Unveiling the Hessian's Connection to the Decision Boundary

TL;DR

This paper reveals that the Hessian's top eigenvectors are linked to the decision boundary complexity in neural networks, enabling new ways to measure and identify well-generalizing minima with wide-margin boundaries.

Contribution

It establishes a novel connection between the Hessian spectrum and decision boundary complexity, introducing new measures and techniques for analyzing neural network minima.

Findings

Hessian top eigenvectors characterize decision boundary properties

Number of Hessian outliers correlates with boundary complexity

Proposed methods accurately identify minima with wide-margin boundaries

Abstract

Understanding the properties of well-generalizing minima is at the heart of deep learning research. On the one hand, the generalization of neural networks has been connected to the decision boundary complexity, which is hard to study in the high-dimensional input space. Conversely, the flatness of a minimum has become a controversial proxy for generalization. In this work, we provide the missing link between the two approaches and show that the Hessian top eigenvectors characterize the decision boundary learned by the neural network. Notably, the number of outliers in the Hessian spectrum is proportional to the complexity of the decision boundary. Based on this finding, we provide a new and straightforward approach to studying the complexity of a high-dimensional decision boundary; show that this connection naturally inspires a new generalization measure; and finally, we develop a novel margin estimation technique which, in combination with the generalization measure, precisely identifies minima with simple wide-margin boundaries. Overall, this analysis establishes the connection between the Hessian and the decision boundary and provides a new method to identify minima with simple wide-margin decision boundaries.