A Geometric Approach of Gradient Descent Algorithms in Neural Networks.
Yacine Chitour, Zhenyu Liao, Romain Couillet · arXiv (Cornell University) · 2018
In this paper, we present an original geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. Built upon a key invariance property induced by the network structure, we propose a conjecture called \emph{overfitting conjecture} stating that, for almost every training data, the corresponding gradient descent trajectory converges to a global minimum, for almost every initial condition. This would imply that, for linear neural networks of an arbitrary number of hidden layers, the solution achieved by simple gradient descent algorithm is equivalent to that of least square estimation. Our first result consists in establishing, in the case of linear networks of arbitrary depth, convergence of gradient descent trajectories to critical points of the loss function. Our second result is the proof of the \emph{overfitting conjecture} in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost every training data).