A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions

In this article we study the stochastic gradient descent (SGD) optimization method in the training of fully connected feedforward artificial neural networks with ReLU activation. The main result of this work proves that the risk of the SGD process converges to zero if the target function under consideration is constant. In the established convergence result the considered artificial neural networks consist of one input layer, one hidden layer, and one output layer (with d∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \in {\mathbb {N}}$$\end{document} neurons on the input layer, H∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in {\mathbb {N}}$$\end{document} neurons on the hidden layer, and one neuron on the output layer). The learning rates of the SGD process are assumed to be sufficiently small, and the input data used in the SGD process to train the artificial neural networks is assumed to be independent and identically distributed.