Gaussian Process Regression is a popular nonparametric regression method based on Bayesian principles that provides uncertainty estimates for its predictions. However, these estimates are of a Bayesian nature, whereas for some important applications, like learning-based control with safety guarantees, frequentist uncertainty bounds are required. Although such rigorous bounds are available for Gaussian Processes, they are too conservative to be useful in applications. This often leads practitioners to replacing these bounds by heuristics, thus breaking all theoretical guarantees. To address this problem, we introduce new uncertainty bounds that are rigorous, yet practically useful at the same time. In particular, the bounds can be explicitly evaluated and are much less conservative than state of the art results. Furthermore, we show that certain model misspecifications lead to only graceful degradation. We demonstrate these advantages and the usefulness of our results for learning-based control with numerical examples.

翻译：高斯进程回归是一种流行的非对称回归方法,它基于贝叶西亚原则,为预测提供了不确定性的估计。然而,这些估算具有巴耶斯性质,而对于一些重要的应用,如以学习为基础的安全保障措施控制,则需要常态不确定性界限。虽然高斯进程有这种严格的界限,但这种界限太保守,无法用于应用。这往往导致从业者用重力取代这些界限,从而打破所有理论保障。为了解决这一问题,我们引入了新的不确定性界限,这些界限是严格的,但实际上在同时有用。特别是,这些界限可以明确评估,而且比艺术成果的状态要保守得多。此外,我们证明某些模型的错误特征只能导致优雅的退化。我们用数字例子来证明这些优势以及我们结果对基于学习的控制的用处。