Fast Objective & Duality Gap Convergence for Non-Convex Strongly-Concave Min-Max Problems with PL Condition
This work addresses a bottleneck in optimization for deep learning applications like AUC maximization, offering incremental improvements in algorithm speed and analysis.
The paper tackles the slow convergence of stochastic algorithms for non-convex strongly-concave min-max problems by leveraging the Polyak-Lojasiewicz condition, achieving improved convergence rates with better dependence on the condition number for both primal objective and duality gaps.
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.