Stochastic Momentum Tracking Push-Pull for Decentralized Optimization over Directed Graphs

Abstract

Decentralized optimization over directed networks is frequently challenged by asymmetric communication and the inherent high variance of stochastic gradients, which collectively cause severe oscillations and hinder algorithmic convergence. To address these challenges, we propose the Stochastic Momentum Tracking Push-Pull (SMTPP) algorithm, which tracks the momentum term rather than raw stochastic gradients within the Push-Pull architecture. This design successfully decouples the variance reduction capacity from the algebraic connectivity of the graph.Although the inherent topology mismatch of directed graphs precludes exact convergence under persistent stochastic noise, SMTPP rigorously compresses this unavoidable steady-state error floor into a minimal neighborhood determined by network connectivity and gradient variance. Furthermore, SMTPP guarantees convergence on any strongly connected directed graph. Extensive experiments on non-convex logistic regression demonstrate that the algorithm is highly robust to network connectivity. By effectively dampening topology-induced oscillations, SMTPP achieves convergence rates and overall performance that closely match those of centralized baselines, regardless of whether the network is sparse or dense.