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

. 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 \ (\) -stationary point is bounded by \ (O (^-4) \), and the number of function value estimates is bounded by \ (O (dₗ+dₘ) \) per iteration. Moreover, we propose a zeroth-order block alternating randomized proximal gradient algorithm (ZO-BAPG) for solving blockwise nonsmooth nonconvex-concave minimax optimization problems; its iteration complexity to obtain an \ (\) -stationary point is bounded by \ (O (^-4) \), and the number of function value estimates per iteration is bounded by \ (O (K dₗ+dₘ) \). 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. Keywordsnonconvex-concave minimax problemzeroth-order algorithmalternating randomized gradient projection algorithmalternating randomized proximal gradient algorithmcomplexity analysismachine learningMSC codes90C4790C2690C30