Quasi-Equivalence of Width and Depth of Neural Networks

While classic studies proved that wide networks allow universal approximation, recent research and successes of deep learning demonstrate the power of the network depth. Based on a symmetric consideration, we investigate if the design of artificial neural networks should have a directional preference, and what the mechanism of interaction is between the width and depth of a network. We address this fundamental question by establishing a quasi-equivalence between the width and depth of ReLU networks. Specifically, we formulate a transformation from an arbitrary ReLU network to a wide network and a deep network for either regression or classification so that an essentially same capability of the original network can be implemented. That is, a deep regression/classification ReLU network has a wide equivalent, and vice versa, subject to an arbitrarily small error. Interestingly, the quasi-equivalence between wide and deep classification ReLU networks is a data-driven version of the De Morgan law

Fundamental Limits of Deep Learning-Based Binary Classifiers Trained with Hinge Loss

Although deep learning (DL) has led to several breakthroughs in many disciplines as diverse as chemistry, computer science, electrical engineering, mathematics, medicine, neuroscience, and physics, a comprehensive understanding of why and how DL is empirically successful remains fundamentally elusive. To attack this fundamental problem and unravel the mysteries behind DL's empirical successes, significant innovations toward a unified theory of DL have been made. These innovations encompass nearly fundamental advances in optimization, generalization, and approximation. Despite these advances, however, no work to date has offered a way to quantify the testing performance of a DL-based algorithm employed to solve a pattern classification problem. To overcome this fundamental challenge in part, this paper exposes the fundamental testing performance limits of DL-based binary classifiers trained with hinge loss. For binary classifiers that are based on deep rectified linear unit (ReLU) feedforward neural networks (FNNs) and ones that are based on deep FNNs with ReLU and Tanh activation, we derive their respective novel asymptotic testing performance limits. The derived testing performance limits are validated by extensive computer experiments