On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers

TL;DR

This paper provides a theoretical analysis of dynamic Hamiltonian Monte Carlo methods, including NUTS, establishing their invariance, irreducibility, aperiodicity, and geometric ergodicity, and improves convergence results for HMC.

Contribution

It introduces a general framework for dynamic HMC, shows NUTS is a special case, and proves new ergodicity and convergence properties under broad conditions.

Findings

NUTS is invariant, irreducible, and aperiodic.

NUTS is geometrically ergodic under certain conditions.

HMC is ergodic without boundedness constraints for Gaussian-like targets.

Abstract

There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution.