The Spectrum of Fisher Information of Deep Networks Achieving Dynamical Isometry

TL;DR

This paper analyzes the spectral distribution of the Fisher information matrix in deep neural networks with dynamical isometry, revealing how the local metric scales with depth and impacts training dynamics.

Contribution

It introduces an algebraic method based on free probability to analyze the Fisher information spectrum in deep networks, especially under dynamical isometry.

Findings

Fisher information spectrum concentrates around the maximum and grows linearly with depth.

The local metric depends linearly on depth despite dynamical isometry.

Optimal learning rate inversely scales with network depth, guided by the Fisher spectrum.

Abstract

The Fisher information matrix (FIM) is fundamental to understanding the trainability of deep neural nets (DNN), since it describes the parameter space's local metric. We investigate the spectral distribution of the conditional FIM, which is the FIM given a single sample, by focusing on fully-connected networks achieving dynamical isometry. Then, while dynamical isometry is known to keep specific backpropagated signals independent of the depth, we find that the parameter space's local metric linearly depends on the depth even under the dynamical isometry. More precisely, we reveal that the conditional FIM's spectrum concentrates around the maximum and the value grows linearly as the depth increases. To examine the spectrum, considering random initialization and the wide limit, we construct an algebraic methodology based on the free probability theory. As a byproduct, we provide an analysis of the solvable spectral distribution in two-hidden-layer cases. Lastly, experimental results verify that the appropriate learning rate for the online training of DNNs is in inverse proportional to depth, which is determined by the conditional FIM's spectrum.