Stochastic Gradient Tracking over Time-Varying Networks: One-Step Lyapunov Analysis

2026-08-17

AI Summary

This study addresses the analytical challenges in establishing convergence for decentralized stochastic gradient tracking over time-varying networks by proposing a single-step Lyapunov analysis framework. By constructing a time-varying quadratic norm and deriving a single-step Lyapunov identity, this approach circumvents traditional window expansion techniques under window mixing conditions, enabling precise recursive characterization of coupled errors. The proposed method effectively resolves critical analytical bottlenecks associated with dynamic topologies. Theoretical results demonstrate that the algorithm achieves a convergence rate matching centralized mini-batch methods, thereby attaining linear speedup. Consequently, this work provides a more concise and rigorous theoretical foundation for distributed optimization in time-varying network environments.

Abstract

We study decentralized stochastic gradient tracking over a time-varying network of $N$ agents under a uniform window-mixing condition. Products of $τ$ consecutive doubly stochastic mixing matrices contract disagreement by a factor $λ<1$, although individual matrices need not contract disagreement strictly and individual communication graphs may be disconnected. We construct a time-varying quadratic norm that turns this window contraction into an exact one-step Lyapunov identity. This leads to coupled one-step recursions for the centroid and disagreement errors, without unrolling the dynamics over communication windows. For smooth strongly convex objectives, the leading stochastic term is $\widetilde{\mathcal O}(1/(NK))$; for smooth convex objectives, it is $\mathcal O(1/\sqrt{NK})$. Both match their centralized mini-batch counterparts and yield linear speedup after a network-dependent transient.

Problem

• Decentralized stochastic gradient tracking
• Time-varying networks
• Lyapunov analysis
• Convergence rate

Innovation

• One-Step Lyapunov Analysis
• Stochastic Gradient Tracking
• Time-Varying Networks
• Window-Mixing Condition
• Linear Speedup