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：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中采样的问题。在此背景下，若无额外密度信息可用，已知动能Langevin扩散的随机中点离散化方法是在高维大条件数问题中最具可扩展性的方法。我们的主要结果是该方法在Wasserstein-2误差上的一个非渐近且易于计算的上界。为更详尽阐述建立该可计算上界的方法，我们对标准Langevin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，并提供重要见解，我们利用这些见解建立了动能Langevin过程中点离散化的改进上界。此外，通过应用这些技术，我们为采用欧拉离散化的动能Langevin过程建立了新的保证，其对条件数的依赖关系优于现有上界。