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

arXiv: https://arxiv.org/abs/2104.09582 · HTML: https://mlsanity.com/paper/2104.09582/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 53 references that are in the map.
- First posted on arXiv 19 Apr 2021.

## Builds on (most cited first, 20 of 20)

- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Every Model Learned by Gradient Descent Is Approximately a Kernel Machine](https://mlsanity.com/paper/2012.00152/) (2020)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [Learning Model Predictive Control for iterative tasks. A Data-Driven Control Framework](https://mlsanity.com/paper/1609.01387/) (2016)
- [Approximation beats concentration? An approximation view on inference with smooth radial kernels](https://mlsanity.com/paper/1801.03437/) (2018)
- [Planning with Learned Dynamics: Probabilistic Guarantees on Safety and Reachability via Lipschitz Constants](https://mlsanity.com/paper/2010.08993/) (2020)
- [Feedback Linearization based on Gaussian Processes with event-triggered Online Learning](https://mlsanity.com/paper/1911.06565/) (2019)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [Probabilistic model predictive safety certification for learning-based control](https://mlsanity.com/paper/1906.10417/) (2019)
- [Set Membership identification of linear systems with guaranteed simulation accuracy](https://mlsanity.com/paper/2001.07628/) (2020)
- [KPC: Learning-Based Model Predictive Control with Deterministic Guarantees](https://mlsanity.com/paper/2011.11303/) (2020)

## Built on by (most cited first, 14 of 14)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Automatic nonlinear MPC approximation with closed-loop guarantees](https://mlsanity.com/paper/2312.10199/) (2023)
- [Safe exploration in reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2503.10352/) (2025)
- [Tighter Confidence Bounds for Sequential Kernel Regression](https://mlsanity.com/paper/2403.12732/) (2024)
- [PACSBO: Probably approximately correct safe Bayesian optimization](https://mlsanity.com/paper/2409.01163/) (2024)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Streaming Generated Gaussian Process Experts for Online Learning and Control: Extended Version](https://mlsanity.com/paper/2508.03679/) (2025)
- [Safe reinforcement learning in uncertain contexts](https://mlsanity.com/paper/2401.05876/) (2024)
- [Derandomizing Simultaneous Confidence Regions for Band-Limited Functions by Improved Norm Bounds and Majority-Voting Schemes](https://mlsanity.com/paper/2506.17764/) (2025)
- [Single Image Inpainting and Super-Resolution with Simultaneous Uncertainty Guarantees by Universal Reproducing Kernels](https://mlsanity.com/paper/2506.23221/) (2025)
- [Optimal uncertainty bounds for multivariate kernel regression under bounded noise: A Gaussian process-based dual function](https://mlsanity.com/paper/2603.16481/) (2026)
- [Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors](https://mlsanity.com/paper/2307.04415/) (2023)
- [On Uniform Error Bounds for Kernel Regression under Non-Gaussian Noise](https://mlsanity.com/paper/2605.09757/) (2026)
- [Mitigating Over-Optimization in PRM-Guided Search in Mathematical Reasoning by Optimizing the Guide](https://mlsanity.com/paper/2608.30051/) (2026)

## Abstract (arXiv)

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.