Nonparametric regression with modified ReLU networks

TL;DR

This paper introduces a modified ReLU neural network model for regression, where weights are transformed by a function before multiplication, achieving near-optimal prediction rates for smooth functions.

Contribution

It proposes a new class of modified ReLU networks with a specific weight transformation function, demonstrating their statistical optimality in regression tasks.

Findings

Achieves near-minimax prediction rates for smooth functions

Provides an example of a suitable weight modification function

Shows empirical risk minimizers perform optimally up to a logarithmic factor

Abstract

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.