Reproduce “Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm” on OpenResearch

About this paper

Zhu Li, Dimitri Meunier, Mattes Mollenhauer, Arthur Gretton

The authors develop learning-rate theory for regularized least squares with vector- or function-valued outputs, including infinite-dimensional Hilbert spaces, when the conditional mean lies outside the hypothesis space. They construct tensor-product interpolation spaces to measure target smoothness and analyze the usual estimator. In a Sobolev setting, squared error has rate n^{-(s-t)/(s+d/2)}, matched by a worst-case lower bound under stated assumptions.

Key insights

• Novel vector-valued interpolation spaces effectively bridge the gap between L2 and hypothesis space norms, enabling the characterization of function smoothness in both well-specified and misspecified settings.
• The established Lq-embedding property demonstrates that target functions satisfying a source condition automatically possess sufficient integrability, negating the need for explicit boundedness assumptions.
• The derived learning rates are independent of the output space dimension, making them applicable to a broad range of infinite-dimensional vector-valued learning problems.

Summary by alphaXiv.