In application areas where data generation is expensive, Gaussian processes
are a preferred supervised learning model due to their high data-efficiency.
Particularly in model-based control, Gaussian processes allow the derivation of
performance guarantees using probabilistic model error bounds. To make these
approaches applicable in practice, two open challenges must be solved i)
Existing error bounds rely on prior knowledge, which might not be available for
many real-world tasks. (ii) The relationship between training data and the
posterior variance, which mainly drives the error bound, is not well understood
and prevents the asymptotic analysis. This article addresses these issues by
presenting a novel uniform error bound using Lipschitz continuity and an
analysis of the posterior variance function for a large class of kernels.
Additionally, we show how these results can be used to guarantee safe control
of an unknown dynamical system and provide numerical illustration examples.