  This paper presents a margin-based multiclass generalization bound for neural
networks that scales with their margin-normalized "spectral complexity": their
Lipschitz constant, meaning the product of the spectral norms of the weight
matrices, times a certain correction factor. This bound is empirically
investigated for a standard AlexNet network trained with SGD on the mnist and
cifar10 datasets, with both original and random labels; the bound, the
Lipschitz constants, and the excess risks are all in direct correlation,
suggesting both that SGD selects predictors whose complexity scales with the
difficulty of the learning task, and secondly that the presented bound is
sensitive to this complexity.
