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Recent results in nonparametric regression show that for deep learning, i.e.,for neural network estimates with many hidden layers, we are able to achievegood rates of convergence even in case of high-dimensional predictor variables,provided suitable assumptions on the structure of the regression function areimposed. The estimates are defined by minimizing the empirical $L_2$ risk overa class of neural networks. In practice it is not clear how this can be doneexactly. In this article we introduce a new neural network regression estimatewhere most of the weights are chosen regardless of the data motivated by somerecent approximation results for neural networks, and which is therefore easyto implement. We show that for this estimate we can derive rates of convergenceresults in case the regression function is smooth. We combine this estimatewith the projection pursuit, where we choose the directions randomly, and weshow that for sufficiently many repititions we get a neural network regressionestimate which is easy to implement and which achieves the one-dimensional rateof convergence (up to some logarithmic factor) in case that the regressionfunction satisfies the assumptions of projection pursuit.

Analysis of the rate of convergence of neural network regression estimates which are easy to implement