Single-Loop Stochastic Projected Damped Extragradient Methods for Stochastic Nonconvex--(Strongly) Concave Minimax Optimization

2026-09-18

AI Summary

该研究提出了单循环随机投影阻尼外梯度方法，用于解决随机非凸-（强）凹极小极大优化问题，并在游戏平稳性和优化平稳性方面提供了复杂度保证。

Abstract

We develop single-loop stochastic projected damped extragradient methods for stochastic nonconvex--(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(κ\varepsilon^{-4})$ and $O(\varepsilon^{-5})$ in the nonconvex--strongly concave and nonconvex--concave settings, respectively, where $κ=L/μ$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(κ^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-9/2})$, respectively. For an $\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(κ\varepsilon^{-4})$ and $O(\varepsilon^{-6})$, while VR-SPDE achieves $O(κ^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.

Problem

• stochastic nonconvex-concave minimax optimization
• game stationarity
• optimization stationarity

Innovation

• Single-Loop
• Stochastic Projected Damped Extragradient
• Variance-Reduced
• Game Stationarity
• Optimization Stationarity