Decentralized Optimization over Time-Varying Row-Stochastic Digraphs

Abstract: Decentralized optimization over directed graphs is essential for applications such as robotic swarms, sensor networks, and distributed learning. In many practical scenarios, the underlying network is a Time-Varying Broadcast Network (TVBN), where only row-stochastic mixing matrices can be constructed due to inaccessible out-degree information. Achieving exact convergence over TVBNs has remained a long-standing open question, as the limiting distribution of time-varying row-stochastic mixing matrices depends on unpredictable future graph realizations, rendering standard bias-correction techniques infeasible. This paper resolves this open question by developing the first algorithm that achieves exact convergence using only time-varying row-stochastic matrices. We propose PULM (Pull-with-Memory), a gossip protocol that attains average consensus with exponential convergence by alternating between row-stochastic mixing and local adjustment. Building on PULM, we develop PULM-DGD, which converges to a stationary solution at $\mathcal{O}(\ln(T)/T)$ for smooth nonconvex objectives. Our results significantly extend decentralized optimization to highly dynamic communication environments.