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

Variance reduction improves the convergence rate of stochastic non-convex optimization under the average smoothness condition. Existing adaptive variance reduction methods are especially designed for average smoothness and do not adapt to standard smoothness. In this paper, we study how one algorithm can adapt to these two settings and the noiseless setting without knowing the smoothness constants or the variance bound. Given the gradient budget $T$, we develop a two-phase method. The first phase compares two independent gradients and switches to normalized momentum when their difference exceeds a threshold. This phase provides both a candidate with standard smooth and noiseless guarantees and also ensures a bounded expected function gap for the second variance reduction phase. The resulting rates are $O(T^{-1/3})$ under average smoothness, $O(T^{-1/4})$ under standard smoothness, and $O(T^{-1/2})$ without noise. When $T$ is unknown, we propose a method with two paths that attain the corresponding rates with logarithmic factors in the stochastic settings. Here, we use a variance reduction path to deal with average smoothness and employ the AdaGrad-Norm for standard smoothness and noiseless setting. Neither method restarts from the initial point or requires a growing number of oracle calls in each round.

We develop single-loop stochastic projected damped extragradient methods for nonconvex--(strongly) concave minimax optimization, with complexity guarantees for both game stationarity (GS) and optimization stationarity (OS). Our approach combines a stochastic projected damped extragradient (SPDE) method with a recursive variance-reduced variant, VR-SPDE, both of which retain a single-loop structure. Under an unbiased stochastic gradient oracle with uniformly bounded variance, SPDE finds an $\varepsilon$-game-stationary point with stochastic first-order oracle (SFO) complexities of $O(κ\varepsilon^{-4})$ and $O(\varepsilon^{-5})$ in the nonconvex--strongly concave and nonconvex--concave settings, respectively, where $κ=L/μ$. Under an additional mean-square Lipschitz condition on the stochastic gradients, VR-SPDE improves these GS complexities to $O(κ^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-9/2})$, respectively. For an $\varepsilon$-optimization-stationary point, SPDE achieves SFO complexities of $O(κ\varepsilon^{-4})$ and $O(\varepsilon^{-6})$, while VR-SPDE achieves $O(κ^{3/2}\varepsilon^{-3})$ and $O(\varepsilon^{-6})$, in the two settings, respectively. These OS guarantees match the best-known bounds achieved by multi-loop methods while preserving a single-loop implementation. To the best of our knowledge, our results provide the best-known SFO complexity guarantees among single-loop stochastic first-order methods for the respective stationarity criteria and problem classes.

We study nonconvex stochastic optimization under the Blum-Gladyshev ($\mathsf{BG}$-0) noise model, where the stochastic gradient variance grows quadratically with the distance from the initialization. We consider this problem under both standard smoothness and the symmetric generalized-smoothness framework, which captures objectives whose local curvature can scale with the gradient norm. We prove that normalized stochastic gradient descent with momentum, using only one stochastic gradient per iteration, converges under $\mathsf{BG}$-0 noise with oracle complexity $O(\varepsilon^{-6})$. This rate holds both for standard smoothness and for $α$-symmetric generalized smoothness, showing that generalized smoothness is rate-neutral for normalized momentum in this setting. We then study a variance-reduced normalized STORM method. Under mean-square smoothness and sharp initialization, the method achieves the minimax optimal $O(\varepsilon^{-4})$ complexity, matching the lower bound. Under expected $α$-symmetric generalized smoothness, the STORM recursion couples gradient-dependent smoothness with distance-dependent noise, leading to complexity $O(\varepsilon^{-(4+α)})$ for $α\in(0,1)$ and $O(\varepsilon^{-5})$ for $α=1$. When the distance-growth parameter in the noise model vanishes, our guarantees recover the standard bounded-variance rates: $O(\varepsilon^{-4})$ for momentum, $O(\varepsilon^{-3})$ for variance reduction, and $O(\varepsilon^{-2})$ in the deterministic case. To our knowledge, these are the first convergence guarantees for normalized methods in non-convex stochastic optimization under $\mathsf{BG}$-0 noise without bounded domains, increasing batch sizes, or explicit anchoring, covering both standard and generalized smoothness regimes.