On the convergence of Hamiltonian Monte Carlo

TL;DR

This paper analyzes the convergence properties of Hamiltonian Monte Carlo, establishing conditions for irreducibility, recurrence, Harris recurrence, and geometric ergodicity of the algorithm.

Contribution

It provides new theoretical results on the convergence and ergodicity of HMC under various conditions on the potential function.

Findings

HMC is irreducible and recurrent under mild conditions.

HMC is Harris recurrent under more stringent conditions.

Conditions for geometric ergodicity of HMC are provided.

Abstract

This paper discusses the irreducibility and geometric ergodicity of the Hamiltonian Monte Carlo (HMC) algorithm. We consider cases where the number of steps of the symplectic integrator is either fixed or random. Under mild conditions on the potential $\F$ associated with target distribution $\pi$, we first show that the Markov kernel associated to the HMC algorithm is irreducible and recurrent. Under more stringent conditions, we then establish that the Markov kernel is Harris recurrent. Finally, we provide verifiable conditions on $\F$ under which the HMC sampler is geometrically ergodic.