding location and flip gives an improvement of $8 \\%$ , on top of the $8 \\%$ already over source only. This is not surprising since those tasks were originally developed for ImageNet pre-training i.e. object recognition in natural scenes – our method is very successful when the task fits the application well. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b0043d90f5b3caf3d8044b974cf121c2498b22feab9e0cb35d9062da3a86420a.jpg",
+        "table_caption": [
+            "Table 3: Test accuracy $( \\% )$ on $\\mathrm { G T A } 5 $ Cityscapes. Our method significantly improves over source only, and also over CyCADA when combined. This indicates that additional self-supervision using our training algorithm further aligns the domains. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"13\">GTA5→ Cityscapes</td><td colspan=\"7\"></td></tr><tr><td></td><td></td><td>seeapra 12.7</td><td>Buiping 39.6</td><td></td><td></td><td></td><td></td><td>trggee</td><td>usrs ogen</td><td>enee</td><td>Eirrin</td><td>3</td><td>uos.iad</td><td></td><td>u</td><td></td><td>1</td><td></td><td></td><td>mooiritte</td><td>aeleir</td><td>mIoU</td></tr><tr><td>Source only Ours</td><td>28.8 69.9</td><td></td><td>22.7</td><td>69.7</td><td>9.4 18.1</td><td>3.5 9.9</td><td>18.1 13.5</td><td>22.7 18.7</td><td>9.4 8.9</td><td>80.9 80.3</td><td>12.4 19.4</td><td>45.8 58.4</td><td></td><td>53.9 53.8</td><td>9.6 2.6</td><td>74.7 75.1</td><td>20.9 13.6</td><td>15.0 5.2</td><td>0.0 0.3</td><td>19.4 8.1</td><td>3.9 1.2</td><td>25.3 28.9</td></tr><tr><td>CyCADA</td><td></td><td>79.1 33.1</td><td>77.923.4</td><td></td><td></td><td>17.332.1</td><td></td><td></td><td>33.3 31.8 81.5</td><td></td><td>26.7</td><td>69.0</td><td>62.8</td><td></td><td></td><td>14.774.520.925.6</td><td></td><td></td><td>6.9</td><td></td><td>18.820.4</td><td>39.5</td></tr><tr><td>Ours + CyCADA</td><td>86.6</td><td>37.8</td><td>80.8</td><td>29.7</td><td></td><td>16.4</td><td>28.9</td><td>30.9</td><td>22.2</td><td>83.8</td><td>37.1</td><td>76.9</td><td>60.1</td><td>7.8</td><td>84.1</td><td></td><td>30.8</td><td>32.1</td><td>1.2</td><td>23.2</td><td>13.3</td><td>41.2</td></tr><tr><td>Oracle</td><td>97.3</td><td>79.8</td><td>88.6</td><td>32.5</td><td></td><td>48.2</td><td>56.3</td><td>63.6</td><td>73.3</td><td>89.0</td><td>58.9</td><td>93.0</td><td>78.2</td><td>55.2</td><td>92.2</td><td></td><td>45.0</td><td>67.3</td><td>39.6</td><td>49.9</td><td>73.6</td><td>67.4</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.2 BENCHMARK FOR SEMANTIC SEGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To experiment with our method in more diverse applications, we also evaluate on a challenging simulation-to-real benchmark for semantic segmentation – $\\mathbf { \\partial } \\cdot \\mathbf { G T A } 5  \\mathbf { \\langle }$ Cityscapes. GTA5 (Richter et al., 2016) contains 24,966 video frames taken from the computer game, where dense segmentation labels are automatically given by the game engine. Cityscapes (Cordts et al., 2016) contains 5,000 video frames taken from real-world dash-cams. The main task is to classify every pixel in an image as one of the 19 classes shared across both datasets, and accuracy is measured by intersection over union (IoU). The best possible results are given as the oracle, when the labels on Cityscapes are available for training, so typical supervised learning methods are applicable. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our results are shown in Table 3, and implementation details in Appendix F. Our self-supervised tasks were designed for classification, where the label on an image depends on its global content, while in segmentation the labels tend to be highly local. Nevertheless, with very little modification, we see significant improvements over the source only baseline. In Appendix G we provide visualizations of the segmentation results, and make qualitative comparisons between those produced by the baseline and our method. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We also experiment with combining our method with another popular unsupervised domain adaptation method – CyCADA (Hoffman et al., 2017), designed specifically for segmentation. Surprisingly, when operating on top of images produced by this already very strong baseline, our method further improves performance. This demonstrates that pixel-level adaptation methods might still benefit from an additional round of adaptation by inducing alignment through self-supervision. We emphasize that these results are obtained with a very simple instantiation of our method, as a start towards the development of self-supervised tasks more suitable for semantic segmentation. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We hope that this work encourages future researchers in unsupervised domain adaptation to consider the study of self-supervision as an alternative to adversarial learning, and researchers in selfsupervision to consider designing tasks and evaluating them in our problem setting. Most selfsupervised tasks today were originally designed for pre-training and evaluated in terms of accuracy gains on a downstream recognition, localization or detection tasks. It will be interesting to see if new self-supervised tasks can arise from the motivation of adaptation, for which alignment is the key objective. Moreover, domain experts could perhaps incorporate their dataset specific knowledge into the design of a self-supervised task specifically for their application. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One additional advantage of our method, not considered in this paper, is that it might be particularly amenable to very small target sample size, when those other methods based on adversarial learning cannot accurately estimate the target distribution. We leave this topic for the future work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            287,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pablo Arbelaez, Michael Maire, Charless Fowlkes, and Jitendra Malik. Contour detection and hierarchical image segmentation. IEEE transactions on pattern analysis and machine intelligence, 33(5):898–916, 2011. ",
+        "bbox": [
+            173,
+            126,
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+            167
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yang Zou, Zhiding Yu, BVK Vijaya Kumar, and Jinsong Wang. Unsupervised domain adaptation for semantic segmentation via class-balanced self-training. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 289–305, 2018. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+            103,
+            263,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL DISCUSSION ON SPECIFIC PAPERS RELATED TO OURS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            142,
+            758,
+            156
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this section we discuss the four papers mentioned in section 2 that are related to ours, but different in training algorithm, philosophy or problem setting. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Deep Reconstruction-Classification Networks (Ghifary et al., 2016). This method works in unsupervised domain adaptation, the same problem setting as ours. It learns a denoising autoencoder on the target, together with the main classifier on the source. However, they use only the target domain for reconstruction, claiming both empirically and theoretically that it is better than using both domains together. In contrast, we use both the source and target for self-supervision. These two algorithmic differences really reflect our fundamental difference in philosophy. They take the target data as analogous to the unlabeled (source) data in semi-supervised learning, so any form of self-supervision suitable for semi-supervised learning is good enough; their theoretical analysis is also directly borrowed from that of semi-supervised learning, which concludes that it is necessary and sufficient to only use the target data for self-supervision. We take data from both domains for self-supervision in order to align their feature distributions; also, both conceptually and empirically, we cannot use reconstruction tasks because they are unsuitable for inducing alignment (see section 3). ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Empirical experiments further support our arguments in three ways: 1) If we were to only use the target for our self-supervision tasks, we would perform barely better than the source only baseline, even on MNIST MNIST-M, the easiest benchmark where we would otherwise observe a huge improvement. 2) If we were to use the reconstruction task i.e. a denoising autoencoder, we would again perform barely better than the source only baseline, and sometimes even worse, as described in section 3. 3) As shown in Table 2, our results are much better than theirs on all of the benchmarks by a large margin. This shows that implementing the philosophy of alignment, developed for unsupervised domain adaptation, is much superior in our problem setting to blithely borrowing from semi-supervised learning. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In addition to the three classes of methods covered in section 2, unsupervised domain adaptation has also been studied using sample-to-sample correspondence through graph matching (Das & George Lee, 2018; Das & Lee, 2018); these methods are less relevant to ours and not discussed in detail. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Using Self-Supervised Learning Can Improve Model Robustness and Uncertainty (Hendrycks et al., 2019). This paper studies the setting of robustness, where there is no sample provided, even unlabeled, from the target domain. Their method jointly trains for the main task and the self-supervised task only on the source domain, really because there is no target data of any kind. Because their problem setting is very challenging (with so little information provided on the target), their primary evaluation is on a dataset of CIFAR-10 images with the addition of 15 types of common corruptions (e.g. impulse noise, JPEG compression, snow weather, motion blur and pixelation), simulated by an algorithm. No idea on unsupervised domain adaptation is mentioned. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Domain Generalization by Solving Jigsaw Puzzles (Carlucci et al., 2019). This paper studies two setting. The first is the robustness setting, exactly the same as in Hendrycks et al. (2019), except that evaluation is done on MNIST MNIST-M and MNIST ${ } ^ { \\prime } {  } S \\mathbf { V } \\mathbf { H } \\mathbf { N }$ . Their baseline is also a method from the robustness community that trains on adversarial examples and uses no target data. Again, because their problem setting is very challenging, the accuracy is low for both their proposed method and the baseline. The second setting is called domain generalization, which is very similar to meta-learning. The goal is to perform well simultaenously on multiple distributions, all labeled. Evaluation is done using the mean accuracy on all the domains. Beside the name, there is little similarity between the setting of unsupervised domain adaptation and domain generalization, which has no unsupervised component. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Boosting Supervision with Self-Supervision for Few-shot Learning (Su et al., 2019). As evident from the title, this paper studies the setting of few-shot learning, where the goal is to perform well on a very small dataset. Again, there is no unsupervised component and little connection to our setting. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Most of our results have already been produced when Carlucci et al. (2019) and Su et al. (2019) were presented at a conference. Hendrycks et al. (2019) was submitted to NeurIPS 2019 and should be considered concurrent work. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL ALGORITHMIC DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In practice and for our experiments, the source and target datasets are often imbalanced in size. If we were to blithely solve for the objective in Equation 2, the domain with a larger dataset would carry more weight for every $\\mathcal { L } _ { k } ( S , T ; \\phi , h _ { k } ) , k = 1 . . . K$ because we are summing over all samples in both datasets. We would like to keep the two sums inside $\\mathcal { L } _ { k }$ roughly balanced such that in terms of features produced by $\\phi$ , neither of the two domains would dominate the other. This is easy to achieve in our implementation through balanced batches. When we need to sample a batch for task $k$ , we simply sample half the batch from the source, and another half from the target, then put them together. In the end, our implementation requires little change on top of an existing supervised learning codebase. Each self-supervised task is defined as a module, and adding a new one only amounts to defining the structural modifications and the loss function. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We optimize the objective in Equation 2 with stochastic gradient descent (SGD). One can simply take a joint step on Equation 2, which covers all the losses for $k = 0$ and $k = 1 . . . K$ . However, the implementation would then have to store the gradients with respect to all $K + 1$ losses together. For memory efficiency, our implementation loops over $k = 1 . . . K$ ; for each self-supervised task $k$ , it samples a batch of combined source and target data (without their original labels), structurally modifies the images by $f _ { k }$ , creates new labels according to the modification, and obtains a loss for $h _ { k }$ and $\\phi$ on this batch. So a gradient step is taken for each $k = 1 . . . K$ , before finally a batch of original source images and labels are sampled for a gradient step on $h _ { 0 }$ and $\\phi$ . In terms of test accuracy, these two implementations make little difference (usually less than $1 \\%$ ), and the choice simply comes down to a time-memory trade-off. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Because our source only baseline on $\\mathrm { S T L } { \\cdot } 1 0  \\mathrm { C I F A R } { \\cdot } 1 0$ is considerably worse than the closest competitor DIRT-T, we borrow from their implementation and use dropout regularization (Srivastava et al., 2014) with $p = 0 . 5$ , only on this benchmark. This makes our source only baseline closer to theirs, so to allow for a fair comparison of the adaptation methods. Without this modification, our accuracy for source only, R, and $\\mathrm { R + L + F }$ are respectively 56.1, 65.6 and $7 4 . 0 \\%$ . ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL DISCUSSION ON THE MEAN DISTANCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section we answer some potential questions about our selection rule, which uses the mean distance, from the perspective of a possibly confused reader. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$Q$ : The motivation of the paper is that methods based on minimax optimization, such as those using kernel-MMD, are difficult to optimize. Since you are using the mean distance (MMD under the linear kernel) for hyper-parameter tuning and early stopping, which is a form of model selection, how are you different from those other methods and why is optimization easy for you? ",
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+    },
+    {
+        "type": "text",
+        "text": "$A$ : Our method is fundamentally different from those other methods based on MMD, and therefore more amenable to optimization in the following two ways: 1) Even though we use the mean distance, which is a form of MMD, we never pose a minimax optimization problem. The model parameters minimize the loss functions of the tasks, which we hope makes the mean distance small (see Figure 3). Model selection using subsection 4.1 also minimizes the mean distance. In our method, all the loss functions, for model parameters and hyper-parameters, work towards the same goal of inducing alignment, so optimization is easier. 2) Even though hyper-parameter tuning and early stopping are a forms of model selection just like training, there is a qualitative difference in the degrees of freedom involved. For subsection 4.1, when performing early stopping for example, optimization amounts to a grid search over the epochs after training is finished, and there is only one degree of freedom. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$Q$ : Why is the mean distance suitable for model selection when it comes to hyper-parameter tuning and early stopping, but not regular training? ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$A$ : Again, we agree that hyper-parameter tuning and early stopping are a forms of model selection. However, unlike model parameters ranging in the hundred of thousands, there are at most a few hyper-parameters and only one parameter for early stopping. Model parameters can easily overfit to the mean distance while those few degrees of freedom we use cannot. This is precisely also the reason why previous works using MMD resort to minimax optimization. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D DETAILS OF THE SIX DATASETS USED FOR OBJECT RECOGNITION ",
+        "text_level": 1,
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1) MNIST (LeCun et al., 1998): greyscale images of handwritten digits 0 to 9; 60,000 samples in the training set and 10,000 in the test set. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "2) MNIST-M (Ganin et al., 2016): constructed by blending MNIST digits with random color patches from the BSDS500 dataset Arbelaez et al. (2011); same training / test set size as MNIST. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "3) SVHN (Netzer et al., 2011): colored images of cropped out house numbers from Google Street View; the task is to classify the digit at the center; 73,257 samples in the training set, 26,032 in the test set and 531,131 easier samples for additional training. ",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "4) USPS: greyscale images of handwritten digits only slightly different from MNIST; 7291 samples in the training set and 2007 in the test set. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "5) CIFAR-10 (Krizhevsky & Hinton, 2009): colored images of 10 classes of centered natural scene objects; 50,000 samples in the training set and 10,000 in the test set. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "6) STL-10 (Coates et al., 2011): colored images of objects only slightly different from CIFAR-10;   \n5000 samples in the training set and 8000 in the test set. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Because CIFAR-10 and STL-10 differ in one class category, we follow common practice (Shu et al., 2018; French et al., 2017; Ghifary et al., 2016) and delete the offending categories, so each of these two datasets actually only has 9 classes. ",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E IMPLEMENTATION DETAILS ON THE OBJECT RECOGNITION BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We use a 26-layer pre-activation ResNet (He et al., 2016) as our test-time model $h _ { 0 } ( \\phi ( x ) )$ , where $h _ { 0 }$ is the last linear layer that makes the predictions, and $\\phi$ is everything before that. For unsupervised domain adaptation, there is no consensus on what base architecture to use among the previous works. Our choice is simply base on the widespread adoption of the ResNet architecture and the ease of implementation. In Table 2 we provide the source only results using our base architecture, and the ones from Shu et al. (2018), our closest competitor. Our source only results are in fact worse than theirs, indicating that our improvements are indeed made through adaptation. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "At training time, the self-supervised heads $h _ { k } , k = 1 . . . K$ are simply linear layers connected to the end of $\\phi$ as discussed in section 4. There is no other modification on the standard ResNet. For all experiments on the object recognition benchmarks, we optimize our model with SGD using weight decay 5e-4 and momentum 0.9, with a batch size of 128. We use an initial learning rate of 0.1 and a two milestone schedule, where the learning rate drops by a factor of 10 at each milestone. All these optimization hyper-parameters are taken directly from the standard literature (He et al., 2016; Huang et al., 2016; Guo et al., 2017) without any modification for our problem setting. We select the total number of epochs and the two milestones based on convergence of the source classifier $h _ { 0 }$ and unsupervised classifiers $h _ { 1 } , . . . , h _ { K }$ . Early stopping is done using the selection heuristic discussed in subsection 4.1. For fair comparison with our baselines, we do not perform data augmentation, following previous works (Hoffman et al., 2017; Sener et al., 2016; Ghifary et al., 2016). ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F IMPLEMENTATION DETAILS ON GTA5 CITYSCAPES ",
+        "text_level": 1,
+        "bbox": [
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+            750,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For our experiments, we initialize our model from the DeepLab-v3 architecture (Chen et al., 2017), pre-trained on ImageNet, as commonly done in the field. Each self-supervised head consists of a global average pooling layer on the pre-logit layer, followed by a single linear layer. To take advantage of the large size of the natural scene images, we use the continuous i.e. regression version of location prediction. The self-supervised head is trained on the square loss, to regress the coordinates (in two dimensions) that the patch is cropped from. Natural for the regression version, instead of cropping from the quadrants like for the classification version on the small datasets, we instead crop out $4 0 0 \\times 4 0 0$ patches taken at random from the segmentation scenes. We optimize our model with SGD using a learning rate of 0.007 for 15,000 iterations, with a batch size of 48. Once again, we use the selection heuristic in subsection 4.1 for early-stopping. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b95329aeb1141d373e0aac7d1097346c51996a0c1192aeba9a892e068e9bbed.jpg",
+        "image_caption": [
+            "Table 4: Qualitative comparison of segmentation results using our method, the baseline, and the ground truth label. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "G QUALITATIVE COMPARISON ON GTA5 CITYSCAPES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We thank the anonymous Reviewer 3 for suggeting us to add these visualizations. This section shows segmentation results produced before and after adaptation(as shown in Table 4), alongside the original input image and the ground truth label. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Qualitatively, from visual inspection of these images, we can see that the quality of produced segmentations improves drastically after adaptation with our method. In particular, we note that location prediction as a self-supervised task helps to prevent nonsensical label configurations, which are often produced by the baseline. For example, the baseline model sometimes predicts “building” or “terrain” for pixels directly in front of a car. These errors are largely corrected for by our proposed adaptation method, leading to more accurate “road” predictions. ",
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+        "page_idx": 15
+    }
+]

parse/train/VRgITLy0l2/VRgITLy0l2_content_list.json

@@ -0,0 +1,1775 @@
+[
+    {
+        "type": "text",
+        "text": "A PRIORI GUARANTEES OF FINITE-TIME CONVERGENCE FOR DEEP NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            236,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we perform Lyapunov based analysis of the loss function to derive an a priori upper bound on the settling time of deep neural networks. While previous studies have attempted to understand deep learning using control theory framework, there is limited work on a priori finite time convergence analysis. Drawing from the advances in analysis of finite-time control of non-linear systems, we provide a priori guarantees of finite-time convergence in a deterministic control theoretic setting. We formulate the supervised learning framework as a control problem where weights of the network are control inputs and learning translates into a tracking problem. An analytical formula for finite-time upper bound on settling time is provided a priori under the assumptions of boundedness of input. Finally, we prove that our loss function is robust against input perturbations. ",
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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": "Over the past decade, Deep neural networks have achieved human-like performance in various machine learning tasks, such as classification, natural language processing and speech recognition. Despite the popularity of deep learning, the underlying theoretical understanding remains relatively less explored. While attempts have been made to develop deep learning theory by drawing inspiration from other related fields such as statistical learning and information theory, a comprehensive theoretical framework is still in an early developmental stage. It is difficult to perform mathematical analysis on Deep neural networks due to the large number of parameters involved. Other problems in deep neural networks revolve around the stability and desired convergence rate of the training. Since the performance of the network depends highly on the training data and the choice of the optimization algorithm, there is no guarantee that the training will converge. Our work attempts to give finite-time convergence guarantees for training of a deep neural network by utilizing an established stabilization framework from control theory. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing works in deep learning theory have attempted to bridge the gap in understanding deep learning dynamics by focusing on simple models of neural networks [Saxe et al. (2013), Li & Yuan (2017), Arora et al. (2018), Jacot et al. (2018)]. This could be attributed to the fact that current state-of-the-art deep learning models are highly complex structures to analyze. Jacot et al. (2018) proved that a multilayer fully-connected network with infinite width converges to a deterministic limit at initialization and the rate of change of weights goes to zero. Saxe et al. (2013) analyzed deep linear networks and proved that these networks, surprisingly, have a rich non-linear structure. The study shows that given the right initial conditions, deep linear networks are a finite amount slower than shallow networks. Following this work, Arora et al. (2018) proved the convergence of gradient descent to global minima for networks with dimensions of every layer being full rank in dimensions. While these studies give important insights into the design of neural network architecture and the behavior of training, their results may need to be modified in order to provide convergence guarantees for the conventional deep neural networks. Du et al. (2018) extended the work of Jacot et al. (2018) further by proving convergence for gradient descent to achieve zero training loss in deep neural networks with residual connections. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When it comes to convergence of certain state variables of a dynamical system, control theory provides a rich mathematical framework which can be utilized for analyzing the non-linear dynamics of deep learning [Liu & Theodorou (2019)]. One of the early works relating deep learning to control theory was of LeCun et al. (1988), which used the concept of optimal control and formulated back-propagation as an optimization problem with non-linear constraints. Non-linear control has gained increasing attention over the past few years in the context of neural networks, especially for recurrent neural networks [Allen-Zhu et al. (2019), Xiao (2017)] and reinforcement learning [Xu et al. (2013), Gupta et al. (2019), Wang et al. (2019), Kaledin et al. (2020)]. A new class of recurrent neural networks, called Zhang Neural Networks (ZNN), was developed that expressed dynamics of the network as a set of ordinary differential equations and used non-linear control to prove global or exponential stability for time-varying Sylvester equation [Zhang et al. (2002), Guo et al. (2011)]. Li et al. (2013) introduced the sign bi-power activation function for Zhang Neural Networks (ZNN) which helps to prove the existence of finite-time convergence property. Haber & Ruthotto (2017) presents deep learning as a parameter estimation problem of non-linear dynamical systems to tackle the exploding and vanishing gradients. ",
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+    {
+        "type": "text",
+        "text": "The focus of this paper is on deriving a priori guarantee of attaining finite-time convergence of training in a supervised learning framework under some assumptions on inputs. The novelty lies in the fact that the weight update is cast as a finite-time control synthesis such that the loss function is proven to be a valid Lyapunov function. The resulting training update is derived as a function of time such that it ensures the convergence of Lyapunov function in finite time. The only assumption used is that the magnitude of atleast one input is greater than zero. Thus, the learning problem is converted into a finite time stabilization problem as studied rigorously in Bhat & Bernstein (2000). The contributions of the proposed study are twofold. First, we propose a Lyapunov candidate function to be used as loss function. Second, we modify the weight update of the neural network in such a way that the supervised training is converted into a dynamical control system. This allows us to use results from Bhat & Bernstein (2000) to a priori guarantee finite time convergence on the training. To the best of our knowledge, a guarantee of finite-time convergence is being studied for the first time in context of training a general multi-layer neural network. The proposed results will enable time bound training that will be useful in real-time applications. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The paper is organized as follows. Section 2 starts with introducing the Lyapunov function from the control theory perspective. Section 2.1 derives the weight update and lyapunov loss function for a single neuron case and proves that it satisfies the conditions required for finite-time stability theorems developed in Bhat & Bernstein (2000) to be applicable. Section 2.2 then proves that a similar result extends to a multi-layer perceptron network under reasonable assumptions on the input. In Section 2.3, we state the equations to compute upper bounds on the convergence time for training neural networks. Section 2.4 provides an extension to the case when bounded perturbations are admitted at the input and convergence guarantees are shown to hold true. In Section 3, some numerical simulations are presented for both single neuron and multi-layer perceptron cases for regression. Section 4 collects conclusions and discusses future scope. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PROPOSED METHOD TO CONVERT SUPERVISED LEARNING INTO DYNAMICAL CONTROL SYSTEM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            626,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "This section motivates the development of a priori bounds on settling time with certain assumptions on the input. The weight update problem for supervised learning in neural networks is similar to the tracking problem of non-linear control systems. The idea is to synthesize a feedback control law based on certain Lyapunov function $V ( x )$ . A Lyapunov function is a positive definite function, i.e. $V ( x ) > 0$ , and its time derivative is negative definite along the given system dynamics, i.e. $\\dot { V } < 0$ . We cast the supervised learning problem as a dynamical system which admits a valid Lyapunov function as the loss function with the weight update is designed as a function of time. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 SINGLE NEURON CASE",
+        "text_level": 1,
+        "bbox": [
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+            792,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We start with a simplistic single neuron case. Let $x \\in \\mathbb { R } ^ { n }$ be the input to the network where $x = [ x _ { 1 } \\quad x _ { 2 } \\quad \\ldots \\quad x _ { n } ] ^ { \\top } , | x _ { i } | < c , i = 1 , 2 , \\cdots n$ $c \\in \\mathsf { \\Gamma } ( 0 , \\infty )$ . Let with i $y ^ { \\star }$ beuts e target oand bias tput and . The de $\\textstyle z = \\sum _ { i = 1 } ^ { n } w _ { i } x _ { i } + b$ holds true for some a priori but arbitrary scalar be the linear comction used here is $w _ { i }$ $x _ { i }$ $b$ $\\mathrm { s i g n } ( x ) =$ $1 , \\forall x > 0 , { \\mathrm { s i g n } } ( x ) = - 1 , \\forall x < 0 , { \\mathrm { s i g n } } ( x ) \\in [ - 1 , 1 ] , x = 0$ . For our analysis, we choose sigmoid function as our activation function, i.e. $\\begin{array} { r } { \\sigma ( z ) \\doteq \\frac { 1 ^ { - } } { 1 + e ^ { - z } } } \\end{array}$ . The output of the neural network is given by $y = \\sigma ( z )$ . Let the error in output be defined as $\\bar { e } = y - y ^ { * }$ . The first objective is to convert our loss function into a candidate Lyapunov function in order to apply the control theoretic principles. Consider a continuous function $E ( \\bar { e } )$ to be a candidate Lyapunov function as follows: ",
+        "bbox": [
+            173,
+            819,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/872ea7e653ce72855c51059ab2110b9dfc5edfb2693a29328a7c922dccf8d82c.jpg",
+        "text": "$$\nE = \\frac { | \\bar { e } | ^ { ( \\alpha + 1 ) } } { ( \\alpha + 1 ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            450,
+            138,
+            545,
+            174
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha \\in ( 0 , 1 )$ is a user-defined parameter. The second objective is to define the temporal rate of weight as the control input to enforce the stability of the origin $\\bar { e } = 0$ as $t \\to \\infty$ . The Lyapunov function in (1) is used to show that it is indeed plausible to achieve this asymptotic stability goal. Taking the temporal derivative of the candidate Lyapunov function (1) produces (by chain rule of differentiation) ",
+        "bbox": [
+            173,
+            179,
+            825,
+            250
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1ba005a9e05d6860ffb82ef99f0c73f5b99e6939dfda56a7aaa9ba5d40f36ac.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } E } { \\mathrm { d } t } = \\frac { \\mathrm { d } E } { \\mathrm { d } \\bar { e } } \\frac { \\mathrm { d } \\bar { e } } { \\mathrm { d } y } \\frac { \\mathrm { d } y } { \\mathrm { d } z } \\frac { \\mathrm { d } z } { \\mathrm { d } w } \\frac { \\mathrm { d } w } { \\mathrm { d } t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            246,
+            584,
+            280
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\frac { \\mathrm { d } E } { \\mathrm { d } \\bar { e } } \\frac { \\mathrm { d } \\bar { e } } { \\mathrm { d } y } \\frac { \\mathrm { d } y } { \\mathrm { d } z } \\frac { \\mathrm { d } z } { \\mathrm { d } w }$ is the weight update for standard gradient descent algorithm and $\\textstyle { \\frac { \\mathrm { d } w } { \\mathrm { d } t } }$ is the additional term we introduce to the weight update equation. Using (1), we get: ",
+        "bbox": [
+            174,
+            284,
+            821,
+            315
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d8ddb5c0e54a95facb67fcd8e2d6b64466a9bf58a7dc5ca76b89fa175a0aa58.jpg",
+        "text": "$$\n\\frac { d E } { d t } = | \\bar { e } | ^ { \\alpha } \\mathrm { s i g n } ( \\bar { e } ) \\left( \\frac { e ^ { - z } } { ( 1 + e ^ { - z } ) ^ { 2 } } \\right) \\left( x _ { 1 } \\dot { w } _ { 1 } + x _ { 2 } \\dot { w } _ { 2 } + \\cdot \\cdot \\cdot + x _ { n } \\dot { w } _ { n } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            320,
+            717,
+            356
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Define, $u _ { 1 } \\triangleq \\dot { w } _ { 1 } , u _ { 2 } \\triangleq \\dot { w } _ { 2 } , \\cdot \\cdot \\cdot , u _ { n } \\triangleq \\dot { w } _ { n }$ , and ",
+        "bbox": [
+            174,
+            363,
+            480,
+            380
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95bff0a6b5a6072ca78c47de10a1a2ff308e4e9bfe527b06c950d11cc385d248.jpg",
+        "text": "$$\nu _ { i } = - k _ { i } \\mathrm { s i g n } ( x _ { i } ) \\mathrm { s i g n } ( \\bar { e } ) e ^ { z } ( 1 + e ^ { - z } ) ^ { 2 } , \\quad i = 1 , 2 , \\cdots , n ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            383,
+            694,
+            402
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $k _ { i } > 0$ , for all $i$ , are tuning parameters to be chosen by the user. It can be noted that all control inputs $u _ { 1 } , u _ { 2 } , \\cdots , u _ { n }$ remain bounded due to boundedness assumption of all the inputs $x _ { i }$ and that of $e ^ { z }$ . Substituting (3) into (2) produces ",
+        "bbox": [
+            173,
+            407,
+            825,
+            450
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa4850938208647c60ecfde4c1a51f500f4d29fa9ba1f8b75901cab64f139946.jpg",
+        "text": "$$\n\\frac { d E } { d t } = - | \\bar { e } | ^ { \\alpha } \\left( \\sum _ { i = 1 } ^ { n } k _ { i } | x _ { i } | \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            455,
+            589,
+            498
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. At least one input of all $x _ { i } , i = 1 , 2 , \\cdots , n$ is non-zero such that $| x _ { j } | > \\gamma > 0$ where $\\gamma$ is a priori known scalar for some integers $j \\in [ 1 , n ]$ . ",
+        "bbox": [
+            171,
+            503,
+            825,
+            534
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It can be noted that Assumption 1 is reasonable for many practical applications in that some inputs will always be nonzero with a known lower bound on its magnitude. First main result of the paper is in order. ",
+        "bbox": [
+            173,
+            542,
+            825,
+            585
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Assuming Assumption 1 holds true, let the output of the neural network be given by $y = \\sigma ( z )$ . Let all the inputs $x _ { i } , i = 1 , 2 , \\cdots , n$ be bounded by some a priori known scalar $a \\in$ $( 0 , \\infty )$ such that $| x _ { i } | < a$ holds true for all $i$ . Then, weight update (3) causes the error $\\bar { e } = y - y ^ { * }$ to converge to zero in finite time. ",
+        "bbox": [
+            173,
+            589,
+            825,
+            646
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof of the theorem is furnished using standard Lyapunov analysis arguments. Consider $E$ defined by (1) as a candidate Lyapunov function. Observing (4), it can be concluded that the right hand side of the temporal derivative of remains negative definite since it involves power terms and norm of inputs. Furthermore, (4) can be rewritten under the assumption 1 as follows: ",
+        "bbox": [
+            173,
+            661,
+            825,
+            718
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/75ed1f2068fff90fdfb171c62f1ca6fe03a1ef78a38952899f9f6f010356c6da.jpg",
+        "text": "$$\n\\frac { d E } { d t } \\leq - k _ { \\operatorname* { m i n } } \\gamma E ^ { \\beta }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            723,
+            562,
+            753
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { k _ { \\operatorname* { m i n } } = \\operatorname* { m i n } ( k _ { i } ) , i = 1 , 2 , \\cdots , n , \\beta = \\frac { \\alpha } { \\alpha + 1 } } \\end{array}$ = αα+1 and |e¯|α = \u0000|e¯|α+1\u0001 αα+1 = Eβ has been utilized. Noting that $E$ is a positive definite function and scalars $k _ { \\mathrm { m i n } }$ and $\\gamma$ are always positive, the proof is complete by applying (Bhat & Bernstein, 2000, Theorem 4.2). □ ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 MULTI NEURON CASE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            824,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a multi-layer perceptron with $N$ layers where the layers are connected in a feed-forward manner (Bishop, 1995, Chapter 4). Let $\\boldsymbol { x } \\in \\mathbb { R } ^ { n }$ be the input to the network where $x \\quad =$ $\\left[ x _ { 1 } \\quad x _ { 2 } \\quad \\cdots \\quad x _ { n } \\right] ^ { \\top }$ and $| x _ { i } | < c , i = 1 , 2 , \\cdot \\cdot \\cdot n$ holds true for some a priori but arbitrary scalar $c \\in$ $( 0 , \\infty )$ . Let $y \\in \\mathbb { R } ^ { m }$ define the multi-neuron output to the network where $y = [ y _ { 1 } \\quad y _ { 2 } \\quad \\cdot \\cdot \\cdot \\quad y _ { m } ]$ . Let $y ^ { \\star } \\in \\mathbb { R } ^ { m }$ define the target output values of the network where $y ^ { \\star } = [ y _ { 1 } ^ { \\star } \\quad y _ { 2 } ^ { \\star } \\quad \\cdot \\cdot \\quad y _ { m } ^ { \\star } ]$ . The error in the output layer can be expressed as $\\bar { e } = [ \\left| y _ { 1 } - y _ { 1 } ^ { \\star } \\right| \\quad \\left| y _ { 2 } - y _ { 2 } ^ { \\star } \\right| \\quad \\cdot \\cdot \\quad \\left| y _ { m } - y _ { m } ^ { \\star } \\right| ]$ . Hence, the scalar candidate Lyapunov function can be written as follows: ",
+        "bbox": [
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+            849,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            102,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64f4032b7201624c5aa7625653cae8979801a2461d12a488e464d381745c1ab3.jpg",
+        "text": "$$\nE = E _ { 1 } + \\cdot \\cdot \\cdot + E _ { m } = { \\frac { | { \\bar { e } } _ { 1 } | ^ { ( \\alpha + 1 ) } } { ( \\alpha + 1 ) } } + \\cdot \\cdot \\cdot + { \\frac { | { \\bar { e } } _ { m } | ^ { ( \\alpha + 1 ) } } { ( \\alpha + 1 ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            148,
+            676,
+            185
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As is usually done in the case of feed-forward networks, consider unit $j$ of layer $l$ that computes its output ",
+        "bbox": [
+            173,
+            188,
+            823,
+            217
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9691f01bede06feadd31f35682c04e1c5b7dfcbc684eb0a9ccfb8b0be9f73f3d.jpg",
+        "text": "$$\na _ { j } ^ { l } = \\sum _ { i } w _ { j i } ^ { l } z _ { i } ^ { l }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            447,
+            215,
+            552,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "using its inputs $z _ { i } ^ { l }$ from layer $l$ where bias parameter has been embedded inside the linear combination and $z _ { j } ^ { l } = \\sigma ( a _ { j } ^ { l - 1 } )$ , where $\\sigma$ is non-linear activation function. The aim of this section is to extend the single neuron case to a multi-neuron one. The simplest way do achieve this is to find sensitivity of $E$ to the weight $w _ { j i } ^ { l }$ of a given layer $l$ , which is given by ",
+        "bbox": [
+            173,
+            253,
+            825,
+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f63aad8b4f807131b36343bf6adc5c3c47de23ef9728037efc4863310985217.jpg",
+        "text": "$$\n\\frac { \\partial E _ { m } } { \\partial w _ { j i } ^ { l } } = \\frac { \\partial E _ { m } } { \\partial a _ { j } ^ { l } } \\frac { \\partial a _ { j } ^ { l } } { \\partial w _ { j i } ^ { l } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            323,
+            568,
+            363
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using standard notation $\\begin{array} { r } { \\delta _ { j } ^ { l } \\triangleq \\frac { \\partial E _ { m } } { \\partial a _ { j } ^ { l } } } \\end{array}$ with (7) results in: ",
+        "bbox": [
+            173,
+            369,
+            529,
+            392
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc73ec32b1af3e1ff5b4c76c492d81131a7a85f47a9250ca4437944c86c871df.jpg",
+        "text": "$$\n\\frac { \\partial E _ { m } } { \\partial w _ { j i } ^ { l } } = \\delta _ { j } ^ { l } z _ { i } ^ { l }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            398,
+            545,
+            436
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is straightforward to compute $\\delta _ { m } ^ { L }$ that belongs to the output layer $L$ as shown below: ",
+        "bbox": [
+            171,
+            444,
+            740,
+            460
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/da38da637cabc28725527e3ab82c015bbdde2c70c07085e5aef3e0dc45d27809.jpg",
+        "text": "$$\n\\delta _ { m } ^ { L } = \\frac { \\partial E _ { m } } { \\partial a _ { m } ^ { L } } = \\sigma ^ { \\prime } ( a _ { m } ^ { L } ) \\frac { \\partial E _ { m } } { \\partial y _ { m } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            467,
+            594,
+            501
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { m } ^ { L }$ is replaced by $y _ { m }$ as it is the output layer. Finally, computation of $\\delta _ { j } ^ { l }$ for all hidden units is given by ",
+        "bbox": [
+            173,
+            508,
+            823,
+            536
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f803de9aa8bf0000ab6be5c5f60207d8625920c71b53efb0241c91687676ce33.jpg",
+        "text": "$$\n\\delta _ { j } ^ { l } = \\frac { \\partial E _ { m } } { \\partial a _ { j } ^ { l } } = \\sum _ { k } \\frac { \\partial E _ { m } } { \\partial a _ { k } ^ { l + 1 } } \\frac { \\partial a _ { k } ^ { l + 1 } } { \\partial a _ { j } ^ { l } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            536,
+            606,
+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where units with a label $k$ includes either hidden layer units or an output unit in layer $l + 1$ . Combining (7), $z _ { j } ^ { l } = \\sigma ( a _ { j } ^ { l } )$ and $\\begin{array} { r } { \\delta _ { j } ^ { l } \\triangleq \\frac { \\partial E _ { m } } { \\partial a _ { j } ^ { l } } } \\end{array}$ produces ",
+        "bbox": [
+            176,
+            580,
+            821,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e77fe2b19c659719860eaa3e4c9d5bd48a6a72e6d392a564b4b5e4d4b518311c.jpg",
+        "text": "$$\n\\delta _ { j } ^ { l } = \\sigma ^ { \\prime } ( a _ { j } ^ { l } ) \\sum _ { k } w _ { k j } ^ { l + 1 } \\delta _ { k } ^ { l + 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            625,
+            586,
+            659
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A slightly modified version of assumption 1 is required before the next result of the paper is presented. ",
+        "bbox": [
+            173,
+            664,
+            823,
+            694
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2. At least one input of all $z _ { i } , i = 1 , 2 , \\cdots , L$ is non-zero such that $\\left| z _ { n } \\right| > \\gamma > 0$ where $\\gamma$ is a priori known scalar for some integers $n \\in [ 1 , L ]$ where $L$ is the number of units in layer $l$ . ",
+        "bbox": [
+            173,
+            698,
+            826,
+            739
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Let the weight update for connecting unit $i$ of layer $l$ to unit $j$ of layer $l + 1$ of $a$ multi-layer neural network be given by ",
+        "bbox": [
+            173,
+            744,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7398c7ad851567a3f672544aa35074a599bae7f105f5f0b152db2e4e6852a4a5.jpg",
+        "text": "$$\n\\begin{array} { r } { \\dot { w } _ { j i } ^ { l } = - k _ { j i } ^ { l } \\mathrm { s i g n } ( \\delta _ { j } ^ { l } z _ { i } ^ { l } ) | \\delta _ { j } ^ { l } z _ { i } ^ { l } | ^ { \\alpha } E ^ { \\beta } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with some scalar $\\beta \\in ( 0 , 1 )$ such that $\\alpha + \\beta < 1$ and $k _ { j i } > 0$ is a tuning parameter. Then the output vector y converges to $y ^ { * }$ in finite time. ",
+        "bbox": [
+            173,
+            808,
+            826,
+            838
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. Consider the candidate Lyapunov function $E$ given by (6). The temporal derivative of the Lyapunov function is given by ",
+        "bbox": [
+            169,
+            853,
+            823,
+            885
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa714f02d5719f8f68af2c6ab1d42ee45ba3df73a2f4954817ab6b80b83466b9.jpg",
+        "text": "$$\n\\dot { E } = \\sum _ { m } \\dot { E } _ { m } = \\sum _ { m } \\frac { \\partial E _ { m } } { \\partial w _ { j i ^ { l } } } \\dot { w } _ { j i } ^ { l }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            890,
+            599,
+            928
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which can be simplified using (13) and (9) as ",
+        "bbox": [
+            176,
+            103,
+            470,
+            118
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c143e106a3225e2588e3de4d29da01723a567a9d88baa6ae02169926fcf01598.jpg",
+        "text": "$$\n\\dot { E } = - E ^ { \\beta } \\sum _ { m } k _ { j i } ^ { l } | \\delta _ { j } ^ { l } z _ { i } ^ { l } | ^ { \\alpha + 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            126,
+            591,
+            160
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using Assumption 2 it is easy to conclude that for some $k _ { \\mathrm { m i n } } = \\operatorname* { m i n } _ { i , j , l } k _ { j i } ^ { l } > 0$ , the following inequality holds true: ",
+        "bbox": [
+            169,
+            167,
+            825,
+            205
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49d712ce85c1baabb5648709694d2d26c38691e128d540752f5885f74906b05e.jpg",
+        "text": "$$\n\\dot { E } \\le - k _ { \\mathrm { m i n } } \\gamma ^ { \\alpha + 1 } E ^ { \\beta }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            204,
+            568,
+            223
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Noting that $E$ is a positive definite function and scalars $k _ { \\mathrm { m i n } }$ , $\\gamma$ are always positive, the proof is complete by applying (Bhat & Bernstein, 2000, Theorem 4.2). □ ",
+        "bbox": [
+            171,
+            229,
+            825,
+            258
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. It can be seen that weight update $( I 3 )$ (or respectively (3)) is in a feedback control form where state $z _ { i }$ and $\\delta _ { j }$ (respectively $x _ { i }$ and $\\bar { e }$ ) are being used for influencing the learning process. ",
+        "bbox": [
+            173,
+            270,
+            825,
+            299
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 SETTLING TIME FOR NEURAL NETWORK TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            318,
+            566,
+            333
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 and 2 prove that using candidate Lyapunov function as the loss function and the temporal rate of change of the loss function as the weight update equation, we can convert supervised learning framework into a control problem. (Bhat & Bernstein, 2000, Theorem 4.2) states that if there is continuous function that is positive definite and it’s rate of change with respect to time is negative definite, then it’s finite time stable equilibrium is at origin. The theorem also gives the settling time when the above conditions are satisfied by the control system. For our case, this means that the Lyapunov based loss function will converge in finite time and time taken to converge is as given below: ",
+        "bbox": [
+            173,
+            343,
+            825,
+            455
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The settling time for Single Neuron case: ",
+        "bbox": [
+            176,
+            462,
+            459,
+            477
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/369cbd0f9d9cb89c30427d18b5299eef7e38545f14f19942ea69ac45f276c817.jpg",
+        "text": "$$\nT \\leq \\frac { 1 } { k _ { m i n } \\gamma ( 1 - \\beta ) } E _ { i n i } ^ { ( 1 - \\beta ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            484,
+            589,
+            518
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The settling time for Multi Neuron case: ",
+        "bbox": [
+            176,
+            532,
+            454,
+            547
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7c5b8287e9049f1c0e5fbc51cc1c1c0e31a0d5b2a670b2c5790f9f18b1dee5e5.jpg",
+        "text": "$$\nT \\leq \\frac { 1 } { k _ { m i n } \\gamma ^ { \\alpha + 1 } ( 1 - \\beta ) } E _ { i n i } ^ { ( 1 - \\beta ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            555,
+            602,
+            588
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma$ is the lower bound on the inputs, $k _ { m i n }$ is the minimum value of tuning parameter $k , \\alpha$ is the scalar used in the Lyapunov loss function and $E _ { i n i }$ is the initial value of loss, i.e. loss at $\\mathrm { { t } } = 0$ . ",
+        "bbox": [
+            176,
+            595,
+            826,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 SENSITIVITY TO PERTURBATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            643,
+            446,
+            657
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This section considers the robustness of training a neural network based on our proposed algorithm. In control theoretic terms, perturbation can be understood either as a disturbance to the process or as modelling uncertainty. When supervised learning is viewed as a control process, perturbation in each neuron of the input layer can be seen as external noise. This perturbation could cause the control system to diverge. In this section, we develop theoretical claims that even with perturbed inputs, training a neural network with the proposed algorithm will converge. The following assumption of the upper bound on the perturbation of inputs is invoked. ",
+        "bbox": [
+            173,
+            669,
+            825,
+            767
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3. There exists an a priori known constant $M \\ > \\ 0$ such that all inputs $x _ { i } , i \\ =$ $1 , 2 , \\cdots , N$ admit additive perturbations $\\Delta x _ { i }$ such that ",
+        "bbox": [
+            173,
+            772,
+            825,
+            801
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b40939002753316b2ec8a4141954ba6a3eadbb259524e33cf0b580dbbe1fc8cd.jpg",
+        "text": "$$\n| \\Delta x _ { i } | \\leq M | x _ { i } | ^ { \\alpha }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            809,
+            555,
+            827
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "for all i where $N$ is the number of inputs. ",
+        "bbox": [
+            173,
+            834,
+            444,
+            849
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following result is in order. ",
+        "bbox": [
+            174,
+            861,
+            382,
+            876
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3. Let assumptions 2 and 3 hold true. Let the weight update for connecting unit i of layer $l$ to unit $j$ of layer $l + 1$ of a multi-layer neural network be given by (13). Then the output vector $y$ converges to $y ^ { * }$ in finite time in the presence of additive perturbations $\\Delta x _ { n } , n \\in \\left[ 1 , L \\right] i \\bar { f } k _ { \\operatorname* { m i n } } > M$ . ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. It can be seen that the perturbation is considered only in inputs. Hence all hidden layer weights are updated as done in the proof of Theorem 2. Hence, the dynamics of learning results in the following revised temporal derivative of Lyapunov function: ",
+        "bbox": [
+            173,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4dc72b6e4df57eda670ac819656a88f05ca5ffbef21977b4d29b2a9113387dc3.jpg",
+        "text": "$$\n\\dot { E } = - E ^ { \\beta } \\sum _ { m } k _ { j i } | \\delta _ { j } z _ { i } | ^ { \\alpha + 1 } + \\sum _ { p = 0 } ^ { n } k _ { 1 p } | \\delta _ { p } x _ { p } | ^ { \\alpha } \\mathrm { s i g n } ( \\delta _ { p } x _ { p } ) \\delta _ { p } \\Delta x _ { p }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            151,
+            710,
+            194
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $k _ { 1 p }$ is the gain parameter for training of all the input layer weights. The expression in (20) can be simplified using Assumption 3 as follows: ",
+        "bbox": [
+            173,
+            200,
+            825,
+            229
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcc361cf180566dce1ec779d8922bc9877e543fe464039ad8a929eb98e15288e.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\dot { E } \\leq - E ^ { \\beta } \\sum _ { m } k _ { j i } | \\delta _ { j } z _ { i } | ^ { \\alpha + 1 } + E ^ { \\beta } \\sum _ { p = 0 } ^ { n } { k _ { 1 p } | \\delta _ { p } x _ { p } | ^ { \\alpha + 1 } M } , } \\\\ { \\displaystyle \\leq - E ^ { \\beta } \\sum _ { m } ( k _ { j i } - M ) | \\delta _ { j } z _ { i } | ^ { \\alpha + 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            234,
+            679,
+            313
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where inputs $z _ { i }$ now collect inputs $x _ { i }$ as well. Similar to the proof of Theorem 2, (21) can be re-written by applying Assumption 2 as follows: ",
+        "bbox": [
+            171,
+            318,
+            825,
+            345
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/529bf585a6d0757875ce3eb6e53b38f5be512bd8cb70f06d1537cdb6e28e0d70.jpg",
+        "text": "$$\n\\dot { E } \\le - ( k _ { \\operatorname* { m i n } } - M ) \\gamma E ^ { \\beta }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            352,
+            581,
+            372
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since $k _ { \\operatorname* { m i n } { } } > M$ , the proof is complete by applying (Bhat & Bernstein, 2000, Theorem 4.2). ",
+        "bbox": [
+            173,
+            377,
+            779,
+            393
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            328,
+            429
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This section presents empirical evidence that the theoretical results derived in the above sections hold for real-life applications. ",
+        "bbox": [
+            173,
+            443,
+            823,
+            473
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Convergence Rate for training. To analyze the training convergence rate of the proposed method, we train a multi-layer perceptron (covered in Section 2.2) to perform regression on the Boston Housing dataset (Harrison Jr & Rubinfeld (1978)). Experiment on single neuron case is covered in Appendix A.2.2 ",
+        "bbox": [
+            173,
+            478,
+            825,
+            535
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training details. We compare our proposed method (Lyapunov based loss and modified weight update equation) with traditional $L _ { 1 }$ , $L _ { 2 }$ loss functions and standard SGD weight update equation. The modifications to the weight update equation for Lyapunov loss are as stated in (15) for multi-neuron case. We plot the training and test loss with respect to time in Figure 1. ",
+        "bbox": [
+            173,
+            541,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The details about the training examples, epochs, learning rate etc. are specified under the description of the figures. We observe that the Lyapunov Loss function converges faster than $L _ { 1 }$ and $L _ { 2 }$ loss functions, demonstrating the results proven extensively in Section 2. This is attributed mainly to the non-Lipschitz weight updates given by (3) and (13). ",
+        "bbox": [
+            174,
+            598,
+            483,
+            709
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The settling time as well as metrics of the trained networks are reported in Table 1. The time taken by the proposed algorithm to converge falls within the a priori upper bound. We also observe that the metrics (Accuracy for single neuron case and rmse (Root mean square error) for multi-neuron case) achieved by the proposed method is better than the baselines. Admittedly, the theoretical upper bound on settling time for single neuron and multi-layer perceptron produced by (5) and (16) are very conservative, yet it is an a priori deterministic upper bound and the aggressive learning proves to be faster than traditional loss functions in the scenarios considered here. ",
+        "bbox": [
+            174,
+            715,
+            483,
+            922
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/94677af0d4160d1d0f013e93407e9bd174fcde65f7757d87a59d03e10d863a68.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of convergence with respect to time for the multi-layer perceptron trained on the Boston Housing dataset. All networks are trained for 4000 epochs with learning rate $= 0 . 0 2$ and $\\alpha = 0 . 6 8$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
+            607,
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+            838
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/d1c2910423b44b4d3f4bd0fa4e11fb7c284f1aee8dfd25da184b713eb0dbe40d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Experiment</td><td rowspan=\"2\">Theoretical Upper Bound</td><td colspan=\"3\">Exp. Convergence Time (in seconds)</td><td colspan=\"3\">Metric</td></tr><tr><td>(in seconds)</td><td>L1</td><td>L2</td><td>Lyap.</td><td>L1</td><td>L2</td><td>Lyap.</td></tr><tr><td>SingleNeuron (acc)</td><td>~3.4e3</td><td>0.065</td><td>0.064</td><td>0.025</td><td>0.95</td><td>0.95</td><td>1.0</td></tr><tr><td>MLP (rmse)</td><td>~ 1.4e10</td><td>4.3e3</td><td>3.4e3</td><td>3.1e3</td><td>0.091</td><td>0.179</td><td>0.085</td></tr></table>",
+        "bbox": [
+            173,
+            101,
+            828,
+            181
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Settling time in seconds for each experiment conducted. We compare the time taken for convergence by three different loss functions, $L _ { 1 }$ , $L _ { 2 }$ and Lyapunov Loss function. The training conditions were similar for individual cases in the experiment. The single neuron case is trained on Iris dataset and the metric used is Accuracy (higher is better). For Multi-layer perceptron, we use Boston Housing dataset and the metric used for this is rmse (lower is better). ",
+        "bbox": [
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+            190,
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+            261
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This can be attributed to the fact that we operate in the realm of non-smooth functions which tend to be more aggressive than the smooth $L _ { 2 }$ function we usually encounter in most learning problems. We can clearly observe that the loss value does not converge to zero. According to (Bhat & Bernstein, 2000, Theorem 5.2), a control system will converge to a non-zero value at steady state, if there is constant perturbation in the input. Usually in real life datasets, there is an inherent bias or constant perturbation, which prevents the loss to converge to zero. Of course, by setting $\\alpha = 0$ , we can reject all persisting disturbances (Orlov (2005)) and force the loss to converge to zero, but this may result in discontinuity in back propagation (discussed in Appendix A.1) and result into large numerical errors. ",
+        "bbox": [
+            173,
+            268,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameter analysis and its effect on training. In this section, we attempt to give an empirical analysis as to how changes in the hyperparameters like $k$ and $\\alpha$ affect the training of the neural network. In order to study the effect of tuning $k$ , we work with the single neuron case for simplicity and ease of understanding. We use Iris dataset for this experiment Dua & Graff (2017). Existing literature in control theory suggests that a monotonic increase in the tuning parameter, $k$ should result in a monotonic decrease in the corresponding loss. In neural network terminology, $k$ can be considered as the ’learning rate’. We can clearly see in Figure 2 that $k$ behaves like learning rate parameter where increasing $k$ makes the learning more aggressive. ",
+        "bbox": [
+            173,
+            401,
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+            512
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/a553b1180d6f11315ea77c381a836a1152ae4f595414a80947015fee38d20db9.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of training and test convergence with respect to time for different values of tuning parameter, $k$ in a single neuron trained on the Iris dataset. We convert the problem into a binary classification problem by considering only two output classes. All networks are trained for 600 epochs with 80 training examples and 20 test examples. The learning rate for all $L _ { 1 }$ , $L _ { 2 }$ and Lyapunov is set to the tuning parameter value $k$ and $\\alpha = 0 . 8$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            529,
+            794,
+            808
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "From Table 2, we observe that the settling time for the training loss of our Lyapunov loss function is always less than $L _ { 1 }$ and $L _ { 2 }$ . We also observe that the Accuracy of our method is either better or similar to the baselines, which shows that the optimized solution achieved by our method is either better or on par with the baselines. For the multi-neuron case, we have defined $k _ { i j } ^ { l }$ for every weight. Similar to adaptive learning rate, we can either tune $k _ { i j } ^ { l }$ for every weight (which can be quite cumbersome for larger neural networks) or we can set it arbitrarily to the same value for all weights. It must be noted that $k _ { i j } ^ { l }$ can be set to any value greater than an a priori known positive constant $M$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c68ae6c0ac1504524f930addb93a6a539078d5fd92d58d6726f02be909a061ad.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Experiment for single neuron case on Iris dataset</td><td rowspan=\"2\">Theoretical Upper Bound (in seconds)</td><td colspan=\"3\">Exp. Convergence Time (in seconds)</td><td colspan=\"3\">Accuracy on Test Set</td></tr><tr><td>L1</td><td>L2</td><td>Lyap.</td><td>L1</td><td>L2</td><td>Lyap.</td></tr><tr><td>k=0.001</td><td>~ 37213.848</td><td>0.312</td><td>0.310</td><td>0.0918</td><td>0.6</td><td>0.95</td><td>0.95</td></tr><tr><td>k = 0.005</td><td>~7442.769</td><td>0.0943</td><td>0.0942</td><td>0.0344</td><td>0.95</td><td>1.0</td><td>1.0</td></tr><tr><td>k= 0.01</td><td>~ 3721.385</td><td>0.0566</td><td>0.0573</td><td>0.0221</td><td>0.95</td><td>1.0</td><td>1.0</td></tr></table>",
+        "bbox": [
+            187,
+            189,
+            810,
+            297
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Settling time in seconds for different values of tuning parameter, $k$ for the single neuron case on Iris dataset. We compare the time taken for convergence by three different loss functions, $L _ { 1 }$ , $L _ { 2 }$ and Lyapunov Loss function. The training conditions were similar for individual cases in the experiment. ",
+        "bbox": [
+            174,
+            308,
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+            363
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We will discuss the impact of different values of $\\alpha$ on the training process of our neural network. For the given experiments, we tested the network for various values of $\\alpha$ ranging between $( 0 , 1 )$ . We observed that the training loss follows equations proved in above sections as long as $\\alpha \\in ( 0 . 5 , 0 . 9 )$ . If $\\alpha$ happens to be too close to zero, the resulting control law becomes discontinuous thereby resulting in numerical instabilities owing to the fact that the current ODE solvers are unable to handle functions that are discontinuous. More details about the case $\\alpha = 0$ is given in Appendix A.1 (Bhat & Bernstein, 2000, Theorem 5.2). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb0cb9a2a0c72346b1743ee9223cfec4cf4ee23758e8756964bd84b532fb7c69.jpg",
+        "image_caption": [
+            "Stability of learning in the presence of amplitude-bounded perturbations. ",
+            "Figure 3: Comparison of training loss convergence with respect to time for different values of input perturbations, $\\Delta x$ for a multi-layer perceptron trained on the IMDB Wiki Faces dataset.The a priori upper bound on input perturbations are as follows: (a) $\\Delta x = 0 . 1$ , (b) $\\Delta x = 0 . 2$ , $( \\mathrm { c } ) \\Delta x = 0 . 3$ Learning rate $k = 0 . 0 0 0 9$ , $\\alpha = 0 . 8$ , epochs $_ { \\mathrm { - } 1 0 0 }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            218,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present results that demonstrate the robustness of proposed algorithm to bounded perturbations in the input while maintaining convergence of training dynamics. For this case, we train a multi-layer perceptron on a regression task of predicting the age given the image of a face. We use IMDB Wiki Faces Dataset Rothe et al. (2015) which has over 0.5 million face images of celebrities with their age and gender labels. Only part of the dataset is used for this experiment, i.e. 20,000 images for training and 4,000 images each for validation and test. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For adding input perturbations, we asssume that an a priori upper bound, $M$ is known on the amplitude of possible input perturbations following Assumption 3. We add input perturbations to each pixel in the image using a randomized uniform distribution ranging from $( - \\Delta x , \\Delta x )$ . Hence, our additive noise remains in the abovementioned a priori bounds. We vary the values of $M$ from (0.1, 0.3) with a 0.1 increment, giving us three training cases. Figure 3 shows that the proposed training loss still manages to achieve steady state error and does not diverge due to perturbations introduced in the input dataset. Since we are dealing with noisy data, the loss values converge to a non-zero value in the steady-state ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Experiments on larger datasets. In this section, we present results to demonstrate the performance of our proposed algorithm on a larger dataset. We use the entire IMDB Wiki Faces Dataset Rothe et al. (2015) with 0.5 million images for this experiment. Training dataset consists of ${ \\sim } 0 . 2$ million images and the test and validation set consists of ${ \\sim } 0 . 1$ million images each. As described in the previous section, a multi-layer perceptron is trained to predict the age from the image of a face. The MLP is trained for 100 epochs with learning rate 0.0005 for all three loss functions and with $\\alpha = 0 . 7$ for Lyapunov loss function. From Figure 4 and Table 3, we can see that our proposed algorithm achieves similar generalization / testing rmse (Root mean square error) as $L _ { 1 }$ and $L _ { 2 }$ baselines while providing finite time convergence guarantees even on large datasets. The convergence time is also within the theoretically derived upper bound. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/5b5329f6c05dd7e9ed3e8faf2625740921023501de47d20852c27b587256edbe.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of training convergence with respect to time for 0.5 million IMDB Wiki dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            294,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/1839bb9925424e9cdc0b0796452e088c320a336c82bbf5966a1e0b30f6447cdb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Experiment</td><td>Theoretical Upper Bound</td><td colspan=\"3\">Exp. Convergence Time (in seconds)</td><td colspan=\"3\">Metric</td></tr><tr><td></td><td>(in seconds)</td><td>L1</td><td>L2</td><td>Lyap.</td><td>L1</td><td>rmse L2</td><td>Lyap.</td></tr><tr><td>IMDBWiki</td><td>8.3e6</td><td>980.40</td><td>712.99</td><td>467.96</td><td>0.414</td><td>0.415</td><td>0.416</td></tr></table>",
+        "bbox": [
+            189,
+            609,
+            810,
+            676
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 3: rmse on IMDB Wiki test dataset for $L _ { 1 }$ , $L _ { 2 }$ and Lyapunov Loss function. ",
+        "bbox": [
+            228,
+            685,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            712,
+            495,
+            729
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper studies the training of a deep neural network from control theory perspective. We pose the supervised learning problem as a control problem by jointly designing loss function as Lyapunov function and weight update as temporal derivative of the Lyapunov function. Control theory principles are then applied to provide guarantees on finite time convergence and settling time of the neural network. Through experiments on benchmark datasets, our proposed method converges within the a priori bounds derived from theory. It is also observed that in some cases our method enforces faster convergence as compared to standard $L _ { 1 }$ and $L _ { 2 }$ loss functions. We also prove that our method is robust to any perturbations in the input and convergence guarantees still hold true. The given a priori guarantees for the convergence time is a desirable result for training networks that are extremely difficult to converge, specifically in Reinforcement Learning. A future scope of this work may be to convert the continuous time analysis framework to discrete time. This study introduces a novel perspective of viewing neural networks as control systems and opens up the field of machine learning research to a plethora of new results that can be derived from control theory. ",
+        "bbox": [
+            173,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
+            299,
+            357
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 IMPACT OF $\\alpha$ ON TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            375,
+            406,
+            388
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The case when $\\alpha = 0$ , as depicted in Figure 5, raises continuity issues. ",
+        "bbox": [
+            173,
+            402,
+            635,
+            417
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/dba33dd8e19f76befd2fd50e3232f785f2f61abf9d083b943e646c81f5a3789d.jpg",
+        "image_caption": [
+            "Figure 5: Depiction of the numerical instability observed when we take $\\alpha = 0$ . Effectively, at this point, the loss function contains a discontinuous signum function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            318,
+            460,
+            653,
+            651
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Consider the left and right limits of the function $f ( \\varrho ) = | \\varrho | ^ { \\alpha } \\mathrm { s i g n } ( \\varrho )$ , where $\\varrho = \\delta _ { j } z _ { i }$ as appearing in (15). It can be seen that $\\begin{array} { r } { \\operatorname* { l i m } _ { \\varrho \\to 0 ^ { - } } f ( \\varrho ) = \\operatorname* { l i m } _ { \\varrho \\to 0 ^ { + } } f ( \\varrho ) = 0 } \\end{array}$ for $\\alpha \\in ( 0 , 1 )$ . However, for $\\alpha = 0$ , $\\begin{array} { r } { \\operatorname* { l i m } _ { \\varrho \\to 0 ^ { - } } f ( \\varrho ) = - 1 } \\end{array}$ and $\\begin{array} { r } { \\operatorname* { l i m } _ { \\varrho \\to 0 ^ { + } } f ( \\varrho ) = 1 } \\end{array}$ . It should be noted that function $f ( \\varrho )$ is non-Lipschitz since $\\partial f / \\partial \\varrho$ tends to infinity in the limit $\\varrho \\to 0$ . We agree that $\\alpha = 0$ reduces the loss function to L1 loss, but the control update becomes purely discontinuous due to the presence of $f ( \\varrho )$ in (15). We do not deal with this case for the reasons of continuity as mentioned in the main paper. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            828,
+            318,
+            843
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            856,
+            393,
+            869
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For our experiments, we use the NVIDIA GEFORCE GTX1080Ti GPU card with 12GB RAM, 4- core CPU with 32GB of RAM for training. The training code is in Python. All networks are trained on data with an $8 0 \\% - 2 0 \\%$ train data-test data split. ",
+        "bbox": [
+            174,
+            881,
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+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2.2 CONVERGENCE EXPERIMENTS ON SINGLE NEURON",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            591,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we perform binary classification on the Iris dataset (Dua & Graff (2017)) as a representative example of single neuron case presented in Section 2.1. ",
+        "bbox": [
+            174,
+            135,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/484caabc86462e795402029c89b573f150b284c3ff0f88ef33e68e42ffbeaa24.jpg",
+        "image_caption": [
+            "Figure 6: Comparison of convergence with respect to time for a single neuron trained on the Iris dataset. We convert the problem into a binary classification problem by considering only two of the three output classes. All networks are trained for 2100 epochs with 80 training examples and 20 test examples, $\\alpha = 0 . 8$ , learning rate $/ \\operatorname { k } = 0 . 0 1$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            196,
+            785,
+            366
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "From the Figure 6, we can clearly see that Lyapunov loss function with control weight update converges much faster towards zero as compared to $L _ { 1 }$ and $L _ { 2 }$ loss with standard gradient descent weight update. This shows that explicitly adding a control weight update (15) that drives the loss towards zero is helpful in reaching convergence faster. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2.3 BOSTON HOUSING DATASET EXPERIMENT WITH PERTURBATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            547,
+            683,
+            563
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The Boston Housing Dataset predicts the price of houses in various areas of Boston Mass, given 14 different relevant attributes, like per capita crime rate and pupil-teacher ratio by town. The 506 examples in the data are divided into a 80-20 training-test split to give us 505 training examples and 101 test examples respectively. We consider three cases here, $M = 0 . 1$ , $M = 0 . 2$ ,and $M = 0 . 3$ . Table 4 presents the theoretical upper bounds for the proposed Lyapunov function’s settling time and experimental results for the training conducted for different upper bounds assumed for the input perturbation. We observe that the settling time for the proposed Lyapunov function is similar to the one observed for $L _ { 2 }$ loss function whereas it performs way better than the $L _ { 1 }$ loss function. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/1823a2b0ba40c65f79ffe1f9f4a8634ad7fd1905520b4384030b47e2ae9a44f6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Experiment for MLP case on Boston</td><td rowspan=\"2\">Theoretical Upper Bound (in seconds)</td><td colspan=\"3\">Exp. Convergence Time to within 10e-9 (in seconds)</td><td colspan=\"3\">Metric rmse</td></tr><tr><td>L1</td><td>L2</td><td>Lyap.</td><td>L1</td><td>L2</td><td>Lyap.</td></tr><tr><td>dataset M=0.1</td><td>~1.97e11</td><td>1277.66</td><td>906.61</td><td>755.59</td><td>0.095</td><td>0.145</td><td>0.092</td></tr><tr><td>M= 0.2</td><td>~ 5.67e10</td><td>1297.11</td><td>1097.62</td><td>914.78</td><td>0.096</td><td>0.146</td><td>0.093</td></tr><tr><td>M= 0.3</td><td>~2.73e10</td><td>1290.48</td><td>1197.53</td><td>998.05</td><td>0.093</td><td>0.147</td><td>0.101</td></tr></table>",
+        "bbox": [
+            174,
+            720,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Settling time in seconds for different values of the upper bound on additive input perturbations, $M$ for the multi layer perceptron case on Boston Housing dataset. We compare the time taken for convergence by three different loss functions, $L _ { 1 }$ , $L _ { 2 }$ and Lyapunov Loss function. The training conditions were similar for individual cases in the experiment. ",
+        "bbox": [
+            173,
+            839,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/11d49f7bb289d1b3734f3bf88fc7219a72fb3e0346b5a0100b0d547b4a55cde2.jpg",
+        "image_caption": [
+            "Figure 7: Pictorial representation of the effect of random additive noise with an upper bound of 0.2 on IMDB Wiki Faces Dataset. In this experiment, we work with monochrome images for the sake of simplicity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            165,
+            787,
+            325
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows us how the noise affects the training images. We take the specific case where $M =$ 0.2. We can clearly observe that the images obtained after adding input perturbations happens to be quite noisy. Figure 8 and Figure 9 show that even when inputs are perturbed, our proposed method converges in finite time. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/97c396dd6181b32b466eaaa868e746b184b69235d7d298d2c9dd76c686d88565.jpg",
+        "image_caption": [
+            "Figure 8: Comparison of training loss convergence with respect to time for different values of input perturbations, $\\Delta x$ for a multi-layer perceptron trained on the IMDB Wiki Faces dataset. The a priori upper bound on input perturbations are as follows: (a) $\\Delta x = 0 . 1$ , (b) $\\Delta x = 0 . 2$ , $( \\mathrm { c } ) \\Delta x = 0 . 3$ , (d) $\\Delta x = 0 . 4$ , (e) $\\Delta x = 0 . 5$ , and (f) $\\Delta x = 1 . 2$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            518,
+            803,
+            854
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/73d5133f391dd03ab63be601efd6eb5e468d1b8b4a9c16d108dbdd1ddc4e42f2.jpg",
+        "image_caption": [
+            "Figure 9: Comparison of test loss convergence with respect to time for different values of input perturbations, $\\Delta x$ for a multi-layer perceptron trained on the IMDB Wiki Faces dataset.The a priori upper bound on input perturbations are as follows: (a) $\\Delta x = 0 . 1$ , (b) $\\Delta x = 0 . 2$ , $( \\mathrm { c } ) \\Delta x = 0 . 3$ , (d) $\\Delta x = 0 . 4$ , (e) $\\Delta x = 0 . 5$ , and (f) $\\Delta x = 1 . 2$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            102,
+            802,
+            438
+        ],
+        "page_idx": 13
+    }
+]