On Irreversible Metropolis Sampling Related to Langevin Dynamics

TL;DR

This paper introduces Hamiltonian assisted Metropolis sampling (HAMS), an irreversible Markov chain Monte Carlo method that generalizes Langevin dynamics, with theoretical analysis and empirical evidence showing its superior performance in high-dimensional sampling tasks.

Contribution

The paper proposes HAMS, a novel irreversible sampling algorithm that unifies and generalizes Langevin-based methods, with theoretical analysis and practical tuning guidelines.

Findings

HAMS converges to Langevin dynamics as step size approaches zero.

Theoretical analysis provides optimal tuning parameters for HAMS.

Numerical experiments show HAMS outperforms existing Langevin-based algorithms.

Abstract

There has been considerable interest in designing Markov chain Monte Carlo algorithms by exploiting numerical methods for Langevin dynamics, which includes Hamiltonian dynamics as a deterministic case. A prominent approach is Hamiltonian Monte Carlo (HMC), where a leapfrog discretization of Hamiltonian dynamics is employed. We investigate a recently proposed class of irreversible sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS), which uses an augmented target density similarly as in HMC, but involves a flexible proposal scheme and a carefully formulated acceptance-rejection scheme to achieve generalized reversibility. We show that as the step size tends to 0, the HAMS proposal satisfies a class of stochastic differential equations including Langevin dynamics as a special case. We provide theoretical results for HAMS under the univariate Gaussian setting, including the stationary variance, the expected acceptance rate, and the spectral radius. From these results, we derive default choices of tuning parameters for HAMS, such that only the step size needs to be tuned in applications. Various relatively recent algorithms for Langevin dynamics are also shown to fall in the class of HAMS proposals up to negligible differences. Our numerical experiments on sampling high-dimensional latent variables confirm that the HAMS algorithms consistently achieve superior performance, compared with several Metropolis-adjusted algorithms based on popular integrators of Langevin dynamics.