Approximation with Random Shallow ReLU Networks with Applications to Model Reference Adaptive Control

TL;DR

This paper proves that shallow ReLU neural networks with randomly initialized weights can approximate smooth functions with quantifiable error bounds, enabling their use in adaptive control applications.

Contribution

It provides the first probabilistic approximation guarantees for random shallow ReLU networks, bridging theory and practical control applications.

Findings

ReLU networks with random weights achieve $L_{ abla} error of $O(m^{-1/2})$ for smooth functions.

Weights and biases can be randomly generated over specific distributions to ensure approximation quality.

Application to model reference adaptive control demonstrates practical utility.

Abstract

Neural networks are regularly employed in adaptive control of nonlinear systems and related methods of reinforcement learning. A common architecture uses a neural network with a single hidden layer (i.e. a shallow network), in which the weights and biases are fixed in advance and only the output layer is trained. While classical results show that there exist neural networks of this type that can approximate arbitrary continuous functions over bounded regions, they are non-constructive, and the networks used in practice have no approximation guarantees. Thus, the approximation properties required for control with neural networks are assumed, rather than proved. In this paper, we aim to fill this gap by showing that for sufficiently smooth functions, ReLU networks with randomly generated weights and biases achieve $L_{\infty}$ error of $O(m^{-1/2})$ with high probability, where $m$ is the number of neurons. It suffices to generate the weights uniformly over a sphere and the biases uniformly over an interval. We show how the result can be used to get approximations of required accuracy in a model reference adaptive control application.