# Nested Expectations with Kernel Quadrature

arXiv: https://arxiv.org/abs/2502.18284 · HTML: https://mlsanity.com/paper/2502.18284/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 102 references that are in the map.
- First posted on arXiv 25 Feb 2025.

## Builds on (most cited first, 30 of 39)

- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Modern Bayesian Experimental Design](https://mlsanity.com/paper/2302.14545/) (2023)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [On Nesting Monte Carlo Estimators](https://mlsanity.com/paper/1709.06181/) (2017)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space](https://mlsanity.com/paper/1910.12799/) (2019)
- [Parallel Bayesian Global Optimization of Expensive Functions](https://mlsanity.com/paper/1602.05149/) (2016)
- [GLASSES: Relieving The Myopia Of Bayesian Optimisation](https://mlsanity.com/paper/1510.06299/) (2015)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Positively Weighted Kernel Quadrature via Subsampling](https://mlsanity.com/paper/2107.09597/) (2021)
- [Convergence Guarantees for Adaptive Bayesian Quadrature Methods](https://mlsanity.com/paper/1905.10271/) (2019)
- [Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees](https://mlsanity.com/paper/1506.02681/) (2015)
- [On the Optimality of Misspecified Kernel Ridge Regression](https://mlsanity.com/paper/2305.07241/) (2023)
- [A Kernel Stein Test for Comparing Latent Variable Models](https://mlsanity.com/paper/1907.00586/) (2019)
- [Active Multi-Information Source Bayesian Quadrature](https://mlsanity.com/paper/1903.11331/) (2019)
- [Sampling-based Nystr"om Approximation and Kernel Quadrature](https://mlsanity.com/paper/2301.09517/) (2023)
- [Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings](https://mlsanity.com/paper/1709.00147/) (2017)
- [Bayesian Distributionally Robust Optimization](https://mlsanity.com/paper/2112.08625/) (2021)
- [(De)-regularized Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/2409.14980/) (2024)
- [Multilevel Monte Carlo estimation of expected information gains](https://mlsanity.com/paper/1811.07546/) (2018)
- [Vector-Valued Control Variates](https://mlsanity.com/paper/2109.08944/) (2021)
- [Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference](https://mlsanity.com/paper/2301.11674/) (2023)
- [Kernel quadrature with randomly pivoted Cholesky](https://mlsanity.com/paper/2306.03955/) (2023)
- [Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization](https://mlsanity.com/paper/2006.07487/) (2020)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)

## Built on by (most cited first, 5 of 5)

- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Multilevel neural simulation-based inference](https://mlsanity.com/paper/2506.06087/) (2025)
- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)

## Abstract (arXiv)

This paper considers the challenging computational task of estimating nested expectations. Existing algorithms, such as nested Monte Carlo or multilevel Monte Carlo, are known to be consistent but require a large number of samples at both inner and outer levels to converge. Instead, we propose a novel estimator consisting of nested kernel quadrature estimators and we prove that it has a faster convergence rate than all baseline methods when the integrands have sufficient smoothness. We then demonstrate empirically that our proposed method does indeed require fewer samples to estimate nested expectations on real-world applications including Bayesian optimisation, option pricing, and health economics.