Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo

Abstract

We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nyström 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.

References in corpus (12)

• Canonical sampling through velocity-rescaling
• Geometric integrators and the Hamiltonian Monte Carlo method
• Coupling and Convergence for Hamiltonian Monte Carlo
• Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs
• An efficient sampling algorithm for Variational Monte Carlo
• A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients
• Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients
• Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo
• Complexity of randomized algorithms for underdamped Langevin dynamics
• Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models
• Couplings for Andersen Dynamics
• Randomized Runge-Kutta method -- stability and convergence under inexact information