The birthday problem and Markov chain Monte Carlo

TL;DR

This paper presents an approximation algorithm for sampling from the stationary distribution of a Markov chain, specifically for random walks on regular graphs, achieving near-optimal expected running time.

Contribution

It introduces a new algorithm that efficiently samples from the stationary distribution with expected time close to the theoretical lower bound.

Findings

Expected running time is O^*(√n * L^2 * mixing time)

Algorithm nearly matches the lower bound of √n for worst-case scenarios

Provides insights into the complexity of sampling from Markov chain stationary distributions

Abstract

We study the problem of generating a sample from the stationary distribution of a Markov chain, given a method to simulate the chain. We give an approximation algorithm for the case of a random walk on a regular graph with n vertices that runs in expected time O^*(\sqrt{n} x L^2-mixing time). This is close to the best possible, since \sqrt{n} is a lower bound on the worst-case expected running time of any algorithm.