High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion.

Abstract

The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proven for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adjusted and unadjusted chains. In particular, for the unadjusted chain, in terms of the dimension d and the desired accuracy ε, the Wasserstein efficiency bounds are of order $\sqrt{d}∕\mathit{\epsilon }$ in the general case, $\sqrt{d∕\mathit{\epsilon }}$ if the Hessian of the potential is Lipschitz, and $d^{1∕4}∕\sqrt{\mathit{\epsilon }}$ in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.