@inproceedings{
mart{\'\i}nez2024symmetries,
title={Symmetries in Overparametrized Neural Networks: A Mean Field View},
author={Javier Maass Mart{\'\i}nez and Joaquin Fontbona},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=L86glqNCUj}
}
We develop a Mean-Field (MF) view of the learning dynamics of overparametrized Artificial Neural Networks (NN) under distributional symmetries of the data w.r.t. the action of a general compact group $G$. We consider for this a class of  generalized shallow NNs given by an ensemble of  $N$ multi-layer units, jointly trained  using stochastic gradient descent (SGD) and possibly symmetry-leveraging (SL) techniques, such as Data Augmentation (DA), Feature Averaging  (FA) or Equivariant Architectures (EA).  We introduce the notions of weakly and strongly invariant  laws (WI and SI) on the parameter space of each single unit, corresponding, respectively, to $G$-invariant distributions, and to distributions supported on parameters fixed by the group action (which encode EA). This allows us to define symmetric models compatible with taking $N\to\infty$ and  give an interpretation of the asymptotic dynamics of DA, FA and EA in terms of Wasserstein Gradient Flows describing their MF limits. When activations respect the group action, we show that, for  symmetric data, DA, FA and freely-trained models obey the exact same MF  dynamic, which stays in the space of WI parameter laws and attains therein the population risk's minimizer. We also provide a counterexample to the general attainability of such an optimum over SI laws.
Despite this,  and quite remarkably, we show that the space of SI laws  is also preserved by these MF distributional dynamics even when freely trained. This sharply contrasts the finite-$N$ setting, in which EAs are generally not preserved by unconstrained SGD. We illustrate the validity of our findings as $N$ gets larger,  in a teacher-student experimental setting, training a student NN to learn from a WI, SI  or arbitrary teacher model through various SL schemes. We lastly deduce a data-driven heuristic to discover the largest subspace of parameters supporting SI distributions for a problem, that could be used for designing EA with minimal generalization error.
Symmetries in Overparametrized Neural Networks: A Mean Field View
Javier Maass Martínez Joaquin Fontbona