Lagrangian Dynamical Monte Carlo

TL;DR

This paper introduces an explicit geometric integrator for Riemannian Manifold Hamiltonian Monte Carlo, transforming it into a Lagrangian framework to enhance computational efficiency and reduce costly implicit calculations.

Contribution

The authors propose a novel explicit integrator that replaces momentum with velocity, converting Riemannian Hamiltonian dynamics into Lagrangian dynamics for improved efficiency.

Findings

The new method reduces computational overhead compared to traditional RMHMC.

Experimental results demonstrate improved overall efficiency of the proposed approach.

Code and data are publicly available for replication.

Abstract

Hamiltonian Monte Carlo (HMC) improves the computational efficiency of the Metropolis algorithm by reducing its random walk behavior. Riemannian Manifold HMC (RMHMC) further improves HMC's performance by exploiting the geometric properties of the parameter space. However, the geometric integrator used for RMHMC involves implicit equations that require costly numerical analysis (e.g., fixed-point iteration). In some cases, the computational overhead for solving implicit equations undermines RMHMC's benefits. To avoid this problem, we propose an explicit geometric integrator that replaces the momentum variable in RMHMC by velocity. We show that the resulting transformation is equivalent to transforming Riemannian Hamilton dynamics to Lagrangian dynamics. Experimental results show that our method improves RMHMC's overall computational efficiency. All computer programs and data sets are available online (http://www.ics.uci.edu/~babaks/Site/Codes.html) in order to allow replications of the results reported in this paper.