Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

TL;DR

This paper analyzes the convergence of deterministic kernel-based quadrature rules in misspecified Sobolev spaces, providing guarantees under conditions on weights and design points, and extending results to Bayesian quadrature robustness.

Contribution

It introduces new convergence guarantees for kernel quadrature in misspecified settings and identifies conditions for robustness and adaptivity in Bayesian quadrature.

Findings

Convergence rates depend on weight sums and design point spacing.

Bayesian quadrature can be robust to misspecification under certain design conditions.

Adaptive convergence rates are achievable for less smooth integrands.

Abstract

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.