Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlo

TL;DR

This paper explores the geometric ergodicity of modified Riemannian manifold Hamiltonian and Lagrangian Monte Carlo methods, proposing a mixture approach with MALA to enhance theoretical guarantees and evaluating performance on Bayesian inference tasks.

Contribution

It introduces a novel mixture kernel combining LMC, RMHMC, and MALA to establish inherited geometric ergodicity and assesses its effectiveness on benchmark problems.

Findings

The mixture kernel exhibits geometric ergodicity under certain conditions.

Modified kernels perform competitively on Bayesian inference benchmarks.

The approach provides a theoretical framework for ergodicity in Riemannian Monte Carlo methods.

Abstract

Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte Carlo (LMC) have emerged as powerful methods of Bayesian inference. Unlike Euclidean Hamiltonian Monte Carlo (EHMC) and the Metropolis-adjusted Langevin algorithm (MALA), the geometric ergodicity of these Riemannian algorithms has not been extensively studied. On the other hand, the manifold Metropolis-adjusted Langevin algorithm (MMALA) has recently been shown to exhibit geometric ergodicity under certain conditions. This work investigates the mixture of the LMC and RMHMC transition kernels with MMALA in order to equip the resulting method with an "inherited" geometric ergodicity theory. We motivate this mixture kernel based on an analogy between single-step HMC and MALA. We then proceed to evaluate the original and modified transition kernels on several benchmark Bayesian inference tasks.