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.

本文旨在解决高斯过程回归（GPR）在存在有界支持噪声时的误差量化问题，提供了更为精确的误差界限。通过引入集中不等式，并假设潜在函数在对应于高斯过程核的再生核希尔伯特空间中具有低复杂度，研究表明这些新界限在神经网络核的GPR（深度核学习）中尤为适用，显著提升了控制系统的安全概率。

在有界支持噪声下的高斯过程回归误差界限及其在安全认证中的应用