This paper presents 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.

中文翻译：

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指定环境下确定性基于核的正交规则的收敛性分析

本文介绍了在错误指定的环境中基于核的正交规则的收敛性分析，重点是Sobolev空间中的确定性正交。特别是，我们要处理错误指定的设置，在这些设置中，测试被积函数不如Sobolev RKHS平滑，后者基于该函数构造正交规则。我们基于正交规则的两个不同假设提供收敛保证：一个基于正交权重，另一个基于设计点。更准确地说，我们表明可以得出收敛速度（i）如果绝对权重的总和保持恒定（或不迅速增加），或者（ii）如果设计点之间的最小距离没有很快减小。由于后者的结果，我们得出了在错误指定的环境中贝叶斯正交的收敛速度。