Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spaces

TL;DR

This paper develops methods for optimal experimental design in reproducing kernel Hilbert spaces to estimate linear functionals with bias, enabling accurate inference in modern applications like differential equations.

Contribution

It generalizes experimental design for linear functionals to RKHSs, allowing for biased estimates and providing algorithms with confidence guarantees.

Findings

Algorithms for bias-aware design in RKHSs

Non-asymptotic confidence sets under sub-Gaussian noise

Certifiable estimation with bounded error

Abstract

Optimal experimental design seeks to determine the most informative allocation of experiments to infer an unknown statistical quantity. In this work, we investigate the optimal design of experiments for {\em estimation of linear functionals in reproducing kernel Hilbert spaces (RKHSs)}. This problem has been extensively studied in the linear regression setting under an estimability condition, which allows estimating parameters without bias. We generalize this framework to RKHSs, and allow for the linear functional to be only approximately inferred, i.e., with a fixed bias. This scenario captures many important modern applications, such as estimation of gradient maps, integrals, and solutions to differential equations. We provide algorithms for constructing bias-aware designs for linear functionals. We derive non-asymptotic confidence sets for fixed and adaptive designs under sub-Gaussian noise, enabling us to certify estimation with bounded error with high probability.