arXiv 1504.075532015-04cs.CRcs.LG

阈值函数的差分隐私发布与学习

Differentially Private Release and Learning of Threshold Functions

Harvard University（哈佛大学）；Georgetown University（乔治城大学）；Tel Aviv University（特拉维夫大学）；Ben-Gurion University（本-古里安大学）；Google, Inc.（谷歌公司）；Simons Investigator（西蒙斯研究者机构）

Mark Bun, Kobbi Nissim, Uri Stemmer, Salil Vadhan

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AI总结：

本文研究了阈值函数差分隐私发布与学习的样本复杂度，通过双向归约内部点问题，证明了随域大小增长的新下界，并给出了改进的指数级上界算法。

摘要：

我们证明了用于发布阈值函数近似答案的 $(ε, δ)$ 差分隐私算法样本复杂度的新上下界。全序域 $X$ 上的阈值函数 $c_x$ 当 $y \le x$ 时取值为 $c_x(y) = 1$，否则取值为 $0$。我们给出了发布阈值且满足 $(ε,δ)$ 差分隐私的首个非平凡下界，证明该任务在无限域 $X$ 上不可能实现，且其样本复杂度需满足 $n \ge Ω(\log^*|X|)$，随域大小增长。受证明该下界所用技术的启发，我们给出了一个使用 $n \le 2^{(1+ o(1))\log^*|X|}$ 个样本发布阈值的算法。这改进了此前 $8^{(1 + o(1))\log^*|X|}$ 的最佳上界（Beimel et al., RANDOM '13）。\n 我们的样本复杂度上下界同样适用于在 Kolmogorov 距离下学习分布以及在差分隐私下正确 PAC 学习阈值的任务。该下界给出了在 $(ε,δ)$ 差分隐私下正确学习一个概念类与无隐私学习之间的首个样本复杂度分离。对于在 $\ell$ 维空间中正确学习阈值，该下界扩展为 $n \ge Ω(\ell \cdot \log^*|X|)$。\n 为了获得上述结果，我们在发布与正确学习阈值及更简单的内部点问题之间给出了双向归约。给定包含 $X$ 中元素的数据库 $D$，内部点问题要求找出一个介于 $D$ 中最小与最大元素之间的元素。我们引入了新的递归构造来界定内部点问题的样本复杂度，并提出了进一步的归约与技术，以证明差分隐私中其他基本问题的不可能性结果。

英文摘要：

We prove new upper and lower bounds on the sample complexity of $(ε, δ)$ differentially private algorithms for releasing approximate answers to threshold functions. A threshold function $c_x$ over a totally ordered domain $X$ evaluates to $c_x(y) = 1$ if $y \le x$, and evaluates to $0$ otherwise. We give the first nontrivial lower bound for releasing thresholds with $(ε,δ)$ differential privacy, showing that the task is impossible over an infinite domain $X$, and moreover requires sample complexity $n \ge Ω(\log^*|X|)$, which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with $n \le 2^{(1+ o(1))\log^*|X|}$ samples. This improves the previous best upper bound of $8^{(1 + o(1))\log^*|X|}$ (Beimel et al., RANDOM '13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with $(ε,δ)$ differential privacy and learning without privacy. For properly learning thresholds in $\ell$ dimensions, this lower bound extends to $n \ge Ω(\ell \cdot \log^*|X|)$. To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database $D$ of elements from $X$, the interior point problem asks for an element between the smallest and largest elements in $D$. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy.

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