  It is well-known that the Hessian of deep loss landscape matters to
optimization, generalization, and even robustness of deep learning. Recent
works empirically discovered that the Hessian spectrum in deep learning has a
two-component structure that consists of a small number of large eigenvalues
and a large number of nearly-zero eigenvalues. However, the theoretical
mechanism or the mathematical behind the Hessian spectrum is still largely
under-explored. To the best of our knowledge, we are the first to demonstrate
that the Hessian spectrums of well-trained deep neural networks exhibit simple
power-law structures. Inspired by the statistical physical theories and the
spectral analysis of natural proteins, we provide a maximum-entropy theoretical
interpretation for explaining why the power-law structure exist and suggest a
spectral parallel between protein evolution and training of deep neural
networks. By conducing extensive experiments, we further use the power-law
spectral framework as a useful tool to explore multiple novel behaviors of deep
learning.
