Sharp phase transitions for exact support recovery under local differential privacy
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We address the problem of variable selection in the Gaussian mean model in 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 α-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 α^2/ d^2≲ 1. However, in the regime nα^2/d^2≫log(nα^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≫ a^* and impossible for a≤ a^*. We show that these results can be improved when allowing for all non-interactive (that act globally on all coordinates) locally α-differentially private mechanisms in the sense that phase transitions occur at lower levels.
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