Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

TL;DR - This paper proves that unadjusted Hamiltonian Monte Carlo and underdamped Langevin samplers exhibit "delocalization of bias," meaning their sampling error can be controlled cheaply for low-dimensional marginals of high-dimensional distributions—reducing the need for costly Metropolis–Hastings correction.

• Extends the delocalization-of-bias phenomenon (previously shown for overdamped Langevin) to unadjusted HMC and underdamped Langevin, avoiding the small step sizes and iteration-complexity blowup that Metropolis–Hastings adjustment imposes.
• Main result: controlling the $W_2$ (Wasserstein-2) bias of any $K$-dimensional marginal of a high-dimensional target needs only $O(\sqrt{K})$ integration steps, up to $\log d$ factors, assuming weak or sparse interactions among variables.
• Handles the technical difficulty of discrete-time integrators via a broadly applicable matrix-polynomial framework characterizing their propagators.
• The underdamped result holds for all large friction parameters, implying the Leimkuhler–Matthews integrator for overdamped Langevin dynamics also shows delocalization of bias.