Distributed Metropolis Sampler with Optimal Parallelism

TL;DR

This paper presents a distributed Metropolis-Hastings algorithm that achieves optimal parallelism and linear speedup for sampling from graphical models, without bias, under weaker conditions than traditional mixing criteria.

Contribution

It introduces a novel asynchronous distributed algorithm for Metropolis chains that ensures unbiased sampling and optimal parallelism under weaker conditions than existing methods.

Findings

Achieves $O(N/n+ ext{log} n)$ rounds for simulating N-step chains.

Ensures unbiased sampling in fully asynchronous message-passing models.

Provides linear speedup for important graphical models like Ising and graph coloring.

Abstract

The Metropolis-Hastings algorithm is a fundamental Markov chain Monte Carlo (MCMC) method for sampling and inference. With the advent of Big Data, distributed and parallel variants of MCMC methods are attracting increased attention. In this paper, we give a distributed algorithm that can correctly simulate sequential single-site Metropolis chains without any bias in a fully asynchronous message-passing model. Furthermore, if a natural Lipschitz condition is satisfied by the Metropolis filters, our algorithm can simulate $N$-step Metropolis chains within $O(N/n+\log n)$ rounds of asynchronous communications, where $n$ is the number of variables. For sequential single-site dynamics, whose mixing requires $\Omega(n\log n)$ steps, this achieves an optimal linear speedup. For several well-studied important graphical models, including proper graph coloring, hardcore model, and Ising model, our condition for linear speedup is weaker than the respective uniqueness (mixing) conditions. The novel idea in our algorithm is to resolve updates in advance: the local Metropolis filters can often be executed correctly before the full information about neighboring spins is available. This achieves optimal parallelism without introducing any bias.