Decentralized Langevin Dynamics

TL;DR

This paper analyzes decentralized Langevin dynamics for Bayesian inference, providing theoretical results on convergence rates and complexity in distributed settings with peer-to-peer communication.

Contribution

It offers the first theoretical analysis of convergence and complexity for decentralized Langevin MCMC algorithms in distributed data scenarios.

Findings

Time complexity to $psilon$-consensus in continuous dynamics

Convergence rate in $L^2$ norm for discrete Euler-Maruyama scheme

Wasserstein convergence rate to stationary distribution

Abstract

Langevin MCMC gradient optimization is a class of increasingly popular methods for estimating a posterior distribution. This paper addresses the algorithm as applied in a decentralized setting, wherein data is distributed across a network of agents which act to cooperatively solve the problem using peer-to-peer gossip communication. We show, theoretically, results in 1) the time-complexity to $\epsilon$-consensus for the continuous time stochastic differential equation, 2) convergence rate in $L^2$ norm to consensus for the discrete implementation as defined by the Euler-Maruyama discretization and 3) convergence rate in the Wasserstein metric to the optimal stationary distribution for the discretized dynamics.