Near-Optimal Algorithms for Minimax Optimization

TL;DR

This paper introduces a near-optimal first-order algorithm for smooth, strongly-convex-strongly-concave minimax problems, achieving the theoretical lower bound on gradient evaluations and extending to various problem settings.

Contribution

It presents the first algorithm matching the lower bound of gradient complexity for minimax problems, using an accelerated proximal point method and solver.

Findings

Achieves $ ilde{O}(\sqrt{\kappa_{ ext{ extbf{x}}}\kappa_{ ext{ extbf{y}}}})$ gradient complexity

Extends to strongly-convex-concave, convex-concave, and nonconvex settings

Outperforms existing methods in gradient complexity

Abstract

This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $\tilde{O}(\kappa_{\mathbf x}+\kappa_{\mathbf y})$ or $\tilde{O}(\min\{\kappa_{\mathbf x}\sqrt{\kappa_{\mathbf y}}, \sqrt{\kappa_{\mathbf x}}\kappa_{\mathbf y}\})$ gradient evaluations, where $\kappa_{\mathbf x}$ and $\kappa_{\mathbf y}$ are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap still remains between these results and the best existing lower bound $\tilde{\Omega}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$. This paper presents the first algorithm with $\tilde{O}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$ gradient complexity, matching the lower bound up to logarithmic factors. Our algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvex-concave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.