We consider deep neural networks with a Lipschitz continuous activation function and with weight matrices of variable widths. We establish a uniform convergence analysis framework in which sufficient conditions on weight matrices and bias vectors together with the Lipschitz constant are provided to ensure uniform convergence of the deep neural networks to a meaningful function as the number of their layers tends to infinity. In the framework, special results on uniform convergence of deep neural networks with a fixed width, bounded widths and unbounded widths are presented. In particular, as convolutional neural networks are special deep neural networks with weight matrices of increasing widths, we put forward conditions on the mask sequence which lead to uniform convergence of resulting convolutional neural networks. The Lipschitz continuity assumption on the activation functions allows us to include in our theory most of commonly used activation functions in applications.

翻译：我们考虑一类具有Lipschitz连续激活函数且权重矩阵宽度可变的深度神经网络。本文建立了一个均匀收敛分析框架，在该框架中给出了关于权重矩阵、偏置向量及Lipschitz常数的充分条件，以确保当网络层数趋于无穷时，深度神经网络均匀收敛于某个有意义的函数。在该框架下，我们分别针对固定宽度、有界宽度及无界宽度的深度神经网络给出了均匀收敛的特定结果。特别地，考虑到卷积神经网络是权重矩阵宽度递增的特殊深度神经网络，我们提出了导致相应卷积神经网络均匀收敛的掩码序列条件。对激活函数施加的Lipschitz连续性假设，使得我们的理论能够涵盖应用中大多数常用的激活函数。