Nonparametric regression with modified ReLU networks

We consider regression estimation with modified ReLU neural networks in which network weight matrices are first modified by a function $\alpha$ before being multiplied by input vectors. We give an example of continuous, piecewise linear function $\alpha$ for which the empirical risk minimizers over the classes of modified ReLU networks with $l_1$ and squared $l_2$ penalties attain, up to a logarithmic factor, the minimax rate of prediction of unknown $\beta$-smooth function.