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 smooth, log-concave 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 $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\left(\frac{d^{1/4}}{ \varepsilon^{1/2}} \right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $\alpha$-th order smoothness, we prove that the mixing time scales as $O \left( \frac{d^{1/4}}{\varepsilon^{1/2}} + \frac{d^{1/2}}{ \varepsilon^{1/(\alpha - 1)}} \right)$.