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：高斯过程回归是一种基于贝叶斯原理的流行非参数回归方法，可为预测提供不确定性估计。然而，这些估计本质上是贝叶斯性质的，但在一些重要应用（如具有安全保证的基于学习的控制）中，需要频率主义的不确定性界。尽管高斯过程存在严格的频率主义界，但这些界过于保守而无法在实际应用中发挥作用。这常导致从业者用启发式方法替代这些界，从而破坏所有理论保证。为解决此问题，我们提出既严格又实际可用的新不确定性界。特别地，这些界可显式计算，且远优于现有最先进结果。进一步，我们证明特定模型误设仅导致性能温和下降。通过数值示例，我们展示了这些优势及其结果在基于学习的控制中的实用性。