IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v87y2023i2d10.1007_s10898-022-01160-0.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01160-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-022-01160-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jingjing Shen & Ziqi Wang & Zi Xu, 2023. "Zeroth-order single-loop algorithms for nonconvex-linear minimax problems," Journal of Global Optimization, Springer, vol. 87(2), pages 551-580, November.
    2. Jonas Bjerg Thomsen & Francesco Ferri & Jens Peter Kofoed & Kevin Black, 2018. "Cost Optimization of Mooring Solutions for Large Floating Wave Energy Converters," Energies, MDPI, vol. 11(1), pages 1-23, January.
    3. Ghadimi, Saeed & Powell, Warren B., 2024. "Stochastic search for a parametric cost function approximation: Energy storage with rolling forecasts," European Journal of Operational Research, Elsevier, vol. 312(2), pages 641-652.
    4. Gabriele Eichfelder & Corinna Krüger & Anita Schöbel, 2017. "Decision uncertainty in multiobjective optimization," Journal of Global Optimization, Springer, vol. 69(2), pages 485-510, October.
    5. Gabriela Simonet & Julie Subervie & Driss Ezzine-De-Blas & Marina Cromberg & Amy Duchelle, 2015. "Paying smallholders not to cut down the amazon forest: impact evaluation of a REDD+ pilot project," Working Papers 1514, Chaire Economie du climat.
    6. Somayeh Moazeni & Warren B. Powell & Boris Defourny & Belgacem Bouzaiene-Ayari, 2017. "Parallel Nonstationary Direct Policy Search for Risk-Averse Stochastic Optimization," INFORMS Journal on Computing, INFORMS, vol. 29(2), pages 332-349, May.
    7. V. Kungurtsev & F. Rinaldi, 2021. "A zeroth order method for stochastic weakly convex optimization," Computational Optimization and Applications, Springer, vol. 80(3), pages 731-753, December.
    8. Hoang Tran & Qiang Du & Guannan Zhang, 2025. "Convergence analysis for a nonlocal gradient descent method via directional Gaussian smoothing," Computational Optimization and Applications, Springer, vol. 90(2), pages 481-513, March.
    9. Jakubik, Johannes & Binding, Adrian & Feuerriegel, Stefan, 2021. "Directed particle swarm optimization with Gaussian-process-based function forecasting," European Journal of Operational Research, Elsevier, vol. 295(1), pages 157-169.
    10. Nikita Kornilov & Alexander Gasnikov & Pavel Dvurechensky & Darina Dvinskikh, 2023. "Gradient-free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact," Computational Management Science, Springer, vol. 20(1), pages 1-43, December.
    11. Christophe Gouel & Nicolas Legrand, 2017. "Estimating the Competitive Storage Model with Trending Commodity Prices," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(4), pages 744-763, June.
    12. Zhao, Jake, 2020. "Accounting for the corporate cash increase," European Economic Review, Elsevier, vol. 123(C).
    13. Hannes Schwarz & Valentin Bertsch & Wolf Fichtner, 2018. "Two-stage stochastic, large-scale optimization of a decentralized energy system: a case study focusing on solar PV, heat pumps and storage in a residential quarter," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 40(1), pages 265-310, January.
    14. Breitmoser, Yves & Valasek, Justin, 2017. "A rationale for unanimity in committees," Discussion Papers, Research Unit: Economics of Change SP II 2017-308, WZB Berlin Social Science Center.
    15. Krese, Gorazd & Lampret, Žiga & Butala, Vincenc & Prek, Matjaž, 2018. "Determination of a Building's balance point temperature as an energy characteristic," Energy, Elsevier, vol. 165(PB), pages 1034-1049.
    16. Tavakol Aghaei, Vahid & Ağababaoğlu, Arda & Bawo, Biram & Naseradinmousavi, Peiman & Yıldırım, Sinan & Yeşilyurt, Serhat & Onat, Ahmet, 2023. "Energy optimization of wind turbines via a neural control policy based on reinforcement learning Markov chain Monte Carlo algorithm," Applied Energy, Elsevier, vol. 341(C).
    17. Andrzej Wędzik & Tomasz Siewierski & Michał Szypowski, 2019. "The Use of Black-Box Optimization Method for Determination of the Bus Connection Capacity in Electric Power Grid," Energies, MDPI, vol. 13(1), pages 1-21, December.
    18. Jean-Jacques Forneron, 2023. "Noisy, Non-Smooth, Non-Convex Estimation of Moment Condition Models," Papers 2301.07196, arXiv.org, revised Feb 2023.
    19. Linas Stripinis & Remigijus Paulavičius, 2023. "Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions," Mathematics, MDPI, vol. 11(13), pages 1-19, June.
    20. Pál, László & Sándor, Zsolt, 2023. "Comparing procedures for estimating random coefficient logit demand models with a special focus on obtaining global optima," International Journal of Industrial Organization, Elsevier, vol. 88(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01160-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.