Sequential Bayesian Quadrature with Hilbert space Gaussian processes

Bayesian quadrature provides a probabilistic approach to numerical integration, but exact sequential Bayesian quadrature becomes increasingly expensive as function evaluations accumulate. We develop a scalable sequential Bayesian quadrature (BQ) method based on Hilbert space Gaussian process (HSGP) approximations of Mat\'ern kernels. Both the quadrature estimator and an integral-targeted variance-reduction acquisition are represented in a finite Laplacian eigenbasis, while a geometry-stabilization constraint guarantees quasi-uniform sequential designs. For integrands in Sobolev spaces, we establish nonasymptotic mean-squared integration-error bounds that allow the target smoothness to differ from that of the Mat\'ern reproducing kernel Hilbert space (RKHS), covering both noiseless and noisy observations. The error bounds explicitly track the feature budget, kernel lengthscale, regularization, sampling resolution, and observation noise, and determine how many features are sufficient to preserve the corresponding full-kernel quadrature rates. They further characterize response-independent parameter sequences that avoid repeated marginal-likelihood evaluation and optimization. Computation is carried out entirely in the finite feature space, substantially reducing the complexity for each integral evaluation relative to full-kernel BQ. Numerical experiments confirm the predicted trade-offs among accuracy, feature dimension, and runtime against full-kernel and approximate Bayesian quadrature baselines.