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arXiv 2011.14881math.STstat.TH
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局部差分隐私下支持恢复的阶段转换

Phase transitions for support recovery under local differential privacy

Cristina Butucea, Amandine Dubois, Adrien Saumard

摘要:

我们解决的问题,变量选择在一个高维的,但稀疏的平均值模型,在额外的约束下,只有私有化的数据可用于推理。原始数据是具有独立项的向量,这些独立项在$\mathbb{R}$上具有对称的强对数凹分布。为此,我们采用了最近的经典极大极小理论的局部$α-$差分隐私的框架的概括。我们提供的下限和上限的收敛速度的预期汉明损失类的最$s$稀疏向量的非零坐标是由一个常数$a>0$从$0$分开。作为推论,我们得出的必要和充分条件(日志因素)的确切恢复和几乎完全恢复。当我们限制我们的注意力,以非交互式的机制,独立地作用于每个坐标我们的下限表明,相反的非私人设置,无论$a$的值在高维政权,$n α^2/ d^2\lesssim 1$的精确和几乎完全恢复是不可能的。然而,在方案$nα^2/d^2\gg \log(d)$中,我们可以表现出临界值$a^*$(达到对数因子),使得所有$a\gg a^*$都可能精确且几乎完全回收,而$a\leq a^*$不可能。我们表明,这些结果可以得到改善时,允许所有的非交互式的(全球所有坐标)本地$α-$差分私人机制的意义上,相变发生在较低的水平。

英文摘要:

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 $α-$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 $n α^2/ d^2\lesssim 1$. However, in the regime $nα^2/d^2\gg \log(d)$ 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 $α-$differentially private mechanisms in the sense that phase transitions occur at lower levels.

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