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Zeroth-order algorithms for nonconvex–strongly-concave minimax problems with improved complexities

Author

Listed:
  • Zhongruo Wang

    (University of California)

  • Krishnakumar Balasubramanian

    (University of California)

  • Shiqian Ma

    (University of California)

  • Meisam Razaviyayn

    (University of Southern California)

Abstract

In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (ZO-GDA) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step Ascent (ZO-GDMSA) algorithm that significantly improves the oracle complexity of ZO-GDA. We then consider stochastic versions of ZO-GDA and ZO-GDMSA, to handle stochastic nonconvex minimax problems. For this case, we provide oracle complexity results under two assumptions on the stochastic gradient: (i) the uniformly bounded variance assumption, which is common in traditional stochastic optimization, and (ii) the Strong Growth Condition (SGC), which has been known to be satisfied by modern over-parameterized machine learning models. We establish that under the SGC assumption, the complexities of the stochastic algorithms match that of deterministic algorithms. Numerical experiments are presented to support our theoretical results.

Suggested Citation

  • Zhongruo Wang & Krishnakumar Balasubramanian & Shiqian Ma & Meisam Razaviyayn, 2023. "Zeroth-order algorithms for nonconvex–strongly-concave minimax problems with improved complexities," Journal of Global Optimization, Springer, vol. 87(2), pages 709-740, November.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01160-0
    DOI: 10.1007/s10898-022-01160-0
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    References listed on IDEAS

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    1. Luis Rios & Nikolaos Sahinidis, 2013. "Derivative-free optimization: a review of algorithms and comparison of software implementations," Journal of Global Optimization, Springer, vol. 56(3), pages 1247-1293, July.
    2. Victor Picheny & Mickael Binois & Abderrahmane Habbal, 2019. "A Bayesian optimization approach to find Nash equilibria," Journal of Global Optimization, Springer, vol. 73(1), pages 171-192, January.
    3. Yurii NESTEROV & Vladimir SPOKOINY, 2017. "Random gradient-free minimization of convex functions," LIDAM Reprints CORE 2851, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Dimitris Bertsimas & Omid Nohadani, 2010. "Robust optimization with simulated annealing," Journal of Global Optimization, Springer, vol. 48(2), pages 323-334, October.
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