  Bayesian optimization has emerged as a highly effective tool for the safe
online optimization of systems, due to its high sample efficiency and noise
robustness. To further enhance its efficiency, reduced physical models of the
system can be incorporated into the optimization process, accelerating it.
These models are able to offer an approximation of the actual system, and
evaluating them is significantly cheaper. The similarity between the model and
reality is represented by additional hyperparameters, which are learned within
the optimization process. Safety is a crucial criterion for online optimization
methods such as Bayesian optimization, which has been addressed by recent works
that provide safety guarantees under the assumption of known hyperparameters.
In practice, however, this does not apply. Therefore, we extend the robust
Gaussian process uniform error bounds to meet the multi-task setting, which
involves the calculation of a confidence region from the hyperparameter
posterior distribution utilizing Markov chain Monte Carlo methods.
Subsequently, the robust safety bounds are employed to facilitate the safe
optimization of the system, while incorporating measurements of the models.
Simulation results indicate that the optimization can be significantly
accelerated for expensive to evaluate functions in comparison to other
state-of-the-art safe Bayesian optimization methods, contingent on the fidelity
of the models.
