Approximation with Random Shallow ReLU Networks with Applications to Model Reference Adaptive Control
This provides theoretical justification for neural network approximations in adaptive control, addressing a gap where such properties were previously assumed rather than proven.
The paper tackles the lack of approximation guarantees for shallow ReLU networks with fixed hidden layers in adaptive control by proving that randomly generated weights and biases achieve $L_{\infty}$ error of $O(m^{-1/2})$ with high probability for smooth functions, where m is the number of neurons, and applies this to model reference adaptive control.
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.