High-Order Langevin Monte Carlo Algorithms

TL;DR

This paper introduces high-order Langevin Monte Carlo algorithms that leverage advanced discretization techniques to improve sampling efficiency and convergence guarantees for large-scale problems with smooth, log-concave densities.

Contribution

It develops $P$-th order Langevin Monte Carlo algorithms for any $P\u2265 3$, providing enhanced convergence rates and better dependence on dimension and accuracy compared to existing methods.

Findings

Convergence guarantees in Wasserstein distance for the proposed algorithms.

Improved mixing time scaling with dimension and accuracy as $P$ increases.

Numerical experiments demonstrating efficiency gains.

Abstract

Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling problems that often arise in data science. We propose Monte Carlo algorithms based on the discretizations of $P$-th order Langevin dynamics for any $P\geq 3$. Our design of $P$-th order Langevin Monte Carlo (LMC) algorithms is by combining splitting and accurate integration methods. We obtain Wasserstein convergence guarantees for sampling from distributions with log-concave and smooth densities. Specifically, the mixing time of the $P$-th order LMC algorithm scales as $O\left(d^{\frac{1}{R}}/\epsilon^{\frac{1}{2R}}\right)$ for $R=4\cdot 1_{\{ P=3\}}+ (2P-1)\cdot 1_{\{ P\geq 4\}}$, which has a better dependence on the dimension $d$ and the accuracy level $\epsilon$ as $P$ grows. Numerical experiments illustrate the efficiency of our proposed algorithms.