  New estimates for the population risk are established for two-layer neural
networks. These estimates are nearly optimal in the sense that the error rates
scale in the same way as the Monte Carlo error rates. They are equally
effective in the over-parametrized regime when the network size is much larger
than the size of the dataset. These new estimates are a priori in nature in the
sense that the bounds depend only on some norms of the underlying functions to
be fitted, not the parameters in the model, in contrast with most existing
results which are a posteriori in nature. Using these a priori estimates, we
provide a perspective for understanding why two-layer neural networks perform
better than the related kernel methods.
