Sharp phase transitions for exact support recovery under local differential privacy

We address the problem of variable selection in the Gaussian mean model in $\mathbb{R}^d$ under the additional constraint that only privatised data are available for inference. 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 $n\alpha^2/d^2\gg \log(n\alpha^2/d^2)\log(d)$ we can exhibit a sharp 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.