Fisher-Rao Metric, Geometry, and Complexity of Neural Networks
This work addresses the need for better capacity measures in deep learning, offering a novel invariant norm that unifies existing approaches, though it is incremental in advancing theoretical understanding.
The authors tackled the problem of understanding neural network capacity by introducing the Fisher-Rao norm, a new invariant measure motivated by Information Geometry, and showed it generalizes existing norms and provides generalization error bounds, with experimental validation on CIFAR-10.
We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks.