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《遍历随机微分方程数值逼近分布的Wasserstein距离估计》一文中提出了基于Wasserstein-2距离分析遍历随机微分方程数值离散非渐近保证的方法。他们分析了具有强二阶精度且每步仅需一次梯度评估的UBU积分器，该积分器产生了理想的非渐近保证，特别是达到与目标分布Wasserstein-2距离为$\epsilon > 0$时所需的步数为$\mathcal{O}(d^{1/4}\epsilon^{-1/2})$。然而，Sanz-Serna与Zygalakis（2021）的局部误差估计中存在错误，具体而言，实现这些复杂度估计需要更强的假设条件。本文旨在调和理论与众多实际应用中观察到的维数依赖性之间的矛盾。