  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.
