Is the distribution-free sample bound for generalization tight?

A general relationship is developed between the two sharp transition points, the statistical capacity which represents the memorization, and the universal sample bound for generalization, for a network composed of random samples drawn from a specific class of distributions. This relationship indicates that generalization happens after memorization. It is shown through one example that the sample complexity needed for generalization can coincide with the capacity point. For the worst case, the sample complexity for generalization can be on the order of the distribution-free bound, whereas, for a more structured case, it can be smaller than the worst case bound. The analysis sheds light on why in practice the number of samples needed for generalization can be smaller than the bound given in term of the VC-dimension.<>