Determine Gaussian-process hyperparameters for safe Bayesian optimization

Determine how to select Gaussian-process hyperparameters for safe Bayesian optimization when the objective function is genuinely unknown, so that safety-preserving optimization can be performed without relying on fixed, application-specific hyperparameters.

Background

Safe Bayesian optimization must select new parameter configurations without violating safety constraints, typically by estimating a safe set using Gaussian-process confidence bounds. The safety guarantees and practical behavior of these methods depend strongly on the Gaussian-process kernel and hyperparameters, including length scales and noise assumptions.

The paper notes that many practical safe Bayesian optimization applications fix these quantities in advance, even though the appropriate values may be difficult to know for an unknown objective. Resolving how to determine them is important for extending safe Bayesian optimization beyond controlled demonstrations and for preserving reliable safety behavior in real applications.