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

We study decentralized stochastic gradient tracking over a time-varying network of $N$ agents under a uniform window-mixing condition. Products of $\tau$ consecutive doubly stochastic mixing matrices contract disagreement by a factor $\lambda <1$, 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 $\tilde{\mathcal{O}}(1\mathrm{/}(NK))$; for smooth convex objectives, it is $\mathcal{O}(1\mathrm{/}\sqrt{NK})$. Both match their centralized mini-batch counterparts and yield linear speedup after a network-dependent transient.