Gaussian Process Regression (GPR) is a powerful and elegant method for learning complex functions from noisy data with a wide range of applications, including in safety-critical domains. Such applications have two key features: (i) they require rigorous error quantification, and (ii) the noise is often bounded and non-Gaussian due to, e.g., physical constraints. While error bounds for applying GPR in the presence of non-Gaussian noise exist, they tend to be overly restrictive and conservative in practice. In this paper, we provide novel error bounds for GPR under bounded support noise. Specifically, by relying on concentration inequalities and assuming that the latent function has low complexity in the reproducing kernel Hilbert space (RKHS) corresponding to the GP kernel, we derive both probabilistic and deterministic bounds on the error of the GPR. We show that these errors are substantially tighter than existing state-of-the-art bounds and are particularly well-suited for GPR with neural network kernels, i.e., Deep Kernel Learning (DKL). Furthermore, motivated by applications in safety-critical domains, we illustrate how these bounds can be combined with stochastic barrier functions to successfully quantify the safety probability of an unknown dynamical system from finite data. We validate the efficacy of our approach through several benchmarks and comparisons against existing bounds. The results show that our bounds are consistently smaller, and that DKLs can produce error bounds tighter than sample noise, significantly improving the safety probability of control systems.

The St. Louis and Kansas City metropolitan areas each face challenges in meeting national ambient air quality standards for particulate matter and ozone. This effort introduces CALIPSO and OMI satellite observations into State air-quality decision-making. By integrating remote sensing data, monitoring will evolve from a sparse, 2-D network into the third (vertical) dimension to allow for modeling comparisons above ground level. In support of the air quality and public health national applications and focusing on air quality compliance, planning, and emissions Inventories, the work proposes to enhance the Missouri State implementation plan (SIP) decision support system with NASA Earth science results to improve decision-making related to ozone and particulate matter. Expected outcomes include improved decision support information providing enhanced decisions, improved air quality management choices, improved air quality, improved health, and lowered societal cost.