# Tail-Sensitive KL and R'enyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings

arXiv: https://arxiv.org/abs/2601.09019 · HTML: https://mlsanity.com/paper/2601.09019/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jan 2026.

## Built on by (most cited first, 5 of 5)

- [Algorithmic warm starts for Hamiltonian Monte Carlo](https://mlsanity.com/paper/2603.22741/) (2026)
- [On couplings for kinetic Langevin diffusions](https://mlsanity.com/paper/2605.31088/) (2026)
- [Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time](https://mlsanity.com/paper/2606.17260/) (2026)
- [Accelerated Mixing Time of Randomized Hamiltonian Monte Carlo](https://mlsanity.com/paper/2607.12902/) (2026)
- [Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin](https://mlsanity.com/paper/2607.15208/) (2026)

## Abstract (arXiv)

Hamiltonian Monte Carlo (HMC) algorithms are among the most widely used sampling methods in high dimensional settings, yet their convergence properties are poorly understood in divergences that quantify relative density mismatch, such as Kullback-Leibler (KL) and R'enyi divergences. These divergences naturally govern acceptance probabilities and warm-start requirements for Metropolis-adjusted Markov chains. In this work, we develop a framework for upgrading Wasserstein convergence guarantees for unadjusted Hamiltonian Monte Carlo (uHMC) to guarantees in tail-sensitive KL and R'enyi divergences. Our approach is based on one-shot couplings, which we use to establish a regularization property of the uHMC transition kernel. This regularization allows Wasserstein-2 mixing-time and asymptotic bias bounds to be lifted to KL divergence, and analogous Orlicz-Wasserstein bounds to be lifted to R'enyi divergence, paralleling earlier work of Bou-Rabee and Eberle (2023) that upgrade Wasserstein-1 bounds to total variation distance via kernel smoothing. As a consequence, our results provide quantitative control of relative density mismatch, clarify the role of discretization bias in strong divergences, and yield principled guarantees relevant both for unadjusted sampling and for generating warm starts for Metropolis-adjusted Markov chains.