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In this article we study the stochastic gradient descent (SGD) optimizationmethod in the training of fully-connected feedforward artificial neuralnetworks with ReLU activation. The main result of this work proves that therisk of the SGD process converges to zero if the target function underconsideration is constant. In the established convergence result the consideredartificial neural networks consist of one input layer, one hidden layer, andone output layer (with $d \in \mathbb{N}$ neurons on the input layer, $H \in\mathbb{N}$ 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 andthe input data used in the SGD process to train the artificial neural networksis assumed to be independent and identically distributed.

zeitschrift für angewandte mathematik und physik

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