Stochastic Optimization

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Period ending 2026-09-14

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Period ending 2026-09-07

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How to Make the Gradient Mapping Small for Constrained Stochastic Min-Max Problems and Beyond
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Sep 8, 2026 · math.OCJ/K move · Enter open · S save

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

Ahmet Alacaoglu

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

\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.

Convex Optimization Stochastic Optimization Stochastic

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