Fisher-Rao Metric, Geometry, and Complexity of Neural Networks

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Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, James Stokes ;
Proceedings of Machine Learning Research, PMLR 89:888-896, 2019.

Abstract

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.

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