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.

翻译：最大似然估计器与控制变量估计器已结合已知信息广泛应用于草图算法及机器学习应用中。我们证明在指数族的特定条件下，最优控制变量估计器将达到与最大似然估计器相同的渐近方差，从而为最大似然估计提供一种期望最大化算法。实验表明，对于二元正态分布的最大似然估计，该期望最大化算法相较于其他求根算法具有更快的计算速度和更好的数值稳定性，且我们预期此优势在满足相同条件的分布中普遍成立。我们进一步阐明该期望最大化算法如何提升基于最大似然估计/控制变量估计算法的可复现性，并演示当控制变量权重已知时，如何通过该算法求解最大似然估计。