We propose an efficient algorithm for finding first-order Nash equilibria in min-max problems of the form $\min_{x \in X}\max_{y\in Y} F(x,y)$, where the objective function is smooth in both variables and concave with respect to $y$; the sets $X$ and $Y$ are convex and "projection-friendly," and $Y$ is compact. Our goal is to find an $(\varepsilon_x,\varepsilon_y)$-first-order Nash equilibrium with respect to a stationarity criterion that is stronger than the commonly used proximal gradient norm. The proposed approach is fairly simple: we perform approximate proximal-point iterations on the primal function, with inexact oracle provided by Nesterov's algorithm run on the regularized function $F(x_t,\cdot)$, $x_t$ being the current primal iterate. The resulting iteration complexity is $O(\varepsilon_x{}^{-2} \varepsilon_y{}^{-1/2})$ up to a logarithmic factor. As a byproduct, the choice $\varepsilon_y = O(\varepsilon_x{}^2)$ allows for the $O(\varepsilon_x{}^{-3})$ complexity of finding an $\varepsilon_x$-stationary point for the standard Moreau envelope of the primal function. Moreover, when the objective is strongly concave with respect to $y$, the complexity estimate for our algorithm improves to $O(\varepsilon_x{}^{-2}{κ_y}^{1/2})$ up to a logarithmic factor, where $κ_y$ is the condition number appropriately adjusted for coupling. In both scenarios, the complexity estimates are the best known so far, and are only known for the (weaker) proximal gradient norm criterion. Meanwhile, our approach is "user-friendly:" (i) the algorithm is built upon running a variant of Nesterov's accelerated algorithm as subroutine and avoids extragradient steps; (ii) the convergence analysis recycles the well-known results on accelerated methods with inexact oracle. Finally, we extend the approach to non-Euclidean proximal geometries.
Unlike convex case, a local equilibrium point of a nonconvex Nash-Cournot oligopolistic equilibrium problem may not be a global one. Finding such a local equilibrium point or even a stationary point of this problem is not an easy task. This paper deals with a numerical method for Nash-Cournot equilibrium models involving nonconvex cost functions. We develop a local method to compute a stationary point of this class of problems. The convergence of the algorithm is proved and its complexity is estimated under certain assumptions. Numerical examples are implemented to illustrate the convergence b…
This paper presents smoothing schemes for obtaining approximate stationary points of unconstrained or linearly-constrained composite nonconvex-concave min-max (and hence nonsmooth) problems by applying well-known algorithms to composite smooth approximations of the original problems. More specifically, in the unconstrained (resp. constrained) case, approximate stationary points of the original problem are obtained by applying, to its composite smooth approximation, an accelerated inexact proximal point (resp. quadratic penalty) method presented in a previous paper by the authors. Iteration com…
We consider Nash-Cournot oligopolistic equilibrium models involving separable concave cost functions. In contrast to the models with linear and convex cost functions, in these models a local equilibrium point may not be a global one. We propose algorithms for finding global and local equilibrium points for the models having separable concave cost functions. The proposed algorithms use the convex envelope of a separable concave cost function over boxes to approximate a concave cost model with an affine cost one. The latter is equivalent to a strongly convex quadratic program that can be solved …
Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative algorithms. In this paper, we are interested in optimization problems of the form F(Kx) + G(x), where K is a fixed linear transformation, while F and G are functions that may be nonconvex and/or nondifferentiable. In particular, if either of the terms are nonconvex, existing alternating minimization techniques may fail to converge; other types of existing appr…
This paper shows the existence of $\mathcal{O}(\frac{1}{n^γ})$-Nash equilibria in $n$-player noncooperative sum-aggregative games in which the players' cost functions, depending only on their own action and the average of all players' actions, are lower semicontinuous in the former while $γ$-Hölder continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of sum-aggregative games, which includes congestion games with $γ$ equal to 1, a gradient-proximal algorithm is used to construct $\mathcal{O}(\frac{1}{n})$-Nash equilibria with at most…
This paper investigates full stability properties for \emph{variational Nash equilibriums} of a system of parametric nonconvex optimal control problems governed by semilinear elliptic partial differential equations. We first obtain some new results on the existence of variational Nash equilibriums for the system of original/parametric nonconvex optimal control problems. Then we establish explicit characterizations of the Lipschitzian and Hölderian full stability of variational Nash equilibriums under perturbations. These results deduce the equivalence between variational Nash equilibriums and …
The technique of semidefinite programming (SDP) relaxation can be used to obtain a nontrivial bound on the optimal value of a nonconvex quadratically constrained quadratic program (QCQP). We explore concave quadratic inequalities that hold for any vector in the integer lattice ${\bf Z}^n$, and show that adding these inequalities to a mixed-integer nonconvex QCQP can improve the SDP-based bound on the optimal value. This scheme is tested using several numerical problem instances of the max-cut problem and the integer least squares problem.
The distributed computation of equilibria and optima has seen growing interest in a broad collection of networked problems. We consider the computation of equilibria of convex stochastic Nash games characterized by a possibly nonconvex potential function. Our focus is on two classes of stochastic Nash games: (P1): A potential stochastic Nash game, in which each player solves a parameterized stochastic convex program; and (P2): A misspecified generalization, where the player-specific stochastic program is complicated by a parametric misspecification. In both settings, exact proximal BR solution…
We extend the Malitsky-Tam forward-reflected-backward (FRB) splitting method for inclusion problems of monotone operators to nonconvex minimization problems. By assuming the generalized concave Kurdyka-Łojasiewicz (KL) property of a quadratic regularization of the objective, we show that the FRB method converges globally to a stationary point of the objective and enjoys finite length property. The sharpness of our approach is guaranteed by virtue of the exact modulus associated with the generalized concave KL property. Numerical experiments suggest that FRB is competitive compared to the Dougl…
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…
We develop a novel and single-loop variance-reduced algorithm to solve a class of stochastic nonconvex-convex minimax problems involving a nonconvex-linear objective function, which has various applications in different fields such as machine learning and robust optimization. This problem class has several computational challenges due to its nonsmoothness, nonconvexity, nonlinearity, and non-separability of the objective functions. Our approach relies on a new combination of recent ideas, including smoothing and hybrid biased variance-reduced techniques. Our algorithm and its variants can achi…
A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems
Shenzhen Research Institute of Big Data（深圳大数据研究院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of ISE, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校工业与系统工程系）
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.
An Efficient Stochastic Algorithm for Decentralized Nonconvex-Strongly-Concave Minimax Optimization
SGIT AI Lab, State Grid Corporation of China（国网公司SGIT AI实验室）；School of Data Science, Fudan University（复旦大学数据科学学院）；Shanghai Key Laboratory for Contemporary Applied Mathematics（上海市现代应用数学重点实验室）
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）
We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. We establish a lower bound of $Ω\left(\sqrtκε^{-2}\right)$ for deterministic oracles, where $ε$ defines the level of approximate stationarity and $κ$ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the $ε$ and $κ$ dependence up to logarithmic factors. For stochastic oracles, we provide a…
This paper takes an initial step to systematically investigate the generalization bounds of algorithms for solving nonconvex-(strongly)-concave (NC-SC/NC-C) stochastic minimax optimization measured by the stationarity of primal functions. We first establish algorithm-agnostic generalization bounds via uniform convergence between the empirical minimax problem and the population minimax problem. The sample complexities for achieving $ε$-generalization are $\tilde{\mathcal{O}}(dκ^2ε^{-2})$ and $\tilde{\mathcal{O}}(dε^{-4})$ for NC-SC and NC-C settings, respectively, where $d$ is the dimension and…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Twente（特文特大学）；Pontificia Universidad Católica de Chile（智利天主教大学）；EPFL（洛桑联邦理工学院）；ETH Zürich（苏黎世联邦理工学院）
This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds of $Ω(\sqrtκΔLε^{-2})$ and $Ω(n+\sqrt{nκ}ΔLε^{-2})$ for the two settings, respectively, where $κ$ is the condition number, $L$ is the smoothness constant, and $Δ$ is the initial gap. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that …
The Hong Kong University of Science and Technology（香港科技大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）
We consider nonconvex-concave minimax optimization problems of the form $\min_{\bf x}\max_{\bf y\in{\mathcal Y}} f({\bf x},{\bf y})$, where $f$ is strongly-concave in $\bf y$ but possibly nonconvex in $\bf x$ and ${\mathcal Y}$ is a convex and compact set. We focus on the stochastic setting, where we can only access an unbiased stochastic gradient estimate of $f$ at each iteration. This formulation includes many machine learning applications as special cases such as robust optimization and adversary training. We are interested in finding an ${\mathcal O}(\varepsilon)$-stationary point of the f…
The Pennsylvania State University（宾夕法尼亚州立大学）；Massachusetts Institute of Technology（麻省理工学院）
Despite the established convergence theory of Optimistic Gradient Descent Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax problems, little is known about the theoretical guarantees of these methods in nonconvex settings. To bridge this gap, for the first time, this paper establishes the convergence of OGDA and EG methods under the nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings by providing a unified analysis through the lens of single-call extra-gradient methods. We further establish lower bounds on the convergence of GDA/OGDA/EG, shedding …
A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is…
This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss-Seidel method and Lasserre type Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss-Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinar's Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method.
In this paper, we consider a distributed learning problem in a subnetwork zero-sum game, where agents are competing in different subnetworks. These agents are connected through time-varying graphs where each agent has its own cost function and can receive information from its neighbors. We propose a distributed mirror descent algorithm for computing a Nash equilibrium and establish a sublinear regret bound on the sequence of iterates when the graphs are uniformly strongly connected and the cost functions are convex-concave. Moreover, we prove its convergence with suitably selected diminishing …
