Uniform Convergence of Deep Neural Networks with Lipschitz Continuous
  Activation Functions and Variable Widths
Yuesheng Xu Haizhang Zhang
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.