Phase transitions for support recovery under local differential privacy

TL;DR

This paper investigates the limits of support recovery in high-dimensional sparse models under local differential privacy constraints, revealing phase transitions and the impact of privacy mechanisms on recoverability.

Contribution

It introduces new bounds and phase transition results for support recovery under local differential privacy, highlighting the effects of different privacy mechanisms.

Findings

Exact and almost full recovery are impossible with coordinate-wise mechanisms in certain regimes.

Global mechanisms enable support recovery at lower privacy levels.

Phase transitions depend on the privacy parameter, sample size, and dimension.

Abstract

We address the problem of variable selection in a high-dimensional but sparse mean model, under the additional constraint that only privatised data are available for inference. The original data are vectors with independent entries having a symmetric, strongly log-concave distribution on $\mathbb{R}$. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local $\alpha-$differential privacy. We provide lower and upper bounds on the rate of convergence for the expected Hamming loss over classes of at most $s$-sparse vectors whose non-zero coordinates are separated from $0$ by a constant $a>0$. As corollaries, we derive necessary and sufficient conditions (up to log factors) for exact recovery and for almost full recovery. When we restrict our attention to non-interactive mechanisms that act independently on each coordinate our lower bound shows that, contrary to the non-private setting, both exact and almost full recovery are impossible whatever the value of $a$ in the high-dimensional regime such that $n \alpha^2/ d^2\lesssim 1$. However, in the regime $n\alpha^2/d^2\gg \log(d)$ we can exhibit a critical value $a^*$ (up to a logarithmic factor) such that exact and almost full recovery are possible for all $a\gg a^*$ and impossible for $a\leq a^*$. We show that these results can be improved when allowing for all non-interactive (that act globally on all coordinates) locally $\alpha-$differentially private mechanisms in the sense that phase transitions occur at lower levels.