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The paper analyzes Langevin dynamics with noise projected onto directions orthogonal to an isometric group action, a model relevant to understanding symmetry effects in stochastic gradient descent for over-parameterized models. The key finding is that when initial and target densities are group-invariant, this projected Langevin dynamics is equivalent in law to standard Langevin dynamics with isotropic diffusion but with an additional drift term related to the negative log volume of the group orbit. This equivalence is proven through a coupling argument involving a third process on the group, identifying the drift as the mean curvature of the orbits, thus revealing a novel form of implicit regularization.
Projected noise in Langevin dynamics induces an implicit regularization equivalent to adding a drift term proportional to the negative log volume of the group orbit, revealing a novel connection between symmetry and regularization.
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.