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.

本研究解决了深度权重归一化模型在优化和泛化方面的理论空白。本文首次提供了权重归一化模型的优化和泛化的理论特性，特别是提出了光滑激活函数下的收敛性和一致性界限。实验结果表明，归一化项与深度神经网络的训练效果密切相关，具有重要的应用潜力。

权重归一化的优化和泛化保证