Function Approximation with Randomly Initialized Neural Networks for Approximate Model Reference Adaptive Control
This work provides a constructive method for function approximation in control systems, addressing a gap in classical theory for widely used activation functions, though it is incremental in extending existing results to more general cases.
The paper tackles the problem of constructing neural networks with approximation guarantees for a variety of activation functions by introducing mollified integral representations, enabling high accuracy via linear combinations of randomly initialized activations without requiring specialized direct representations.
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.