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

arXiv: https://arxiv.org/abs/2104.12384 · HTML: https://mlsanity.com/paper/2104.12384/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 26 Apr 2021.

## Builds on (most cited first, 18 of 18)

- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Non-asymptotic bounds for sampling algorithms without log-concavity](https://mlsanity.com/paper/1808.07105/) (2018)
- [Geometric integrators and the Hamiltonian Monte Carlo method](https://mlsanity.com/paper/1711.05337/) (2017)
- [Analysis of Optimization Algorithms via Integral Quadratic Constraints: Nonstrongly Convex Problems](https://mlsanity.com/paper/1705.03615/) (2017)
- [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://mlsanity.com/paper/1906.08530/) (2019)
- [The connections between Lyapunov functions for some optimization algorithms and differential equations](https://mlsanity.com/paper/2009.00673/) (2020)
- [Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs](https://mlsanity.com/paper/1903.01138/) (2019)
- [Implicit Langevin Algorithms for Sampling From Log-concave Densities](https://mlsanity.com/paper/1903.12322/) (2019)
- [Contractivity of Runge-Kutta methods for convex gradient systems](https://mlsanity.com/paper/1909.09971/) (2019)

## Built on by (most cited first, 8 of 8)

- [Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamics](https://mlsanity.com/paper/2410.19780/) (2024)
- [Momentum Particle Maximum Likelihood](https://mlsanity.com/paper/2312.07335/) (2023)
- [Squared Wasserstein-2 Distance for Efficient Reconstruction of Stochastic Differential Equations](https://mlsanity.com/paper/2401.11354/) (2024)
- [Single-seed generation of Brownian paths and integrals for adaptive and high order SDE solvers](https://mlsanity.com/paper/2405.06464/) (2024)
- [Randomised Splitting Methods and Stochastic Gradient Descent](https://mlsanity.com/paper/2504.04274/) (2025)
- [Correction to "Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations"](https://mlsanity.com/paper/2402.08711/) (2024)
- [Random Reshuffling for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2501.16055/) (2025)
- [Kinetic Langevin Splitting Schemes for Constrained Sampling](https://mlsanity.com/paper/2603.23397/) (2026)

## Abstract (arXiv)

We present a framework that allows for the non-asymptotic study of the 2-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyse a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a d--dimensional strongly log-concave distribution with condition number kappa, the algorithm is shown to produce with an Obig(kappa5/4 d1/4ε-1/2 big) complexity samples from a distribution that, in Wasserstein distance, is at most ε>0 away from the target distribution.