We study the question of how to compute a point in the convex hull of an input set $S$ of $n$ points in ${\mathbb R}^d$ in a differentially private manner. This question, which is trivial non-privately, turns out to be quite deep when imposing differential privacy. In particular, it is known that the input points must reside on a fixed finite subset $G\subseteq{\mathbb R}^d$, and furthermore, the size of $S$ must grow with the size of $G$. Previous works focused on understanding how $n$ needs to grow with $|G|$, and showed that $n=O\left(d^{2.5}\cdot8^{\log^*|G|}\right)$ suffices (so $n$ does not have to grow significantly with $|G|$). However, the available constructions exhibit running time at least $|G|^{d^2}$, where typically $|G|=X^d$ for some (large) discretization parameter $X$, so the running time is in fact $\Omega(X^{d^3})$. In this paper we give a differentially private algorithm that runs in $O(n^d)$ time, assuming that $n=\Omega(d^4\log X)$. To get this result we study and exploit some structural properties of the Tukey levels (the regions $D_{\ge k}$ consisting of points whose Tukey depth is at least $k$, for $k=0,1,...$). In particular, we derive lower bounds on their volumes for point sets $S$ in general position, and develop a rather subtle mechanism for handling point sets $S$ in degenerate position (where the deep Tukey regions have zero volume). A naive approach to the construction of the Tukey regions requires $n^{O(d^2)}$ time. To reduce the cost to $O(n^d)$, we use an approximation scheme for estimating the volumes of the Tukey regions (within their affine spans in case of degeneracy), and for sampling a point from such a region, a scheme that is based on the volume estimation framework of Lov\'asz and Vempala (FOCS 2003) and of Cousins and Vempala (STOC 2015). Making this framework differentially private raises a set of technical challenges that we address.

中文翻译：

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如何私下在凸包中找到一个点

我们研究了如何以差异私有的方式计算 ${\mathbb R}^d$ 中 $n$ 个点的输入集 $S$ 的凸包中的一个点的问题。这个问题在非私密情况下是微不足道的，但在施加差异隐私时却显得非常深刻。特别是，已知输入点必须位于固定的有限子集 $G\subseteq{\mathbb R}^d$ 上，而且 $S$ 的大小必须随着 $G$ 的大小而增长。以前的工作侧重于理解 $n$ 需要如何随 $|G|$ 增长，并表明 $n=O\left(d^{2.5}\cdot8^{\log^*|G|}\right)$就足够了（所以 $n$ 不必随着 $|G|$ 显着增长）。然而，可用的结构表现出至少 $|G|^{d^2}$ 的运行时间，其中对于某些（大）离散化参数 $X$，通常 $|G|=X^d$，因此运行时间在事实 $\Omega(X^{d^3})$。在本文中，我们给出了一个在 $O(n^d)$ 时间内运行的差分私有算法，假设 $n=\Omega(d^4\log X)$。为了得到这个结果，我们研究并利用了 Tukey 层的一些结构特性（区域 $D_{\ge k}$ 由 Tukey 深度至少为 $k$ 的点组成，对于 $k=0,1，... $）。特别是，我们为一般位置的点集 $S$ 推导出了它们的体积下限，并开发了一种相当微妙的机制来处理退化位置的点集 $S$（其中深 Tukey 区域的体积为零）。构建 Tukey 区域的简单方法需要 $n^{O(d^2)}$ 时间。为了将成本降低到 $O(n^d)$，我们使用近似方案来估计 Tukey 区域的体积（在退化的情况下在它们的仿射跨度内），并从这样的区域采样一个点，一种基于 Lov\'asz 和 Vempala（FOCS 2003）以及 Cousins 和 Vempala（STOC 2015）的体积估计框架的方案。使这个框架差异化私有会引发一系列我们要解决的技术挑战。