High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm
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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 ε > 0 in Wasserstein distance from the target distribution in O(d^1/3/ε^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/ε^2/3 + d^1/2/ε^1/(α - 1)).
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