The Sample Complexity of One-Hidden-Layer Neural Networks
This work provides theoretical insights into sample complexity for neural networks, addressing a foundational problem in machine learning theory, though it is incremental in refining existing bounds.
The paper investigates uniform convergence bounds for one-hidden-layer neural networks, showing that controlling the Frobenius norm of weights is sufficient for guarantees, while spectral norm control is insufficient in general but works in specific cases like smooth activations or convolutional networks.
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.