Georgia Institute of Technology College of Sciences

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 \varepsilon in Wasserstein distance from the target distribution in O(d^{1/3}/ \varepsilon^{2/3}) steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with α-th order smoothness, we prove that the mixing time scales as O (d^{1/3} / \varepsilon^{2/3} + d^{1/2} / \varepsilon^{1 / (\alpha - 1)} ). Our high-order Langevin diffusion reduces the problem of log-concave sampling to numerical integration along a fixed deterministic path, which makes it possible for further improvements in high-dimensional MCMC problems. Joint work with Yi-An Ma, Martin J, Wainwright, Peter L. Bartlett and Michael I. Jordan.