On the implicit regularization of Langevin dynamics with projected noise

TL;DR

This paper investigates how symmetry influences Langevin dynamics with projected noise, revealing a new implicit regularization effect linked to the geometry of group orbits and their curvature.

Contribution

It introduces a novel analysis of Langevin dynamics with projected noise, showing its equivalence to isotropic diffusion plus a drift related to group orbit volume and curvature.

Findings

Langevin dynamics with projected noise is equivalent to isotropic Langevin with an additional drift.

The additional drift is proportional to the negative log volume of the group orbit.

The drift is identified as the mean curvature of the group orbits.

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.