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

arXiv: https://arxiv.org/abs/2310.07399 · HTML: https://mlsanity.com/paper/2310.07399/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 11 Oct 2023.

## Builds on (most cited first, 11 of 11)

- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Geometric integrators and the Hamiltonian Monte Carlo method](https://mlsanity.com/paper/1711.05337/) (2017)
- [On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method](https://mlsanity.com/paper/2011.03176/) (2020)
- [Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.00887/) (2021)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [Connecting the Dots: Numerical Randomized Hamiltonian Monte Carlo with State-Dependent Event Rates](https://mlsanity.com/paper/2005.01285/) (2020)
- [Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration](https://mlsanity.com/paper/2211.11003/) (2022)

## Built on by (most cited first, 1 of 1)

- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)

## Abstract (arXiv)

We introduce 5/2- and 7/2-order L2-accurate randomized Runge-Kutta-Nystr"om 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 L2-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.