Hamiltonian Monte Carlo (HMC) and its variants are among the most widely used algorithms for sampling from probability distributions. Despite their popularity, quantitative convergence guarantees for unadjusted HMC remain limited, especially in divergences that provide strong relative-density control such as KL divergence and Rényi divergence. In this talk, we establish regularization properties for unadjusted HMC via one-shot couplings, which enable Wasserstein convergence guarantees to be upgraded to guarantees in KL and Rényi divergence. As a consequence, we obtain finite-time KL and Rényi guarantees for unadjusted HMC with the velocity Verlet integrator, including mixing-time bounds for the unadjusted Markov chain and quantitative bounds on its asymptotic bias. Our results also extend to other discretization schemes and can be combined with Metropolis-adjusted HMC to remove asymptotic bias. Based on joint work with Nawaf Bou-Rabee and Andre Wibisono [link: https://arxiv.org/abs/2601.09019].
