A third-order Langevin MCMC algorithm is proven to sample from smooth log-concave distributions in O(d^(1/4)/epsilon^(1/2)) iterations for generalized linear model potentials, improving on the earlier d^(1/3) barrier.
Coupling and Convergence for Hamiltonian Monte Carlo
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abstract
Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal target distributions are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given error are a direct consequence of contractivity. These bounds show that HMC can overcome diffusive behaviour if the duration of the Hamiltonian dynamics is adjusted appropriately.
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High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm
A third-order Langevin MCMC algorithm is proven to sample from smooth log-concave distributions in O(d^(1/4)/epsilon^(1/2)) iterations for generalized linear model potentials, improving on the earlier d^(1/3) barrier.