  In this brief we investigate the generalization properties of a
recently-proposed class of non-parametric activation functions, the kernel
activation functions (KAFs). KAFs introduce additional parameters in the
learning process in order to adapt nonlinearities individually on a per-neuron
basis, exploiting a cheap kernel expansion of every activation value. While
this increase in flexibility has been shown to provide significant improvements
in practice, a theoretical proof for its generalization capability has not been
addressed yet in the literature. Here, we leverage recent literature on the
stability properties of non-convex models trained via stochastic gradient
descent (SGD). By indirectly proving two key smoothness properties of the
models under consideration, we prove that neural networks endowed with KAFs
generalize well when trained with SGD for a finite number of steps.
Interestingly, our analysis provides a guideline for selecting one of the
hyper-parameters of the model, the bandwidth of the scalar Gaussian kernel. A
short experimental evaluation validates the proof.
