Distributed Zeroth-Order Optimization with Rademacher Perturbations and Momentum Gradient Tracking

Abstract

Zeroth-order (ZO) optimization is indispensable for complex non-convex tasks where explicit gradients are computationally prohibitive or strictly inaccessible. For deploying ZO methods over distributed heterogeneous networks, the gradient tracking technique is often employed to eliminate structural data biases. However, the inherent variance of derivative-free estimators is also amplified. To overcome this problem, we propose Zeroth-Order Momentum Gradient Tracking (ZO-MGT), which integrates momentum-based variance reduction with dynamic gradient tracking. Specifically, ZO-MGT that requires exactly two function queries per iteration can avoid costly batch sampling and prevent variance explosion, while eliminating structural biases. Moreover, by utilizing Rademacher perturbations, it preserves optimal query efficiency and enables bitwise hardware acceleration. We theoretically analyze the convergence of ZO-MGT and establish an $\mathcal{O}(1/T)$ convergence rate. Furthermore, we prove that a large momentum factor can aggressively suppress the heterogeneity-induced bias floor at a remarkable quadratic rate of $\mathcal{O}((1-β)^2)$. Numerical experiments under extreme data heterogeneity verify that ZO-MGT can effectively overcome traditional tracking failures with accelerated convergence guarantees, while achieving significantly tighter consensus.