On the implicit regularization of Langevin dynamics with projected noise

Govind Menon, Austin Stromme, Adrien Vacher
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:5159-5187, 2026.

Abstract

We study Langevin dynamics with noise projected onto the directions orthogonal to an isometric group action. This mathematical model is introduced to shed new light on the effects of symmetry on stochastic gradient descent for over-parametrized models. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, Langevin dynamics with projected noise is equivalent in law to Langevin dynamics with isotropic diffusion but with an additional drift term proportional to the negative log volume of the group orbit. We prove this result by constructing a coupling of the two processes via a third process on the group itself, and identify the additional drift as the mean curvature of the orbits.

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-menon26a, title = {On the implicit regularization of Langevin dynamics with projected noise}, author = {Menon, Govind and Stromme, Austin and Vacher, Adrien}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {5159--5187}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/menon26a/menon26a.pdf}, url = {https://proceedings.mlr.press/v336/menon26a.html}, abstract = {We study Langevin dynamics with noise projected onto the directions orthogonal to an isometric group action. This mathematical model is introduced to shed new light on the effects of symmetry on stochastic gradient descent for over-parametrized models. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, Langevin dynamics with projected noise is equivalent in law to Langevin dynamics with isotropic diffusion but with an additional drift term proportional to the negative log volume of the group orbit. We prove this result by constructing a coupling of the two processes via a third process on the group itself, and identify the additional drift as the mean curvature of the orbits.} }
Endnote
%0 Conference Paper %T On the implicit regularization of Langevin dynamics with projected noise %A Govind Menon %A Austin Stromme %A Adrien Vacher %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-menon26a %I PMLR %P 5159--5187 %U https://proceedings.mlr.press/v336/menon26a.html %V 336 %X We study Langevin dynamics with noise projected onto the directions orthogonal to an isometric group action. This mathematical model is introduced to shed new light on the effects of symmetry on stochastic gradient descent for over-parametrized models. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, Langevin dynamics with projected noise is equivalent in law to Langevin dynamics with isotropic diffusion but with an additional drift term proportional to the negative log volume of the group orbit. We prove this result by constructing a coupling of the two processes via a third process on the group itself, and identify the additional drift as the mean curvature of the orbits.
APA
Menon, G., Stromme, A. & Vacher, A.. (2026). On the implicit regularization of Langevin dynamics with projected noise. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:5159-5187 Available from https://proceedings.mlr.press/v336/menon26a.html.

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