Reproduce “Optimal Rates for Regularized Conditional Mean Embedding Learning” on OpenResearch
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About this paper
Zhu Li, Dimitri Meunier, Mattes Mollenhauer, Arthur Gretton
The authors analyze regularized least-squares estimation of conditional mean embeddings using a vector-valued interpolation space that allows misspecified targets, including infinite-dimensional output RKHSs. Their high-probability squared L2 error rate depends on input eigenvalue decay and target smoothness, approaching n^-1 for Gaussian kernels in a qualified setting. A matching lower bound holds when β≥α; β<α remains open.
Key insights
- A novel vector-valued interpolation space, [G]_alpha, rigorously characterizes the target Conditional Mean Embedding (CME) operator, extending analysis to 'misspecified' settings where the CME does not strictly belong to the standard hypothesis space.
- The conventional definition of 'well-specifiedness' for CMEs (conditional expectation belonging to H_X) does not necessarily imply that the CME itself belongs to the hypothesis space G used by the KRR estimator.
- Optimal learning rates are achieved through a detailed bias-variance decomposition and advanced concentration inequalities within the framework of vector-valued Reproducing Kernel Hilbert Spaces.
Summary by alphaXiv.
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