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We study a decentralized variant of stochastic approximation, a data-drivenapproach for finding the root of an operator under noisy measurements. Anetwork of agents, each with its own operator and data observations,cooperatively find the fixed point of the aggregate operator over adecentralized communication graph. Our main contribution is to provide afinite-time analysis of this decentralized stochastic approximation method whenthe data observed at each agent are sampled from a Markov process; this lack ofindependence makes the iterates biased and (potentially) unbounded. Underfairly standard assumptions, we show that the convergence rate of the proposedmethod is essentially the same as if the samples were independent, differingonly by a log factor that accounts for the mixing time of the Markov processes.The key idea in our analysis is to introduce a novel Razumikhin-Lyapunovfunction, motivated by the one used in analyzing the stability of delayedordinary differential equations. We also discuss applications of the proposedmethod on a number of interesting learning problems in multi-agent systems.

Finite-Time Convergence Rates of Decentralized Stochastic Approximation with Applications in Multi-Agent and Multi-Task Learning