Error Bounds For Gaussian Process Regression Under Bounded Support Noise With Applications To Safety Certification
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.
Code (0)
등록된 구현이 없습니다.
Tasks
GPRSimilar Papers 제목 키워드 기반
Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control
Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provi…
Gaussian ProcessesregressionOn Uniform Error Bounds for Kernel Regression under Non-Gaussian Noise
Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critic…
Towards safe Bayesian optimization with Wiener kernel regression
Bayesian Optimization (BO) is a data-driven strategy for minimizing/maximizing black-box functions based on probabilistic surrogate models. In the presence of safety constraints, the performance of BO crucially relies on…
Bayesian OptimizationregressionGeneralization Errors and Learning Curves for Regression with Multi-task Gaussian Processes
We provide some insights into how task correlations in multi-task Gaussian process (GP) regression affect the generalization error and the learning curve. We analyze the asymmetric two-task case, where a secondary task …
Gaussian ProcessesregressionUniform Error and Posterior Variance Bounds for Gaussian Process Regression with Application to Safe Control
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 d…
Gaussian Processesregression