# A Quadrature Rule combining Control Variates and Adaptive Importance Sampling

arXiv: https://arxiv.org/abs/2205.11890 · HTML: https://mlsanity.com/paper/2205.11890/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 40 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 10 of 10)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Provable Bayesian Inference via Particle Mirror Descent](https://mlsanity.com/paper/1506.03101/) (2015)
- [Layered Adaptive Importance Sampling](https://mlsanity.com/paper/1505.04732/) (2015)
- [Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff](https://mlsanity.com/paper/2110.15590/) (2021)
- [Asymptotic optimality of adaptive importance sampling](https://mlsanity.com/paper/1806.00989/) (2018)
- [Empirical Variance Minimization with Applications in Variance Reduction and Optimal Control](https://mlsanity.com/paper/1712.04667/) (2017)
- [Risk bounds when learning infinitely many response functions by ordinary linear regression](https://mlsanity.com/paper/2006.09223/) (2020)

## Built on by (most cited first, 2 of 2)

- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates](https://mlsanity.com/paper/2402.01493/) (2024)

## Abstract (arXiv)

Driven by several successful applications such as in stochastic gradient descent or in Bayesian computation, control variates have become a major tool for Monte Carlo integration. However, standard methods do not allow the distribution of the particles to evolve during the algorithm, as is the case in sequential simulation methods. Within the standard adaptive importance sampling framework, a simple weighted least squares approach is proposed to improve the procedure with control variates. The procedure takes the form of a quadrature rule with adapted quadrature weights to reflect the information brought in by the control variates. The quadrature points and weights do not depend on the integrand, a computational advantage in case of multiple integrands. Moreover, the target density needs to be known only up to a multiplicative constant. Our main result is a non-asymptotic bound on the probabilistic error of the procedure. The bound proves that for improving the estimate's accuracy, the benefits from adaptive importance sampling and control variates can be combined. The good behavior of the method is illustrated empirically on synthetic examples and real-world data for Bayesian linear regression.