A Simple Parallel and Distributed Sampling Technique: Local Glauber Dynamics

TL;DR

This paper introduces a simple, parallelizable local sampling algorithm based on Glauber dynamics that efficiently samples from Gibbs distributions under Dobrushin's condition, significantly improving convergence speed over previous methods.

Contribution

It proposes a new local update rule for Glauber dynamics enabling fast parallel sampling with $O( ext{log} n)$ mixing time under Dobrushin's condition, nearly matching sequential thresholds.

Findings

Achieves $O( ext{log} n)$ mixing time under Dobrushin's condition.

Improves over LubyGlauber and LocalMetropolis algorithms in convergence speed.

Samples proper colorings with $eta riangle$ colors in $O( ext{log} n)$ rounds.

Abstract

\emph{Sampling} constitutes an important tool in a variety of areas: from machine learning and combinatorial optimization to computational physics and biology. A central class of sampling algorithms is the \emph{Markov Chain Monte Carlo} method, based on the construction of a Markov chain with the desired sampling distribution as its stationary distribution. Many of the traditional Markov chains, such as the \emph{Glauber dynamics}, do not scale well with increasing dimension. To address this shortcoming, we propose a simple local update rule based on the Glauber dynamics that leads to efficient parallel and distributed algorithms for sampling from Gibbs distributions. Concretely, we present a Markov chain that mixes in $O(\log n)$ rounds when Dobrushin's condition for the Gibbs distribution is satisfied. This improves over the \emph{LubyGlauber} algorithm by Feng, Sun, and Yin [PODC'17], which needs $O(\Delta \log n)$ rounds, and their \emph{LocalMetropolis} algorithm, which converges in $O(\log n)$ rounds but requires a considerably stronger mixing condition. Here, $n$ denotes the number of nodes in the graphical model inducing the Gibbs distribution, and $\Delta$ its maximum degree. In particular, our method can sample a uniform proper coloring with $\alpha \Delta$ colors in $O(\log n)$ rounds for any $\alpha>2$, which almost matches the threshold of the sequential Glauber dynamics and improves on the $\alpha>2 +\sqrt{2}$ threshold of Feng et al.