Finite-Time Convergence Rates of Decentralized Local Markovian Stochastic Approximation

Markovian stochastic approximation has recently aroused a great deal of interest in many fields; however, it is not well understood in decentralized settings. Decentralized Markovian stochastic approximation is far more challenging than its single-agent counterpart due to the complex coupling structure between decentralized communication and Markovian noise-corrupted local updates. In this paper, a decentralized local markovian stochastic approximation (DLMSA) algorithm has been proposed and attains a near-optimal convergence rate. Specifically, we first provide a local variant of decentralized Markovian stochastic approximation so that each agent performs multiple local updates and then periodically communicate with its neighbors. Furthermore, we propose DLMSA with compressed communication (C-DLMSA) for further reducing the communication overhead. In this way, each agent only needs to communicate compressed information (e.g., sign compression) with its neighbors. We show that C-DLMSA enjoys the same convergence rate as that of the original DLMSA. Finally, we verify our theoretical results by applying our methods to solve multi-task reinforcement learning problems over multi-agent systems.