Optimal nonparametric inference via deep neural network

Deep neural network is a state-of-art method in modern science and technology. Much statistical literature have been devoted to understanding its performance nonparametric estimation, whereas the results are suboptimal due redundant logarithmic sacrifice. In this paper, we show that such log-factors not necessary. We derive upper bounds for L 2 minimax risk estimation. Sufficient conditions on architectures provided become optimal (without log-sacrifice). Our proof relies an explicitly constructed estimator based tensor product B-splines. also asymptotic distributions relating hypothesis testing procedure. The procedure further proved as under suitable architectures.