Tuning of MCMC with Langevin, Hamiltonian, and other stochastic autoregressive proposals

TL;DR

This paper provides a unified theoretical analysis of stochastic autoregressive proposals in Metropolis-Hastings MCMC, offering guidelines for optimal tuning of Langevin and Hamiltonian dynamics-based algorithms in high dimensions.

Contribution

It unifies and extends theoretical analysis of Langevin and Hamiltonian proposals, offering new tuning guidelines for high-dimensional MCMC.

Findings

Optimal Langevin step count is at least three.

Guidelines for choosing the number of Hamiltonian steps.

Recommendations for selecting the mass matrix in HMC.

Abstract

Proposals for Metropolis-Hastings MCMC derived by discretizing Langevin diffusion or Hamiltonian dynamics are examples of stochastic autoregressive proposals that form a natural wider class of proposals with equivalent computability. We analyze Metropolis-Hastings MCMC with stochastic autoregressive proposals applied to target distributions that are absolutely continuous with respect to some Gaussian distribution to derive expressions for expected acceptance probability and expected jump size, as well as measures of computational cost, in the limit of high dimension. Thus, we are able to unify existing analyzes for these classes of proposals, and to extend the theoretical results that provide useful guidelines for tuning the proposals for optimal computational efficiency. For the simplified Langevin algorithm we find that it is optimal to take at least three steps of the proposal before the Metropolis-Hastings accept-reject step, and for Hamiltonian/hybrid Monte Carlo we provide new guidelines for the optimal number of integration steps and criteria for choosing the optimal mass matrix.