High-Dimensional Differentially-Private EM Algorithm: Methods and Near-Optimal Statistical Guarantees

TL;DR

This paper introduces a framework for differentially private EM algorithms in high-dimensional latent variable models, achieving near-optimal statistical guarantees and extending to low-dimensional cases with practical validation.

Contribution

It presents a novel general framework for differentially private EM algorithms with theoretical guarantees in high-dimensional settings, applicable to multiple models.

Findings

Achieves near-optimal convergence rates under privacy constraints.

Extends techniques to classic low-dimensional latent models.

Validates methods through simulations and real data analysis.

Abstract

In this paper, we develop a general framework to design differentially private expectation-maximization (EM) algorithms in high-dimensional latent variable models, based on the noisy iterative hard-thresholding. We derive the statistical guarantees of the proposed framework and apply it to three specific models: Gaussian mixture, mixture of regression, and regression with missing covariates. In each model, we establish the near-optimal rate of convergence with differential privacy constraints, and show the proposed algorithm is minimax rate optimal up to logarithm factors. The technical tools developed for the high-dimensional setting are then extended to the classic low-dimensional latent variable models, and we propose a near rate-optimal EM algorithm with differential privacy guarantees in this setting. Simulation studies and real data analysis are conducted to support our results.