In this paper, we focus on the nonconvex-strongly-concave minimax optimization problem (MCC), where the inner maximization subproblem contains constraints that couple the primal variable of the outer minimization problem. We prove that by introducing the dual variable of the inner maximization subproblem, (MCC) has the same first-order minimax points as a nonconvex-strongly-concave minimax optimization problem without coupled constraints (MOL). We then extend our focus to a class of nonconvex-strongly-concave minimax optimization problems (MM) that generalize (MOL). By performing the partial forward-backward envelope to the primal variable of the inner maximization subproblem, we propose a minimization problem (MMPen), where its objective function is explicitly formulated. We prove that the first-order stationary points of (MMPen) coincide with the first-order minimax points of (MM). Therefore, various efficient minimization methods and their convergence guarantees can be directly employed to solve (MM), hence solving (MCC) through (MOL). Preliminary numerical experiments demonstrate the great potential of our proposed approach.
By applying the perturbation function approach, we propose the Lagrangian and the conjugate duals for minimization problems of the sum of two, generally nonconvex, functions. The main tools are the $Φ$-convexity theory and minimax theorems for $Φ$-convex functions. We provide conditions ensuring zero duality gap and introduce $Φ$-Karush-Kuhn-Tucker conditions that characterize solutions to primal and dual problems. We also discuss the relationship between the dual problems introduced in the present investigation and some conjugate-type duals existing in the literature.
In this work we study a special minimax problem where there are linear constraints that couple both the minimization and maximization decision variables. The problem is a generalization of the traditional saddle point problem (which does not have the coupling constraint), and it finds applications in wireless communication, game theory, transportation, just to name a few. We show that the considered problem is challenging, in the sense that it violates the classical max-min inequality, and that it is NP-hard even under very strong assumptions (e.g., when the objective is strongly convex-strong…
Minimax single facility location problems in multidimensional space with Chebyshev distance are examined within the framework of idempotent algebra. The aim of the study is twofold: first, to give a new algebraic solution to the location problems, and second, to extend the area of application of idempotent algebra. A new algebraic approach based on investigation of extremal properties of eigenvalues for irreducible matrices is developed to solve multidimensional problems that involve minimization of functionals defined on idempotent vector semimodules. Furthermore, an unconstrained location pr…
Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature o…
Minimax optimization has seen a surge in interest with the advent of modern applications such as GANs, and it is inherently more challenging than simple minimization. The difficulty is exacerbated by the training data residing at multiple edge devices or \textit{clients}, especially when these clients can have heterogeneous datasets and local computation capabilities. We propose a general federated minimax optimization framework that subsumes such settings and several existing methods like Local SGDA. We show that naive aggregation of heterogeneous local progress results in optimizing a mismat…
In this article a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worst-case sense, i.e. the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative convex relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov type and a barrier regularization scheme are developed,…
Iterative Minimax Games with Coupled Linear Constraints
Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China（上海大学数学系）；LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China（中国科学院数学与系统科学研究院）；Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, People’s Republic of China（上海大学数学中心）
Minimax problems have gained tremendous attentions across the optimization and machine learning community recently. In this paper, we introduce a new quasi-Newton method for minimax problems, which we call $J$-symmetric quasi-Newton method. The method is obtained by exploiting the $J$-symmetric structure of the second-order derivative of the objective function in minimax problem. We show that the Hessian estimation (as well as its inverse) can be updated by a rank-2 operation, and it turns out that the update rule is a natural generalization of the classic Powell symmetric Broyden (PSB) method…
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）
ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学习系统中心）；Amazon（亚马逊）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy…
Several learning problems involve solving min-max problems, e.g., empirical distributional robust learning or learning with non-standard aggregated losses. More specifically, these problems are convex-linear problems where the minimization is carried out over the model parameters $w\in\mathcal{W}$ and the maximization over the empirical distribution $p\in\mathcal{K}$ of the training set indexes, where $\mathcal{K}$ is the simplex or a subset of it. To design efficient methods, we let an online learning algorithm play against a (combinatorial) bandit algorithm. We argue that the efficiency of s…
The success of minimax learning problems of generative adversarial networks (GANs) has been observed to depend on the minimax optimization algorithm used for their training. This dependence is commonly attributed to the convergence speed and robustness properties of the underlying optimization algorithm. In this paper, we show that the optimization algorithm also plays a key role in the generalization performance of the trained minimax model. To this end, we analyze the generalization properties of standard gradient descent ascent (GDA) and proximal point method (PPM) algorithms through the le…
A class of convex constrained minimization problems over polyhedral cones for geometry-dependent quadratic objective functions is considered in a functional analysis framework. Shape differentiability of the primal minimization problem needs a bijective property for mapping of the primal cone. This restrictive assumption is relaxed to bijection of the dual cone within the Lagrangian formulation as a primal-dual minimax problem. In this paper, we give results on primal-dual shape sensitivity analysis that extends the class of shape-differentiable problems supported by explicit formula of the sh…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Princeton University（普林斯顿大学）
Smooth game optimization has recently attracted great interest in machine learning as it generalizes the single-objective optimization paradigm. However, game dynamics is more complex due to the interaction between different players and is therefore fundamentally different from minimization, posing new challenges for algorithm design. Notably, it has been shown that negative momentum is preferred due to its ability to reduce oscillation in game dynamics. Nevertheless, the convergence rate of negative momentum was only established in simple bilinear games. In this paper, we extend the analysis …
Local SGD is a promising approach to overcome the communication overhead in distributed learning by reducing the synchronization frequency among worker nodes. Despite the recent theoretical advances of local SGD in empirical risk minimization, the efficiency of its counterpart in minimax optimization remains unexplored. Motivated by large scale minimax learning problems, such as adversarial robust learning and training generative adversarial networks (GANs), we propose local Stochastic Gradient Descent Ascent (local SGDA), where the primal and dual variables can be trained locally and averaged…
Minimax optimization has served as the backbone of many machine learning (ML) problems. Although the convergence behavior of optimization algorithms has been extensively studied in the minimax settings, their generalization guarantees in stochastic minimax optimization problems, i.e., how the solution trained on empirical data performs on unseen testing data, have been relatively underexplored. A fundamental question remains elusive: What is a good metric to study generalization of minimax learners? In this paper, we aim to answer this question by first showing that primal risk, a universal me…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；ETH Zürich（苏黎世联邦理工学院）；University of Washington（华盛顿大学）
We study the bilinearly coupled minimax problem: $\min_{x} \max_{y} f(x) + y^\top A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions and admit first-order gradient oracles. Surprisingly, no known first-order algorithms have hitherto achieved the lower complexity bound of $Ω((\sqrt{\frac{L_x}{μ_x}} + \frac{\|A\|}{\sqrt{μ_x μ_y}} + \sqrt{\frac{L_y}{μ_y}}) \log(\frac1{\varepsilon}))$ for solving this problem up to an $\varepsilon$ primal-dual gap in the general parameter regime, where $L_x, L_y,μ_x,μ_y$ are the corresponding smoothness and strongly convexity constants. We cl…
University of Pennsylvania（宾夕法尼亚大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g., convex-concave minimax optimization is now understood to a significant extent. Nevertheless, minimax problems extend far beyond the continuous domain to mixed continuous-discrete domains or even fully discrete domains. In this paper, we study mixed continuous-discrete minimax problems where the minimization is over a continuous variable belonging to Euclidean space an…
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…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）
In this paper, we revisit the smooth and strongly-convex-strongly-concave minimax optimization problem. Zhang et al. (2021) and Ibrahim et al. (2020) established the lower bound $Ω\left(\sqrt{κ_xκ_y} \log \frac{1}ε\right)$ on the number of gradient evaluations required to find an $ε$-accurate solution, where $κ_x$ and $κ_y$ are condition numbers for the strong convexity and strong concavity assumptions. However, the existing state-of-the-art methods do not match this lower bound: algorithms of Lin et al. (2020) and Wang and Li (2020) have gradient evaluation complexity $\mathcal{O}\left( \sqrt…
In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a {proximal step} in the nonsmooth component of the coupling function. We consider the situations…
This paper studies a structured compound stochastic program (SP) involving multiple expectations coupled by nonconvex and nonsmooth functions. We present a successive convex-programming based sampling algorithm and establish its subsequential convergence. We describe stationarity properties of the limit points for several classes of the compound SP. We further discuss probabilistic stopping rules based on the computable error-bound for the algorithm. We present several risk measure minimization problems that can be formulated as such a compound stochastic program; these include generalized dev…
George Washington University（乔治华盛顿大学）；University of Southern California（南加州大学）；University of North Carolina, Chapel Hill（北卡罗来纳大学教堂山分校）
This paper has two main goals: (a) establish several statistical properties---consistency, asymptotic distributions, and convergence rates---of stationary solutions and values of a class of coupled nonconvex and nonsmoothempirical risk minimization problems, and (b) validate these properties by a noisy amplitude-based phase retrieval problem, the latter being of much topical interest.Derived from available data via sampling, these empirical risk minimization problems are the computational workhorse of a population risk model which involves the minimization of an expected value of a random func…
We consider minimization problems with bisubmodular objective functions. We propose valid inequalities, namely the poly-bimatroid inequalities, and provide a complete linear description of the convex hull of the epigraph of a bisubmodular function. Furthermore, we develop a cutting plane algorithm for constrained bisubmodular minimization based on the poly-bimatroid inequalities. Our computational experiments on the minimization subproblem in robust coupled sensor placement show that our algorithm can solve highly non-linear problems that do not admit compact mixed-integer linear formulations.
This paper is concerned with a numerical simulation of shape optimization in a two-dimensional viscous incompressible flow governed by Navier--Stokes equations with mixed boundary conditions containing the pressure. The minimization problem of total dissipated energy was established in the fluid domain. We derive the structures of shape gradient of the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally a gradient type algorithm is effectively used for our problem.
This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random B…
This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of fourth-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical duality-triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial …
In this paper, we present a semi-proximal alternating direction method of multipliers (ADMM) for solving $3$-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed semi-proximal ADMM for the step-length $τ\in (0, (1+\sqrt{5})/2)$ and the penalty parameter $σ\in (0, +\infty)$. In particular, if $σ>0$ is smaller than a certain threshold and the first and third linear operators in the linear equa…
Western Norway University of Applied Sciences（挪威西部应用科技大学）；University of Bergen（卑尔根大学）
The paper presents a fully coupled TV-Stokes model, and propose an algorithm based on alternating minimization of the objective functional whose first iteration is exactly the modified TV-Stokes model proposed earlier. The model is a generalization of the second order Total Generalized Variation model. A convergence analysis is given.
Coupled Tensor-Matrix Recovery via Proximal Alternating Linearized Minimization, with an Application to Workforce Skill and Small-Business Health Estimation
In compressed sensing, the l0-norm minimization of sparse signal reconstruction is NP-hard. Recent work shows that compared with the best convex relaxation (l1-norm), nonconvex penalties can better approximate the l0-norm and can reconstruct the signal based on fewer observations. In this paper, the original problem is relaxed by using minimax concave penalty (MCP). Then alternating direction method of multipliers (ADMM) and modified iterative hard thresholding method are used to solve the problem. Under certain reasonable assumptions, the global convergence of the proposed method is proved. T…
Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机学研究所）；Mannheim University（曼海姆大学）；Technion - Israel Institute of Technology（以色列理工学院）
