On Safety in Safe Bayesian Optimization

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-\b{eta}-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.