Life sciences · Preprint

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

arXiv · September 8, 2026

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AI Summary

This is a theoretical computer science preprint establishing improved first-order oracle complexity bounds for constrained stochastic min-max optimization problems. The authors prove that the gradient mapping complexity can be improved from Õ(ε⁻⁴) to Õ(ε⁻²), bridging a gap with unconstrained problem complexity. This work has not been peer reviewed and is of interest to the optimization and machine learning theory community, not to clinical practice.

What was studied

Preprint.

Key findings

Prior best-known complexity for constrained convex-concave min-max problems: Õ(ε⁻⁴) Improved complexity proved in this paper: Õ(ε⁻²) Matches near-optimal complexity of unconstrained case: Õ(ε⁻²)

Safety

Safety was not reported in the material analysed. Check the source before drawing any conclusion about harm.

Who this may help

The source did not state who this applies to in practice.

Plain-language summary

This is a theoretical algorithmic analysis paper establishing complexity bounds for optimization problems; it presents mathematical results without empirical validation, clinical data, or real-world application evidence.

Reported figures

Quoted from the source exactly as published.

Prior gradient mapping complexity boundÕ(ε⁻⁴)

Unconstrained problem complexity (baseline)Õ(ε⁻²)

Improved constrained complexity (this work)Õ(ε⁻²)

Caveats

This is a preprint and has not been peer reviewed.
Work is theoretical and algorithmic; no empirical validation or real-world application data provided.
Intended audience is the optimization and theoretical machine learning community, not clinical practitioners.
No figures, tables, or numerical experiments are reported in the abstract.

Evidence assessment

Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.

Study design

insufficient

Theoretical mathematical paper; no empirical study design applicable.

Directness

insufficient

Addresses algorithmic complexity, not clinical or applied outcomes.

Consistency

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No comparison to conflicting results reported within the source.

Risk of bias

insufficient

Mathematical proofs do not involve empirical bias assessment.

Sample adequacy

insufficient

No empirical sample; theoretical derivation only.

Source 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.

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Ahmet Alacaoglu