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：遍历随机微分方程数值逼近的分布》一文中提出了一种分析方法，用于研究遍历随机微分方程数值离散化在Wasserstein-2距离下的非渐近保证。他们分析了强二阶且每步仅需一次梯度评估的UBU积分器，该积分器具有理想的非渐近性质，特别地，仅需$\mathcal{O}(d^{1/4}\epsilon^{-1/2})$步即可使Wasserstein-2距离达到目标分布的$\epsilon>0$邻域内。然而，Sanz-Serna与Zygalakis（2021）的局部误差估计存在缺陷：实现上述复杂度估计需要更强的假设条件。本文旨在调和该理论结果与诸多实际应用中观察到的维度依赖关系。