Title: Fisher Information and Natural Gradient Learning of Random Deep Networks,
Abstract: A deep neural network is a hierarchical nonlinear model transforming input
signals to output signals. Its input-output relation is considered to be
stochastic, being described for a given input by a parameterized conditional
probability distribution of outputs. The space of parameters consisting of
weights and biases is a Riemannian manifold, where the metric is defined by the
Fisher information matrix. The natural gradient method uses the steepest
descent direction in a Riemannian manifold, so it is effective in learning,
avoiding plateaus. It requires inversion of the Fisher information matrix,
however, which is practically impossible when the matrix has a huge number of
dimensions. Many methods for approximating the natural gradient have therefore
been introduced. The present paper uses statistical neurodynamical method to
reveal the properties of the Fisher information matrix in a net of random
connections under the mean field approximation. We prove that the Fisher
information matrix is unit-wise block diagonal supplemented by small order
terms of off-block-diagonal elements, which provides a justification for the
quasi-diagonal natural gradient method by Y. Ollivier. A unitwise
block-diagonal Fisher metrix reduces to the tensor product of the Fisher
information matrices of single units. We further prove that the Fisher
information matrix of a single unit has a simple reduced form, a sum of a
diagonal matrix and a rank 2 matrix of weight-bias correlations. We obtain the
inverse of Fisher information explicitly. We then have an explicit form of the
natural gradient, without relying on the numerical matrix inversion, which
drastically speeds up stochastic gradient learning.