On kernel-based statistical learning in the mean field limit
Christian Fiedler Michael Herty Sebastian Trimpe
In many applications of machine learning, a large number of variables are
considered. Motivated by machine learning of interacting particle systems, we
consider the situation when the number of input variables goes to infinity.
First, we continue the recent investigation of the mean field limit of kernels
and their reproducing kernel Hilbert spaces, completing the existing theory.
Next, we provide results relevant for approximation with such kernels in the
mean field limit, including a representer theorem. Finally, we use these
kernels in the context of statistical learning in the mean field limit,
focusing on Support Vector Machines. In particular, we show mean field
convergence of empirical and infinite-sample solutions as well as the
convergence of the corresponding risks. On the one hand, our results establish
rigorous mean field limits in the context of kernel methods, providing new
theoretical tools and insights for large-scale problems. On the other hand, our
setting corresponds to a new form of limit of learning problems, which seems to
have not been investigated yet in the statistical learning theory literature.