A Deeper Look at the Hessian Eigenspectrum of Deep Neural Networks and its Applications to Regularization

TL;DR

This paper investigates the layerwise Hessian eigenspectra of deep neural networks, revealing similarities across layers and proposing a regularizer based on the Hessian trace that improves generalization by encouraging flatter minima.

Contribution

It introduces a layerwise loss landscape analysis, shows the Hessian eigenspectrum similarity across layers, and proposes a novel Hessian trace regularizer for better generalization.

Findings

Layerwise Hessian eigenspectra are similar to the entire Hessian.

Middle layers' Hessian eigenspectrum closely resembles the overall spectrum.

Hessian trace decreases as training progresses, correlating with improved generalization.

Abstract

Loss landscape analysis is extremely useful for a deeper understanding of the generalization ability of deep neural network models. In this work, we propose a layerwise loss landscape analysis where the loss surface at every layer is studied independently and also on how each correlates to the overall loss surface. We study the layerwise loss landscape by studying the eigenspectra of the Hessian at each layer. In particular, our results show that the layerwise Hessian geometry is largely similar to the entire Hessian. We also report an interesting phenomenon where the Hessian eigenspectrum of middle layers of the deep neural network are observed to most similar to the overall Hessian eigenspectrum. We also show that the maximum eigenvalue and the trace of the Hessian (both full network and layerwise) reduce as training of the network progresses. We leverage on these observations to propose a new regularizer based on the trace of the layerwise Hessian. Penalizing the trace of the Hessian at every layer indirectly forces Stochastic Gradient Descent to converge to flatter minima, which are shown to have better generalization performance. In particular, we show that such a layerwise regularizer can be leveraged to penalize the middlemost layers alone, which yields promising results. Our empirical studies on well-known deep nets across datasets support the claims of this work