VAMO: Efficient Zeroth-Order Variance Reduction for SGD with Faster Convergence

Abstract

Optimizing large-scale nonconvex problems, common in deep learning, demands balancing rapid convergence with computational efficiency. First-order (FO) optimizers, which serve as today's baselines, provide fast convergence and good generalization but often incur high computation and memory costs due to the large size of modern models. Conversely, zeroth-order (ZO) algorithms reduce this burden using estimated gradients, yet their slow convergence in high-dimensional settings limits practicality. We introduce VAMO (VAriance-reduced Mixed-gradient Optimizer), a stochastic variance-reduced method that extends mini-batch SGD with full-batch ZO gradients under an SVRG-style framework. VAMO's hybrid design utilizes a two-point ZO estimator to achieve a dimension-agnostic convergence rate of $\mathcal{O}(1/T + 1/b)$, where $T$ is the number of iterations and $b$ is the batch-size, surpassing the dimension-dependent slowdown of purely ZO methods and significantly improving over SGD's $\mathcal{O}(1/\sqrt{T})$ rate. Additionally, we propose a multi-point variant that mitigates the $O(1/b)$ error by adjusting the number of estimation points to balance convergence and cost. Importantly, VAMO achieves these gains with smaller dynamic memory requirements than many FO baselines, making it particularly attractive for edge deployment. Experiments including traditional neural network training and LLM finetuning confirm that VAMO not only outperforms established FO and ZO methods, but also does so with a light memory footprint.