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

arXiv: https://arxiv.org/abs/1709.00147 · HTML: https://mlsanity.com/paper/1709.00147/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2017.

## Built on by (most cited first, 16 of 16)

- [Positively Weighted Kernel Quadrature via Subsampling](https://mlsanity.com/paper/2107.09597/) (2021)
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- [Nonlinear Meta-Learning Can Guarantee Faster Rates](https://mlsanity.com/paper/2307.10870/) (2023)
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- [Bayesian Numerical Integration with Neural Networks](https://mlsanity.com/paper/2305.13248/) (2023)
- [A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Lifting](https://mlsanity.com/paper/2404.12219/) (2024)
- [Sparse solutions of the kernel herding algorithm by improved gradient approximation](https://mlsanity.com/paper/2105.07900/) (2021)
- [Impact of Computation in Integral Reinforcement Learning for Continuous-Time Control](https://mlsanity.com/paper/2402.17375/) (2024)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Interpolation with the polynomial kernels](https://mlsanity.com/paper/2212.07658/) (2022)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)
- [Hierarchical Bayesian Quadrature](https://mlsanity.com/paper/2607.10793/) (2026)

## Abstract (arXiv)

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.