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

TL;DR

This paper develops a convex optimization method to compute tight, finite-sample uncertainty bounds for unknown functions in reproducing kernel Hilbert spaces, accommodating bounded noise without independence assumptions.

Contribution

It introduces a novel convex optimization framework for deriving out-of-sample bounds in RKHS, applicable under broad noise conditions and without data independence assumptions.

Findings

Bounds are computed via parametric quadratically constrained linear programs.

The approach is shown to be tight and versatile across scenarios.

Numerical experiments demonstrate practical effectiveness and advantages over existing methods.

Abstract

The problem of establishing out-of-sample bounds for the values of an unkonwn 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 independence 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, properties of our approach are established and its relationship with another methods is studied. Numerical experiments are presented to exemplify how the theory can be applied in a number of scenarios, and to contrast it with other closed-form alternatives.