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

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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 \(\varepsilon \)-first-order Nash equilibrium is \({\mathcal {O}}\left( \varepsilon ^{-3} \right) \) and the number of function value estimation per iteration is bounded by \({\mathcal {O}}\left( d_{x}\varepsilon ^{-2} \right) \). 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 \({\mathcal {O}}\left( K^{\frac{3}{2}} \varepsilon ^{-3} \right) \) and the number of function value estimation is bounded by \({\mathcal {O}}\left( d_{x}\varepsilon ^{-2} \right) \) per iteration. The numerical results indicate the efficiency of the proposed algorithms.

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Jingjing Shen, Ziqi Wang & Zi Xu

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  1. Jingjing Shen
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  2. Ziqi Wang
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  3. Zi Xu
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Correspondence to Zi Xu.

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This work is supported by National Natural Science Foundation of China under the Grant 12071279 and by General Project of Shanghai Natural Science Foundation (No. 20ZR1420600)

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Shen, J., Wang, Z. & Xu, Z. Zeroth-order single-loop algorithms for nonconvex-linear minimax problems. J Glob Optim 87, 551–580 (2023). https://doi.org/10.1007/s10898-022-01169-5

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