Breaking the Stochasticity Barrier: An Adaptive Variance-Reduced Method for Variational Inequalities

• URL: http://arxiv.org/abs/2601.23034v1
• Date: Fri, 30 Jan 2026 14:43:07 GMT
• Title: Breaking the Stochasticity Barrier: An Adaptive Variance-Reduced Method for Variational Inequalities
• Abstract summary: We propose VR-A-A (VarianceReduced-Ascent with Armijo) as a novel algorithm for non-con optimization tasks.<n>We show that our method effectively suppresses limit cycles and accelerates convergence with reduced dependence on manual learning scheduling.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Stochastic non-convex non-concave optimization, formally characterized as Stochastic Variational Inequalities (SVIs), presents unique challenges due to rotational dynamics and the absence of a global merit function. While adaptive step-size methods (like Armijo line-search) have revolutionized convex minimization, their application to this setting is hindered by the Stochasticity Barrier: the noise in gradient estimation masks the true operator curvature, triggering erroneously large steps that destabilize convergence. In this work, we propose VR-SDA-A (Variance-Reduced Stochastic Descent-Ascent with Armijo), a novel algorithm that integrates recursive momentum (STORM) with a rigorous Same-Batch Curvature Verification mechanism. We introduce a theoretical framework based on a Lyapunov potential tracking the Operator Norm, proving that VR- SDA-A achieves an oracle complexity of O(epsilon -3) for finding an epsilon-stationary point in general Lipschitz continuous operators. This matches the optimal rate for non-convex minimization while uniquely enabling automated step-size adaptation in the saddle-point setting. We validate our approach on canonical rotational benchmarks and non-convex robust regression tasks, demonstrating that our method effectively suppresses limit cycles and accelerates convergence with reduced dependence on manual learning rate scheduling.

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