Recently, minimax optimization received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods. However, the nonconvexities and nonconcavities present in these problems, prevents the application of typical Gradient Descent-Ascent, which is known to diverge even in bilinear problems. Recently, it was shown that the Proximal Point Method (PPM) converges linearly for a family of nonconvex-nonconcave problems. In this paper, we study the convergence of a damped version of Extra-Gradient Method (EGM) which avoids potentially costly proximal computations, only relying on gradient evaluation. We show that EGM converges linearly for smooth minimax optimization problem satisfying the same nonconvex-nonconcave condition needed by PPM.
This paper considers a class of nonsmooth nonconvex-nonconcave min-max problems in machine learning and games. We first provide sufficient conditions for the existence of global minimax points and local minimax points. Next, we establish the first-order and second-order optimality conditions for local minimax points by using directional derivatives. These conditions reduce to smooth min-max problems with Fr{é}chet derivatives. We apply our theoretical results to generative adversarial networks (GANs) in which two neural networks contest with each other in a game. Examples are used to illustrat…
Modern minimax problems, such as generative adversarial network and adversarial training, are often under a nonconvex-nonconcave setting, and developing an efficient method for such setting is of interest. Recently, two variants of the extragradient (EG) method are studied in that direction. First, a two-time-scale variant of the EG, named EG+, was proposed under a smooth structured nonconvex-nonconcave setting, with a slow $\mathcal{O}(1/k)$ rate on the squared gradient norm, where $k$ denotes the number of iterations. Second, another variant of EG with an anchoring technique, named extra anc…
Minimax problems, such as generative adversarial network, adversarial training, and fair training, are widely solved by a multi-step gradient descent ascent (MGDA) method in practice. However, its convergence guarantee is limited. In this paper, inspired by the primal-dual hybrid gradient method, we propose a new semi-anchoring (SA) technique for the MGDA method. This makes the MGDA method find a stationary point of a structured nonconvex-nonconcave composite minimax problem; its saddle-subdifferential operator satisfies the weak Minty variational inequality condition. The resulting method, na…
Shanghai University（上海大学）；Georgia Institute of Technology（佐治亚理工学院）
Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an $\varepsilon$-stationary po…
Cornell University（康奈尔大学）；Google Research（谷歌研究院）；University of Chicago（芝加哥大学）
Unlike nonconvex optimization, where gradient descent is guaranteed to converge to a local optimizer, algorithms for nonconvex-nonconcave minimax optimization can have topologically different solution paths: sometimes converging to a solution, sometimes never converging and instead following a limit cycle, and sometimes diverging. In this paper, we study the limiting behaviors of three classic minimax algorithms: gradient descent ascent (GDA), alternating gradient descent ascent (AGDA), and the extragradient method (EGM). Numerically, we observe that all of these limiting behaviors can arise i…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Nonconvex minimax problems appear frequently in emerging machine learning applications, such as generative adversarial networks and adversarial learning. Simple algorithms such as the gradient descent ascent (GDA) are the common practice for solving these nonconvex games and receive lots of empirical success. Yet, it is known that these vanilla GDA algorithms with constant step size can potentially diverge even in the convex setting. In this work, we show that for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-Łojasiewicz inequality, the alternating gradi…
Google Research（谷歌研究院）；Cornell University（康奈尔大学）；University of Chicago（芝加哥大学）
Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties this poses. In this paper, we study the classic proximal point method (PPM) applied to nonconvex-nonconcave minimax problems. We find that a classic generalization of the Moreau envelope by Attouch and Wets provides key insights. Critically, we show this envelope not only smooths the objective but can convexify and…
Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered the very basic question: what is a proper definition of local optima of a minimax optimization problem, and proposed a proper de…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Massachusetts Institute of Technology（麻省理工学院）；UC Berkeley（加州大学伯克利分校）
The use of min-max optimization in adversarial training of deep neural network classifiers and training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in these applications. Unfortunately, recent results have established that even approximate first-order stationary points of such objectives are intractable, even under smoothness conditions, motivating the study of min-max objectives with additional structure. We introduce a new class of structured nonconvex-nonconcave min-max optimization problems, proposing a …
KU Leuven（鲁汶大学）；Télécom Paris（巴黎电信学院）；Institut Polytechnique de Paris（巴黎理工学院）
This paper introduces a new extragradient-type algorithm for a class of nonconvex-nonconcave minimax problems. It is well-known that finding a local solution for general minimax problems is computationally intractable. This observation has recently motivated the study of structures sufficient for convergence of first order methods in the more general setting of variational inequalities when the so-called weak Minty variational inequality (MVI) holds. This problem class captures non-trivial structures as we demonstrate with examples, for which a large family of existing algorithms provably conv…
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research, India（微软印度研究院）
Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises---"what is a proper definition of local optima?" Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and …
Massachusetts Institute of Technology（麻省理工学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Berkeley（加州大学伯克利分校）
Min-max optimization problems involving nonconvex-nonconcave objectives have found important applications in adversarial training and other multi-agent learning settings. Yet, no known gradient descent-based method is guaranteed to converge to (even local notions of) min-max equilibrium in the nonconvex-nonconcave setting. For all known methods, there exist relatively simple objectives for which they cycle or exhibit other undesirable behavior different from converging to a point, let alone to some game-theoretically meaningful one~\cite{flokas2019poincare,hsieh2021limits}. The only known conv…
Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems
Shanghai University（上海大学）；Newtouch Center for Mathematics of Shanghai University（上海大学纽崔克数学中心）；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）
University of Michigan（密歇根大学）；Tata Consultancy Services Research（塔塔咨询服务公司研究院）
This study develops a fixed-time convergent saddle point dynamical system for solving min-max problems under a relaxation of standard convexity-concavity assumption. In particular, it is shown that by leveraging the dynamical systems viewpoint of an optimization algorithm, accelerated convergence to a saddle point can be obtained. Instead of requiring the objective function to be strongly-convex--strongly-concave (as necessitated for accelerated convergence of several saddle-point algorithms), uniform fixed-time convergence is guaranteed for functions satisfying only the two-sided Polyak-Łojas…
A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China（应用数学系，香港理工大学）；Institute for Math & AI, Wuhan (IMAI), Wuhan university, China（数学与人工智能研究院，武汉大学）；Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY, USA（数学科学系，伦斯勒理工学院）
In this paper, we perform sensitivity analysis for the maximal value function which is the optimal value function for a parametric maximization problem. Our aim is to study various subdifferentials for the maximal value function. We obtain upper estimates of Fréchet, limiting, and horizon subdifferentials of the maximal value function by using some sensitivity analysis techniques sophisticatedly. The derived upper estimates depend only on the union of all solutions and not on its convex hull or only one solution from the solution set. Finally, we apply the derived results to develop some new n…
Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization
Min-Max Optimisation for Nonconvex-Nonconcave Functions Using a Random Zeroth-Order Extragradient Algorithm
School of Engineering Australian National University（工程学院澳大利亚国立大学）；CIICADA Lab（CIICADA实验室）；Department of Electrical Engineering Polytechnique Montréal（电气工程系蒙特利尔理工学院）；GERAD & Mila（GERAD与Mila）；Department of Mathematics and Industrial Engineering Polytechnique Montréal（数学与工业工程系蒙特利尔理工学院）；GERAD；Department of Electrical and Electronic Engineering University of Melbourne（电气与电子工程系墨尔本大学）
Min-max optimization of an objective function $f: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is an important model for robustness in an adversarial setting, with applications to many areas including optimization, economics, and deep learning. In many applications $f$ may be nonconvex-nonconcave, and finding a global min-max point may be computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max optimization model. However, many of the alternative models have solution points which are only guaranteed to…
Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain
School of Mathematics & Industrial & Systems Engineering (ISyE), Atlanta, USA（数学与工业与系统工程学院（ISyE），美国亚特兰大）；Viterbi School of Engineering, University of Southern California, Los Angeles, USA（维特比工程学院，美国南加州大学，洛杉矶）
It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems, numerical algorithms for nonsmooth minimax problems with joint linear constraints are very rare. This paper aims to discuss optimality conditions and develop practical numerical algorithms for minimax problems with joint linear constraints. First of all, we use the properties of proximal mapping and KKT system to establish optimality conditions. Secondly, we propose a framework of alternating coordinate algorithm for the minimax problem and analyze its convergence properties. Thirdly, we dev…
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By leveraging the momentum acceleration technique, we pr…
