  The Fisher information matrix (FIM) is a fundamental quantity to represent
the characteristics of a stochastic model, including deep neural networks
(DNNs). The present study reveals novel statistics of FIM that are universal
among a wide class of DNNs. To this end, we use random weights and large width
limits, which enables us to utilize mean field theories. We investigate the
asymptotic statistics of the FIM's eigenvalues and reveal that most of them are
close to zero while the maximum eigenvalue takes a huge value. Because the
landscape of the parameter space is defined by the FIM, it is locally flat in
most dimensions, but strongly distorted in others. Moreover, we demonstrate the
potential usage of the derived statistics in learning strategies. First, small
eigenvalues that induce flatness can be connected to a norm-based capacity
measure of generalization ability. Second, the maximum eigenvalue that induces
the distortion enables us to quantitatively estimate an appropriately sized
learning rate for gradient methods to converge.

Machine Learning;Disordered Systems and Neural Networks;Machine Learning