This paper discusses the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximate solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution is an analysis showing that these methods yield state-of-the-art convergence rates, both theoretically and practically. Next, we extend the HPD algorithm with linesearch proposed by Malitsky and Pock in 2018 to the setting where the dual space has a Bregman divergence, and the dual function is relatively strongly convex to the Bregman's kernel. This extension yields a new method for OT and WB problems based on smoothing of the objective that also achieves state-of-the-art convergence rates. Finally, we introduce a new Bregman divergence based on a scaled entropy function that makes the algorithm numerically stable and reduces the smoothing, leading to sparse solutions of OT and WB problems. We complement our findings with numerical experiments and comparisons.
We develop a novel stochastic primal dual splitting method with Bregman distances for solving a structured composite problems involving infimal convolutions in non-Euclidean spaces. The sublinear convergence in expectation of the primal-dual gap is proved under mild conditions on stepsize for the general case. The linear convergence rate is obtained under additional condition like the strong convexity relative to Bregman functions.
In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures from computational and statistical sides. The statistical focus is estimating the sample size of measures necessary to calculate an approximation for Fréchet mean (barycenter) of a probability distribution with a given precision. For empirical risk minimization approaches, the question of the regularization is also studied together with proposing a new regularization which contributes to the better complexity bounds in comparison with quadratic regularization. The computational focus is developing a…
In this paper, we focus on computational aspects of the Wasserstein barycenter problem. We propose two algorithms to compute Wasserstein barycenters of $m$ discrete measures of size $n$ with accuracy $\e$. The first algorithm, based on mirror prox with a specific norm, meets the complexity of celebrated accelerated iterative Bregman projections (IBP), namely $\widetilde O(mn^2\sqrt n/\e)$, however, with no limitations in contrast to the (accelerated) IBP, which is numerically unstable under small regularization parameter. The second algorithm, based on area-convexity and dual extrapolation, im…
This work is concerned with the optimization of nonconvex, nonsmooth composite optimization problems, whose objective is a composition of a nonlinear mapping and a nonsmooth nonconvex function, that can be written as an infimal convolution (inf-conv). To tackle this problem class we propose to reformulate the problem exploiting its inf-conv structure and derive a block coordinate descent scheme on a Bregman augmented Lagrangian, that can be implemented asynchronously. We prove convergence of our scheme to stationary points of the original model for a specific choice of the penalty parameter. O…
We study a stochastic first order primal-dual method for solving convex-concave saddle point problems over real reflexive Banach spaces using Bregman divergences and relative smoothness assumptions, in which we allow for stochastic error in the computation of gradient terms within the algorithm. We show ergodic convergence in expectation of the Lagrangian optimality gap with a rate of O(1/k) and that every almost sure weak cluster point of the ergodic sequence is a saddle point in expectation under mild assumptions. Under slightly stricter assumptions, we show almost sure weak convergence of t…
We study the Bregman Augmented Lagrangian method (BALM) for solving convex problems with linear constraints. For classical Augmented Lagrangian method, the convergence rate and its relation with the proximal point method is well-understood. However, the convergence rate for BALM has not yet been thoroughly studied in the literature. In this paper, we analyze the convergence rates of BALM in terms of the primal objective as well as the feasibility violation. We also develop, for the first time, an accelerated Bregman proximal point method, that improves the convergence rate from $O(1/\sum_{k=0}…
We provide an overview of primal-dual algorithms for nonsmooth and non-convex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence.
We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validat…
We revisit the operator splitting schemes proposed in a recent work of [Some extensions of the operator splitting schemes based on Lagrangian and primal-dual: A unified proximal point analysis, Feng Xue, Optimization, 2022, doi: 10.1080/02331934.2022.2057309], and further analyze the convergence of the generalized Bregman distance and the primal-dual gap of these algorithms within a unified proximal point framework. The possibility of reduction to a simple resolvent is also discussed by exploiting the structure and possible degeneracy of the underlying metric.
We introduce a stochastic coordinate extension of the first-order primal-dual method studied by Cohen and Zhu (1984) and Zhao and Zhu (2018) to solve Composite Optimization with Composite Cone-constraints (COCC). In this method, we randomly choose a block of variables based on the uniform distribution. The linearization and Bregman-like function (core function) to that randomly selected block allow us to get simple parallel primal-dual decomposition for COCC. We obtain almost surely convergence and O(1/t) expected convergence rate in this work. The high probability complexity bound is also der…
Linear constrained convex programming has many practical applications, including support vector machine and machine learning portfolio problems. We propose the randomized primal-dual coordinate (RPDC) method, a randomized coordinate extension of the first-order primal-dual method by Cohen and Zhu, 1984 and Zhao and Zhu, 2019, to solve linear constrained convex programming. We randomly choose a block of variables based on a uniform distribution, linearize, and apply a Bregman-like function (core function) to the selected block to obtain simple parallel primal-dual decomposition. We then establi…
In this paper, we propose a variance-reduced primal-dual algorithm with Bregman distance for solving convex-concave saddle-point problems with finite-sum structure and nonbilinear coupling function. This type of problems typically arises in machine learning and game theory. Based on some standard assumptions, the algorithm is proved to converge with oracle complexity of $O\left(\frac{\sqrt n}ε\right)$ and $O\left(\frac{n}{\sqrt ε}+\frac{1}{ε^{1.5}}\right)$ using constant and non-constant parameters, respectively where $n$ is the number of function components. Compared with existing methods, ou…
Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers and generalized minimizers are explicitly constructed from solutions of modified dual problems, …
We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method…
Block coordinate descent (BCD) methods and their variants have been widely used in coping with large-scale nonconstrained optimization problems in many fields such as imaging processing, machine learning, compress sensing and so on. For problem with coupling constraints, Nonlinear convex cone programs (NCCP) are important problems with many practical applications, but these problems are hard to solve by using existing block coordinate type methods. This paper introduces a stochastic primal-dual coordinate (SPDC) method for solving large-scale NCCP. In this method, we randomly choose a block of…
We consider distributed optimization problems in which a group of agents are to collaboratively seek the global optimum through peer-to-peer communication networks. The problem arises in various application areas, such as resource allocation, sensor fusion and distributed learning. We propose a general efficient distributed algorithm--termed Distributed Forward-Backward Bregman Splitting (D-FBBS)--to simultaneously solve the above primal problem as well as its dual based on Bregman method and operator splitting. The proposed algorithm allows agents to communicate asynchronously and thus lends …
VinAI Research（VinAI研究院）；Massachusetts Institute of Technology（麻省理工学院）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
We consider robust variants of the standard optimal transport, named robust optimal transport, where marginal constraints are relaxed via Kullback-Leibler divergence. We show that Sinkhorn-based algorithms can approximate the optimal cost of robust optimal transport in $\widetilde{\mathcal{O}}(\frac{n^2}{\varepsilon})$ time, in which $n$ is the number of supports of the probability distributions and $\varepsilon$ is the desired error. Furthermore, we investigate a fixed-support robust barycenter problem between $m$ discrete probability distributions with at most $n$ number of supports and deve…
Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems
National Center for Applied Mathematics in Chongqing, Chongqing Normal University（重庆师范大学应用数学中心）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）
We study the complexity of approximating Wassertein barycenter of $m$ discrete measures, or histograms of size $n$ by contrasting two alternative approaches, both using entropic regularization. The first approach is based on the Iterative Bregman Projections (IBP) algorithm for which our novel analysis gives a complexity bound proportional to $\frac{mn^2}{\varepsilon^2}$ to approximate the original non-regularized barycenter. Using an alternative accelerated-gradient-descent-based approach, we obtain a complexity proportional to $\frac{mn^{2.5}}{\varepsilon} $. As a byproduct, we show that the…
Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis
Department of ACMS University of Notre Dame（ACMS系，诺特大学）；Department of Mathematics University of California（数学系，加州大学）；Department of Mathematics University of Southern California（数学系，南加州大学）
We propose a new self-adaptive, double-loop smoothing algorithm to solve composite, nonsmooth, and constrained convex optimization problems. Our algorithm is based on Nesterov's smoothing technique via general Bregman distance functions. It self-adaptively selects the number of iterations in the inner loop to achieve a desired complexity bound without requiring the accuracy a priori as in variants of Augmented Lagrangian methods (ALM). We prove $\BigO{\frac{1}{k}}$-convergence rate on the last iterate of the outer sequence for both unconstrained and constrained settings in contrast to ergodic …
This paper proposes a novel family of primal-dual-based distributed algorithms for smooth, convex, multi-agent optimization over networks that uses only gradient information and gossip communications. The algorithms can also employ acceleration on the computation and communications. We provide a unified analysis of their convergence rate, measured in terms of the Bregman distance associated to the saddle point reformation of the distributed optimization problem. When acceleration is employed, the rate is shown to be optimal, in the sense that it matches (under the proposed metric) existing com…
We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes $(δ,L)$ inexact oracle and relative smoothness condition. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [Nesterov, 2018]. The second one is approximating optimal…
We consider a multimarginal optimal transport, which includes as a particular case the Wasserstein barycenter problem. In this problem one has to find an optimal coupling between $m$ probability measures, which amounts to finding a tensor of the order $m$. We propose an accelerated method based on accelerated alternating minimization and estimate its complexity to find the approximate solution to the problem. We use entropic regularization with sufficiently small regularization parameter and apply accelerated alternating minimization to the dual problem. A novel primal-dual analysis is used to…
In this paper, we study the nonexpansive properties of metric resolvent, and present a convergence rate analysis for the associated fixed-point iterations (Banach-Picard and Krasnosel'skii-Mann types). Equipped with a variable metric, we develop the global ergodic and non-ergodic iteration-complexity bounds in terms of both solution distance and objective value. A byproduct of our expositions also extends the proximity operator and Moreau's decomposition identity to arbitrary variable metric. It is further shown that many classes of the first-order operator splitting algorithms, including alte…
INRIA, École Normale Supérieure（法国国家信息与自动化研究所，巴黎高等师范学院）；MILA, Université de Montréal（米拉人工智能研究中心，蒙特利尔大学）；CMCC, Federal University of ABC（气候变化、海洋和陆地交叉研究中心，ABC联邦大学）
The optimal transport (OT) problem is a classical optimization problem having the form of linear programming. Machine learning applications put forward new computational challenges in its solution. In particular, the OT problem defines a distance between real-world objects such as images, videos, texts, etc., modeled as probability distributions. In this case, the large dimension of the corresponding optimization problem does not allow applying classical methods such as network simplex or interior-point methods. This challenge was overcome by introducing entropic regularization and using the e…
Convex optimization models find interesting applications, especially in signal/image processing and compressive sensing. We study some augmented convex models, which are perturbed by strongly convex functions, and propose a dual gradient algorithm. The proposed algorithm includes the linearized Bregman algorithm and the singular value thresholding algorithm as special cases. Based on fundamental properties of proximal operators, we present a concise approach to establish the convergence of both primal and dual sequences, improving the results in the existing literature.
