A Geometric Modeling of Occam's Razor in Deep Learning

TL;DR

This paper introduces a geometric information-theoretic approach to explain why deep neural networks perform well despite their high parameter count, linking simplicity to better generalization through manifold complexity analysis.

Contribution

It proposes a novel geometric framework using singular semi-Riemannian geometry to analyze the complexity of DNNs via the Fisher information matrix.

Findings

Derives complexity measures based on singularity analysis of the Fisher information matrix.

Provides bounds on model complexity that explain DNNs' generalization capabilities.

Links the simplicity of models to their effective description length.

Abstract

Why do deep neural networks (DNNs) benefit from very high dimensional parameter spaces? Their huge parameter complexities vs stunning performance in practice is all the more intriguing and not explainable using the standard theory of model selection for regular models. In this work, we propose a geometrically flavored information-theoretic approach to study this phenomenon. With the belief that simplicity is linked to better generalization, as grounded in the theory of minimum description length, the objective of our analysis is to examine and bound the complexity of DNNs. We introduce the locally varying dimensionality of the parameter space of neural network models by considering the number of significant dimensions of the Fisher information matrix, and model the parameter space as a manifold using the framework of singular semi-Riemannian geometry. We derive model complexity measures which yield short description lengths for deep neural network models based on their singularity analysis thus explaining the good performance of DNNs despite their large number of parameters.