English

Optimal Sampling for Kernel Quadrature on Unbounded Domains

Computation 2026-05-19 v1 Methodology

Abstract

Kernel quadrature is widely used to approximate integrals of smooth functions, with worst-case error typically decaying at the minimax rate nα/dn^{-\alpha/d} for smoothness α\alpha in dimension dd. Existing rate-optimal methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and less robust in practice. In this work, we study randomized quadrature methods with a focus on robustness rather than kernel-specific optimality. We construct an explicit, nn-dependent sampling distribution that achieves minimax rates for worst-case error over smoothness classes without requiring knowledge of the kernel. This kernel-agnostic design improves robustness while retaining optimal rates. Our analysis includes unbounded sampling measures such as Gaussian and Student-tt distributions, extending beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal randomized quadrature.

Keywords

Cite

@article{arxiv.2605.18134,
  title  = {Optimal Sampling for Kernel Quadrature on Unbounded Domains},
  author = {Edoardo Bandoni and Christian Robert and Julien Stoehr},
  journal= {arXiv preprint arXiv:2605.18134},
  year   = {2026}
}