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.

本文针对具有强烈对数凹密度的平滑目标分布的采样问题进行探究，借助随机中点离散化方法，建立可计算的Wasserstein-2误差的上界，并基于中点离散化的Langevin扩散过程进行分析以明确其基本原理和提供有价值的见解，进而建立起更改进的上界以改进Euler离散化的Langevin扩散过程。

强对数凹分布的朗之万蒙特卡罗方法：随机中点重访