# Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

arXiv: https://arxiv.org/abs/2607.15208 · HTML: https://mlsanity.com/paper/2607.15208/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 51 references that are in the map.
- First posted on arXiv 16 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?](https://mlsanity.com/paper/2304.04724/) (2023)
- [Unbiased Kinetic Langevin Monte Carlo with Inexact Gradients](https://mlsanity.com/paper/2311.05025/) (2023)
- [Tail-Sensitive KL and R'enyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings](https://mlsanity.com/paper/2601.09019/) (2026)
- [Stein's method for marginals on large graphical models](https://mlsanity.com/paper/2410.11771/) (2024)
- [Algorithmic warm starts for Hamiltonian Monte Carlo](https://mlsanity.com/paper/2603.22741/) (2026)
- [A hierarchical entropy method for the delocalization of bias in high-dimensional Langevin Monte Carlo](https://mlsanity.com/paper/2509.08619/) (2025)

## Abstract (arXiv)

Unadjusted samplers such as unadjusted Hamiltonian Monte Carlo and underdamped Langevin are well-known to be biased. Metropolis--Hastings adjustment has been conventionally incorporated into Hamiltonian Monte Carlo to eliminate the bias. However, this adjustment can significantly increase the iteration complexity due to the small step size required for reasonable Metropolis acceptance rates. In this work, we extend the delocalization of bias phenomenon, previously established for the overdamped Langevin algorithm, to these two unadjusted algorithms. We show that to control the W2 bias of any K-dimensional marginal of a high-dimensional distribution, O(√K) integration steps suffice up to log d terms, assuming either weak or sparse interactions among variables. The discrete-time integrators here introduce technical difficulties beyond those of the overdamped setting, which we address through a broadly applicable matrix-polynomial framework that characterizes their propagators. Our result for the underdamped Langevin algorithm is valid for all large friction parameters, implying that the Leimkuhler-Matthews integrator for the overdamped Langevin dynamics also exhibits delocalization of bias.