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

TL;DR

This paper analyzes the efficiency of a high-dimensional underdamped Langevin MCMC sampler using a standard splitting scheme, providing dimension-free convergence results and explicit efficiency bounds for both unadjusted and Metropolis-adjusted chains.

Contribution

It establishes the first dimension-free contraction and convergence rates for the discrete underdamped Langevin chain with a classical integrator, including explicit efficiency bounds.

Findings

Dimension-free Wasserstein contraction proven.

Efficiency bounds scale with dimension and accuracy, e.g., rac{rac{rac{rac{d}{\u007varepsilon}}}

Results align with known bounds for other kinetic Langevin schemes.

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. \nv{In particular, for the unadjusted chain,} in terms of the dimension $d$ and the desired accuracy $\varepsilon$, the Wasserstein efficiency bounds are of order $\sqrt d / \varepsilon$ in the general case, $\sqrt{d/\varepsilon}$ if the Hessian of the potential is Lipschitz, and $d^{1/4}/\sqrt\varepsilon$ in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.