It is well known that data-driven regression smoothing parameters h based on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow as n-1/10; that is, for a bandwidth h0 optimizing the averaged squared error, n1/10 (h - h0)/h0 tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidths h0. This (nonrandom) smoothing parameter can be approximated much faster. We use the technique of double smoothing to show that there is an h such that, under certain conditions, n1/2(h - h0)/h0 tends to an asymptotic normal distribution