How regularization affects the critical points in linear networks

This paper is concerned with the problem of representing and learning a linear transformation using a linear neural network. In recent years, there is a growing interest in the study of such networks, in part due to the successes of deep learning. The main question of this body of research (and also of our paper) is related to the existence and optimality properties of the critical points of the mean-squared loss function. An additional primary concern of our paper pertains to the robustness of these critical points in the face of (a small amount of) regularization. An optimal control model is introduced for this purpose and a learning algorithm (backprop with weight decay) derived for the same using the Hamilton's formulation of optimal control.