{"ID":2879020,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.17545","arxiv_id":"2508.17545","title":"High-Order Langevin Monte Carlo Algorithms","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 $O\\left(d^{\\frac{1}{R}}/ε^{\\frac{1}{2R}}\\right)$ 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 $ε$ as $P$ grows. Numerical experiments illustrate the efficiency of our proposed algorithms.","short_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...","url_abs":"https://arxiv.org/abs/2508.17545","url_pdf":"https://arxiv.org/pdf/2508.17545v1","authors":"[\"Thanh Dang\",\"Mert Gurbuzbalaban\",\"Mohammad Rafiqul Islam\",\"Nian Yao\",\"Lingjiong Zhu\"]","published":"2025-08-24T22:37:44Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.PR\"]","methods":"[]","has_code":false}