Differentially Private Algorithms for Learning Mixtures of Separated Gaussians

TL;DR

This paper introduces new differentially private algorithms for learning high-dimensional, well-separated Gaussian mixture models, achieving near-optimal sample complexity without requiring strong prior bounds.

Contribution

It presents a differentially private algorithm that matches non-private sample complexity and removes the need for strong a priori parameter bounds.

Findings

Sample complexity matches non-private algorithms

No need for strong a priori bounds

Applicable to high-dimensional, well-separated mixtures

Abstract

Learning the parameters of Gaussian mixture models is a fundamental and widely studied problem with numerous applications. In this work, we give new algorithms for learning the parameters of a high-dimensional, well separated, Gaussian mixture model subject to the strong constraint of differential privacy. In particular, we give a differentially private analogue of the algorithm of Achlioptas and McSherry. Our algorithm has two key properties not achieved by prior work: (1) The algorithm's sample complexity matches that of the corresponding non-private algorithm up to lower order terms in a wide range of parameters. (2) The algorithm does not require strong a priori bounds on the parameters of the mixture components.