On the implicit regularization of Langevin dynamics with projected noise
2026-02-12 • Artificial Intelligence
Artificial Intelligence
AI summaryⓘ
The authors study a mathematical model called Langevin dynamics where randomness is restricted to directions that don’t change symmetry. They show that, under symmetric starting and target conditions, this special version behaves like the usual Langevin dynamics but with an extra effect pushing the process based on the shape of symmetrical paths. This new effect is linked to a geometric property called mean curvature. Their work helps explain how symmetry affects learning algorithms in complex models.
Langevin dynamicsisometric group actionimplicit regularizationstochastic gradient descentsymmetryisotropic diffusiondrift termgroup orbitmean curvaturecoupling of stochastic processes
Authors
Govind Menon, Austin J. Stromme, Adrien Vacher
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.