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Recent results show that estimates defined by over-parametrized deep neuralnetworks learned by applying gradient descent to a regularized empirical $L_2$risk are universally consistent and achieve good rates of convergence. In thispaper, we show that the regularization term is not necessary to obtain similarresults. In the case of a suitably chosen initialization of the network, asuitable number of gradient descent steps, and a suitable step size we showthat an estimate without a regularization term is universally consistent forbounded predictor variables. Additionally, we show that if the regressionfunction is H\"older smooth with H\"older exponent $1/2 \leq p \leq 1$, the$L_2$ error converges to zero with a convergence rate of approximately$n^{-1/(1+d)}$. Furthermore, in case of an interaction model, where theregression function consists of a sum of H\"older smooth functions with $d^*$components, a rate of convergence is derived which does not depend on the inputdimension $d$.

Analysis of the expected $L_2$ error of an over-parametrized deep neural network estimate learned by gradient descent without regularization