Order-Optimal Error Bounds for Noisy Kernel-Based Bayesian Quadrature

In this paper, we study the sample complexity of {\em noisy Bayesian quadrature} (BQ), in which we seek to approximate an integral based on noisy black-box queries to the underlying function. We consider functions in a {\em Reproducing Kernel Hilbert Space} (RKHS) with the Mat\'ern-$\nu$ kernel, focusing on combinations of the parameter $\nu$ and dimension $d$ such that the RKHS is equivalent to a Sobolev class. In this setting, we provide near-matching upper and lower bounds on the best possible average error. Specifically, we find that when the black-box queries are subject to Gaussian noise having variance $\sigma^2$, any algorithm making at most $T$ queries (even with adaptive sampling) must incur a mean absolute error of $\Omega(T^{-\frac{\nu}{d}-1} + \sigma T^{-\frac{1}{2}})$, and there exists a non-adaptive algorithm attaining an error of at most $O(T^{-\frac{\nu}{d}-1} + \sigma T^{-\frac{1}{2}})$. Hence, the bounds are order-optimal, and establish that there is no adaptivity gap in terms of scaling laws.