A method for analyzing non-asymptotic guarantees of numerical discretizations of ergodic SDEs in Wasserstein-2 distance is presented by Sanz-Serna and Zygalakis in ``Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations". They analyze the UBU integrator which is strong order two and only requires one gradient evaluation per step, resulting in desirable non-asymptotic guarantees, in particular $\mathcal{O}(d^{1/4}\epsilon^{-1/2})$ steps to reach a distance of $\epsilon > 0$ in Wasserstein-2 distance away from the target distribution. However, there is a mistake in the local error estimates in Sanz-Serna and Zygalakis (2021), in particular, a stronger assumption is needed to achieve these complexity estimates. This note reconciles the theory with the dimension dependence observed in practice in many applications of interest.

翻译：Sanz-Serna与Zygalakis在《Wasserstein距离估计在遍历随机微分方程数值逼近分布中的应用》一文中，提出了一种在Wasserstein-2距离下分析遍历SDE数值离散化非渐近保证的方法。他们分析了UBU积分器——该积分器具有二阶强收敛阶，且每步仅需一次梯度计算，从而获得了理想的非渐近保证，具体而言达到与目标分布Wasserstein-2距离ε>0所需步数为$\mathcal{O}(d^{1/4}\epsilon^{-1/2})$。然而，Sanz-Serna与Zygalakis（2021）中的局部误差估计存在一处错误：为实现该复杂度估计，需要更强的假设条件。本注记通过理论修正，使其与众多实际应用中所观察到的维度依赖性现象相吻合。