Kernel quadrature is widely used to approximate integrals of smooth functions, with worst-case error typically decaying at the minimax rate $n^{-α/d}$ for smoothness $α$ 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.

翻译：核求积广泛用于逼近光滑函数的积分，其最坏情况误差通常以极小化最大速率 $n^{-α/d}$ 衰减，其中 $α$ 为光滑度，$d$ 为维度。现有速率最优方法通常依赖于针对特定核设计的确定性点集，导致其对模型误设敏感且实践中鲁棒性较差。本研究聚焦随机化求积方法，以鲁棒性而非核特定最优性为核心。我们构造了一种显式的、依赖 $n$ 的抽样分布，无需核先验知识即可在光滑类上的最坏情况误差中达到极小化最大速率。这种与核无关的设计在保持最优速率的同时增强了鲁棒性。我们的分析涵盖高斯分布和学生 $t$ 分布等无界抽样测度，将适用范围扩展至紧致域之外。这些结果既提供了理论保证，也为实现鲁棒且速率最优的随机化求积提供了实用方案。