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This article contributes to the current statistical theory of deep neuralnetworks (DNNs). It was shown that DNNs are able to circumvent the so--calledcurse of dimensionality in case that suitable restrictions on the structure ofthe regression function hold. In most of those results the tuning parameter isthe sparsity of the network, which describes the number of non-zero weights inthe network. This constraint seemed to be the key factor for the good rate ofconvergence results. Recently, the assumption was disproved. In particular, itwas shown that simple fully connected DNNs can achieve the same rate ofconvergence. Those fully connected DNNs are based on the unbounded ReLUactivation function. In this article we extend the results to smooth activationfunctions, i.e., to the sigmoid activation function. It is shown thatestimators based on fully connected DNNs with sigmoid activation function alsoachieve the minimax rates of convergence (up to $\ln n$-factors). In our resultthe number of hidden layers is fixed, the number of neurons per layer tends toinfinity for sample size tending to infinity and a bound for the weights in thenetwork is given.

Analysis of the rate of convergence of fully connected deep neural network regression estimates with smooth activation function