# Neural Control Variates

arXiv: https://arxiv.org/abs/2006.01524 · HTML: https://mlsanity.com/paper/2006.01524/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 30 of 34)

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- [Amortized Variational Inference: A Systematic Review](https://mlsanity.com/paper/2209.10888/) (2022)
- [NeuMIP: Multi-Resolution Neural Materials](https://mlsanity.com/paper/2104.02789/) (2021)
- [Inverse Global Illumination using a Neural Radiometric Prior](https://mlsanity.com/paper/2305.02192/) (2023)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [Accelerating Unbiased LLM Evaluation via Synthetic Feedback](https://mlsanity.com/paper/2502.10563/) (2025)
- [Path Guiding Using Spatio-Directional Mixture Models](https://mlsanity.com/paper/2111.13094/) (2021)
- [Online Neural Path Guiding with Normalized Anisotropic Spherical Gaussians](https://mlsanity.com/paper/2303.08064/) (2023)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)
- [When can Regression-Adjusted Control Variates Help? Rare Events, Sobolev Embedding and Minimax Optimality](https://mlsanity.com/paper/2305.16527/) (2023)
- [Leveraging neural control variates for enhanced precision in lattice field theory](https://mlsanity.com/paper/2312.08228/) (2023)
- [Neural Rendering and Its Hardware Acceleration: A Review](https://mlsanity.com/paper/2402.00028/) (2024)
- [Neural Product Importance Sampling via Warp Composition](https://mlsanity.com/paper/2409.18974/) (2024)
- [Vertex Features for Neural Global Illumination](https://mlsanity.com/paper/2508.07852/) (2025)
- [Prediction-Enhanced Monte Carlo: A Machine Learning View on Control Variate](https://mlsanity.com/paper/2412.11257/) (2024)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [Differentiable Neural Radiosity](https://mlsanity.com/paper/2201.13190/) (2022)
- [Physics-informed neural networks for pathloss prediction](https://mlsanity.com/paper/2211.12986/) (2022)
- [Example-Based Sampling with Diffusion Models](https://mlsanity.com/paper/2302.05116/) (2023)
- [Training neural control variates using correlated configurations](https://mlsanity.com/paper/2505.07719/) (2025)
- [Neural Cone Radiosity for Interactive Global Illumination with Glossy Materials](https://mlsanity.com/paper/2509.07522/) (2025)
- [Hypothesis Testing for Progressive Kernel Estimation and VCM Framework](https://mlsanity.com/paper/2504.04411/) (2025)
- [Geometric Integration for Neural Control Variates](https://mlsanity.com/paper/2509.15538/) (2025)

## Abstract (arXiv)

We propose neural control variates (NCV) for unbiased variance reduction in parametric Monte Carlo integration. So far, the core challenge of applying the method of control variates has been finding a good approximation of the integrand that is cheap to integrate. We show that a set of neural networks can face that challenge: a normalizing flow that approximates the shape of the integrand and another neural network that infers the solution of the integral equation. We also propose to leverage a neural importance sampler to estimate the difference between the original integrand and the learned control variate. To optimize the resulting parametric estimator, we derive a theoretically optimal, variance-minimizing loss function, and propose an alternative, composite loss for stable online training in practice. When applied to light transport simulation, neural control variates are capable of matching the state-of-the-art performance of other unbiased approaches, while providing means to develop more performant, practical solutions. Specifically, we show that the learned light-field approximation is of sufficient quality for high-order bounces, allowing us to omit the error correction and thereby dramatically reduce the noise at the cost of negligible visible bias.