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.

翻译：本文研究非凸-凹极小极大问题的零阶算法，该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，我们提出一种零阶交替随机梯度投影算法，其获得$\varepsilon$-稳定点的迭代复杂度界为$\mathcal{O}(\varepsilon^{-4})$，每轮迭代的函数值估计次数界为$\mathcal{O}(d_{x}+d_{y})$。此外，针对分块非光滑非凸-凹极小极大优化问题，我们提出一种零阶块交替随机近端梯度算法，其获得$\varepsilon$-稳定点的迭代复杂度界为$\mathcal{O}(\varepsilon^{-4})$，每轮迭代的函数值估计次数界为$\mathcal{O}(K d_{x}+d_{y})$。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。基于数据投毒攻击问题与分布式非凸稀疏主成分分析问题的数值实验结果验证了所提算法的有效性。