Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach

The problem of establishing out-of-sample bounds for the values of an unknown ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein, along with an observational model where outputs are corrupted by bounded measurement noise. The noise can originate from any compactly supported distribution, and no independent assumptions are made on the available data. In this setting, we show how computing tight, finite-sample uncertainty bounds amounts to solving parametric quadratically constrained linear programs. Next, the properties of our approach are established, and its relationship with another method is studied. Numerical experiments are presented to exemplify how the theory can be applied in various scenarios and to contrast it with other closed-form alternatives.