Min-max optimization is emerging as a key framework for analyzing problems of robustness to strategically and adversarially generated data. We propose a random reshuffling-based gradient free Optimistic Gradient Descent-Ascent algorithm for solving convex-concave min-max problems with finite sum structure.
We prove that the algorithm enjoys the same convergence rate as that of zeroth-order algorithms for convex minimization problems. We further specialize the algorithm to solve distributionally robust, decision-dependent learning problems, where gradient information is not readily available. Through illustrative simulations, we observe that our proposed approach learns models that are simultaneously robust against adversarial distribution shifts and strategic decisions from the data sources, and outperforms existing methods from the strategic classification literature.
We study the role that a finite timescale separation parameter $τ$ has on gradient descent-ascent in two-player non-convex, non-concave zero-sum games where the learning rate of player 1 is denoted by $γ_1$ and the learning rate of player 2 is defined to be $γ_2=τγ_1$. Existing work analyzing the role of timescale separation in gradient descent-ascent has primarily focused on the edge cases of players sharing a learning rate ($τ=1$) and the maximizing player approximately converging between each update of the minimizing player ($τ\rightarrow \infty$). For the parameter choice of $τ=1$, it is k…
University of California, Davis（加州大学戴维斯分校）；University of Southern California（南加州大学）
In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (\texttt{ZO-GDA}) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step A…
Zeroth-Order primal-dual Alternating Projection Gradient Algorithms for Nonconvex Minimax Problems with Coupled linear Constraints
Department of Mathematics, College of Sciences, Shanghai University（上海大学数学系）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Newtouch Center for Mathematics of Shanghai University（上海大学数学中心）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
This paper defines a strong convertible nonconvex(SCN) function for solving the unconstrained optimization problems with the nonconvex or nonsmooth(nondifferentiable) function. First, many examples of SCN function are given, where the SCN functions are nonconvex or nonsmooth. Second, the operational properties of the SCN functions are proved, including addition, multiplication, compound operations and so on. Third, the SCN forms of some special functions common in machine learning and engineering applications are presented respectively where these SCN function optimization problems can be tran…
Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems
Shanghai University（上海大学）；Newtouch Center for Mathematics of Shanghai University（上海大学新触数学中心）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The Ohio State University（俄亥俄州立大学）；Princeton University（普林斯顿大学）
Many important machine learning applications amount to solving minimax optimization problems, and in many cases there is no access to the gradient information, but only the function values. In this paper, we focus on such a gradient-free setting, and consider the nonconvex-strongly-concave minimax stochastic optimization problem. In the literature, various zeroth-order (i.e., gradient-free) minimax methods have been proposed, but none of them achieve the potentially feasible computational complexity of $\mathcal{O}(ε^{-3})$ suggested by the stochastic nonconvex minimization theorem. In this pa…
Solving the Offline and Online Min-Max Problem of Non-smooth Submodular-Concave Functions: A Zeroth-Order Approach
School of Engineering, Australian National University, Canberra, Australia（澳大利亚国立大学工程学院，澳大利亚堪培拉）；Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, Australia（墨尔本大学电气与电子工程系，澳大利亚墨尔本）；Department of Mechanical Engineering, University of Texas at Dallas, Texas, USA（德克萨斯大学达拉斯分校机械工程系，美国德克萨斯）
Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions
The Fuqua School of Business, Duke University（德克萨斯大学福克商学院）；Department of Statistics, The Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院统计系）；Department of Statistics and Department of Mathematics, University of Washington（华盛顿大学统计系和数学系）
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Higher School of Economics（高等经济学院）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）；Caucasus Mathematical Center, Adyghe State University（阿迪格国立大学高加索数学中心）
Saddle-point problems have recently gained increased attention from the machine learning community, mainly due to applications in training Generative Adversarial Networks using stochastic gradients. At the same time, in some applications only a zeroth-order oracle is available. In this paper, we propose several algorithms to solve stochastic smooth (strongly) convex-concave saddle-point problems using zeroth-order oracles and estimate their convergence rate and its dependence on the dimension $n$ of the variable. In particular, our analysis shows that in the case when the feasible set is a dir…
Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants
Joint Center for Quantum Information and Computer Science（量子信息与计算机科学联合中心）；University of Maryland（马里兰大学）；Center on Frontiers of Computing Studies（计算前沿研究中心）；Peking University（北京大学）；Simons Institute（西蒙斯研究所）；UC Berkeley（加州大学伯克利分校）；Pennsylvania State University（宾夕法尼亚州立大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Saddle-point optimization problems are an important class of optimization problems with applications to game theory, multi-agent reinforcement learning and machine learning. A majority of the rich literature available for saddle-point optimization has focused on the offline setting. In this paper, we study nonstationary versions of stochastic, smooth, strongly-convex and strongly-concave saddle-point optimization problem, in both online (or first-order) and multi-point bandit (or zeroth-order) settings. We first propose natural notions of regret for such nonstationary saddle-point optimization…
First-Order Optimization under Uniform Nondegeneracy: Geometry, Computation, and Information
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Peking University（北京大学数学科学学院北京国际数学研究中心定量生物学中心机器学习研究中心）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）
