Couplings for Multinomial Hamiltonian Monte Carlo

TL;DR

This paper develops couplings for multinomial Hamiltonian Monte Carlo, improving meeting times and robustness, enabling more practical and efficient unbiased Monte Carlo estimation in high-dimensional Bayesian inference.

Contribution

It introduces couplings for multinomial HMC based on optimal transport, providing theoretical bounds and demonstrating improved performance over existing methods.

Findings

Smaller meeting times for coupled multinomial HMC.

More robust to step size and trajectory length choices.

Enables wider practical use of coupled HMC methods.

Abstract

Hamiltonian Monte Carlo (HMC) is a popular sampling method in Bayesian inference. Recently, Heng & Jacob (2019) studied Metropolis HMC with couplings for unbiased Monte Carlo estimation, establishing a generic parallelizable scheme for HMC. However, in practice a different HMC method, multinomial HMC, is considered as the go-to method, e.g. as part of the no-U-turn sampler. In multinomial HMC, proposed states are not limited to end-points as in Metropolis HMC; instead points along the entire trajectory can be proposed. In this paper, we establish couplings for multinomial HMC, based on optimal transport for multinomial sampling in its transition. We prove an upper bound for the meeting time - the time it takes for the coupled chains to meet - based on the notion of local contractivity. We evaluate our methods using three targets: 1,000 dimensional Gaussians, logistic regression and log-Gaussian Cox point processes. Compared to Heng & Jacob (2019), coupled multinomial HMC generally attains a smaller meeting time, and is more robust to choices of step sizes and trajectory lengths, which allows re-use of existing adaptation methods for HMC. These improvements together paves the way for a wider and more practical use of coupled HMC methods.