Theory-to-Practice Gap for Neural Networks and Neural Operators

Abstract:This work studies the sampling complexity of learning with ReLU neural networks and neural operators. For mappings belonging to relevant approximation spaces, we derive upper bounds on the best-possible convergence rate of any learning algorithm, with respect to the number of samples. In the finite-dimensional case, these bounds imply a gap between the parametric and sampling complexities of learning, known as the \emph{theory-to-practice gap}. In this work, a unified treatment of the theory-to-practice gap is achieved in a general $L^p$-setting, while at the same time improving available bounds in the literature. Furthermore, based on these results the theory-to-practice gap is extended to the infinite-dimensional setting of operator learning. Our results apply to Deep Operator Networks and integral kernel-based neural operators, including the Fourier neural operator. We show that the best-possible convergence rate in a Bochner $L^p$-norm is bounded by rates of order $1/p$.

Machine Learning (cs.LG); Functional Analysis (math.FA)

arXiv:2503.18219 [cs.LG]

(or arXiv:2503.18219v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2503.18219 Focus to learn more arXiv-issued DOI via DataCite

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