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We investigate the sample complexity of bounded two-layer neural networksusing different activation functions.  In particular, we consider the class  $$ \mathcal{H} = \left\{\textbf{x}\mapsto \langle \textbf{v}, \sigma \circW\textbf{b} + \textbf{b} \rangle : \textbf{b}\in\mathbb{R}^d, W \in\mathbb{R}^{\mathcal{T}\times d}, \textbf{v} \in\mathbb{R}^{\mathcal{T}}\right\} $$  where the spectral norm of $W$ and $\textbf{v}$ is bounded by $O(1)$, theFrobenius norm of $W$ is bounded from its initialization by $R > 0$, and$\sigma$ is a Lipschitz activation function.  We prove that if $\sigma$ is element-wise, then the sample complexity of$\mathcal{H}$ has only logarithmic dependency in width and that this complexityis tight, up to logarithmic factors.  We further show that the element-wise property of $\sigma$ is essential for alogarithmic dependency bound in width, in the sense that there existnon-element-wise activation functions whose sample complexity is linear inwidth, for widths that can be up to exponential in the input dimension.  For the upper bound, we use the recent approach for norm-based bounds namedApproximate Description Length (ADL) by arXiv:1910.05697.  We further develop new techniques and tools for this approach that willhopefully inspire future works.

On the Sample Complexity of Two-Layer Networks: Lipschitz vs. Element-Wise Lipschitz Activation