Does the Data Induce Capacity Control in Deep Learning?

This paper studies how the dataset may be the cause of the anomalous generalization performance of deep networks. We show that the data correlation matrix of typical classification datasets has an eigenspectrum where, after a sharp initial drop, a large number of small eigenvalues are distributed uniformly over an exponentially large range. This structure is mirrored in a network trained on this data: we show that the Hessian and the Fisher Information Matrix (FIM) have eigenvalues that are spread uniformly over exponentially large ranges. We call such eigenspectra "sloppy" because sets of weights corresponding to small eigenvalues can be changed by large magnitudes without affecting the loss. Networks trained on atypical, non-sloppy synthetic data do not share these traits. We show how this structure in the data can give to non-vacuous PAC-Bayes generalization bounds analytically; we also construct data-distribution dependent priors that lead to accurate bounds using numerical optimization.