High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

Abstract

We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of d-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most > 0 in Wasserstein distance from the target distribution in O(d^1/4 ^1/2 ) steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with -th order smoothness, we prove that the mixing time scales as O (d^1/4^1/2 + d^1/2^1/( - 1) ).

Tasks

Reproductions