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

TL;DR

This paper analyzes the finite-time convergence rates of decentralized stochastic approximation algorithms in multi-agent systems, accounting for Markovian data sampling and introducing a novel Lyapunov function for stability analysis.

Contribution

It provides the first finite-time convergence analysis for decentralized stochastic approximation with Markovian data, using a Razumikhin-Lyapunov approach.

Findings

Convergence rate matches that of independent samples up to a log factor.

The analysis applies to various multi-agent learning problems.

The method handles biased and unbounded iterates due to Markov sampling.

Abstract

We study a decentralized variant of stochastic approximation, a data-driven approach for finding the root of an operator under noisy measurements. A network of agents, each with its own operator and data observations, cooperatively find the fixed point of the aggregate operator over a decentralized communication graph. Our main contribution is to provide a finite-time analysis of this decentralized stochastic approximation method when the data observed at each agent are sampled from a Markov process; this lack of independence makes the iterates biased and (potentially) unbounded. Under fairly standard assumptions, we show that the convergence rate of the proposed method is essentially the same as if the samples were independent, differing only 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-Lyapunov function, motivated by the one used in analyzing the stability of delayed ordinary differential equations. We also discuss applications of the proposed method on a number of interesting learning problems in multi-agent systems.