We revisit the problem of sampling from a target distribution that has a smooth strongly log-concave density everywhere in $\mathbb R^p$. In this context, if no additional density information is available, the randomized midpoint discretization for the kinetic Langevin diffusion is known to be the most scalable method in high dimensions with large condition numbers. Our main result is a nonasymptotic and easy to compute upper bound on the Wasserstein-2 error of this method. To provide a more thorough explanation of our method for establishing the computable upper bound, we conduct an analysis of the midpoint discretization for the vanilla Langevin process. This analysis helps to clarify the underlying principles and provides valuable insights that we use to establish an improved upper bound for the kinetic Langevin process with the midpoint discretization. Furthermore, by applying these techniques we establish new guarantees for the kinetic Langevin process with Euler discretization, which have a better dependence on the condition number than existing upper bounds.

翻译：摘要：我们重新研究了从 $\mathbb R^p$ 中具有光滑强对数凹密度的目标分布进行采样的问题。在此背景下，若无额外密度信息可用，基于动能的朗之万扩散的随机中点离散化方法在高维大条件数场景中具有最佳可扩展性。我们的主要成果是建立了该方法在Wasserstein-2误差下的非渐近且易于计算的上界。为更详尽阐释构建可计算上界的方法，我们首先对经典朗之万过程的中点离散化进行了分析。该分析有助于阐明基本原理，并为我们建立基于中点离散化的动能朗之万过程的改进上界提供重要洞见。此外，通过应用这些技术，我们为欧拉离散化下的动能朗之万过程建立了新的保证，该保证在条件数依赖性上优于现有上界。