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Zeroth-order single-loop algorithms for nonconvex-linear minimax problems

Author

Listed:
  • Jingjing Shen

    (Shanghai University)

  • Ziqi Wang

    (Shanghai University)

  • Zi Xu

    (Shanghai University)

Abstract

Nonconvex minimax problems have attracted significant interest in machine learning and many other fields in recent years. In this paper, we propose a new zeroth-order alternating randomized gradient projection algorithm to solve smooth nonconvex-linear problems and its iteration complexity to find an $$\varepsilon $$ ε -first-order Nash equilibrium is $${\mathcal {O}}\left( \varepsilon ^{-3} \right) $$ O ε - 3 and the number of function value estimation per iteration is bounded by $${\mathcal {O}}\left( d_{x}\varepsilon ^{-2} \right) $$ O d x ε - 2 . Furthermore, we propose a zeroth-order alternating randomized proximal gradient algorithm for block-wise nonsmooth nonconvex-linear minimax problems and its corresponding iteration complexity is $${\mathcal {O}}\left( K^{\frac{3}{2}} \varepsilon ^{-3} \right) $$ O K 3 2 ε - 3 and the number of function value estimation is bounded by $${\mathcal {O}}\left( d_{x}\varepsilon ^{-2} \right) $$ O d x ε - 2 per iteration. The numerical results indicate the efficiency of the proposed algorithms.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01169-5
    DOI: 10.1007/s10898-022-01169-5
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    References listed on IDEAS

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    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. Weiwei Pan & Jingjing Shen & Zi Xu, 2021. "An efficient algorithm for nonconvex-linear minimax optimization problem and its application in solving weighted maximin dispersion problem," Computational Optimization and Applications, Springer, vol. 78(1), pages 287-306, January.
    3. 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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