Smoothing Meets Perturbation: Unified and Tight Analysis for Nonconvex-Concave Minimax Optimization

Abstract

This paper studies smooth nonconvex-concave minimax optimization and two acceleration mechanisms for single-loop first-order methods: dual perturbation and smoothing. Although both techniques improve convergence guarantees, their relative advantages remain unclear due to the distinction between game stationarity (GS) and optimization stationarity (OS). We provide a tight characterization of their iteration complexities under both notions. We show that smoothing accelerates convergence to both GS and OS, whereas dual perturbation improves the rate only for GS and does not accelerate OS. Matching lower bounds based on hard instances establish the tightness of these rates. Motivated by this separation, we propose Perturbed Smoothed GDA, a single-loop method combining both techniques. It improves the complexity for GS over existing single-loop methods while preserving the state-of-the-art rate for OS, and further admits asymptotic convergence to 0-GS, which is not available for vanilla Smoothed GDA.