Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems

Ostrovskii, Dmitrii; Lowy, Andrew M.; Razaviyayn, Meisam

doi:10.48550/arxiv.2002.07919

Cited by 10 publications

(27 citation statements)

References 5 publications

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“…The structure presented in nonconvex-concave zero-sum games makes it possible to achieve global finite-time convergence guarantees to -approximate stationary points of the objective function f (•, •) and the best-response function Φ(•) = max y f (•, y). A significant number of papers in the past few years investigate the rates of convergence that can be obtained for this problem [26,29,36,37,38,39,50,51,52,58,64]. The best known existing results in the deterministic setting show that -approximate stationary points of the functions f (•, •) and Φ(•) can be obtained with gradient complexities of O( −2.5 ) [37,51] and O( −3 ) [29,37,58,64], respectively.…”

Section: Related Workmentioning

confidence: 99%

“…A significant number of papers in the past few years investigate the rates of convergence that can be obtained for this problem [26,29,36,37,38,39,50,51,52,58,64]. The best known existing results in the deterministic setting show that -approximate stationary points of the functions f (•, •) and Φ(•) can be obtained with gradient complexities of O( −2.5 ) [37,51] and O( −3 ) [29,37,58,64], respectively. Moreover, the latter notion of an -approximate stationarity point can be obtained using O( −6 ) gradient calls in the stochastic setting of nonconvex-concave zero-sum games [52].…”

Section: Related Workmentioning

confidence: 99%

“…In the deterministic nonconvex-concave problem, a number of algorithms with provable guarantees toapproximate stationary points of the function f (•, •) have been shown with gradient complexities of O( −4 ) [38], O( −3.5 ) [50], and O( −2.5 ) [37,51]. Similarly, in this class of problems, there exist results on algorithms that guarantee convergence to an -approximate stationary points of the function Φ(•) with gradient complexities O( −6 ) [26,36,52] and O( −3 ) [29,37,58,64].…”

Section: A Detailed Related Workmentioning

confidence: 99%

“…A majority of these classical results have been focused on convex-concave functions, and heavily rely on the minimax theorem, i.e., min x max y f (x, y) = max y min x f (x, y), which no longer holds beyond the convex-concave setting. Recent line of works [36,37,50,51,58] address the nonconvex-concave setting where f is nonconvex in x but concave in y by proposing meaningful optimality notions and designing computationally efficient algorithms to find such points. A crucial property heavily exploited in this setting is that the inner maximization over y given a fixed x can be computed efficiently, which unfortunately does not extend to the nonconvex-nonconcave setting.…”

Section: Introductionmentioning

confidence: 99%

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Minimax Optimization with Smooth Algorithmic Adversaries

Fiez,

Jin,

Netrapalli

et al. 2021

Preprint

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This paper considers minimax optimization minx maxy f (x, y) in the challenging setting where f can be both nonconvex in x and nonconcave in y. Though such optimization problems arise in many machine learning paradigms including training generative adversarial networks (GANs) and adversarially robust models, many fundamental issues remain in theory, such as the absence of efficiently computable optimality notions, and cyclic or diverging behavior of existing algorithms. Our framework sprouts from the practical consideration that under a computational budget, the max-player can not fully maximize f (x, •) since nonconcave maximization is NP-hard in general. So, we propose a new algorithm for the min-player to play against smooth algorithms deployed by the adversary (i.e., the max-player) instead of against full maximization. Our algorithm is guaranteed to make monotonic progress (thus having no limit cycles), and to find an appropriate "stationary point" in a polynomial number of iterations. Our framework covers practical settings where the smooth algorithms deployed by the adversary are multi-step stochastic gradient ascent, and its accelerated version. We further provide complementing experiments that confirm our theoretical findings and demonstrate the effectiveness of the proposed approach in practice.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: A Detailed Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimax Optimization with Smooth Algorithmic Adversaries

Fiez,

Jin,

Netrapalli

et al. 2021

Preprint

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“…One intensively studied type is nested-loop algorithms. Various algorithms of this type have been proposed in [46,41,50,22,42,54]. Lin et al [30] propose a class of accelerated algorithms for smooth nonconvex-concave minimax problems with the complexity bound of Õ ε −2.5 which owns the best iteration complexity till now.…”

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confidence: 99%

Zeroth-Order Alternating Randomized Gradient Projection Algorithms for General Nonconvex-Concave Minimax Problems

Xu¹,

Wang²,

Shen³

et al. 2021

Preprint

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In this paper, we study zeroth-order algorithms for nonconvex-concave minimax problems, which have attracted widely attention in machine learning, signal processing and many other fields in recent years. We propose a zeroth-order alternating randomized gradient projection (ZO-AGP) algorithm for smooth nonconvex-concave minimax problems, and its iteration complexity to obtain an ε-stationary point is bounded by O(ε −4 ), and the number of function value estimation is bounded by O(dxε −4 + dyε −6 ) per iteration. Moreover, we propose a zeroth-order block alternating randomized proximal gradient algorithm (ZO-BAPG) for solving block-wise nonsmooth nonconvexconcave minimax optimization problems, and the iteration complexity to obtain an ε-stationary point is bounded by O(ε −4 ) and the number of function value estimation per iteration is bounded by O(Kdxε −4 + dyε −6 ). To the best of our knowledge, this is the first time that zeroth-order algorithms with iteration complexity gurantee are developed for solving both general smooth and block-wise nonsmooth nonconvex-concave minimax problems. Numerical results on data poisoning attack problem validate the efficiency of the proposed algorithms.

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Solving Non-Convex Non-Differentiable Min-Max Games Using Proximal Gradient Method

Barazandeh

Razaviyayn

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Min-max saddle point games appear in a wide range of applications in machine leaning and signal processing. Despite their wide applicability, theoretical studies are mostly limited to the special convex-concave structure. While some recent works generalized these results to special smooth non-convex cases, our understanding of non-smooth scenarios is still limited. In this work, we study special form of non-smooth min-max games when the objective function is (strongly) convex with respect to one of the player's decision variable. We show that a simple multi-step proximal gradient descent-ascent algorithm converges to -first-order Nash equilibrium of the min-max game with the number of gradient evaluations being polynomial in 1/ . We will also show that our notion of stationarity is stronger than existing ones in the literature. Finally, we evaluate the performance of the proposed algorithm through adversarial attack on a LASSO estimator.Keywords-Non-convex min-max games, First-order Nash equilibria, Proximal gradient descent ascent IntroductionNon-convex min-max saddle point games appear in a wide range of applications such as training Generative Adversarial Networks [1,2,3,4], fair statistical inference [5,6,7], and training robust neural networks and systems [8,9,10]. In such a game, the goal is to solve the optimization problem of the formwhich can be considered as a two player game where one player aims at increasing the objective, while the other tries to minimize the objective. Using game theoretic point of view, we may aim for finding Nash equilibria [11] in which no player can do better off by unilaterally changing its strategy. Unfortunately, finding/checking such Nash equilibria is hard in general [12] for non-convex objective functions. Moreover, such Nash equilibria might not even exist. Therefore, many works focus on special cases such as convex-concave problems where f (θ, .) is concave for any given θ and f (., α) is convex for any given α. Under this assumption, different algorithms such as optimistic mirror descent [13,14,15,16], Frank-Wolfe algorithm [17,18] and Primal-Dual method [19] have been studied.In the general non-convex settings, [20] considers the weakly convex-concave case and proposes a primal-dual based approach for finding approximate stationary solutions. More recently, the research works [21,22,23,24] * This arXiv submission includes the details of the proofs for the paper accepted for publication in the proceeding of the 45 th International Conference on Acoustics, Speech, and Signal Processing (ICASSP). arXiv:2003.08093v1 [math.OC] 18 Mar 2020examine the min-max problem in non-convex-(strongly)-concave cases and proposed first-order algorithms for solving them. Some of the results have been accelerated in the "Moreau envelope regime" by the recent interesting work [25]. This work first starts by studying the problem in smooth strongly convex-concave and convex-concave settings, and proposes an algorithm based on the combination of Mirror-Prox [26] and Nesterov's accelerated gra...

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Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems

Cited by 10 publications

References 5 publications

Minimax Optimization with Smooth Algorithmic Adversaries

Minimax Optimization with Smooth Algorithmic Adversaries

Zeroth-Order Alternating Randomized Gradient Projection Algorithms for General Nonconvex-Concave Minimax Problems

Solving Non-Convex Non-Differentiable Min-Max Games Using Proximal Gradient Method

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