High-Order Langevin Monte Carlo Algorithms

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≥ 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 Oleft(d^frac1R/ε^frac12Rright) for R=4· 1_ P=3+ (2P-1)· 1_ P≥ 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.