Archive for contraction rate

optimal sampling for kernel quadrature on unbounded domains

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , on May 22, 2026 by xi'an

My PhD student Edoardo Bandoni, along with Julien Stoehr and myself, completed a paper on validating (Bayesian) kernel quadrature with unbounded domains of integration. Which connects with probabilistic numerics, since the integrand is modelled as a Gaussian process. And RKHS methods. As opposed to Monte Carlo estimators, quadrature methods approximate integrals of smooth functions with worst-case error decaying at a minimax rate α/d for smoothness α in dimension d. Existing rate-optimal quadrature methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and thus less robust in practice. This paper studies instead randomised quadrature methods, with a focus on robustness rather than on kernel-specific optimality. We construct an explicit, n-dependent, sampling distribution that achieves minimax rates for worst-case errors over smoothness classes without requiring knowledge of the kernel. This kernel-agnostic design does improve robustness while retaining optimal rates and extends Briol et al.  (2019) to the unbounded case. Which cannot always be easily handled by a change of variables. Our result thus mostly covers 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, randomised quadrature. [The above is mostly stated in the abstract.]

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