It’s All In The (Exponential) Family: An Equivalence Between Maximum Likelihood Estimation and Control Variates For Sketching Algorithms

Keegan Kang, Kerong Wang, Ding Zhang, Rameshwar Pratap, Bhisham Dev Verma, Benedict H. W. Wong

TL;DR: We show an equivalence between maximum likelihood estimation and control variates which are two fundamental estimators, leading to some surprising implications re: reproducibility and finding MLEs via CV weights

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.

Submission Number: 1581