To understand the dynamics of optimization in deep neural networks, we develop a tool to study the evolution of the entire Hessian spectrum throughout the optimization process. Using this, we study a number of hypotheses concerning smoothness, curvature, and sharpness in the deep learning literature. We then thoroughly analyze a crucial structural feature of the spectra: in non-batch normalized networks, we observe the rapid appearance of large isolated eigenvalues in the spectrum, along with a surprising concentration of the gradient in the corresponding eigenspaces. In batch normalized networks, these two effects are almost absent. We characterize these effects, and explain how they affect optimization speed through both theory and experiments. As part of this work, we adapt advanced tools from numerical linear algebra that allow scalable and accurate estimation of the entire Hessian spectrum of ImageNet-scale neural networks; this technique may be of independent interest in other applications.
During the preparation of this paper, @cite_29 appeared on Arxiv which briefly introduces the same spectrum estimation methodology and studies the Hessian on small subsamples of MNIST and CIFAR-10 at the end of the training. In comparison, we provide a detailed exposition, error analysis and validation of the estimator in Section , and present optimization results on full datasets, up to and including ImageNet.
@cite_29
2900531695
Previous works observed the spectrum of the Hessian of the training loss of deep neural networks. However, the networks considered were of minuscule size. We apply state-of-the-art tools in modern high-dimensional numerical linear algebra to approximate the spectrum of the Hessian of deep nets with tens of millions of parameters. Our results corroborate previous findings, based on small-scale networks, that the Hessian exhibits 'spiked' behavior, with several outliers isolated from a continuous bulk. However we find that the bulk does not follow a simple Marchenko-Pastur distribution, as previously suggested, but rather a heavier-tailed distribution. Finally, we document the dynamics of the outliers and the bulk with varying sample size.