Near-Optimal Pure Single-Loop Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization

2026-09-17

AI Summary

本文提出了一种单循环阻尼外梯度方法，用于解决强凸-强凹极小极大优化问题，并证明了该方法具有最优的条件数阶次。

Abstract

We study smooth strongly convex--strongly concave minimax optimization with general nonlinear coupling in the deterministic unconstrained setting. We propose a pure single-loop damped extragradient method with fixed parameters and two new full-gradient evaluations per iteration after one initialization query. The method uses an auxiliary feedback recursion and requires no inner solves, accuracy schedules, or staged restarts. We establish last-iterate linear convergence and show that reducing the squared Euclidean distance to the saddle point to an $\varepsilon$ fraction of its initial value requires $O(\sqrt{κ_xκ_y}\log(2κ_xκ_y/\varepsilon))$ full-gradient queries, where $κ_x=L/μ_x$ and $κ_y=L/μ_y$. This bound attains the optimal condition-number order up to logarithmic factors through fixed explicit updates. Numerical experiments demonstrate the effectiveness of the method.

Problem

• minimax optimization
• strongly convex-strongly concave
• extragradient method

Innovation

• damped extragradient method
• single-loop
• linear convergence
• condition-number order