# Stagewise Safe Bayesian Optimization with Gaussian Processes

arXiv: https://arxiv.org/abs/1806.07555 · HTML: https://mlsanity.com/paper/1806.07555/

## Key facts

- Cited by 79 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2018.

## Built on by (most cited first, 30 of 79)

- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Optimal Order Simple Regret for Gaussian Process Bandits](https://mlsanity.com/paper/2108.09262/) (2021)
- [Joint Entropy Search for Multi-objective Bayesian Optimization](https://mlsanity.com/paper/2210.02905/) (2022)
- [Neural Contextual Bandits without Regret](https://mlsanity.com/paper/2107.03144/) (2021)
- [Convergence and sample complexity of natural policy gradient primal-dual methods for constrained MDPs](https://mlsanity.com/paper/2206.02346/) (2022)
- [Best Arm Identification with Safety Constraints](https://mlsanity.com/paper/2111.12151/) (2021)
- [On Kernelized Multi-Armed Bandits with Constraints](https://mlsanity.com/paper/2203.15589/) (2022)
- [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)
- [Active Learning with Safety Constraints](https://mlsanity.com/paper/2206.11183/) (2022)
- [Principled Preferential Bayesian Optimization](https://mlsanity.com/paper/2402.05367/) (2024)
- [A Human-Centered Safe Robot Reinforcement Learning Framework with Interactive Behaviors](https://mlsanity.com/paper/2302.13137/) (2023)
- [On Safety in Safe Bayesian Optimization](https://mlsanity.com/paper/2403.12948/) (2024)
- [Meta-Learning Priors for Safe Bayesian Optimization](https://mlsanity.com/paper/2210.00762/) (2022)
- [Distributed Contextual Linear Bandits with Minimax Optimal Communication Cost](https://mlsanity.com/paper/2205.13170/) (2022)
- [UCB-based Algorithms for Multinomial Logistic Regression Bandits](https://mlsanity.com/paper/2103.11489/) (2021)
- [Safe Active Learning for Time-Series Modeling with Gaussian Processes](https://mlsanity.com/paper/2402.06276/) (2024)
- [Near-Optimal Multi-Agent Learning for Safe Coverage Control](https://mlsanity.com/paper/2210.06380/) (2022)
- [Safe and Efficient Model-free Adaptive Control via Bayesian Optimization](https://mlsanity.com/paper/2101.07825/) (2021)
- [Active learning for distributionally robust level-set estimation](https://mlsanity.com/paper/2102.04000/) (2021)
- [VABO: Violation-Aware Bayesian Optimization for Closed-Loop Control Performance Optimization with Unmodeled Constraints](https://mlsanity.com/paper/2110.07479/) (2021)
- [Pure Exploration in Bandits with Linear Constraints](https://mlsanity.com/paper/2306.12774/) (2023)
- [Bayesian Optimization with Formal Safety Guarantees via Online Conformal Prediction](https://mlsanity.com/paper/2306.17815/) (2023)
- [Safe exploration in reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2503.10352/) (2025)
- [Rectified Pessimistic-Optimistic Learning for Stochastic Continuum-armed Bandit with Constraints](https://mlsanity.com/paper/2211.14720/) (2022)
- [Information-Theoretic Safe Exploration with Gaussian Processes](https://mlsanity.com/paper/2212.04914/) (2022)
- [Tuning Legged Locomotion Controllers via Safe Bayesian Optimization](https://mlsanity.com/paper/2306.07092/) (2023)
- [Learning Regions of Interest for Bayesian Optimization with Adaptive Level-Set Estimation](https://mlsanity.com/paper/2307.13371/) (2023)
- [Directional Optimism for Safe Linear Bandits](https://mlsanity.com/paper/2308.15006/) (2023)
- [Safe Guaranteed Exploration for Non-linear Systems](https://mlsanity.com/paper/2402.06562/) (2024)

## Abstract (arXiv)

Enforcing safety is a key aspect of many problems pertaining to sequential decision making under uncertainty, which require the decisions made at every step to be both informative of the optimal decision and also safe. For example, we value both efficacy and comfort in medical therapy, and efficiency and safety in robotic control. We consider this problem of optimizing an unknown utility function with absolute feedback or preference feedback subject to unknown safety constraints. We develop an efficient safe Bayesian optimization algorithm, StageOpt, that separates safe region expansion and utility function maximization into two distinct stages. Compared to existing approaches which interleave between expansion and optimization, we show that StageOpt is more efficient and naturally applicable to a broader class of problems. We provide theoretical guarantees for both the satisfaction of safety constraints as well as convergence to the optimal utility value. We evaluate StageOpt on both a variety of synthetic experiments, as well as in clinical practice. We demonstrate that StageOpt is more effective than existing safe optimization approaches, and is able to safely and effectively optimize spinal cord stimulation therapy in our clinical experiments.