Stochastic Gradient Descent in Hilbert Scales: Smoothness, Preconditioning and Earlier Stopping

When solving nonparametric least-squares problems in an RKHS we face the problem that the unknown solution may not have the expected smoothness (regularity) implied by the kernel. Then the question arises whether the use of such mis-specified kernels still allows for good reconstructions yielding errors of optimal order. Although it is a commonly accepted fact that the regularity inherent in the solution has an impact on accuracy and convergence of learning algorithms, there are only poor precise mathematical investigations in the framework of learning in RKHSs using SGD. Mathematically, smoothness can be expressed in various different ways. Classically, the concept of source conditions proved to be useful, expressing the target function as element of the domain of a differential operator, see e.g.