A General Limitation on Monte Carlo Algorithms of Metropolis Type

TL;DR

This paper establishes a fundamental lower bound on the autocorrelation time for any Metropolis-type Monte Carlo algorithm, linking it to the specific heat and the distance between target and proposal distributions, especially impacting non-local algorithms.

Contribution

It proves a universal lower bound on autocorrelation time for Metropolis algorithms based on distributional distance, applicable to local and non-local moves.

Findings

Lower bound on autocorrelation time proportional to specific heat.

Bound applies regardless of move locality.

Non-local algorithms are significantly affected by this limitation.

Abstract

We prove that for any Monte Carlo algorithm of Metropolis type, the autocorrelation time of a suitable ``energy''-like observable is bounded below by a multiple of the corresponding ``specific heat''. This bound does not depend on whether the proposed moves are local or non-local; it depends only on the distance between the desired probability distribution $\pi$ and the probability distribution $\pi^{(0)}$ for which the proposal matrix satisfies detailed balance. We show, with several examples, that this result is particularly powerful when applied to non-local algorithms.