Optimization and Generalization Guarantees for Weight Normalization

TL;DR

This paper provides the first theoretical analysis of optimization and generalization for deep neural networks with weight normalization, establishing convergence guarantees and generalization bounds.

Contribution

It offers the first theoretical characterizations of optimization and generalization for WeightNorm models, including spectral norm bounds and convergence guarantees.

Findings

Spectral norm of Hessian depends on network width and normalization terms.

Training convergence guaranteed under certain conditions.

Generalization bound is independent of width and sublinear in depth.

Abstract

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.