# Vector-Valued Control Variates

arXiv: https://arxiv.org/abs/2109.08944 · HTML: https://mlsanity.com/paper/2109.08944/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2021.

## Built on by (most cited first, 13 of 13)

- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Hierarchical Bayesian Modelling for Knowledge Transfer Across Engineering Fleets via Multitask Learning](https://mlsanity.com/paper/2204.12404/) (2022)
- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [On the hierarchical Bayesian modelling of frequency response functions](https://mlsanity.com/paper/2307.06263/) (2023)
- [Nested Expectations with Kernel Quadrature](https://mlsanity.com/paper/2502.18284/) (2025)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [Training neural control variates using correlated configurations](https://mlsanity.com/paper/2505.07719/) (2025)
- [Inter-turbine Modelling of Wind-Farm Power using Multi-task Learning](https://mlsanity.com/paper/2502.14527/) (2025)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [Amortised adaptive importance sampling](https://mlsanity.com/paper/2610.16736/) (2026)

## Abstract (arXiv)

Control variates are variance reduction tools for Monte Carlo estimators. They can provide significant variance reduction, but usually require a large number of samples, which can be prohibitive when sampling or evaluating the integrand is computationally expensive. Furthermore, there are many scenarios where we need to compute multiple related integrals simultaneously or sequentially, which can further exacerbate computational costs. In this paper, we propose vector-valued control variates, an extension of control variates which can be used to reduce the variance of multiple Monte Carlo estimators jointly. This allows for the transfer of information across integration tasks, and hence reduces the need for a large number of samples. We focus on control variates based on kernel interpolants and our novel construction is obtained through a generalised Stein identity and the development of novel matrix-valued Stein reproducing kernels. We demonstrate our methodology on a range of problems including multifidelity modelling, Bayesian inference for dynamical systems, and model evidence computation through thermodynamic integration.