On the fundamental limitations of multiproposal Markov chain Monte Carlo algorithms

TL;DR

This paper establishes fundamental theoretical limits on the efficiency gains of multiproposal MCMC algorithms, showing they cannot significantly outperform single proposal methods, especially for log-concave distributions, regardless of parallelization.

Contribution

It proves that multiproposal MCMC algorithms are inherently limited in speed-up potential, with at most linear or logarithmic improvements, depending on the distribution class.

Findings

Multiproposal schemes cannot surpass a factor of K speed-up over single proposals.

For log-concave distributions, the maximum speed-up is logarithmic in K.

Numerical simulations confirm the theoretical limitations.

Abstract

We study multiproposal Markov chain Monte Carlo algorithms, such as Multiple-try or generalised Metropolis-Hastings schemes, which have recently received renewed attention due to their amenability to parallel computing. First, we prove that no multiproposal scheme can speed-up convergence relative to the corresponding single proposal scheme by more than a factor of $K$, where $K$ denotes the number of proposals at each iteration. This result applies to arbitrary target distributions and it implies that serial multiproposal implementations are always less efficient than single proposal ones. Secondly, we consider log-concave distributions over Euclidean spaces, proving that, in this case, the speed-up is at most logarithmic in $K$, which implies that even parallel multiproposal implementations are fundamentally limited in the computational gain they can offer. Crucially, our results apply to arbitrary multiproposal schemes and purely rely on the two-step structure of the associated kernels (i.e. first generate $K$ candidate points, then select one among those). Our theoretical findings are validated through numerical simulations.