  Gaussian processes have become a promising tool for various safety-critical
settings, since the posterior variance can be used to directly estimate the
model error and quantify risk. However, state-of-the-art techniques for
safety-critical settings hinge on the assumption that the kernel
hyperparameters are known, which does not apply in general. To mitigate this,
we introduce robust Gaussian process uniform error bounds in settings with
unknown hyperparameters. Our approach computes a confidence region in the space
of hyperparameters, which enables us to obtain a probabilistic upper bound for
the model error of a Gaussian process with arbitrary hyperparameters. We do not
require to know any bounds for the hyperparameters a priori, which is an
assumption commonly found in related work. Instead, we are able to derive
bounds from data in an intuitive fashion. We additionally employ the proposed
technique to derive performance guarantees for a class of learning-based
control problems. Experiments show that the bound performs significantly better
than vanilla and fully Bayesian Gaussian processes.
