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

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

doi:10.1137/20m1337600

Cited by 22 publications

(7 citation statements)

References 13 publications

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“…If f is further assumed to be smooth, Zhao [2020] developed a variant of AIPP-S algorithm which only requires an inexact gradient of ρ ξ at each iteration and attains the total gradient complexity of Õ(ǫ −3 ). On the other hand, the stationarity of f (•, •) is proposed for quantifying the efficiency in nonconvex-concave minimax optimization [Lu et al, 2019, Nouiehed et al, 2019, Kong and Monteiro, 2019, Ostrovskii et al, 2020. Using this notion of stationarity, Kong and Monteiro [2019] attains the rate of O(ǫ −2.5 ) but requires the exact gradient of ρ ξ at each iteration.…”

Section: References Gradient Complexitymentioning

confidence: 99%

“…Using this notion of stationarity, Kong and Monteiro [2019] attains the rate of O(ǫ −2.5 ) but requires the exact gradient of ρ ξ at each iteration. Without this assumption, the current state-of-the-art rate is Õ(ǫ −2.5 ) achieved by our Algorithm 5 and the algorithm proposed by a concurrent work [Ostrovskii et al, 2020]. Both algorithms are based on constructing an auxiliary function f ǫ,y and applying an accelerated solver for minimax proximal steps.…”

Section: References Gradient Complexitymentioning

confidence: 99%

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Near-Optimal Algorithms for Minimax Optimization

Lin¹,

Jin²,

Jordan³

2020

Preprint

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This paper resolves a longstanding open question pertaining to the design of near-optimal firstorder algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current stateof-the-art first-order algorithms find an approximate Nash equilibrium using Õ(κ x + κ y ) [Tseng, 1995] et al., 2019] gradient evaluations, where κ x and κ y are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap remains between these results and the best existing lower bound Ω( √ κ x κ y ) due to Zhang et al. [2019]. This paper presents the first algorithm with Õ( √ κ x κ y ) gradient complexity, matching the lower bound up to logarithmic factors. Our new algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvexconcave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.

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Section: References Gradient Complexitymentioning

confidence: 99%

Section: References Gradient Complexitymentioning

confidence: 99%

Near-Optimal Algorithms for Minimax Optimization

Lin¹,

Jin²,

Jordan³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Similar to their minimization counterparts, non-convex constraints have been widely applicable to the minmax optimization as well [Heusel et al, 2017, Daskalakis and Panageas, 2018, Balduzzi et al, 2018, Jin et al, 2020. Recently there has been significant effort in proving tighter results either under more structured assumptions [Thekumprampil et al, 2019, Nouiehed et al, 2019, Lu et al, 2020, Azizian et al, 2020, Diakonikolas, 2020, Golowich et al, 2020, Lin et al, 2020c,b, Liu et al, 2021, Ostrovskii et al, 2021, Kong and Monteiro, 2021, and/or obtaining last-iterate convergence guarantees [Daskalakis and Panageas, 2018, 2019, Adolphs et al, 2019, Liang and Stokes, 2019, Gidel et al, 2019, Mazumdar et al, 2020, Mokhtari et al, 2020, Lin et al, 2020c, Hamedani and Aybat, 2021, Abernethy et al, 2021, Cai et al, 2022 for computing min-max solutions in convex-concave settings. Nonetheless, the analysis of the iteration complexity in the general non-convex non-concave setting is still in its infancy [Vlatakis-Gkaragkounis et al, 2019.…”

Section: Introductionmentioning

confidence: 99%

First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces

Jordan¹,

Lin²,

Vlatakis-Gkaragkounis³

2022

Preprint

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From optimal transport to robust dimensionality reduction, a plethora of machine learning applications can be cast into the min-max optimization problems over Riemannian manifolds. Though many min-max algorithms have been analyzed in the Euclidean setting, it has proved elusive to translate these results to the Riemannian case. Zhang et al. [2022] have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions. Inspired by this result, we study whether a performance gap between Riemannian and optimal Euclidean space convex-concave algorithms is necessary. We answer this question in the negative-we prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate convergence at a linear rate in the geodesically strongly-convex-concave case, matching the Euclidean result. Our results also extend to the stochastic or non-smooth case where RCEG and Riemanian gradient ascent descent (RGDA) achieve near-optimal convergence rates up to factors depending on curvature of the manifold.

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“…Recent years have witnessed a surge of research interest in designing algorithms for solving minimaxstructured nonconvex problems, e.g., [12,13,15,18,20,21,23,34,36,40,44,[46][47][48]. Different from most of the existing works, we are interested in designing a decentralized algorithm for solving (1.1).…”

mentioning

confidence: 99%

A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems

Zhang

et al. 2023

Math. Program.

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Minimax problems have recently attracted a lot of research interests. A few efforts have been made to solve decentralized nonconvex strongly-concave (NCSC) minimax-structured optimization; however, all of them focus on smooth problems with at most a constraint on the maximization variable. In this paper, we make the first attempt on solving composite NCSC minimax problems that can have convex nonsmooth terms on both minimization and maximization variables. Our algorithm is designed based on a novel reformulation of the decentralized minimax problem that introduces a multiplier to absorb the dual consensus constraint. The removal of dual consensus constraint enables the most aggressive (i.e., local maximization instead of a gradient ascent step) dual update that leads to the benefit of taking a larger primal stepsize and better complexity results. In addition, the decoupling of the nonsmoothness and consensus on the dual variable eases the analysis of a decentralized algorithm; thus our reformulation creates a new way for interested researchers to design new (and possibly more efficient) decentralized methods on solving NCSC minimax problems. We show a global convergence result of the proposed algorithm and an iteration complexity result to produce a (near) stationary point of the reformulation. Moreover, a relation is established between the (near) stationarities of the reformulation and the original formulation. With this relation, we show that when the dual regularizer is smooth, our algorithm can have lower complexity results (with reduced dependence on a condition number) than existing ones to produce a near-stationary point of the original formulation. Numerical experiments are conducted on a distributionally robust logistic regression to demonstrate the performance of the proposed algorithm.

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

Cited by 22 publications

References 13 publications

Near-Optimal Algorithms for Minimax Optimization

Near-Optimal Algorithms for Minimax Optimization

First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces

A unified single-loop alternating gradient projection algorithm for nonconvex–concave and convex–nonconcave minimax problems

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