Randomized Runge-Kutta-Nystr\"om Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo

TL;DR

This paper develops high-order randomized Runge-Kutta-Nyström methods for unadjusted Hamiltonian and kinetic Langevin Monte Carlo, providing theoretical accuracy bounds and demonstrating improved efficiency in high-dimensional sampling tasks.

Contribution

Introduction of 5/2- and 7/2-order $L^2$-accurate randomized Runge-Kutta-Nyström methods tailored for unadjusted Hamiltonian-based MCMC algorithms, with proven accuracy bounds.

Findings

Methods achieve superior efficiency in high-dimensional sampling.

Established quantitative $L^2$-accuracy bounds under Lipschitz conditions.

Numerical experiments confirm improved performance over existing samplers.

Abstract

We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nystr\"{o}m methods, tailored for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers, such as unadjusted Hamiltonian Monte Carlo and unadjusted kinetic Langevin Monte Carlo. We establish quantitative $5/2$-order $L^2$-accuracy upper bounds under gradient and Hessian Lipschitz assumptions on the potential energy function. The numerical experiments demonstrate the superior efficiency of the proposed unadjusted samplers on a variety of well-behaved, high-dimensional target distributions.