Sobolev Norm Learning Rates for Conditional Mean Embeddings

TL;DR

This paper introduces new learning rates for conditional mean embeddings using RKHS interpolation theory, enabling better understanding of convergence in complex, infinite-dimensional settings.

Contribution

It derives explicit, adaptive convergence rates for the estimator in misspecified settings, broadening the applicability of conditional mean embeddings.

Findings

Achieves uniform convergence rates in the output RKHS under certain conditions.

Provides explicit, adaptive learning rates for the estimator.

Extends the theoretical understanding of conditional mean embeddings in complex spaces.

Abstract

We develop novel learning rates for conditional mean embeddings by applying the theory of interpolation for reproducing kernel Hilbert spaces (RKHS). We derive explicit, adaptive convergence rates for the sample estimator under the misspecifed setting, where the target operator is not Hilbert-Schmidt or bounded with respect to the input/output RKHSs. We demonstrate that in certain parameter regimes, we can achieve uniform convergence rates in the output RKHS. We hope our analyses will allow the much broader application of conditional mean embeddings to more complex ML/RL settings involving infinite dimensional RKHSs and continuous state spaces.