Regularizing the optimal transport (OT) problem has proven crucial for OT theory to impact the field of machine learning. For instance, it is known that regularizing OT problems with entropy leads to faster computations and better differentiation using the Sinkhorn algorithm, as well as better sample complexity bounds than classic OT. In this work we depart from this practical perspective and propose a new interpretation of regularization as a robust mechanism, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial. This incidentally gives access to a robust dissimilarity measure on the ground space, which can in turn be used in other applications. We propose algorithms to compute this robust cost, and illustrate the interest of this approach empirically.
We investigate the use of entropy-regularized optimal transport (EOT) cost in developing generative models to learn implicit distributions. Two generative models are proposed. One uses EOT cost directly in an one-shot optimization problem and the other uses EOT cost iteratively in an adversarial game. The proposed generative models show improved performance over contemporary models for image generation on MNSIT.
We introduce an independence criterion based on entropy regularized optimal transport. Our criterion can be used to test for independence between two samples. We establish non-asymptotic bounds for our test statistic and study its statistical behavior under both the null hypothesis and the alternative hypothesis. The theoretical results involve tools from U-process theory and optimal transport theory. We also offer a random feature type approximation for large-scale problems, as well as a differentiable program implementation for deep learning applications. We present experimental results on e…
Rapidly growing product lines and services require a finer-granularity forecast that considers geographic locales. However the open question remains, how to assess the quality of a spatio-temporal forecast? In this manuscript we introduce a metric to evaluate spatio-temporal forecasts. This metric is based on an Opti- mal Transport (OT) problem. The metric we propose is a constrained OT objec- tive function using the Gini impurity function as a regularizer. We demonstrate through computer experiments both the qualitative and the quantitative charac- teristics of the Gini regularized OT problem…
Optimal Transport Regularized Divergences: Application to Adversarial Robustness
Department of Mathematics（数学系）；Texas State University（德克萨斯州立大学）；School of Information Systems and Management（信息系统与管理学院）；University of South Florida（佛罗里达州立大学）
LITIS EA4108；University of Rouen Normandy（诺曼底鲁昂大学）；INSA, University of Rouen Normandy（诺曼底鲁昂大学INSA）；Criteo AI Lab, Criteo Paris（Criteo巴黎Criteo人工智能实验室）
We introduce in this paper a novel strategy for efficiently approximating the Sinkhorn distance between two discrete measures. After identifying neglectable components of the dual solution of the regularized Sinkhorn problem, we propose to screen those components by directly setting them at that value before entering the Sinkhorn problem. This allows us to solve a smaller Sinkhorn problem while ensuring approximation with provable guarantees. More formally, the approach is based on a new formulation of dual of Sinkhorn divergence problem and on the KKT optimality conditions of this problem, wh…
Learning meaningful representations using deep neural networks involves designing efficient training schemes and well-structured networks. Currently, the method of stochastic gradient descent that has a momentum with dropout is one of the most popular training protocols. Based on that, more advanced methods (i.e., Maxout and Batch Normalization) have been proposed in recent years, but most still suffer from performance degradation caused by small perturbations, also known as adversarial examples. To address this issue, we propose manifold regularized networks (MRnet) that utilize a novel train…
This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We dev…
Optimal transport (OT) distances are finding evermore applications in machine learning and computer vision, but their wide spread use in larger-scale problems is impeded by their high computational cost. In this work we develop a family of fast and practical stochastic algorithms for solving the optimal transport problem with an entropic penalization. This work extends the recently developed Greenkhorn algorithm, in the sense that, the Greenkhorn algorithm is a limiting case of this family. We also provide a simple and general convergence theorem for all algorithms in the class, with rates tha…
Istituto Italiano di Tecnologia（意大利技术研究院）；University of Novi Sad（诺维萨德大学）；University College London（伦敦大学学院）
In this work we propose a batch version of the Greenkhorn algorithm for multimarginal regularized optimal transport problems. Our framework is general enough to cover, as particular cases, some existing algorithms like Sinkhorn and Greenkhorn algorithm for the bi-marginal setting, and (greedy) MultiSinkhorn for multimarginal optimal transport. We provide a complete convergence analysis, which is based on the properties of the iterative Bregman projections (IBP) method with greedy control. Global linear rate of convergence and explicit bound on the iteration complexity are obtained. When specia…
Université Sorbonne Paris Nord（巴黎北索邦大学）；LaMSN - La Maison des Sciences Numériques（数字科学之家）
Semi-supervised learning provides an effective paradigm for leveraging unlabeled data to improve a model's performance. Among the many strategies proposed, graph-based methods have shown excellent properties, in particular since they allow to solve directly the transductive tasks according to Vapnik's principle and they can be extended efficiently for inductive tasks. In this paper, we propose a novel approach for the transductive semi-supervised learning, using a complete bipartite edge-weighted graph. The proposed approach uses the regularized optimal transport between empirical measures def…
RIKEN AIP（理化学研究所先进智能项目中心）；The University of Akron（阿克伦大学）
Optimal transport (OT) theory provides powerful tools to compare probability measures. However, OT is limited to nonnegative measures having the same mass, and suffers serious drawbacks about its computation and statistics. This leads to several proposals of regularized variants of OT in the recent literature. In this work, we consider an \textit{entropy partial transport} (EPT) problem for nonnegative measures on a tree having different masses. The EPT is shown to be equivalent to a standard complete OT problem on a one-node extended tree. We derive its dual formulation, then leverage this to…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；IBM Research（IBM研究院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potential…
Aalto University（阿尔托大学）；University of Oxford（牛津大学）
Particle Filtering (PF) methods are an established class of procedures for performing inference in non-linear state-space models. Resampling is a key ingredient of PF, necessary to obtain low variance likelihood and states estimates. However, traditional resampling methods result in PF-based loss functions being non-differentiable with respect to model and PF parameters. In a variational inference context, resampling also yields high variance gradient estimates of the PF-based evidence lower bound. By leveraging optimal transport ideas, we introduce a principled differentiable particle filter …
Global pooling is one of the most significant operations in many machine learning models and tasks, which works for information fusion and structured data (like sets and graphs) representation. However, without solid mathematical fundamentals, its practical implementations often depend on empirical mechanisms and thus lead to sub-optimal, even unsatisfactory performance. In this work, we develop a novel and generalized global pooling framework through the lens of optimal transport. The proposed framework is interpretable from the perspective of expectation-maximization. Essentially, it aims at…
University of Southern California（南加州大学）；University of Toronto（多伦多大学）
Generative Adversarial Networks (GANs) are one of the most practical methods for learning data distributions. A popular GAN formulation is based on the use of Wasserstein distance as a metric between probability distributions. Unfortunately, minimizing the Wasserstein distance between the data distribution and the generative model distribution is a computationally challenging problem as its objective is non-convex, non-smooth, and even hard to compute. In this work, we show that obtaining gradient information of the smoothed Wasserstein GAN formulation, which is based on regularized Optimal Tr…
Ecole Polytechnique（巴黎综合理工学院）；Data61, the Australian National University & the University of Sydney（Data61、澳大利亚国立大学与悉尼大学）；The Australian National University & Data61（澳大利亚国立大学与Data61）；Ecole Polytechnique & Sony CS Labs, Inc.（巴黎综合理工学院与索尼计算机科学实验室公司）
Optimal transport is a powerful framework for computing distances between probability distributions. We unify the two main approaches to optimal transport, namely Monge-Kantorovitch and Sinkhorn-Cuturi, into what we define as Tsallis regularized optimal transport (\trot). \trot~interpolates a rich family of distortions from Wasserstein to Kullback-Leibler, encompassing as well Pearson, Neyman and Hellinger divergences, to name a few. We show that metric properties known for Sinkhorn-Cuturi generalize to \trot, and provide efficient algorithms for finding the optimal transportation plan with fo…
Following [21, 23], the present work investigates a new relative entropy-regularized algorithm for solving the optimal transport on a graph problem within the randomized shortest paths formalism. More precisely, a unit flow is injected into a set of input nodes and collected from a set of output nodes while minimizing the expected transportation cost together with a paths relative entropy regularization term, providing a randomized routing policy. The main advantage of this new formulation is the fact that it can easily accommodate edge flow capacity constraints which commonly occur in real-wo…
Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences. Such problems can be used to form an examplar measure out of various probability measures, as in the Wasserstein barycenter problem, or to carry out parametric inference and density fitting, where the loss is measured in terms of an optimal transport cost to the measure of observations. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein dist…
In this paper, we extend the recently proposed multivariate rank energy distance, based on the theory of optimal transport, for statistical testing of distributional similarity, to soft rank energy distance. Being differentiable, this in turn allows us to extend the rank energy to a subspace robust rank energy distance, dubbed Projected soft-Rank Energy distance, which can be computed via optimization over the Stiefel manifold. We show via experiments that using projected soft rank energy one can trade-off the detection power vs the false alarm via projections onto an appropriately selected lo…
Modern machine learning algorithms perform poorly on adversarially manipulated data. Adversarial risk quantifies the error of classifiers in adversarial settings; adversarial classifiers minimize adversarial risk. In this paper, we analyze adversarial risk and adversarial classifiers from an optimal transport perspective. We show that the optimal adversarial risk for binary classification with 0-1 loss is determined by an optimal transport cost between the probability distributions of the two classes. We develop optimal transport plans (probabilistic couplings) for univariate distributions suc…
Université Sorbonne Paris Nord（巴黎北索邦大学）；CNRS（法国国家科学研究中心）；LaMSN, La Maison des Sciences Numériques, USPN（数字科学之家实验室（巴黎北索邦大学））
Collaborative learning has recently achieved very significant results. It still suffers, however, from several issues, including the type of information that needs to be exchanged, the criteria for stopping and how to choose the right collaborators. We aim in this paper to improve the quality of the collaboration and to resolve these issues via a novel approach inspired by Optimal Transport theory. More specifically, the objective function for the exchange of information is based on the Wasserstein distance, with a bidirectional transport of information between collaborators. This formulation …
Imitation-regularized Optimal Transport on Networks: Provable Robustness and Application to Logistics Planning
Toyota Central R&D Labs., Inc.（丰田中央研究实验室有限公司）；Graduate School of Informatics, Kyoto University（京都大学信息科学研究生院）；Frontier Research Center, Toyota Motor Corporation（丰田电机公司前沿研究中心）
Efficient Imputation for Patch-based Missing Single-cell Data via Cluster-regularized Optimal Transport
Department of Computer Science, Stony Brook University（计算机科学系，石板溪大学）；Department of Computer Science, Emory University（计算机科学系，埃默里大学）；Department of Population Health Sciences, Cornell University（流行病学与公共卫生系，康奈尔大学）；Department of Computer Science and Engineering, Cornell University（计算机科学与工程系，康奈尔大学）；Department of Biomedical Informatics, Stony Brook University（生物医学信息学系，石板溪大学）
Imitation learning holds tremendous promise in learning policies efficiently for complex decision making problems. Current state-of-the-art algorithms often use inverse reinforcement learning (IRL), where given a set of expert demonstrations, an agent alternatively infers a reward function and the associated optimal policy. However, such IRL approaches often require substantial online interactions for complex control problems. In this work, we present Regularized Optimal Transport (ROT), a new imitation learning algorithm that builds on recent advances in optimal transport based trajectory-mat…
Many state-of-the-art machine learning models such as deep neural networks have recently shown to be vulnerable to adversarial perturbations, especially in classification tasks. Motivated by adversarial machine learning, in this paper we investigate the robustness of sparse regression models with strongly correlated covariates to adversarially designed measurement noises. Specifically, we consider the family of ordered weighted $\ell_1$ (OWL) regularized regression methods and study the case of OSCAR (octagonal shrinkage clustering algorithm for regression) in the adversarial setting. Under a …
Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport
LMAM and School of Mathematical Sciences, Peking University（北京大学应用数学系和数学科学学院）；Center for Machine Learning Research, Peking University（北京大学机器学习研究中心）；NELBDA, Peking University（北京大学NELBDA）；Center for Quantitative Biology, Peking University（北京大学定量生物学中心）；AI for Science Institute, Beijing（北京人工智能科学研究院）
University of Technology Sydney（悉尼科技大学）；University of New South Wales（新南威尔士大学）
Graph embedding is an effective method to represent graph data in a low dimensional space for graph analytics. Most existing embedding algorithms typically focus on preserving the topological structure or minimizing the reconstruction errors of graph data, but they have mostly ignored the data distribution of the latent codes from the graphs, which often results in inferior embedding in real-world graph data. In this paper, we propose a novel adversarial graph embedding framework for graph data. The framework encodes the topological structure and node content in a graph to a compact representa…
Injecting adversarial examples during training, known as adversarial training, can improve robustness against one-step attacks, but not for unknown iterative attacks. To address this challenge, we first show iteratively generated adversarial images easily transfer between networks trained with the same strategy. Inspired by this observation, we propose cascade adversarial training, which transfers the knowledge of the end results of adversarial training. We train a network from scratch by injecting iteratively generated adversarial images crafted from already defended networks in addition to o…
In this paper, we investigate algorithms for anomaly detection. Previous anomaly detection methods focus on modeling the distribution of non-anomalous data provided during training. However, this does not necessarily ensure the correct detection of anomalous data. We propose a new Regularized Cycle Consistent Generative Adversarial Network (RCGAN) in which deep neural networks are adversarially trained to better recognize anomalous samples. This approach is based on leveraging a penalty distribution with a new definition of the loss function and novel use of discriminator networks. It is based…
Vanderbilt University（范德堡大学）；Washington University in St. Louis（圣路易斯华盛顿大学）
Despite the considerable success of convolutional neural networks in a broad array of domains, recent research has shown these to be vulnerable to small adversarial perturbations, commonly known as adversarial examples. Moreover, such examples have shown to be remarkably portable, or transferable, from one model to another, enabling highly successful black-box attacks. We explore this issue of transferability and robustness from two dimensions: first, considering the impact of conventional $l_p$ regularization as well as replacing the top layer with a linear support vector machine (SVM), and s…
McGill University（麦吉尔大学）；University of Minnesota（明尼苏达大学）
In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networks. The adversarial robustness regularization combines adversarial training, which we show to be equivalent to Total Variation regularization, with Lipschitz regularization. We demonstrate empirically that the regularized models are more robust, and that gradient norms of images …
University of Louisville（路易斯维尔大学）；University of Social Science（社会科学大学）
The two key players in Generative Adversarial Networks (GANs), the discriminator and generator, are usually parameterized as deep neural networks (DNNs). On many generative tasks, GANs achieve state-of-the-art performance but are often unstable to train and sometimes miss modes. A typical failure mode is the collapse of the generator to a single parameter configuration where its outputs are identical. When this collapse occurs, the gradient of the discriminator may point in similar directions for many similar points. We hypothesize that some of these shortcomings are in part due to primitive a…
Carleton University（卡尔顿大学）；University of Ottawa（渥太华大学）
Recent advances on unsupervised domain adaptation (UDA) rely on adversarial learning to disentangle the explanatory and transferable features for domain adaptation. However, there are two issues with the existing methods. First, the discriminability of the latent space cannot be fully guaranteed without considering the class-aware information in the target domain. Second, samples from the source and target domains alone are not sufficient for domain-invariant feature extracting in the latent space. In order to alleviate the above issues, we propose a dual mixup regularized learning (DMRL) meth…
Cornell University（康奈尔大学）；Alibaba Group Inc.（阿里巴巴集团）
Recently smoothing deep neural network based classifiers via isotropic Gaussian perturbation is shown to be an effective and scalable way to provide state-of-the-art probabilistic robustness guarantee against $\ell_2$ norm bounded adversarial perturbations. However, how to train a good base classifier that is accurate and robust when smoothed has not been fully investigated. In this work, we derive a new regularized risk, in which the regularizer can adaptively encourage the accuracy and robustness of the smoothed counterpart when training the base classifier. It is computationally efficient a…
Tsinghua University（清华大学）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）；The Hong Kong Polytechnic University（香港理工大学）
Improving the performance of click-through rate (CTR) prediction remains one of the core tasks in online advertising systems. With the rise of deep learning, CTR prediction models with deep networks remarkably enhance model capacities. In deep CTR models, exploiting users' historical data is essential for learning users' behaviors and interests. As existing CTR prediction works neglect the importance of the temporal signals when embed users' historical clicking records, we propose a time-aware attention model which explicitly uses absolute temporal signals for expressing the users' periodic be…
Monash University（莫纳什大学）；University of Technology Sydney（悉尼科技大学）；City University of Hong Kong（香港城市大学）
Graph embedding aims to transfer a graph into vectors to facilitate subsequent graph analytics tasks like link prediction and graph clustering. Most approaches on graph embedding focus on preserving the graph structure or minimizing the reconstruction errors for graph data. They have mostly overlooked the embedding distribution of the latent codes, which unfortunately may lead to inferior representation in many cases. In this paper, we present a novel adversarially regularized framework for graph embedding. By employing the graph convolutional network as an encoder, our framework embeds the to…
Université Paul Sabatier（保罗·萨巴捷大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Institut de Recherche en Informatique de Toulouse（图卢兹计算机科学研究所）
We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreo…
IBM Research（IBM研究院）；IBM Haifa Labs（IBM海法实验室）；Haifa University（海法大学）
As machine learning algorithms continue to improve, there is an increasing need for explaining why a model produces a certain prediction for a certain input. In recent years, several methods for model interpretability have been developed, aiming to provide explanation of which subset regions of the model input is the main reason for the model prediction. In parallel, a significant research community effort is occurring in recent years for developing adversarial example generation methods for fooling models, while not altering the true label of the input,as it would have been classified by a hu…
Syracuse University（雪城大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Duke University（杜克大学）；The Ohio State University（俄亥俄州立大学）
Despite the growing interest in generative adversarial networks (GANs), training GANs remains a challenging problem, both from a theoretical and a practical standpoint. To address this challenge, in this paper, we propose a novel way to exploit the unique geometry of the real data, especially the manifold information. More specifically, we design a method to regularize GAN training by adding an additional regularization term referred to as manifold regularizer. The manifold regularizer forces the generator to respect the unique geometry of the real data manifold and generate high quality data.…
The Australian National University（澳大利亚国立大学）；Data61（Data61研究所）；Microsoft Research Cambridge（微软剑桥研究院）
Entropic regularization of policies in Reinforcement Learning (RL) is a commonly used heuristic to ensure that the learned policy explores the state-space sufficiently before overfitting to a local optimal policy. The primary motivation for using entropy is for exploration and disambiguating optimal policies; however, the theoretical effects are not entirely understood. In this work, we study the more general regularized RL objective and using Fenchel duality; we derive the dual problem which takes the form of an adversarial reward problem. In particular, we find that the optimal policy found …
The University of Sydney（悉尼大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to adversarial classification models proposed earlier and to maximum-margin classifiers. We also provide a reformulation of the distributionally robust model for linear classification, and show it is equivalent to minimizing a regularized ramp loss objective. Numerical experiments show that, despite the nonconvexity of this formulation, st…
Hausdorff Center for Mathematics（豪斯多夫数学中心）；University of Bonn（波恩大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；North Carolina State University（北卡罗来纳州立大学）
We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $L^1+$ (nonlocal) $\operatorname{TV}$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original probl…
This work studies how the introduction of the entropic regularization term in unbalanced Optimal Transport (OT) models may alter their homogeneity with respect to the input measures. We observe that in common settings (including balanced OT and unbalanced OT with Kullback-Leibler divergence to the marginals), although the optimal transport cost itself is not homogeneous, optimal transport plans and the so-called Sinkhorn divergences are indeed homogeneous. However, homogeneity does not hold in more general Unbalanced Regularized Optimal Transport (UROT) models, for instance those using the Tot…
