  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.
