Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings
Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings
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DOI：
10.1007/s10208-018-09407-7
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发表时间：
2017-09
影响因子：
3
通讯作者：
Motonobu Kanagawa;Bharath K. Sriperumbudur;K. Fukumizu
中科院分区：
文献类型：
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作者：
Motonobu Kanagawa;Bharath K. Sriperumbudur;K. Fukumizu
本文给出了基于核的求积规则在错误设置下的收敛性分析，重点讨论了Sobolev空间中的确定性求积。特别是，我们处理错误的设置，测试被积函数是不太顺利的索博列夫RKHS的基础上，一个正交规则的构造。我们提供的收敛保证基于两个不同的假设正交规则：一个正交权重和其他设计点。更准确地说，我们表明，收敛速度可以推导出（i）如果绝对权重的总和保持不变（或不迅速增加），或（ii）如果设计点之间的最小距离不会很快减少。作为后者的结果，我们得出的贝叶斯正交在错误指定的设置的收敛速度。我们揭示了一个条件的设计点，使贝叶斯求积鲁棒误指定，并表明，在此条件下，它可以自适应地实现最佳收敛速度的Sobolev空间的较低的顺序（即，测试被积函数的未知光滑性），在被积函数上的稍微更强的正则性条件下。
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.