Function Approximation with Randomly Initialized Neural Networks for Approximate Model Reference Adaptive Control

TL;DR

This paper introduces a new method for function approximation using randomly initialized neural networks, leveraging mollified integral representations to achieve guarantees across various activation functions.

Contribution

It develops mollified integral representations that enable approximation guarantees for random neural networks with multiple activation functions.

Findings

Provides approximation guarantees for various activation functions.

Enables construction of neural networks with random initialization for function approximation.

Extends classical neural network approximation theory with new integral techniques.

Abstract

Classical results in neural network approximation theory show how arbitrary continuous functions can be approximated by networks with a single hidden layer, under mild assumptions on the activation function. However, the classical theory does not give a constructive means to generate the network parameters that achieve a desired accuracy. Recent results have demonstrated that for specialized activation functions, such as ReLUs and some classes of analytic functions, high accuracy can be achieved via linear combinations of randomly initialized activations. These recent works utilize specialized integral representations of target functions that depend on the specific activation functions used. This paper defines mollified integral representations, which provide a means to form integral representations of target functions using activations for which no direct integral representation is currently known. The new construction enables approximation guarantees for randomly initialized networks for a variety of widely used activation functions.