Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems

• URL: http://arxiv.org/abs/2108.00473v5
• Date: Thu, 25 Jan 2024 15:15:45 GMT
• Title: Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems
• Authors: Zi Xu, Ziqi Wang, Jingjing Shen, Yuhong Dai
• Abstract summary: In this paper, we propose an algorithm for nonsmooth zeroth-order minimax problems. We show that it can be used to attack nonconcave minimax problems.
• Score: 9.173866646584031
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study zeroth-order algorithms for nonconvex-concave minimax problems, which have attracted widely 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, and its iteration complexity to obtain an $\varepsilon$-stationary point is bounded by $\mathcal{O}(\varepsilon^{-4})$, and the number of function value estimation is bounded by $\mathcal{O}(d_{x}+d_{y})$ per iteration. Moreover, we propose a zeroth-order block alternating randomized proximal gradient algorithm (ZO-BAPG) for solving block-wise nonsmooth nonconvex-concave minimax optimization problems, and the iteration complexity to obtain an $\varepsilon$-stationary point is bounded by $\mathcal{O}(\varepsilon^{-4})$ and the number of function value estimation per iteration is bounded by $\mathcal{O}(K d_{x}+d_{y})$. To the best of our knowledge, this is the first time that zeroth-order algorithms with iteration complexity gurantee are developed for solving both general smooth and block-wise nonsmooth nonconvex-concave minimax problems. Numerical results on data poisoning attack problem and distributed nonconvex sparse principal component analysis problem validate the efficiency of the proposed algorithms.

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