How to Make the Gradient Mapping Small for Constrained Stochastic Min-Max Problems and Beyond

2026-09-08

AI Summary

研究了约束或正则化凸凹最小-最大优化问题中如何减小梯度映射的大小，通过使用Blum-Gladyshev假设，将复杂度改进至$\widetilde{O}(\varepsilon^{-2})$。

Abstract

We study the stochastic first-order oracle complexity for constrained or regularized convex-concave min-max optimization and stochastic monotone variational inequalities. We focus on the case when suboptimality is measured in terms of the gradient mapping, also known as, forward-backward or natural residual, an optimality notion that generalizes the gradient norm for unconstrained problems. In this setting, under standard unbiased oracle access with now-standard variance assumptions, the best-known complexity for making the norm of the gradient mapping less than $\varepsilon$ is $\widetilde{O}(\varepsilon^{-4})$, compared to the near-optimal $\widetilde{O}(\varepsilon^{-2})$ that is established in the unconstrained case. We bridge this gap to improve the gradient mapping complexity for constrained convex-concave min-max problems to $\widetilde{O}(\varepsilon^{-2})$. We then extend to prove the same complexity for problems without the bounded variance, by using the Blum-Gladyshev assumption.

Problem

• stochastic min-max optimization
• gradient mapping
• convex-concave
• variational inequalities
• oracle complexity

Innovation

• gradient mapping
• convex-concave min-max optimization
• stochastic monotone variational inequalities
• Blum-Gladyshev assumption