  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.
