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

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.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2609.08380 [math.OC]

(or arXiv:2609.08380v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2609.08380 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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