Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems

TL;DR

This paper introduces the first zeroth-order algorithms with proven iteration complexity guarantees for a class of nonconvex-nonconcave minimax problems satisfying the Polyak-ojasiewicz condition, applicable in deterministic and stochastic settings.

Contribution

The paper proposes two novel zeroth-order algorithms, ZO-AGDA and ZO-VRAGDA, with theoretical complexity bounds for solving NC-PL minimax problems, filling a gap in zeroth-order optimization.

Findings

ZO-AGDA achieves (\u03b5^{-2}) queries for () stationary points.

ZO-VRAGDA achieves () queries with variance reduction.

First zeroth-order algorithms with iteration complexity guarantees for NC-PL minimax problems.

Abstract

In this paper, we consider a class of nonconvex-nonconcave minimax problems, i.e., NC-PL minimax problems, whose objective functions satisfy the Polyak-\L ojasiewicz (PL) condition with respect to the inner variable. We propose a zeroth-order alternating gradient descent ascent (ZO-AGDA) algorithm and a zeroth-order variance reduced alternating gradient descent ascent (ZO-VRAGDA) algorithm for solving NC-PL minimax problem under the deterministic and the stochastic setting, respectively. The total number of function value queries to obtain an $\epsilon$-stationary point of ZO-AGDA and ZO-VRAGDA algorithm for solving NC-PL minimax problem is upper bounded by $\mathcal{O}(\varepsilon^{-2})$ and $\mathcal{O}(\varepsilon^{-3})$, respectively. To the best of our knowledge, they are the first two zeroth-order algorithms with the iteration complexity gurantee for solving NC-PL minimax problems.