The Sample Complexity of One-Hidden-Layer Neural Networks

TL;DR

This paper investigates the sample complexity and uniform convergence bounds for one-hidden-layer neural networks, highlighting the importance of norm constraints and identifying specific conditions where spectral norm control suffices.

Contribution

It demonstrates that Frobenius norm control is generally necessary for uniform convergence, but spectral norm control can suffice in certain smooth or convolutional network settings.

Findings

Frobenius norm control ensures uniform convergence regardless of network width.

Spectral norm control is sufficient for smooth activation functions and some convolutional networks.

Sample complexity depends on parameters like patch overlap and number of patches in convolutional networks.

Abstract

We study norm-based uniform convergence bounds for neural networks, aiming at a tight understanding of how these are affected by the architecture and type of norm constraint, for the simple class of scalar-valued one-hidden-layer networks, and inputs bounded in Euclidean norm. We begin by proving that in general, controlling the spectral norm of the hidden layer weight matrix is insufficient to get uniform convergence guarantees (independent of the network width), while a stronger Frobenius norm control is sufficient, extending and improving on previous work. Motivated by the proof constructions, we identify and analyze two important settings where (perhaps surprisingly) a mere spectral norm control turns out to be sufficient: First, when the network's activation functions are sufficiently smooth (with the result extending to deeper networks); and second, for certain types of convolutional networks. In the latter setting, we study how the sample complexity is additionally affected by parameters such as the amount of overlap between patches and the overall number of patches.