Fast Objective & Duality Gap Convergence for Non-Convex Strongly-Concave Min-Max Problems with PL Condition

TL;DR

This paper introduces a new stochastic algorithm framework for non-convex strongly-concave min-max problems under the PL condition, achieving faster convergence rates for objective and duality gaps, with theoretical analysis and empirical validation.

Contribution

It proposes a generic proximal stage-based stochastic method leveraging the PL condition, with improved convergence rates and a novel Lyapunov function analysis for non-convex min-max problems.

Findings

Faster convergence rates for primal and duality gaps.

Theoretical analysis with a new Lyapunov function.

Empirical results confirm effectiveness in deep learning tasks.

Abstract

This paper focuses on stochastic methods for solving smooth non-convex strongly-concave min-max problems, which have received increasing attention due to their potential applications in deep learning (e.g., deep AUC maximization, distributionally robust optimization). However, most of the existing algorithms are slow in practice, and their analysis revolves around the convergence to a nearly stationary point.We consider leveraging the Polyak-Lojasiewicz (PL) condition to design faster stochastic algorithms with stronger convergence guarantee. Although PL condition has been utilized for designing many stochastic minimization algorithms, their applications for non-convex min-max optimization remain rare. In this paper, we propose and analyze a generic framework of proximal stage-based method with many well-known stochastic updates embeddable. Fast convergence is established in terms of both the primal objective gap and the duality gap. Compared with existing studies, (i) our analysis is based on a novel Lyapunov function consisting of the primal objective gap and the duality gap of a regularized function, and (ii) the results are more comprehensive with improved rates that have better dependence on the condition number under different assumptions. We also conduct deep and non-deep learning experiments to verify the effectiveness of our methods.