On the convergence of the Metropolis algorithm with fixed-order updates for multivariate binary probability distributions

TL;DR

This paper analyzes the convergence issues of the Metropolis algorithm with fixed-order updates for multivariate binary distributions and proposes a modified operator that guarantees convergence.

Contribution

It introduces a modified Metropolis transition operator that ensures irreducibility and convergence in fixed-order update settings for multivariate binary distributions.

Findings

The standard Metropolis algorithm may not converge due to non-irreducibility.

The modified operator guarantees convergence in fixed-order update scenarios.

Experimental results show similar or improved convergence performance with the modified operator.

Abstract

The Metropolis algorithm is arguably the most fundamental Markov chain Monte Carlo (MCMC) method. But the algorithm is not guaranteed to converge to the desired distribution in the case of multivariate binary distributions (e.g., Ising models or stochastic neural networks such as Boltzmann machines) if the variables (sites or neurons) are updated in a fixed order, a setting commonly used in practice. The reason is that the corresponding Markov chain may not be irreducible. We propose a modified Metropolis transition operator that behaves almost always identically to the standard Metropolis operator and prove that it ensures irreducibility and convergence to the limiting distribution in the multivariate binary case with fixed-order updates. The result provides an explanation for the behaviour of Metropolis MCMC in that setting and closes a long-standing theoretical gap. We experimentally studied the standard and modified Metropolis operator for models were they actually behave differently. If the standard algorithm also converges, the modified operator exhibits similar (if not better) performance in terms of convergence speed.