# On the Sampling Problem for Kernel Quadrature

arXiv: https://arxiv.org/abs/1706.03369 · HTML: https://mlsanity.com/paper/1706.03369/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2017.

## Built on by (most cited first, 7 of 7)

- [Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling](https://mlsanity.com/paper/2311.13548/) (2023)
- [Introduction To Gaussian Process Regression In Bayesian Inverse Problems, With New ResultsOn Experimental Design For Weighted Error Measures](https://mlsanity.com/paper/2302.04518/) (2023)
- [Nested Expectations with Kernel Quadrature](https://mlsanity.com/paper/2502.18284/) (2025)
- [Sparse solutions of the kernel herding algorithm by improved gradient approximation](https://mlsanity.com/paper/2105.07900/) (2021)
- [Policy Gradient with Kernel Quadrature](https://mlsanity.com/paper/2310.14768/) (2023)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)

## Abstract (arXiv)

The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio s/d, where s and d encode the smoothness and dimension of the integrand. However, an empirical investigation reveals that the rate constant C is highly sensitive to the distribution of the random points. In contrast to standard Monte Carlo integration, for which optimal importance sampling is well-understood, the sampling distribution that minimises C for Kernel Quadrature does not admit a closed form. This paper argues that the practical choice of sampling distribution is an important open problem. One solution is considered; a novel automatic approach based on adaptive tempering and sequential Monte Carlo. Empirical results demonstrate a dramatic reduction in integration error of up to 4 orders of magnitude can be achieved with the proposed method.