Hamiltonian Monte Carlo with Asymmetrical Momentum Distributions

TL;DR

This paper introduces a new convergence analysis for Hamiltonian Monte Carlo (HMC) with asymmetric momentum distributions, proposing a modified AD-HMC algorithm that achieves geometric convergence under weaker conditions.

Contribution

It presents a novel convergence framework for HMC with general auxiliary distributions and introduces AD-HMC, a modified algorithm that overcomes symmetry constraints.

Findings

AD-HMC achieves geometric convergence in Wasserstein distance.

The analysis extends to leapfrog integrator with Metropolis-Hastings rejection.

Numerical experiments show AD-HMC outperforms traditional HMC with Gaussian auxiliaries.

Abstract

Existing rigorous convergence guarantees for the Hamiltonian Monte Carlo (HMC) algorithm use Gaussian auxiliary momentum variables, which are crucially symmetrically distributed. We present a novel convergence analysis for HMC utilizing new dynamical and probabilistic arguments. The convergence is rigorously established under significantly weaker conditions, which among others allow for general auxiliary distributions. In our framework, we show that plain HMC with asymmetrical momentum distributions breaks a key self-adjointness requirement. We propose a modified version of HMC, that we call the Alternating Direction HMC (AD-HMC), which overcomes this difficulty. Sufficient conditions are established under which AD-HMC exhibits geometric convergence in Wasserstein distance. The geometric convergence analysis is extended to when the Hamiltonian motion is approximated by the leapfrog symplectic integrator, where an additional Metropolis-Hastings rejection step is required. Numerical experiments suggest that AD-HMC can generalize a popular dynamic auxiliary scheme to show improved performance over HMC with Gaussian auxiliaries.