Paul Scharnhorst

Paul Scharnhorst, Emilio T. Maddalena, Yuning Jiang et al.

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.

Emilio T. Maddalena, Paul Scharnhorst, Colin N. Jones

We consider the problem of reconstructing a function from a finite set of noise-corrupted samples. Two kernel algorithms are analyzed, namely kernel ridge regression and $\varepsilon$-support vector regression. By assuming the ground-truth function belongs to the reproducing kernel Hilbert space of the chosen kernel, and the measurement noise affecting the dataset is bounded, we adopt an approximation theory viewpoint to establish \textit{deterministic}, finite-sample error bounds for the two models. Finally, we discuss their connection with Gaussian processes and two numerical examples are provided. In establishing our inequalities, we hope to help bring the fields of non-parametric kernel learning and system identification for robust control closer to each other.