Weight normalization (WeightNorm) is widely used in practice for the training of deep neural networks and modern deep learning libraries have built-in implementations of it. In this paper, we provide the first theoretical characterizations of both optimization and generalization of deep WeightNorm models with smooth activation functions. For optimization, from the form of the Hessian of the loss, we note that a small Hessian of the predictor leads to a tractable analysis. Thus, we bound the spectral norm of the Hessian of WeightNorm networks and show its dependence on the network width and weight normalization terms--the latter being unique to networks without WeightNorm. Then, we use this bound to establish training convergence guarantees under suitable assumptions for gradient decent. For generalization, we use WeightNorm to get a uniform convergence based generalization bound, which is independent from the width and depends sublinearly on the depth. Finally, we present experimental results which illustrate how the normalization terms and other quantities of theoretical interest relate to the training of WeightNorm networks.

Generative Adversarial Networks (GANs) based semi-supervised learning (SSL) approaches are shown to improve classification performance by utilizing a large number of unlabeled samples in conjunction with limited labeled samples. However, their performance still lags behind the state-of-the-art non-GAN based SSL approaches. We identify that the main reason for this is the lack of consistency in class probability predictions on the same image under local perturbations. Following the general literature, we address this issue via label consistency regularization, which enforces the class probability predictions for an input image to be unchanged under various semantic-preserving perturbations. In this work, we introduce consistency regularization into the vanilla semi-GAN to address this critical limitation. In particular, we present a new composite consistency regularization method which, in spirit, leverages both local consistency and interpolation consistency. We demonstrate the efficacy of our approach on two SSL image classification benchmark datasets, SVHN and CIFAR-10. Our experiments show that this new composite consistency regularization based semi-GAN significantly improves its performance and achieves new state-of-the-art performance among GAN-based SSL approaches.