Phase transitions for support recovery under local differential privacy

Understand

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 $a>0$.