Before revision:  Numerical experiments validate the theory and extend the scope of the paper beyond the Mat\'ern setting .
After revision:  Numerical experiments validate the theory and extend the scope of the learning of the regularity parameter in the Mat\'ern-like field, for which consistency results have been very scarce in the spatial statistics literature. Moreover, we conduct extensive numerical experiments beyond the Mat\'ern-like model, comparing the two algorithms further. These experiments demonstrate learning of other hierarchical parameters, such as amplitude and lengthscale; they also illustrate the setting of model misspecification in which the kernel flow approach could show superior performance to the more traditional empirical Bayes approach .