On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems

TL;DR

This paper analyzes the convergence of two-time-scale gradient descent ascent algorithms for nonconvex-concave minimax problems, providing the first nonasymptotic complexity results and explaining their practical success in training GANs.

Contribution

It offers the first nonasymptotic analysis of two-time-scale GDA for nonconvex-concave minimax problems, demonstrating its efficiency in finding stationary points.

Findings

GDA can find stationary points efficiently in nonconvex-concave problems.

Two-time-scale GDA outperforms single-step methods in convergence.

Results explain GDA's success in training GANs and similar applications.

Abstract

We consider nonconvex-concave minimax problems, $\min_{\mathbf{x}} \max_{\mathbf{y} \in \mathcal{Y}} f(\mathbf{x}, \mathbf{y})$, where $f$ is nonconvex in $\mathbf{x}$ but concave in $\mathbf{y}$ and $\mathcal{Y}$ is a convex and bounded set. One of the most popular algorithms for solving this problem is the celebrated gradient descent ascent (GDA) algorithm, which has been widely used in machine learning, control theory and economics. Despite the extensive convergence results for the convex-concave setting, GDA with equal stepsize can converge to limit cycles or even diverge in a general setting. In this paper, we present the complexity results on two-time-scale GDA for solving nonconvex-concave minimax problems, showing that the algorithm can find a stationary point of the function $\Phi(\cdot) := \max_{\mathbf{y} \in \mathcal{Y}} f(\cdot, \mathbf{y})$ efficiently. To the best our knowledge, this is the first nonasymptotic analysis for two-time-scale GDA in this setting, shedding light on its superior practical performance in training generative adversarial networks (GANs) and other real applications.