Sample-Efficient Private Learning of Mixtures of Gaussians

TL;DR

This paper presents improved sample complexity bounds for differentially private learning of Gaussian mixtures, achieving near-optimal results especially in high dimensions and for univariate cases, using novel techniques.

Contribution

It introduces new sample complexity bounds for private Gaussian mixture learning, improving previous results and providing the first optimal bounds for univariate mixtures.

Findings

Sample complexity for high-dimensional mixtures is roughly $kd^2 + k^{1.5} d^{1.75} + k^2 d$.

Achieves linear-in-$k$ sample complexity for univariate Gaussian mixtures.

Provides algorithms utilizing inverse sensitivity, sample compression, and sumset volume bounding techniques.

Abstract

We study the problem of learning mixtures of Gaussians with approximate differential privacy. We prove that roughly $kd^2 + k^{1.5} d^{1.75} + k^2 d$ samples suffice to learn a mixture of $k$ arbitrary $d$-dimensional Gaussians up to low total variation distance, with differential privacy. Our work improves over the previous best result [AAL24b] (which required roughly $k^2 d^4$ samples) and is provably optimal when $d$ is much larger than $k^2$. Moreover, we give the first optimal bound for privately learning mixtures of $k$ univariate (i.e., $1$-dimensional) Gaussians. Importantly, we show that the sample complexity for privately learning mixtures of univariate Gaussians is linear in the number of components $k$, whereas the previous best sample complexity [AAL21] was quadratic in $k$. Our algorithms utilize various techniques, including the inverse sensitivity mechanism [AD20b, AD20a, HKMN23], sample compression for distributions [ABDH+20], and methods for bounding volumes of sumsets.