  We investigate the local spectral statistics of the loss surface Hessians of
artificial neural networks, where we discover excellent agreement with Gaussian
Orthogonal Ensemble statistics across several network architectures and
datasets. These results shed new light on the applicability of Random Matrix
Theory to modelling neural networks and suggest a previously unrecognised role
for it in the study of loss surfaces in deep learning. Inspired by these
observations, we propose a novel model for the true loss surfaces of neural
networks, consistent with our observations, which allows for Hessian spectral
densities with rank degeneracy and outliers, extensively observed in practice,
and predicts a growing independence of loss gradients as a function of distance
in weight-space. We further investigate the importance of the true loss surface
in neural networks and find, in contrast to previous work, that the exponential
hardness of locating the global minimum has practical consequences for
achieving state of the art performance.
