Breaking the Stochasticity Barrier: An Adaptive Variance-Reduced Method for Variational Inequalities
Yungi Jeong, Takumi Otsuka
TL;DR
This work addresses stochastic variational inequalities (SVIs) in non-convex non-concave settings where rotational dynamics impede adaptive step-size methods. It introduces VR-SDA-A, a variance-reduced adaptive method that couples the STORM estimator with a Same-Batch Curvature Verification to stabilize steps without relying on the Strong Growth Condition. The authors prove convergence to an $\epsilon$-stationary point with oracle complexity $\mathcal{O}(\epsilon^{-3})$ by leveraging a Lyapunov potential that ties merit-function progress to estimator variance. Empirical results on canonical bilinear systems and non-convex robust optimization demonstrate improved stability and faster convergence with reduced hyperparameter tuning, highlighting potential impact on adversarial training and multi-agent learning. Overall, the approach advances parameter-free, scalable optimization for challenging SVIs by decoupling variance control from curvature-aware adaptation.
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.