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

TL;DR

This paper introduces the Fisher-Rao norm, a new invariant capacity measure for deep neural networks based on Information Geometry, providing theoretical insights and empirical validation on CIFAR-10.

Contribution

The paper proposes the Fisher-Rao norm as a novel, invariant capacity measure and characterizes its properties, linking it to existing complexity measures and analyzing its impact on generalization.

Findings

Fisher-Rao norm is invariant and encompasses existing measures.

Theoretical bounds on generalization error using the new measure.

Empirical results on CIFAR-10 support the theoretical claims.

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.