On the Mixing Time of Unadjusted Hamiltonian Monte Carlo in KL Divergence

We prove a mixing time bound for the Unadjusted Hamiltonian Monte Carlo (UHMC) algorithm in KL divergence. Our proof is based on using a one-shot coupling to show a one-step regularity result for the UHMC transition kernel, which consequently allows us to extend Wasserstein-$2$ mixing guarantees for UHMC to KL divergence assuming second and third order smoothness. Our results imply concrete mixing time guarantees in KL divergence in popular settings, e.g., strongly log-concave target distributions, and we also apply our results to study the contraction of mutual information along UHMC.