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

Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widths