Analytic Insights into Structure and Rank of Neural Network Hessian Maps

TL;DR

This paper provides a theoretical analysis of the Hessian matrix in neural networks, revealing its rank deficiency and structural properties, which enhances understanding of model redundancy and informs network design.

Contribution

We develop exact formulas and bounds for the Hessian rank in deep linear networks and extend these insights to nonlinear networks, offering a structural understanding of Hessian properties.

Findings

Exact formulas for Hessian rank in deep linear networks

Bounds on Hessian rank applicable to nonlinear networks

Insights into how architecture affects Hessian rank deficiency

Abstract

The Hessian of a neural network captures parameter interactions through second-order derivatives of the loss. It is a fundamental object of study, closely tied to various problems in deep learning, including model design, optimization, and generalization. Most prior work has been empirical, typically focusing on low-rank approximations and heuristics that are blind to the network structure. In contrast, we develop theoretical tools to analyze the range of the Hessian map, providing us with a precise understanding of its rank deficiency as well as the structural reasons behind it. This yields exact formulas and tight upper bounds for the Hessian rank of deep linear networks, allowing for an elegant interpretation in terms of rank deficiency. Moreover, we demonstrate that our bounds remain faithful as an estimate of the numerical Hessian rank, for a larger class of models such as rectified and hyperbolic tangent networks. Further, we also investigate the implications of model architecture (e.g.~width, depth, bias) on the rank deficiency. Overall, our work provides novel insights into the source and extent of redundancy in overparameterized networks.