Optimal Sampling for Kernel Quadrature on Unbounded Domains

TL;DR

This paper introduces a robust, kernel-agnostic randomized sampling method for kernel quadrature on unbounded domains that achieves minimax error rates without kernel knowledge.

Contribution

It proposes an explicit, $n$-dependent sampling distribution that is robust and rate-optimal, extending to unbounded measures like Gaussian and Student-$t$ distributions.

Findings

Achieves minimax rates for worst-case error over smoothness classes.

Works with unbounded sampling measures such as Gaussian and Student-$t$.

Provides a practical recipe for robust, rate-optimal randomized quadrature.

Abstract

Kernel quadrature is widely used to approximate integrals of smooth functions, with worst-case error typically decaying at the minimax rate $n^{-\alpha/d}$ for smoothness $\alpha$ in dimension $d$. 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, $n$-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-$t$ distributions, extending beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal randomized quadrature.