Hessian eigenvalue patterns explained by hidden symmetry in deep learning
Eigenvalues of the Hessian in Deep Learning: The Origin of Symmetry and Its Breaking
Machine Learning
Summary
Deep learning models have complex structures, and a key to understanding their behavior lies in something called the Hessian matrix. The authors found that the pattern of Hessian eigenvalues can be explained by revealing a hidden, highly symmetric version of the model. This symmetry creates clear groups of eigenvalues, and breaking the symmetry leads to the clusters and outliers seen in real trained models. Their framework applies broadly to various neural network types, including convolutional and transformer models.
What this means in practice
- •For deep learning engineers: Design and analyze neural network training by accounting for hidden symmetries that shape Hessian eigenvalue patterns and influence optimization dynamics.
- •For machine learning platform developers: Improve diagnostics tools for model training by incorporating spectral insights derived from symmetry and its breaking in Hessian structures.
A theory result. No direct application yet.
Authors
Yossi Arjevani
Abstract
Hessian spectra at trained models in deep learning exhibit a persistent pattern: eigenvalues organize into distinct clusters, including a large bulk near zero and a few isolated outliers. This paper shows that a natural account of these spectral phenomena emerges when the original setting is understood as a departure from a nearby, otherwise hidden, highly symmetric reference. Modifications, including changes to the architecture, data distribution, or parameter metric, expose a nearby reference configuration whose Hessian exhibits rich invariances-ones not accounted for by weight symmetries. There, symmetry enables a precise description of the spectra, forcing high-dimensional kernels and eigenvalues of large multiplicity. Returning to the original configuration breaks the Hessian symmetry and thereby produces the observed hierarchy of clusters and outliers. The framework is developed in some generality, with a detailed analysis of three-layer ReLU networks and applications to convolutional, graph, and transformer models, as well as to the NTK. The same mechanism is further shown to yield analogous spectral structures in layerwise Hessians and the Gauss-Newton matrix.