Zigzag path connects two Monte Carlo samplers: Hamiltonian counterpart to a piecewise deterministic Markov process

TL;DR

This paper establishes a theoretical connection between zigzag samplers and a Hamiltonian Monte Carlo variant with Laplace momentum, showing how the Hamiltonian version can outperform the Markovian zigzag in exploring complex, high-dimensional distributions.

Contribution

It introduces a Hamiltonian zigzag sampler, linking it to the Markovian zigzag process, and proves its convergence as momentum refreshes become frequent, highlighting potential advantages in sampling efficiency.

Findings

Hamiltonian zigzag converges to Markovian zigzag with frequent momentum refreshes.

Hamiltonian zigzag better explores highly correlated parameters.

Performance demonstrated on high-dimensional Bayesian models.

Abstract

Zigzag and other piecewise deterministic Markov process samplers have attracted significant interest for their non-reversibility and other appealing properties for Bayesian posterior computation. Hamiltonian Monte Carlo is another state-of-the-art sampler, exploiting fictitious momentum to guide Markov chains through complex target distributions. We establish an important connection between the zigzag sampler and a variant of Hamiltonian Monte Carlo based on Laplace-distributed momentum. The position and velocity component of the corresponding Hamiltonian dynamics travels along a zigzag path paralleling the Markovian zigzag process; however, the dynamics is non-Markovian in this position-velocity space as the momentum component encodes non-immediate pasts. This information is partially lost during a momentum refreshment step, in which we preserve its direction but re-sample magnitude. In the limit of increasingly frequent momentum refreshments, we prove that Hamiltonian zigzag converges strongly to its Markovian counterpart. This theoretical insight suggests that, when retaining full momentum information, Hamiltonian zigzag can better explore target distributions with highly correlated parameters by suppressing the diffusive behavior of Markovian zigzag. We corroborate this intuition by comparing performance of the two zigzag cousins on high-dimensional truncated multivariate Gaussians, including a 11,235-dimensional target arising from a Bayesian phylogenetic multivariate probit modeling of HIV virus data.