{"id":1013,"date":"2026-01-15T03:09:15","date_gmt":"2026-01-15T03:09:15","guid":{"rendered":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?p=1013"},"modified":"2026-01-15T03:10:43","modified_gmt":"2026-01-15T03:10:43","slug":"tail-sensitive-convergence-guarantees-for-unadjusted-hamiltonian-monte-carlo","status":"publish","type":"post","link":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/tail-sensitive-convergence-guarantees-for-unadjusted-hamiltonian-monte-carlo\/","title":{"rendered":"Tail-Sensitive Convergence Guarantees for Unadjusted Hamiltonian Monte Carlo"},"content":{"rendered":"

I\u2019m pleased to share a new paper with Siddharth Mitra<\/strong> and Andre Wibisono<\/strong> (both at Yale), titled:<\/p>\n

Tail-Sensitive KL and R\u00e9nyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings.<\/a> <\/i><\/p>\n

This work develops a framework for tail-sensitive convergence analysis of unadjusted Hamiltonian Monte Carlo (HMC) in Kullback\u2013Leibler (KL) and R\u00e9nyi divergence. These divergences measure relative density mismatch<\/b><\/span> and therefore govern Metropolis acceptance behavior: small errors in the tails\u2014where the target density is tiny\u2014can dominate acceptance probabilities, even when total variation distance is small.<\/p>\n

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Why tail-sensitive divergences matter<\/b><\/h3>\n

Most theoretical analyses of Markov chain mixing time focus on metrics such as total variation distance or Wasserstein distance. While these are useful in many settings, they do not directly control relative density mismatch in the tails of the target distribution. For Metropolis-adjusted Markov chains, including Metropolis-adjusted HMC, acceptance probabilities depend on ratios<\/i> of densities rather than additive differences in probability mass.<\/p>\n

As a result, good performance requires controlling tail behavior. Divergences such as KL and R\u00e9nyi, which measure multiplicative discrepancies between densities, are therefore the natural metrics in this context.<\/p>\n

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What we prove<\/b><\/h3>\n

Our main results provide a framework for lifting Wasserstein convergence guarantees (often easier to establish) into KL and R\u00e9nyi convergence bounds for unadjusted HMC. This is achieved by introducing and analyzing one-shot couplings that establish a regularization property of the unadjusted HMC transition kernel.<\/p>\n

Concretely, we show that:<\/p>\n