On the Geometric Ergodicity of Hamiltonian Monte Carlo

TL;DR

This paper analyzes the conditions under which Hamiltonian Monte Carlo achieves geometric ergodicity, providing theoretical insights into how the target distribution's properties influence convergence rates.

Contribution

It establishes general conditions for geometric ergodicity of Hamiltonian Monte Carlo, including cases with position-dependent integration times and broad tail behaviors.

Findings

Conditions for geometric ergodicity depend on the gradient pointing towards the center and its growth rate.

Allowing variable integration times broadens the class of distributions with geometric ergodicity.

Guidelines for choosing integration times in practice are derived.

Abstract

We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both position-independent and position-dependent integration times. In the former case we find that the conditions for geometric ergodicity are essentially a gradient of the log-density which asymptotically points towards the centre of the space and grows no faster than linearly. In an idealised scenario in which the integration time is allowed to change in different regions of the space, we show that geometric ergodicity can be recovered for a much broader class of tail behaviours, leading to some guidelines for the choice of this free parameter in practice.