Mean Square Error Analysis of Stochastic Runge-Kutta Integrators
Abstract
We analyze the mean square error of stochastic Runge-Kutta integrators for overdamped Langevin dynamics whose potential is strongly convex outside a bounded region. A decomposition splits the local error into a mean-zero term and a smaller remainder, and the discrete Poisson equation turns their moments into a bound on the error of a time average. We carry this out for two stochastic Runge-Kutta integrators proposed by Yang & Wang (2026), both of strong order 3/2 and weak order $2$, which evaluate the gradient of the potential and no higher derivative. We show that their laws approach the law of the exact solution at second order in Wasserstein-1 distance, up to a logarithm, uniformly in the number of steps. For a test function with bounded derivatives up to third order, we prove that the mean square error over N steps with step size h is 𝒪 ( 1/Nh + h4 ), which is the optimal order in the discretization. Experiments measure the strong and weak orders and the sampling bias on a nonconvex potential in ℝ2, and compare the integrators on a diffusion model of CIFAR-10.
Related
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BibTeX
@misc{ye2026mean,
title = {{Mean Square Error Analysis of Stochastic Runge-Kutta Integrators}},
author = {Xuda Ye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.11528},
eprint = {2609.11528},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11528},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mean Square Error Analysis of Stochastic Runge-Kutta Integrators
AU - Xuda Ye
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.11528
UR - https://arxiv.org/abs/2609.11528
AB - We analyze the mean square error of stochastic Runge-Kutta integrators for overdamped Langevin dynamics whose potential is strongly convex outside a bounded region. A decomposition splits the local error into a mean-zero term and a smaller remainder, and the discrete Poisson equation turns their moments into a bound on the error of a time average. We carry this out for two stochastic Runge-Kutta integrators proposed by Yang & Wang (2026), both of strong order 3/2 and weak order $2$, which evaluate the gradient of the potential and no higher derivative. We show that their laws approach the law of the exact solution at second order in Wasserstein-1 distance, up to a logarithm, uniformly in the number of steps. For a test function with bounded derivatives up to third order, we prove that the mean square error over N steps with step size h is 𝒪 ( 1/Nh + h^4 ), which is the optimal order in the discretization. Experiments measure the strong and weak orders and the sampling bias on a nonconvex potential in ℝ^2, and compare the integrators on a diffusion model of CIFAR-10.
ER -