Noether: The More Things Change, the More Stay the Same
Abstract
Symmetries have proven to be important ingredients in the analysis of neural networks. So far their use has mostly been implicit or seemingly coincidental. We undertake a systematic study of the role that symmetry plays. In particular, we clarify how symmetry interacts with the learning algorithm. The key ingredient in our study is played by Noether's celebrated theorem which, informally speaking, states that symmetry leads to conserved quantities (e.g., conservation of energy or conservation of momentum). In the realm of neural networks under gradient descent, model symmetries imply restrictions on the gradient path. E.g., we show that symmetry of activation functions leads to boundedness of weight matrices, for the specific case of linear activations it leads to balance equations of consecutive layers, data augmentation leads to gradient paths that have "momentum"type restrictions, and time symmetry leads to a version of the Neural Tangent Kernel. Symmetry alone does not specify the optimization path, but the more symmetries are contained in the model the more restrictions are imposed on the path. Since symmetry also implies overparametrization, this in effect implies that some part of this overparametrization is cancelled out by the existence of the conserved quantities. Symmetry can therefore be thought of as one further important tool in understanding the performance of neural networks under gradient descent.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.05508
 Bibcode:
 2021arXiv210405508G
 Keywords:

 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 35 pages, 6 figures