Phase transitions for support recovery under local differential privacy

Abstract

We address the problem of variable selection in a high-dimensional but sparse mean model, under the additional constraint that only privatized 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 generalization 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.