mlsanityData to 7 Oct 2026

Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression

arXiv 2105.02796 ↗First posted 6 May 2021cs.LG cs.SY eess.SYcited by 49

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Abstract (arXiv)

Gaussian Process Regression is a popular nonparametric regression method based on Bayesian principles that provides uncertainty estimates for its predictions. However, these estimates are of a Bayesian nature, whereas for some important applications, like learning-based control with safety guarantees, frequentist uncertainty bounds are required. Although such rigorous bounds are available for Gaussian Processes, they are too conservative to be useful in applications. This often leads practitioners to replacing these bounds by heuristics, thus breaking all theoretical guarantees. To address this problem, we introduce new uncertainty bounds that are rigorous, yet practically useful at the same time. In particular, the bounds can be explicitly evaluated and are much less conservative than state of the art results. Furthermore, we show that certain model misspecifications lead to only graceful degradation. We demonstrate these advantages and the usefulness of our results for learning-based control with numerical examples.

Key facts. Cited by 49 papers in the map (data through 7 Oct 2026). Builds on 17 of its 39 references that are in the map. First posted on arXiv 6 May 2021.

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17 of 39 references.

References · 17

  1. 2017On Kernelized Multi-armed Banditscited by 348
  2. 2017Safe Model-based Reinforcement Learning with Stability Guaranteescited by 347
  3. 2018Gaussian Processes and Kernel Methods: A Review on Connections and Equivalencescited by 188
  4. 2015High Dimensional Bayesian Optimisation and Bandits via Additive Modelscited by 137
  5. 2018Learning-based Model Predictive Control for Safe Explorationcited by 116
  6. 2015Safe Controller Optimization for Quadrotors with Gaussian Processescited by 77
  7. 2019Uniform Error Bounds for Gaussian Process Regression with Application to Safe Controlcited by 64
  8. 2017Linearly constrained Gaussian processescited by 41
  9. 2016Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processescited by 38
  10. 2017Scalable Log Determinants for Gaussian Process Kernel Learningcited by 34
  11. 2020Deterministic error bounds for kernel-based learning techniques under bounded noisecited by 24
  12. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
5 more
  1. 2015Large-scale Log-determinant Computation through Stochastic Chebyshev Expansionscited by 17
  2. 2020Learning Constrained Dynamics with Gauss Principle adhering Gaussian Processescited by 13
  3. 2018Provably Robust Learning-Based Approach for High-Accuracy Tracking Control of Lagrangian Systemscited by 9
  4. 2018Mean Square Prediction Error of Misspecified Gaussian Process Modelscited by 4
  5. 2018Scalable Gaussian Process Inference with Finite-data Mean and Variance Guaranteescited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 4; 2023: 12; 2024: 12; 2025: 13; 2026: 5.

49 papers.

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  1. 2021Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applicationscited by 17
  2. 2022GoSafeOpt: Scalable Safe Exploration for Global Optimization of Dynamical Systemscited by 17
  3. 2024Bayesian optimization as a flexible and efficient design framework for sustainable process systemscited by 15
  4. 2024On Safety in Safe Bayesian Optimizationcited by 15
  5. 2023Guaranteed Coverage Prediction Intervals with Gaussian Process Regressioncited by 12
  6. 2023Automatic nonlinear MPC approximation with closed-loop guaranteescited by 12
  7. 2021Safe Policy Learning through Extrapolation: Application to Pre-trial Risk Assessmentcited by 10
  8. 2025Safe exploration in reproducing kernel Hilbert spacescited by 10
  9. 2022Information-Theoretic Safe Exploration with Gaussian Processescited by 9
  10. 2023Tuning Legged Locomotion Controllers via Safe Bayesian Optimizationcited by 8
  11. 2024PACSBO: Probably approximately correct safe Bayesian optimizationcited by 8
  12. 2023BO-Muse: A human expert and AI teaming framework for accelerated experimental designcited by 6
37 more
  1. 2024Safe and Stable Closed-Loop Learning for Neural-Network-Supported Model Predictive Controlcited by 6
  2. 2023Towards Safe Multi-Task Bayesian Optimizationcited by 5
  3. 2024Towards safe Bayesian optimization with Wiener kernel regressioncited by 4
  4. 2025The Hardness of Validating Observational Studies with Experimental Datacited by 4
  5. 2025Optimal kernel regression bounds under energy-bounded noisecited by 4
  6. 2023Information-theoretic Analysis of Test Data Sensitivity in Uncertaintycited by 3
  7. 2023Robust Bayesian Satisficingcited by 3
  8. 2023Exact Inference for Continuous-Time Gaussian Process Dynamicscited by 3
  9. 2023Lipschitz and H"older Continuity in Reproducing Kernel Hilbert Spacescited by 3
  10. 2024Data-Driven Abstractions via Binary-Tree Gaussian Processes for Formal Verificationcited by 3
  11. 2024On Foundation Models for Dynamical Systems from Purely Synthetic Datacited by 3
  12. 2021Learning-enhanced robust controller synthesis with rigorous statistical and control-theoretic guaranteescited by 2
  13. 2022Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functionscited by 2
  14. 2024Meta-Analysis with Untrusted Datacited by 2
  15. 2023Uncertainty-aware transfer across tasks using hybrid model-based successor feature reinforcement learningcited by 1
  16. 2024Safe reinforcement learning in uncertain contextscited by 1
  17. 2025An Analysis of Safety Guarantees in Multi-Task Bayesian Optimizationcited by 1
  18. 2025Robust Satisficing Gaussian Process Bandits Under Adversarial Attackscited by 1
  19. 2025Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphscited by 1
  20. 2025A Hierarchical Surrogate Model for Efficient Multi-Task Parameter Learning in Closed-Loop Controlcited by 1
  21. 2025Safe Bayesian optimization across noise models via scenario programmingcited by 1
  22. 2026Optimal uncertainty bounds for multivariate kernel regression under bounded noise: A Gaussian process-based dual functioncited by 1
  23. 2026Safe learning-based control via function-based uncertainty quantificationcited by 1
  24. 2026Koopman operator theory: fundamentals, control, and applicationscited by 1
  25. 2022Excess risk analysis for epistemic uncertainty with application to variational inferencecited by 0
  26. 2023Error Bounds for Kernel-Based Linear System Identification with Unknown Hyperparameterscited by 0
  27. 2023Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errorscited by 0
  28. 2024Information-Theoretic Safe Bayesian Optimizationcited by 0
  29. 2024Efficient Two-Stage Gaussian Process Regression Via Automatic Kernel Search and Subsamplingcited by 0
  30. 2024Vector Optimization with Gaussian Process Banditscited by 0
  31. 2025Kernel conditional tests from learning-theoretic boundscited by 0
  32. 2025A robust and adaptive MPC formulation for Gaussian process modelscited by 0
  33. 2025Information Preserving Line Search via Bayesian Optimizationcited by 0
  34. 2025Towards safe control parameter tuning in distributed multi-agent systemscited by 0
  35. 2025Practical Global and Local Bounds in Gaussian Process Regression via Chainingcited by 0
  36. 2026On Uniform Error Bounds for Kernel Regression under Non-Gaussian Noisecited by 0
  37. 2026Covariance-Boosted Gaussian Processes for Spatiotemporal Irregularitiescited by 0