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Theoretical guarantees for stochastic gradient Langevin dynamics

  • Daniel Paulin
  • Peter A. Whalley
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Abstract

We prove asymptotic bias bounds for stochastic gradient Langevin dynamics in Wasserstein distance of order two. We assume that the negative log-density is strongly convex with a Lipschitz gradient, and that the stochastic gradient estimator is unbiased with an error satisfying a mean-square Lipschitz condition. The bounds are of order h under a fourth moment assumption on the stochastic gradient error and of order h1/2 under only a second moment assumption, where h is the stepsize. A spiked-noise example shows that a second moment assumption alone is insufficient for a bound of order h that is uniform over noise distributions with a fixed variance.

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Cite

BibTeXRIS
BibTeX
@misc{paulin2026theoretical,
  title = {{Theoretical guarantees for stochastic gradient Langevin dynamics}},
  author = {Daniel Paulin and Peter A. Whalley},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01651},
  eprint = {2610.01651},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01651},
}
RIS
TY  - UNPB
TI  - Theoretical guarantees for stochastic gradient Langevin dynamics
AU  - Daniel Paulin
AU  - Peter A. Whalley
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01651
UR  - https://arxiv.org/abs/2610.01651
AB  - We prove asymptotic bias bounds for stochastic gradient Langevin dynamics in Wasserstein distance of order two. We assume that the negative log-density is strongly convex with a Lipschitz gradient, and that the stochastic gradient estimator is unbiased with an error satisfying a mean-square Lipschitz condition. The bounds are of order h under a fourth moment assumption on the stochastic gradient error and of order h^1/2 under only a second moment assumption, where h is the stepsize. A spiked-noise example shows that a second moment assumption alone is insufficient for a bound of order h that is uniform over noise distributions with a fixed variance.
ER  -