论文标题
平均基于内核的贝叶斯正交误差界限
On Average-Case Error Bounds for Kernel-Based Bayesian Quadrature
论文作者
论文摘要
在本文中，我们研究了{\ em贝叶斯正交}（bq）的误差界限，重点是嘈杂的设置，随机算法和平均案例性能度量。我们试图近似{\ em重现的内核Hilbert Space}（RKHS）中函数的积分，尤其是专注于Matérn-$ν$和平方指数（SE）内核，并带有来自功能的样本，其函数可能会被高斯噪声损坏。我们提供了一个两步的元容量，它是将平均案例正交误差与$ l^2 $ function近似误差联系起来的一般工具。当专门针对Matérn内核时，我们恢复了现有的近乎最佳错误率，同时避免了反复采样点的现有方法。当专门针对其他设置时，我们将获得新的平均案例结果，用于设置，包括带有噪声的SE内核和具有错误指定的Matérn内核。最后，我们提出了与现有算法无关的下限，与现有算法具有更大的通用性和/或提供不同的证据。
In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the Matérn-$ν$ and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the $L^2$-function approximation error. When specialized to the Matérn kernel, we recover an existing near-optimal error rate while avoiding the existing method of repeatedly sampling points. When specialized to other settings, we obtain new average-case results for settings including the SE kernel with noise and the Matérn kernel with misspecification. Finally, we present algorithm-independent lower bounds that have greater generality and/or give distinct proofs compared to existing ones.
