Wasserstein Bounds for Euler and Randomized Midpoint Langevin Sampling with Noisy Gradients
Abstract
Stochastic-gradient Langevin methods are sampling algorithms that replace the gradient of the log probability density by a noisy, possibly biased evaluation. We study how such gradient-evaluation errors affect the Euler and randomized midpoint discretizations of Langevin dynamics by comparing each noisy chain with its exact-gradient counterpart. For continuously differentiable, strongly log-concave targets with globally Lipschitz gradients, we derive comparison bounds under a sub-Gaussian condition on the gradient noise. The resulting 2-Wasserstein bounds hold uniformly over iterations, and are linear in both the step size and the L2 norm of the evaluation bias, with a square-root dependence on dimensionality. To the best of our knowledge, no comparable 2-Wasserstein bounds have previously been established for randomized midpoint with noisy gradients. For Euler, our results complement recent Wasserstein bounds by allowing biased evaluations, and by not requiring additional regularity assumptions on the evaluation errors.
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BibTeX
@misc{xiong2026wasserstein,
title = {{Wasserstein Bounds for Euler and Randomized Midpoint Langevin Sampling with Noisy Gradients}},
author = {Yifei Xiong and Vinayak Rao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37858},
eprint = {2610.37858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37858},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wasserstein Bounds for Euler and Randomized Midpoint Langevin Sampling with Noisy Gradients
AU - Yifei Xiong
AU - Vinayak Rao
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37858
UR - https://arxiv.org/abs/2610.37858
AB - Stochastic-gradient Langevin methods are sampling algorithms that replace the gradient of the log probability density by a noisy, possibly biased evaluation. We study how such gradient-evaluation errors affect the Euler and randomized midpoint discretizations of Langevin dynamics by comparing each noisy chain with its exact-gradient counterpart. For continuously differentiable, strongly log-concave targets with globally Lipschitz gradients, we derive comparison bounds under a sub-Gaussian condition on the gradient noise. The resulting 2-Wasserstein bounds hold uniformly over iterations, and are linear in both the step size and the L^2 norm of the evaluation bias, with a square-root dependence on dimensionality. To the best of our knowledge, no comparable 2-Wasserstein bounds have previously been established for randomized midpoint with noisy gradients. For Euler, our results complement recent Wasserstein bounds by allowing biased evaluations, and by not requiring additional regularity assumptions on the evaluation errors.
ER -