Training invariances and the low-rank phenomenon: beyond linear networks

TL;DR

This paper extends the understanding of training invariances and low-rank phenomena from linear networks to nonlinear ReLU networks, revealing conditions under which weights converge to low-rank structures.

Contribution

It generalizes previous linear network results to nonlinear ReLU networks, identifying local invariances and conditions for low-rank convergence in deep learning models.

Findings

Weights in certain submatrices converge to low-rank structures.

Local invariances hold for neurons with stable activation patterns.

Full matrix invariance does not generally hold for ReLU networks.

Abstract

The implicit bias induced by the training of neural networks has become a topic of rigorous study. In the limit of gradient flow and gradient descent with appropriate step size, it has been shown that when one trains a deep linear network with logistic or exponential loss on linearly separable data, the weights converge to rank-1 matrices. In this paper, we extend this theoretical result to the last few linear layers of the much wider class of nonlinear ReLU-activated feedforward networks containing fully-connected layers and skip connections. Similar to the linear case, the proof relies on specific local training invariances, sometimes referred to as alignment, which we show to hold for submatrices where neurons are stably-activated in all training examples, and it reflects empirical results in the literature. We also show this is not true in general for the full matrix of ReLU fully-connected layers. Our proof relies on a specific decomposition of the network into a multilinear function and another ReLU network whose weights are constant under a certain parameter directional convergence.