Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms

TL;DR

This paper introduces Marathon, an open source library for analyzing Markov Chain Monte Carlo algorithms, focusing on estimating mixing times and comparing bounds to actual performance in sampling problems.

Contribution

The paper presents Marathon, a software tool that evaluates the quality of mixing time bounds for MCMC algorithms, aiding in practical analysis of sampling efficiency.

Findings

Spectral bounds closely approximate actual mixing times.

Canonical path bounds are significantly overestimated, especially for larger instances.

Marathon enables detailed analysis of Markov chain properties.

Abstract

In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects. The running time of such an algorithm depends on the total mixing time of the underlying Markov chain and is unknown in general. For some Markov chains, upper bounds on this total mixing time exist but are too large to be applicable in practice. We try to answer the question, whether the total mixing time is close to its upper bounds, or if there is a significant gap between them. In doing so, we present the software library marathon which is designed to support the analysis of MCMC based sampling algorithms. The main application of this library is to compute properties of so-called state graphs which represent the structure of Markov chains. We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations. In a set of experiments, we compute the total mixing time and several of its bounds for a large number of input instances. We find that the upper bound gained by the famous canonical path method is several magnitudes larger than the total mixing time and deteriorates with growing input size. In contrast, the spectral bound is found to be a precise approximation of the total mixing time.