On the convergence of the Metropolis algorithm with fixed-order updates for multivariate binary probability distributions
This addresses a theoretical gap for practitioners using MCMC methods in fields like statistical physics and machine learning, though it is incremental as it modifies an existing algorithm.
The paper tackled the problem of the Metropolis algorithm potentially failing to converge for multivariate binary distributions with fixed-order updates by proposing a modified operator that ensures irreducibility and convergence, and experimentally showed it performs similarly or better in convergence speed when the standard algorithm works.
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.