On the Dissipation of Ideal Hamiltonian Monte Carlo Sampler

TL;DR

This paper investigates how variable integration time and partial velocity refreshment can reduce dissipation in Ideal Hamiltonian Monte Carlo, improving efficiency on quadratic potentials and exploring randomized integrators for better simulation under certain conditions.

Contribution

It introduces novel methods of variable integration time and partial velocity refreshment to enhance HMC efficiency, with theoretical analysis on quadratic potentials and higher order regularity.

Findings

Efficiency improved by a √κ factor in Wasserstein-2 distance.

Randomized integrators benefit Hamiltonian dynamics simulation.

Partial velocity refreshment reduces dissipative behavior.

Abstract

We report on what seems to be an intriguing connection between variable integration time and partial velocity refreshment of Ideal Hamiltonian Monte Carlo samplers, both of which can be used for reducing the dissipative behavior of the dynamics. More concretely, we show that on quadratic potentials, efficiency can be improved through these means by a $\sqrt{\kappa}$ factor in Wasserstein-2 distance, compared to classical constant integration time, fully refreshed HMC. We additionally explore the benefit of randomized integrators for simulating the Hamiltonian dynamics under higher order regularity conditions.