论文标题
使用不足的Langevin动力学的公正估计
Unbiased Estimation using Underdamped Langevin Dynamics
论文作者
论文摘要
在这项工作中，我们考虑了对具有非负Lebesgue密度的概率度量的预期估计，并且在正常化常数方面是已知的点。我们专注于通过失业不足的Langevin Dynamics开发一种无偏见的方法，由于统计和机器学习的应用，该方法已被证明很受欢迎。具体来说，可以构建动力学{以便随着时间的流逝，他们}承认了感兴趣的概率作为固定度量。 {在许多情况下，在实践中使用了不足的Langevin动力学的时间消费版，仅使用固定数量的迭代运行。}我们根据\ cite {ub_grad，disc_model}的双随机估计而开发了一种新颖的方案，仅需要访问时间限制的动态。 {所提出的方案旨在消除用于运行有限数量迭代的动力学的偏置偏差和偏置。根据标准假设，我们证明我们的估计器具有有限的差异，并且预期成本有限，或者具有很高的可能性。为了说明我们的理论发现，我们提供了验证我们理论的数值实验，其中包括贝叶斯统计和统计物理学的具有挑战性的例子。
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.
