  Exciting new work on the generalization bounds for neural networks (NN) given
by Neyshabur et al. , Bartlett et al. closely depend on two
parameter-depenedent quantities: the Lipschitz constant upper-bound and the
stable rank (a softer version of the rank operator). This leads to an
interesting question of whether controlling these quantities might improve the
generalization behaviour of NNs. To this end, we propose stable rank
normalization (SRN), a novel, optimal, and computationally efficient
weight-normalization scheme which minimizes the stable rank of a linear
operator. Surprisingly we find that SRN, inspite of being non-convex problem,
can be shown to have a unique optimal solution. Moreover, we show that SRN
allows control of the data-dependent empirical Lipschitz constant, which in
contrast to the Lipschitz upper-bound, reflects the true behaviour of a model
on a given dataset. We provide thorough analyses to show that SRN, when applied
to the linear layers of a NN for classification, provides striking
improvements-11.3% on the generalization gap compared to the standard NN along
with significant reduction in memorization. When applied to the discriminator
of GANs (called SRN-GAN) it improves Inception, FID, and Neural divergence
scores on the CIFAR 10/100 and CelebA datasets, while learning mappings with
low empirical Lipschitz constants.
