On the implicit regularization of Langevin dynamics with projected noise
This provides theoretical insight into how symmetries affect optimization in machine learning, though it is incremental as it builds on existing Langevin dynamics models.
The paper tackles the problem of understanding implicit regularization in stochastic gradient descent for over-parametrized models with symmetries, showing that Langevin dynamics with projected noise introduces an additional drift term proportional to the negative log volume of group orbits, equivalent to isotropic diffusion with this 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.