# It's all In the (Exponential) Family: An Equivalence between Maximum Likelihood Estimation and Control Variates for Sketching Algorithms

arXiv: https://arxiv.org/abs/2601.22378 · HTML: https://mlsanity.com/paper/2601.22378/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 48 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 8 of 8)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Hutch++: Optimal Stochastic Trace Estimation](https://mlsanity.com/paper/2010.09649/) (2020)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Estimating the Spectral Density of Large Implicit Matrices](https://mlsanity.com/paper/1802.03451/) (2018)
- [Regularized Zero-Variance Control Variates](https://mlsanity.com/paper/1811.05073/) (2018)
- [Stochastic diagonal estimation: probabilistic bounds and an improved algorithm](https://mlsanity.com/paper/2201.10684/) (2022)
- [Improving textitTug-of-War sketch using Control-Variates method](https://mlsanity.com/paper/2203.02432/) (2022)
- [Euclidean distance compression via deep random features](https://mlsanity.com/paper/2403.01327/) (2024)

## Abstract (arXiv)

Maximum likelihood estimators (MLE) and control variate estimators (CVE) have been used in conjunction with known information across sketching algorithms and applications in machine learning. We prove that under certain conditions in an exponential family, an optimal CVE will achieve the same asymptotic variance as the MLE, giving a fixed point algorithm for the MLE. Experiments show the fixed point algorithm is faster and numerically stable compared to other root finding algorithms for the MLE for the bivariate Normal distribution, and we expect this to hold across distributions satisfying these conditions. We show how this algorithm leads to reproducibility for algorithms using MLE / CVE, and demonstrate how the algorithm leads to finding the MLE when the CV weights are known.