The Spectrum of Fisher Information of Deep Networks Achieving Dynamical Isometry
This provides theoretical insights into trainability for deep learning practitioners, though it is incremental as it builds on known concepts like dynamical isometry.
The paper investigates the spectral distribution of the Fisher information matrix (FIM) in deep neural networks achieving dynamical isometry, revealing that the FIM's spectrum concentrates and grows linearly with depth, and experimentally shows that the optimal learning rate is inversely proportional to depth.
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.