A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case

Greg Ongie, Rebecca Willett, Daniel Soudry, Nathan Srebro

TL;DR: We characterize the space of functions realizable as a ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded.

Abstract: We give a tight characterization of the (vectorized Euclidean) norm of weights required to realize a function $f:\mathbb{R}\rightarrow \mathbb{R}^d$ as a single hidden-layer ReLU network with an unbounded number of units (infinite width), extending the univariate characterization of Savarese et al. (2019) to the multivariate case.

Keywords: inductive bias, regularization, infinite-width networks, ReLU networks