Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks

The optimization foundations of deep linear networks have received significant attention lately. However, due to the non-convexity and hierarchical structure, analyzing the regularized loss of deep linear networks remains a challenging task. In this work, we study the local geometric landscape of the regularized squared loss of deep linear networks, providing a deeper understanding of its optimization properties. Specifically, we characterize the critical point set and establish an error-bound property for all critical points under mild conditions. Notably, we identify the sufficient and necessary conditions under which the error bound holds. To support our theoretical findings, we conduct numerical experiments demonstrating that gradient descent exhibits linear convergence when optimizing the regularized loss of deep linear networks.