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

TL;DR

This paper introduces novel zeroth-order algorithms for nonconvex-concave minimax problems, providing iteration complexity guarantees and demonstrating effectiveness on practical machine learning tasks.

Contribution

It develops the first zeroth-order algorithms with iteration complexity bounds for both smooth and nonsmooth nonconvex-concave minimax problems.

Findings

Algorithms achieve $ ilde{O}(rac{1}{\varepsilon^4})$ iteration complexity.

Function value estimation per iteration is bounded by problem dimensions.

Numerical experiments validate the algorithms' efficiency in real tasks.

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.