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.

翻译：采用动能朗之万动力学进行模拟，是从仅知其非归一化密度的分布中采样的常用方法。针对动能朗之万动力学的多种离散化方案已被研究，由此产生的算法统称为动能朗之万蒙特卡洛（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（简称指数积分器）先前已在强对数凹和对数利普希茨光滑势能条件下，通过同步Wasserstein耦合策略进行了研究。然而，现有分析对参数施加的限制，无法解释KLMC在不同参数选择下的行为。特别地，所有已知结果在过阻尼区域均失效，暗示指数积分器在过阻尼极限下会退化。本研究重新审视了采用指数积分器的KLMC的同步Wasserstein耦合分析。通过改进分析，我们在更宽松的参数限制条件下获得了Wasserstein收缩结果及渐近偏差界，这证实只要施加适当的时间加速，指数积分器能够在过阻尼区域内稳定模拟动能朗之万动力学。