Optimal uncertainty bounds for multivariate kernel regression under bounded noise: A Gaussian process-based dual function

TL;DR

This paper introduces a tight, distribution-free uncertainty bound for multivariate kernel regression that improves scalability and integration into control tasks, based on a duality approach similar to Gaussian process confidence bounds.

Contribution

It presents a novel, unconstrained duality-based formulation for uncertainty bounds in multivariate kernel regression, overcoming limitations of existing methods.

Findings

The bound is tight and distribution-free.

It generalizes many existing uncertainty bounds.

Application demonstrated in quadrotor dynamics learning.

Abstract

Non-conservative uncertainty bounds are essential for making reliable predictions about latent functions from noisy data--and thus, a key enabler for safe learning-based control. In this domain, kernel methods such as Gaussian process regression are established techniques, thanks to their inherent uncertainty quantification mechanism. Still, existing bounds either pose strong assumptions on the underlying noise distribution, are conservative, do not scale well in the multi-output case, or are difficult to integrate into downstream tasks. This paper addresses these limitations by presenting a tight, distribution-free bound for multi-output kernel-based estimates. It is obtained through an unconstrained, duality-based formulation, which shares the same structure of classic Gaussian process confidence bounds and can thus be straightforwardly integrated into downstream optimization pipelines. We show that the proposed bound generalizes many existing results and illustrate its application using an example inspired by quadrotor dynamics learning.