标题： 动力学朗之万蒙特卡罗在随机指数欧拉离散化下的分析，从欠阻尼到过阻尼 显示英文标题

标题： Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped

摘要： 模拟动力朗之万动力学是从分布中进行采样的流行方法，其中仅可获得其未归一化密度。 已经考虑了动力朗之万动力学的各种离散化方法，其中得到的算法统称为动力朗之万蒙特卡洛（KLMC）或欠阻尼朗之万蒙特卡洛。 具体而言，随机指数欧拉离散化，或简称为指数积分器，之前已在强对数凸和对数利普希茨光滑势下通过同步Wasserstein耦合策略进行了研究。 然而，现有的分析对参数施加了限制，这些限制无法解释KLMC在不同参数选择下的行为。 特别是，所有已知的结果在过阻尼区域都不成立，这表明指数积分器在过阻尼极限下会退化。 在本工作中，我们重新审视使用指数积分器的动力朗之万蒙特卡洛的同步Wasserstein耦合分析。 我们的改进分析结果在参数上具有较弱的限制，得到了Wasserstein收缩和渐近偏差的界，这表明只要应用适当的时间加速，指数积分器就能够稳定地模拟过阻尼区域的动力朗之万动力学。 显示英文摘要

摘要： Simulating the kinetic Langevin dynamics is a popular approach for sampling from distributions, where only their unnormalized densities are available. Various discretizations of the kinetic Langevin dynamics have been considered, where the resulting algorithm is collectively referred to as the kinetic Langevin Monte Carlo (KLMC) or underdamped Langevin Monte Carlo. Specifically, the stochastic exponential Euler discretization, or exponential integrator for short, has previously been studied under strongly log-concave and log-Lipschitz smooth potentials via the synchronous Wasserstein coupling strategy. Existing analyses, however, impose restrictions on the parameters that do not explain the behavior of KLMC under various choices of parameters. In particular, all known results fail to hold in the overdamped regime, suggesting that the exponential integrator degenerates in the overdamped limit. In this work, we revisit the synchronous Wasserstein coupling analysis of KLMC with the exponential integrator. Our refined analysis results in Wasserstein contractions and bounds on the asymptotic bias that hold under weaker restrictions on the parameters, which assert that the exponential integrator is capable of stably simulating the kinetic Langevin dynamics in the overdamped regime, as long as proper time acceleration is applied.