Tail-Sensitive KL and Rényi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings

TL;DR

This work develops a framework for tail-sensitive convergence analysis of unadjusted Hamiltonian Monte Carlo (uHMC) by upgrading Wasserstein convergence to Kullback-Leibler (KL) and Rényi divergences through one-shot couplings. The core idea is a regularization property of the uHMC-v kernel via one-shot maps, together with cross-regularization against the exact Hamiltonian dynamics, enabling end-to-end KL and Rényi guarantees. The results include mixing-time and asymptotic-bias bounds for strongly log-concave targets using velocity Verlet and stratified integrators, with modularity that allows different kernels to drive Wasserstein contraction and regularization. Practically, these tail-sensitive guarantees support warm-start generation for Metropolis-adjusted chains, quantify discretization bias in strong divergences, and provide gradient-complexity estimates for achieving prescribed accuracy in KL and Rényi divergences, advancing theoretical understanding of HMC-based sampling in high dimensions.

Abstract

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 Rényi 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 Rényi 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 Rényi 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.

Figures (2)

• Figure 1: Distributions $\pi = \mathcal{N}(0,1)$ plotted as [] and $\mu = 0.99\mathcal{N}(0,1) + 0.01\mathcal{N}(10,1)$ plotted as [] depicted on a (a) linear--$y$ axis, and (b) semilog--$y$ axis. For these distributions, $\mathsf{TV}(\mu, \pi) \leq 0.01$, $\mathsf{KL}(\mu \mathbin{\|} \pi) \geq 0.4$, and $\mathsf{R}_2(\mu \mathbin{\|} \pi) \geq 90$. The linear--$y$ scale shows the large overlap (small TV distance) and the semilog--$y$ scale highlights the discrepancy in the tails (large KL divergence and Rényi--$2$ divergence).
• Figure 2: One-shot couplings. (a) To obtain regularization bounds for uHMC-v, the initial velocities are coupled such that $\tilde{q}_{T,h}(x,v) = \tilde{q}_{T, h}(y, \varphi_{x,y}(v))$. (b) To obtain cross-regularization bounds for uHMC-v, the initial velocities are coupled such that $\tilde{q}_{T,h}(x,v) = q_T(y, \Phi_{x,y}(v))$.

Theorems & Definitions (100)

• Definition 1: Wasserstein--$p$ distance
• Definition 2: Orlicz norm
• Remark 1
• Remark 2
• Lemma 1
• proof
• Definition 3: Orlicz-Wasserstein distance
• Definition 4: KL divergence
• Definition 5: Rényi--$q$ divergence
• Remark 3: Reversibility