Optimal Recovery from Inaccurate Data in Hilbert Spaces: Regularize, but what of the Parameter?
This work solves a specific theoretical gap in deterministic function learning for researchers in approximation theory and optimal recovery, but it is incremental as it builds directly on prior results.
The paper addresses the problem of selecting the optimal regularization parameter in Hilbert-space optimal recovery from inaccurate data, providing explicit solutions for both local (Chebyshev centers) and global (linear algorithms) scenarios, and showing that any parameter is optimal when observational functionals have orthonormal representers.
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.