Distributed metropolis sampler with optimal parallelism

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 faithfully simulates sequential single-site Metropolis chains without introducing any bias. When a natural Lipschitz condition for the the Metropolis filters is satisfied, the algorithm can faithfully 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 Ω(n log n) steps, this achieves an optimal linear speedup. For several well-studied graphical models, including proper graph coloring, hardcore model, and Ising model, our condition for linear speedup is much weaker than the uniqueness conditions for the respective models. The novel idea in our algorithm is to resolve updates in advance: the local Metropolis filters can be executed correctly before the full information about neighboring spins is available. This achieves optimal parallelism without introducing any bias.