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

arXiv: https://arxiv.org/abs/2603.16481 · HTML: https://mlsanity.com/paper/2603.16481/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 27 references that are in the map.
- First posted on arXiv 17 Mar 2026.

## Builds on (most cited first, 16 of 16)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)
- [Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach](https://mlsanity.com/paper/2104.09582/) (2021)
- [Learning-based Symbolic Abstractions for Nonlinear Control Systems](https://mlsanity.com/paper/2004.01879/) (2020)
- [Set Membership identification of linear systems with guaranteed simulation accuracy](https://mlsanity.com/paper/2001.07628/) (2020)
- [Error Bounds For Gaussian Process Regression Under Bounded Support Noise With Applications To Safety Certification](https://mlsanity.com/paper/2408.09033/) (2024)
- [Gaussian processes for dynamics learning in model predictive control](https://mlsanity.com/paper/2502.02310/) (2025)
- [Guarantees for data-driven control of nonlinear systems using semidefinite programming: A survey](https://mlsanity.com/paper/2306.16042/) (2023)
- [Towards safe Bayesian optimization with Wiener kernel regression](https://mlsanity.com/paper/2411.02253/) (2024)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Kernel-based Learning for Safe Control of Discrete-Time Unknown Systems under Incomplete Observations](https://mlsanity.com/paper/2405.00822/) (2024)

## Built on by (most cited first, 1 of 1)

- [Bayesian Optimization with Structured Measurements: A Vector-Valued RKHS Framework](https://mlsanity.com/paper/2605.09775/) (2026)

## Abstract (arXiv)

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 directly apply in the multi-output case, or are difficult to integrate into downstream tasks. This paper addresses these limitations by presenting a tight, deterministic bound for multi-output functions in Reproducing Kernel Hilbert Spaces (RKHSs) subject to bounded noise. It is obtained through an unconstrained, duality-based formulation, which shares the same structure as classic Gaussian process confidence bounds, and can thus be straightforwardly integrated into downstream optimization pipelines. We show that the proposed bound generalizes existing results and illustrate its application using an example inspired by quadrotor dynamics learning.