Unveiling the Hessian's Connection to the Decision Boundary
This work addresses the challenge of identifying well-generalizing minima in neural networks, which is crucial for improving model reliability and performance in machine learning applications.
The paper tackles the problem of understanding generalization in deep learning by linking the Hessian's eigenvectors to the decision boundary complexity, showing that the number of outliers in the Hessian spectrum is proportional to this complexity and leading to a new generalization measure and margin estimation technique.
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.