Differentially Private (Gradient) Expectation Maximization Algorithm with Statistical Guarantees

TL;DR

This paper introduces the first differentially private (Gradient) EM algorithm with finite sample statistical guarantees, applicable to key models like GMM, MRM, and RMC, ensuring privacy without sacrificing statistical accuracy.

Contribution

It develops the first DP (Gradient) EM algorithm with finite sample guarantees, extending privacy-preserving EM to multiple models with near-optimal error bounds.

Findings

Near-optimal estimation error for GMM under DP.

First finite sample guarantees for MRM and RMC.

Numerical experiments validate theoretical results.

Abstract

(Gradient) Expectation Maximization (EM) is a widely used algorithm for estimating the maximum likelihood of mixture models or incomplete data problems. A major challenge facing this popular technique is how to effectively preserve the privacy of sensitive data. Previous research on this problem has already lead to the discovery of some Differentially Private (DP) algorithms for (Gradient) EM. However, unlike in the non-private case, existing techniques are not yet able to provide finite sample statistical guarantees. To address this issue, we propose in this paper the first DP version of (Gradient) EM algorithm with statistical guarantees. Moreover, we apply our general framework to three canonical models: Gaussian Mixture Model (GMM), Mixture of Regressions Model (MRM) and Linear Regression with Missing Covariates (RMC). Specifically, for GMM in the DP model, our estimation error is near optimal in some cases. For the other two models, we provide the first finite sample statistical guarantees. Our theory is supported by thorough numerical experiments.