# Correction to "Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations"

arXiv: https://arxiv.org/abs/2402.08711 · HTML: https://mlsanity.com/paper/2402.08711/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 4 references that are in the map.
- First posted on arXiv 13 Feb 2024.

## Builds on (most cited first, 3 of 3)

- [When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?](https://mlsanity.com/paper/2304.04724/) (2023)
- [Unbiased Kinetic Langevin Monte Carlo with Inexact Gradients](https://mlsanity.com/paper/2311.05025/) (2023)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)

## Built on by (most cited first, 2 of 2)

- [Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamics](https://mlsanity.com/paper/2410.19780/) (2024)
- [Scalable Logistic Gaussian Process Density Regression with Kinetic Langevin Sampling](https://mlsanity.com/paper/2610.09591/) (2026)

## Abstract (arXiv)

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 O(d1/4ε-1/2) steps to reach a distance of ε > 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.