OptEMA: Adaptive Exponential Moving Average for Stochastic Optimization with Zero-Noise Optimality

Abstract

The Exponential Moving Average (EMA) is a cornerstone of widely used optimizers such as Adam. However, existing theoretical analyses of Adam-style methods have notable limitations: their guarantees can remain suboptimal in the zero-noise regime, rely on restrictive boundedness conditions (e.g., bounded gradients or objective gaps), use constant or open-loop stepsizes, or require prior knowledge of Lipschitz constants. To overcome these bottlenecks, we introduce OptEMA and analyze two novel variants: OptEMA-M, which applies an adaptive, decreasing EMA coefficient to the first-order moment with a fixed second-order decay, and OptEMA-V, which swaps these roles. At the heart of these variants is a novel Corrected AdaGrad-Norm stepsize. This formulation renders OptEMA closed-loop and Lipschitz-free, meaning its effective stepsizes are strictly trajectory-dependent and require no parameterization via the Lipschitz constant. Under standard stochastic gradient descent (SGD) assumptions, namely smoothness, a lower-bounded objective, and unbiased gradients with bounded variance, we establish rigorous convergence guarantees. Both variants achieve a noise-adaptive convergence rate of $\widetilde{\mathcal{O}}(T^{-1/2}+\sigma^{1/2} T^{-1/4})$ for the average gradient norm, where $\sigma$ is the noise level. Crucially, the Corrected AdaGrad-Norm stepsize plays a central role in enabling the noise-adaptive guarantees: in the zero-noise regime ($\sigma=0$), our bounds automatically reduce to the nearly optimal deterministic rate $\widetilde{\mathcal{O}}(T^{-1/2})$ without any manual hyperparameter retuning.