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.

本研究探讨了深度学习和神经网络中的的决策边界复杂度与Hessian矩阵特征值之间的关联，并提出了一种新的边界复杂度的衡量方法和边界宽度的估计技术。

揭示Hessian与决策边界的关联