Title: Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians,
Abstract: We consider deep classifying neural networks. We expose a structure in the
derivative of the logits with respect to the parameters of the model, which is
used to explain the existence of outliers in the spectrum of the Hessian.
Previous works decomposed the Hessian into two components, attributing the
outliers to one of them, the so-called Covariance of gradients. We show this
term is not a Covariance but a second moment matrix, i.e., it is influenced by
means of gradients. These means possess an additive two-way structure that is
the source of the outliers in the spectrum. This structure can be used to
approximate the principal subspace of the Hessian using certain "averaging"
operations, avoiding the need for high-dimensional eigenanalysis. We
corroborate this claim across different datasets, architectures and sample
sizes.