发表机构
Department of Mathematics, University of California, Los Angeles, CA 90095, USA; Courant Institute, New York University, NY 10012, USA（加州大学洛杉矶分校数学系; 纽约大学Courant研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究将偏差离域化现象从过阻尼朗之万算法扩展到无调整哈密顿蒙特卡罗和欠阻尼朗之万算法，表明在特定假设下控制高维分布K维边际偏差\(O(\sqrt{K})\)积分步即可，用矩阵多项式框架解决离散时间积分器技术难题。
AI 中文摘要
无调整采样器如无调整哈密顿蒙特卡罗和欠阻尼朗之万存在偏差。传统上通过Metropolis - Hastings调整消除偏差，但这会因合理接受率所需小步长而显著增加迭代复杂度。本文将过阻尼朗之万算法中已有的偏差离域化现象扩展到这两种无调整算法。表明在变量间弱或稀疏交互假设下，控制高维分布任何K维边际的\(W_2\)偏差，\(O(\sqrt{K})\)积分步就足够，离散时间积分器带来技术难题，通过矩阵多项式框架解决。欠阻尼朗之万算法结果对所有大摩擦参数有效。
英文摘要
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.
