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 validate the efficiency of the proposed algorithms.

翻译：本文研究了非凸-凹极小极大问题的零阶算法，这类问题近年来在机器学习、信号处理及众多其他领域受到广泛关注。针对光滑非凸-凹极小极大问题，我们提出了零阶交替随机梯度投影（ZO-AGP）算法，其达到$\varepsilon$-稳定点的迭代复杂度上界为$\mathcal{O}(\varepsilon^{-4})$，且每次迭代的函数估值次数上界为$\mathcal{O}(d_{x}+d_{y})$。此外，针对分块非光滑非凸-凹极小极大优化问题，我们提出了零阶分块交替随机近端梯度算法（ZO-BAPG），其达到$\varepsilon$-稳定点的迭代复杂度上界为$\mathcal{O}(\varepsilon^{-4})$，且每次迭代的函数估值次数上界为$\mathcal{O}(K d_{x}+d_{y})$。据我们所知，这是首次为求解一般光滑与分块非光滑非凸-凹极小极大问题，开发出具有迭代复杂度保证的零阶算法。在数据投毒攻击问题上的数值结果验证了所提算法的有效性。