2022/02/22 by Xu Cai, Cai, Xu, Chi Thanh Lam +3
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Information Theory (cs.IT) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Mathematical Approximation and Integration #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2202.10615
openalex publication_date 2022/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study error bounds for \em Bayesian quadrature (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a \em Reproducing Kernel Hilbert Space (RKHS), particularly focusing on the Matérn-ν and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the L2-function approximation error. When specialized to the Matérn kernel, we recover an existing near-optimal error rate while avoiding the existing method of repeatedly sampling points. When specialized to other settings, we obtain new average-case results for settings including the SE kernel with noise and the Matérn kernel with misspecification. Finally, we present algorithm-independent lower bounds that have greater generality and/or give distinct proofs compared to existing ones.