# PACSBO: Probably approximately correct safe Bayesian optimization

arXiv: https://arxiv.org/abs/2409.01163 · HTML: https://mlsanity.com/paper/2409.01163/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 30 references that are in the map.
- First posted on arXiv 2 Sep 2024.

## Builds on (most cited first, 17 of 17)

- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Benchmarking Deep Reinforcement Learning for Continuous Control](https://mlsanity.com/paper/1604.06778/) (2016)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Bayesian Optimization with Safety Constraints: Safe and Automatic Parameter Tuning in Robotics](https://mlsanity.com/paper/1602.04450/) (2016)
- [Stagewise Safe Bayesian Optimization with Gaussian Processes](https://mlsanity.com/paper/1806.07555/) (2018)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Automatic LQR Tuning Based on Gaussian Process Global Optimization](https://mlsanity.com/paper/1605.01950/) (2016)
- [GoSafe: Globally Optimal Safe Robot Learning](https://mlsanity.com/paper/2105.13281/) (2021)
- [GoSafeOpt: Scalable Safe Exploration for Global Optimization of Dynamical Systems](https://mlsanity.com/paper/2201.09562/) (2022)
- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)
- [Deep Kernels for Optimizing Locomotion Controllers](https://mlsanity.com/paper/1707.09062/) (2017)
- [Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach](https://mlsanity.com/paper/2104.09582/) (2021)
- [Automatic nonlinear MPC approximation with closed-loop guarantees](https://mlsanity.com/paper/2312.10199/) (2023)
- [Learning-based Symbolic Abstractions for Nonlinear Control Systems](https://mlsanity.com/paper/2004.01879/) (2020)
- [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, 8 of 8)

- [Safe exploration in reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2503.10352/) (2025)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Safety and optimality in learning-based control at low computational cost](https://mlsanity.com/paper/2505.08026/) (2025)
- [Safe Bayesian optimization across noise models via scenario programming](https://mlsanity.com/paper/2512.11580/) (2025)
- [Safe learning-based control via function-based uncertainty quantification](https://mlsanity.com/paper/2604.01173/) (2026)
- [Towards safe control parameter tuning in distributed multi-agent systems](https://mlsanity.com/paper/2508.13608/) (2025)
- [Asynchronous Cooperative Online Learning for Multi-Robot Control under Computational Delays](https://mlsanity.com/paper/2608.29562/) (2026)
- [Event-triggered Control and Online Learning for Networked Systems under Computational Delays](https://mlsanity.com/paper/2608.29576/) (2026)

## Abstract (arXiv)

Safe Bayesian optimization (BO) algorithms promise to find optimal control policies without knowing the system dynamics while at the same time guaranteeing safety with high probability. In exchange for those guarantees, popular algorithms require a smoothness assumption: a known upper bound on a norm in a reproducing kernel Hilbert space (RKHS). The RKHS is a potentially infinite-dimensional space, and it is unclear how to, in practice, obtain an upper bound of an unknown function in its corresponding RKHS. In response, we propose an algorithm that estimates an upper bound on the RKHS norm of an unknown function from data and investigate its theoretical properties. Moreover, akin to Lipschitz-based methods, we treat the RKHS norm as a local rather than a global object, and thus reduce conservatism. Integrating the RKHS norm estimation and the local interpretation of the RKHS norm into a safe BO algorithm yields PACSBO, an algorithm for probably approximately correct safe Bayesian optimization, for which we provide numerical and hardware experiments that demonstrate its applicability and benefits over popular safe BO algorithms.