In this work we consider the unbiased estimation of expectations w.r.t.~probability measures that have non-negative Lebesgue density, and which are known point-wise up-to a normalizing constant. We focus upon developing an unbiased method via the underdamped Langevin dynamics, which has proven to be popular of late due to applications in statistics and machine learning. Specifically in continuous-time, the dynamics can be constructed {so that as the time goes to infinity they} admit the probability of interest as a stationary measure. {In many cases, time-discretized versions of the underdamped Langevin dynamics are used in practice which are run only with a fixed number of iterations.} We develop a novel scheme based upon doubly randomized estimation as in \cite{ub_grad,disc_model}, which requires access only to time-discretized versions of the dynamics. {The proposed scheme aims to remove the dicretization bias and the bias resulting from running the dynamics for a finite number of iterations}. We prove, under standard assumptions, that our estimator is of finite variance and either has finite expected cost, or has finite cost with a high probability. To illustrate our theoretical findings we provide numerical experiments which verify our theory, which include challenging examples from Bayesian statistics and statistical physics.

翻译：本文考虑对具有非负勒贝格密度且仅已知点态形式（归一化常数未知）的概率测度进行期望的无偏估计。我们聚焦于通过欠阻尼朗之万动力学发展一种无偏方法——该方法因在统计学和机器学习中的应用而近期备受关注。具体而言，在连续时间框架下，可构建该动力学系统使其在时间趋于无穷时以目标概率为平稳测度。实际应用中，常采用固定迭代步数的欠阻尼朗之万动力学时间离散化版本。我们基于文献\cite{ub_grad,disc_model}中的双重随机化估计方法，提出一种仅需时间离散化动力学访问权限的新型方案。该方案旨在消除离散化偏差以及有限迭代步数运行动力学产生的偏差。在标准假设下，我们证明所提估计量具有有限方差，且其期望计算成本有限，或具有高概率意义下的有限计算成本。为验证理论结果，我们提供了数值实验——涵盖贝叶斯统计与统计物理中的典型挑战性案例。