集中和局部模型的聚类算法
Clustering Algorithms for the Centralized and Local Models
摘要：
我们重新审视了寻找具有差分隐私的最小封闭球的问题：给定欧几里得空间$\mathbb{R}^d$和整数$t\leq n$中的一组$n$点，目标是找到一个最小半径$r_{opt}$的球，该球至少包含$t$输入点。这个问题的动机是它的各种应用差分隐私，包括样本和聚合技术，私人数据探索，聚类。在没有隐私问题的情况下，最小封闭球具有多项式时间近似方案（PTAS），其计算半径几乎为$r_{opt}$的球（该问题是NP-难精确求解的）。相反，在差分隐私下，直到这项工作，只有$O(\sqrt{\log n})$ -近似算法是已知的。我们提供了新的构造差分隐私算法的最小封闭球实现恒定因子近似$r_{opt}$在集中式模型（其中一个可信的馆长收集的敏感信息，并分析它与差分隐私）和本地模型（其中每个受访者随机她的答案的数据馆长，以保护她的隐私）。我们演示了如何使用我们的算法作为一个积木近似$k$ -意味着在这两个模型。
英文摘要：
We revisit the problem of finding a minimum enclosing ball with differential privacy: Given a set of $n$ points in the Euclidean space $\mathbb{R}^d$ and an integer $t\leq n$, the goal is to find a ball of the smallest radius $r_{opt}$ enclosing at least $t$ input points. The problem is motivated by its various applications to differential privacy, including the sample and aggregate technique, private data exploration, and clustering. Without privacy concerns, minimum enclosing ball has a polynomial time approximation scheme (PTAS), which computes a ball of radius almost $r_{opt}$ (the problem is NP-hard to solve exactly). In contrast, under differential privacy, until this work, only a $O(\sqrt{\log n})$-approximation algorithm was known. We provide new constructions of differentially private algorithms for minimum enclosing ball achieving constant factor approximation to $r_{opt}$ both in the centralized model (where a trusted curator collects the sensitive information and analyzes it with differential privacy) and in the local model (where each respondent randomizes her answers to the data curator to protect her privacy). We demonstrate how to use our algorithms as a building block for approximating $k$-means in both models.
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