The Computational Complexity of Estimating Convergence Time

TL;DR

This paper investigates the computational difficulty of estimating convergence times in Markov Chain Monte Carlo algorithms, showing that the problem is computationally hard in various formal complexity classes.

Contribution

It proves that determining convergence time in MCMC is computationally hard, with results placing the problem in SZK-hard, coNP-hard, and PSPACE-complete classes.

Findings

Hardness results for convergence testing in various complexity classes

Shows the problem is SZK-hard, coNP-hard, and PSPACE-complete

Highlights the computational limits of convergence diagnostics

Abstract

An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. Even in cases where the convergence time is known to be polynomial, the theoretical bounds are often too crude to be practical. Thus, practitioners like to carry out some form of statistical analysis in order to assess convergence. This has led to the development of a number of methods known as convergence diagnostics which attempt to diagnose whether the Markov chain is far from stationarity. We study the problem of testing convergence in the following settings and prove that the problem is hard in a computational sense: Given a Markov chain that mixes rapidly, it is hard for Statistical Zero Knowledge (SZK-hard) to distinguish whether starting from a given state, the chain is close to stationarity by time t or far from stationarity at time ct for a constant c. We show the problem is in AM intersect coAM. Second, given a Markov chain that mixes rapidly it is coNP-hard to distinguish whether it is close to stationarity by time t or far from stationarity at time ct for a constant c. The problem is in coAM. Finally, it is PSPACE-complete to distinguish whether the Markov chain is close to stationarity by time t or far from being mixed at time ct for c at least 1.