Tight lower bound of sparse covariance matrix estimation in the local differential privacy model

Abstract

In this paper, we study the sparse covariance matrix estimation problem in the local differential privacy model, and give a lower bound of Ω(s2log⁡pnϵ2) on the ϵ non-interactive private minimax risk in the metric of squared spectral norm, where s is the row sparsity of the underlying covariance matrix, n is the sample size, and p is the dimensionality of the data. We show that the lower bound is actually tight, as it matches a previous upper bound. Our main technique for achieving this lower bound is a general framework, called General Private Assouad Lemma, which is a considerable generalization of the previous private Assouad lemma and can be used as a general method for bounding the private minimax risk of matrix-related estimation problems.