Randomized Hamiltonian Monte Carlo

TL;DR

This paper introduces a randomized Hamiltonian Monte Carlo (RHMC) method where durations are random, demonstrating improved ergodicity and efficiency, especially in Gaussian cases, with extensions to larger time steps.

Contribution

The paper proposes and analyzes a novel randomized HMC algorithm with exponential durations, proving its geometric ergodicity and regular sampling efficiency in Gaussian settings.

Findings

RHMC is geometrically ergodic under standard conditions.

Sampling efficiency of RHMC is regular in Gaussian cases.

Numerical experiments confirm efficiency in non-Gaussian distributions.

Abstract

Tuning the durations of the Hamiltonian flow in Hamiltonian Monte Carlo (also called Hybrid Monte Carlo) (HMC) involves a tradeoff between computational cost and sampling quality, which is typically challenging to resolve in a satisfactory way. In this article we present and analyze a randomized HMC method (RHMC), in which these durations are i.i.d. exponential random variables whose mean is a free parameter. We focus on the small time step size limit, where the algorithm is rejection-free and the computational cost is proportional to the mean duration. In this limit, we prove that RHMC is geometrically ergodic under the same conditions that imply geometric ergodicity of the solution to underdamped Langevin equations. Moreover, in the context of a multi-dimensional Gaussian distribution, we prove that the sampling efficiency of RHMC, unlike that of constant duration HMC, behaves in a regular way. This regularity is also verified numerically in non-Gaussian target distributions. Finally we suggest variants of RHMC for which the time step size is not required to be small.