Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings
This work addresses theoretical gaps in numerical integration for applied mathematicians and statisticians, but it is incremental as it extends existing analysis to misspecified settings.
This paper tackles the problem of analyzing convergence rates for kernel-based quadrature rules when the integrand's smoothness is misspecified, showing that convergence can be guaranteed under conditions on quadrature weights or design points, and revealing that Bayesian quadrature can adaptively achieve optimal rates under specific conditions.
This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.