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

TL;DR

This paper establishes an equivalence between maximum likelihood estimation and control variate estimators within exponential families, introducing an EM algorithm that improves stability and speed in computing MLEs for sketching algorithms.

Contribution

It proves that optimal control variate estimators can match MLE asymptotic variance in exponential families, leading to an EM algorithm for efficient MLE computation.

Findings

EM algorithm is faster than traditional methods for MLE in bivariate Normal distribution

The EM algorithm is numerically stable and reproducible

The approach generalizes to distributions satisfying exponential family conditions

Abstract

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 an Expectation-Maximization (EM) algorithm for the MLE. Experiments show the EM 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 the EM algorithm leads to reproducibility for algorithms using MLE / CVE, and demonstrate how the EM algorithm leads to finding the MLE when the CV weights are known.