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DERIVATIVE-FREE alternating PROJECTION algorithmS FOR GENERAL NONCONVEX-CONCAVE MINIMAX PROBLEMS

SIAM JOURNAL ON OPTIMIZATION 2024年第2期34卷 1879-1908页

作者： Xu, Zi Wang, Ziqi Shen, Jingjing Dai, Yuhong Shanghai Univ Dept Math Shanghai 200444 Peoples R China Shanghai Univ Newtouch Ctr Math Shanghai 200444 Peoples R China Chinese Acad Sci LSEC ICMSEC Acad Math & Syst Sci Beijing 100190 Peoples R China

In this paper, we study zeroth -order algorithms for nonconvex-concave minimax problems, which have attracted much attention in machine learning, signal processing, and many other fields in recent years. We propose a zeroth -order alternating randomized gradient projection (ZO-AGP) algorithm for smooth nonconvex-concave minimax problems;its iteration complexity to obtain an \varepsilon -stationary point is bounded by O( \varepsilon - 4 ), and the number of function value estimates is bounded by O( d x + d y ) per iteration. Moreover, we propose a zeroth -order block alternating randomized proximal gradient algorithm (ZO-BAPG) for solving blockwise nonsmooth nonconvexconcave minimax optimization problems;its iteration complexity to obtain an \varepsilon -stationary point is bounded by O( \varepsilon - 4 ), and the number of function value estimates per iteration is bounded by O( Kd x + d y ). To the best of our knowledge, this is the first time zeroth -order algorithms with iteration complexity guarantee are developed for solving both general smooth and blockwise nonsmooth nonconvex-concave minimax problems. Numerical results on the data poisoning attack problem and the distributed nonconvex sparse principal component analysis problem validate the efficiency of the proposed algorithms.

关键词： nonconvex-concave minimax problem zeroth-order algorithm alternating random- ized gradient projection algorithm alternating randomized proximal gradient algorithm complexity analysis machine learning

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

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JOURNAL OF GLOBAL OPTIMIZATION 2023年第2-4期87卷 551-580页

作者： Shen, Jingjing Wang, Ziqi Xu, Zi Shanghai Univ Dept Math Shanghai 200444 Peoples R China

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 e-first-order Nash equilibrium is O(epsilon(-3)) and the number of function value estimation per iteration is bounded by O(d(x)epsilon(-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 O(K-3/2 epsilon(-3)) and the number of function value estimation is bounded by O(d(x) epsilon(-2))per iteration. The numerical results indicate the efficiency of the proposed algorithms.

关键词： Nonconvex-linear minimax problem Zeroth-order algorithm alternating randomized gradient projection algorithm alternating randomized proximal gradient algorithm Complexity analysis Machine learning

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