Private estimation algorithms for stochastic block models and mixture models

TL;DR

This paper develops efficient differentially private algorithms for high-dimensional stochastic block models and Gaussian mixture models, nearly matching non-private statistical guarantees with improved computational efficiency.

Contribution

It introduces the first efficient private algorithms for stochastic block model recovery and Gaussian mixture center estimation with minimal separation conditions.

Findings

First efficient private algorithms for stochastic block models with strong guarantees.

Private Gaussian mixture learning with minimal separation at O(k^{1/t}√t).

Sample complexity and runtime are polynomial in key parameters.

Abstract

We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient $(\epsilon, \delta)$-differentially private algorithm for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. For the latter, we design an $(\epsilon, \delta)$-differentially private algorithm that recovers the centers of the $k$-mixture when the minimum separation is at least $ O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior work required minimum separation at least $O(\sqrt{k})$ as well as an explicit upper bound on the Euclidean norm of the centers.