Improved Convergence for Decentralized Stochastic Optimization with Biased Gradients

Abstract

Decentralized stochastic optimization has emerged as a fundamental paradigm for large-scale machine learning. However, practical implementations often rely on biased gradient estimators arising from communication compression or inexact local oracles, which severely degrade convergence in the presence of data heterogeneity. To address the challenge, we propose Decentralized Momentum Tracking with Biased Gradients (Biased-DMT), a novel decentralized algorithm designed to operate reliably under biased gradient information. We establish a comprehensive convergence theory for Biased-DMT in nonconvex settings and show that it achieves linear speedup with respect to the number of agents. The theoretical analysis shows that Biased-DMT decouples the effects of network topology from data heterogeneity, enabling robust performance even in sparse communication networks. Notably, when the gradient oracle introduces only absolute bias, the proposed method eliminates the structural heterogeneity error and converges to the exact physical error floor. For the case of relative bias, we further characterize the convergence limit and show that the remaining error is an unavoidable physical consequence of locally injected noise. Extensive numerical experiments corroborate our theoretical analysis and demonstrate the practical effectiveness of Biased-DMT across a range of decentralized learning scenarios.