Optimal Recovery from Inaccurate Data in Hilbert Spaces: Regularize, but what of the Parameter?

TL;DR

This paper investigates optimal regularization parameters in Hilbert space recovery problems with noisy data, providing explicit solutions in local and global scenarios and revealing cases where any parameter is optimal.

Contribution

It offers new methods for determining regularization parameters in optimal recovery, including explicit equations and semidefinite programs, filling previous gaps in the theory.

Findings

Explicit equations for regularization parameters in local scenarios.

Semidefinite programming approach for global scenarios.

Any regularization parameter is optimal with orthonormal representers.

Abstract

In Optimal Recovery, the task of learning a function from observational data is tackled deterministically by adopting a worst-case perspective tied to an explicit model assumption made on the functions to be learned. Working in the framework of Hilbert spaces, this article considers a model assumption based on approximability. It also incorporates observational inaccuracies modeled via additive errors bounded in $\ell_2$. Earlier works have demonstrated that regularization provide algorithms that are optimal in this situation, but did not fully identify the desired hyperparameter. This article fills the gap in both a local scenario and a global scenario. In the local scenario, which amounts to the determination of Chebyshev centers, the semidefinite recipe of Beck and Eldar (legitimately valid in the complex setting only) is complemented by a more direct approach, with the proviso that the observational functionals have orthonormal representers. In the said approach, the desired parameter is the solution to an equation that can be resolved via standard methods. In the global scenario, where linear algorithms rule, the parameter elusive in the works of Micchelli et al. is found as the byproduct of a semidefinite program. Additionally and quite surprisingly, in case of observational functionals with orthonormal representers, it is established that any regularization parameter is optimal.