Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

2026-07-16

AI Summary

This work addresses the bias inherent in unadjusted Hamiltonian Monte Carlo (HMC) and underdamped Langevin samplers, which, while computationally efficient, deviate from the target distribution—a deviation typically corrected at substantial cost via Metropolis–Hastings acceptance steps. The authors extend the recently observed “bias delocalization” phenomenon from overdamped Langevin dynamics to both HMC and underdamped Langevin algorithms. They introduce an analytical framework based on matrix-polynomial discrete-time integrators and leverage high-dimensional Wasserstein distance theory to show that, under mild or sparsity-based assumptions on variable interactions, controlling the $W_2$ bias of any $K$-dimensional marginal distribution requires only $O(\sqrt{K})$ integration steps (ignoring logarithmic dependence on dimension $d$). Moreover, they establish that bias delocalization holds for underdamped Langevin dynamics at arbitrarily large friction coefficients as well as for the Leimkuhler–Matthews integrator.

Abstract

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 \emph{delocalization of bias} phenomenon, previously established for the overdamped Langevin algorithm, to these two unadjusted algorithms. We show that to control the $W_2$ bias of any $K$-dimensional marginal of a high-dimensional distribution, $O(\sqrt{K})$ 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.

Problem

• bias delocalization
• Hamiltonian Monte Carlo
• underdamped Langevin
• high-dimensional sampling
• unadjusted samplers

Innovation

• delocalization of bias
• unadjusted Hamiltonian Monte Carlo
• underdamped Langevin
• matrix-polynomial framework
• Wasserstein-2 bias