# On Safety in Safe Bayesian Optimization

arXiv: https://arxiv.org/abs/2403.12948 · HTML: https://mlsanity.com/paper/2403.12948/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 74 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Builds on (most cited first, 30 of 33)

- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Bayesian Optimization with Safety Constraints: Safe and Automatic Parameter Tuning in Robotics](https://mlsanity.com/paper/1602.04450/) (2016)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Linear Stochastic Bandits Under Safety Constraints](https://mlsanity.com/paper/1908.05814/) (2019)
- [Stagewise Safe Bayesian Optimization with Gaussian Processes](https://mlsanity.com/paper/1806.07555/) (2018)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Safe Exploration in Finite Markov Decision Processes with Gaussian Processes](https://mlsanity.com/paper/1606.04753/) (2016)
- [Adaptive and Safe Bayesian Optimization in High Dimensions via One-Dimensional Subspaces](https://mlsanity.com/paper/1902.03229/) (2019)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Bandits and Experts in Metric Spaces](https://mlsanity.com/paper/1312.1277/) (2013)
- [Conservative Bandits](https://mlsanity.com/paper/1602.04282/) (2016)
- [Misspecified Gaussian Process Bandit Optimization](https://mlsanity.com/paper/2111.05008/) (2021)
- [No-Regret Bayesian Optimization with Unknown Hyperparameters](https://mlsanity.com/paper/1901.03357/) (2019)
- [A General Framework for Constrained Bayesian Optimization using Information-based Search](https://mlsanity.com/paper/1511.09422/) (2015)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Global optimization of Lipschitz functions](https://mlsanity.com/paper/1703.02628/) (2017)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [GoSafe: Globally Optimal Safe Robot Learning](https://mlsanity.com/paper/2105.13281/) (2021)
- [On the Sublinear Regret of GP-UCB](https://mlsanity.com/paper/2307.07539/) (2023)
- [GoSafeOpt: Scalable Safe Exploration for Global Optimization of Dynamical Systems](https://mlsanity.com/paper/2201.09562/) (2022)
- [Safe Learning and Optimization Techniques: Towards a Survey of the State of the Art](https://mlsanity.com/paper/2101.09505/) (2021)
- [Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posed](https://mlsanity.com/paper/2203.09179/) (2022)
- [Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions](https://mlsanity.com/paper/2001.10965/) (2020)
- [Robot Learning with Crash Constraints](https://mlsanity.com/paper/2010.08669/) (2020)
- [Cautious Bayesian Optimization for Efficient and Scalable Policy Search](https://mlsanity.com/paper/2011.09445/) (2020)
- [Automatic nonlinear MPC approximation with closed-loop guarantees](https://mlsanity.com/paper/2312.10199/) (2023)
- [Provably Robust Learning-Based Approach for High-Accuracy Tracking Control of Lagrangian Systems](https://mlsanity.com/paper/1804.01031/) (2018)
- [Lazily Adapted Constant Kinky Inference for Nonparametric Regression and Model-Reference Adaptive Control](https://mlsanity.com/paper/1701.00178/) (2016)

## Built on by (most cited first, 15 of 15)

- [ActSafe: Active Exploration with Safety Constraints for Reinforcement Learning](https://mlsanity.com/paper/2410.09486/) (2024)
- [Safe exploration in reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2503.10352/) (2025)
- [PACSBO: Probably approximately correct safe Bayesian optimization](https://mlsanity.com/paper/2409.01163/) (2024)
- [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)
- [Future Aware Safe Active Learning of Time Varying Systems using Gaussian Processes](https://mlsanity.com/paper/2405.10581/) (2024)
- [An Analysis of Safety Guarantees in Multi-Task Bayesian Optimization](https://mlsanity.com/paper/2503.08555/) (2025)
- [Safe and Near-Optimal Control with Online Dynamics Learning](https://mlsanity.com/paper/2509.16650/) (2025)
- [Optimal uncertainty bounds for multivariate kernel regression under bounded noise: A Gaussian process-based dual function](https://mlsanity.com/paper/2603.16481/) (2026)
- [Safe learning-based control via function-based uncertainty quantification](https://mlsanity.com/paper/2604.01173/) (2026)
- [Safe Bayesian Optimization for Complex Control Systems via Additive Gaussian Processes](https://mlsanity.com/paper/2408.16307/) (2024)
- [Kernel conditional tests from learning-theoretic bounds](https://mlsanity.com/paper/2506.03898/) (2025)
- [On Uniform Error Bounds for Kernel Regression under Non-Gaussian Noise](https://mlsanity.com/paper/2605.09757/) (2026)
- [A Human-in-the-Loop Bayesian Optimization Framework for Constraint-Aware Bioprocess Development](https://mlsanity.com/paper/2606.19230/) (2026)
- [Computationally efficient safe exploration in reinforcement learning](https://mlsanity.com/paper/2609.22919/) (2026)

## Abstract (arXiv)

Optimizing an unknown function under safety constraints is a central task in robotics, biomedical engineering, and many other disciplines, and increasingly safe Bayesian Optimization (BO) is used for this. Due to the safety critical nature of these applications, it is of utmost importance that theoretical safety guarantees for these algorithms translate into the real world. In this work, we investigate three safety-related issues of the popular class of SafeOpt-type algorithms. First, these algorithms critically rely on frequentist uncertainty bounds for Gaussian Process (GP) regression, but concrete implementations typically utilize heuristics that invalidate all safety guarantees. We provide a detailed analysis of this problem and introduce Real-beta-SafeOpt, a variant of the SafeOpt algorithm that leverages recent GP bounds and thus retains all theoretical guarantees. Second, we identify assuming an upper bound on the reproducing kernel Hilbert space (RKHS) norm of the target function, a key technical assumption in SafeOpt-like algorithms, as a central obstacle to real-world usage. To overcome this challenge, we introduce the Lipschitz-only Safe Bayesian Optimization (LoSBO) algorithm, which guarantees safety without an assumption on the RKHS bound, and empirically show that this algorithm is not only safe, but also exhibits superior performance compared to the state-of-the-art on several function classes. Third, SafeOpt and derived algorithms rely on a discrete search space, making them difficult to apply to higher-dimensional problems. To widen the applicability of these algorithms, we introduce Lipschitz-only GP-UCB (LoS-GP-UCB), a variant of LoSBO applicable to moderately high-dimensional problems, while retaining safety.