An entropic approach for Hamiltonian Monte Carlo: the idealized case

TL;DR

This paper provides an entropic analysis of an idealized Hamiltonian Monte Carlo method, establishing convergence and regularization properties, with implications for simulated annealing and connections to Langevin diffusion.

Contribution

It introduces a novel entropic framework for analyzing Hamiltonian Monte Carlo, linking discrete and continuous dynamics and exploring dependence on the log-Sobolev constant.

Findings

Quantitative long-time entropic convergence established.

Short-time regularization properties demonstrated.

Results connect discrete HMC to continuous Langevin diffusion.

Abstract

Quantitative long-time entropic convergence and short-time regularization are established for an idealized Hamiltonian Monte Carlo chain which alternatively follows an Hamiltonian dynamics for a fixed time and then partially or totally refreshes its velocity with an auto-regressive Gaussian step. These results, in discrete time, are the analogous of similar results for the continuous-time kinetic Langevin diffusion, and the latter can be obtained from our bounds in a suitable limit regime. The dependency in the log-Sobolev constant of the target measure is sharp and is illustrated on a mean-field case and on a low-temperature regime, with an application to the simulated annealing algorithm. The practical unadjusted algorithm is briefly discussed.