Structure of universal formulas

Published: 21 Sept 2023, Last Modified: 02 Nov 2023NeurIPS 2023 posterEveryoneRevisionsBibTeX
Keywords: VC-dimension, neural networks, activation functions, approximation, polynomials, algebraic functions
TL;DR: We study which parameterized elementary functions of bounded complexity can approximate arbitrary continuous functions
Abstract: By universal formulas we understand parameterized analytic expressions that have a fixed complexity, but nevertheless can approximate any continuous function on a compact set. There exist various examples of such formulas, including some in the form of neural networks. In this paper we analyze the essential structural elements of these highly expressive models. We introduce a hierarchy of expressiveness classes connecting the global approximability property to the weaker property of infinite VC dimension, and prove a series of classification results for several increasingly complex functional families. In particular, we introduce a general family of polynomially-exponentially-algebraic functions that, as we prove, is subject to polynomial constraints. As a consequence, we show that fixed-size neural networks with not more than one layer of neurons having transcendental activations (e.g., sine or standard sigmoid) cannot in general approximate functions on arbitrary finite sets. On the other hand, we give examples of functional families, including two-hidden-layer neural networks, that approximate functions on arbitrary finite sets, but fail to do that on the whole domain of definition.
Submission Number: 7423