We study a variant of the canonical $k$-center problem over a set of vertices in a metric space, where the underlying distances are apriori unknown. Instead, we can query an oracle which provides noisy/incomplete estimates of the distance between any pair of vertices. We consider two oracle models: Dimension Sampling where each query to the oracle returns the distance between a pair of points in one dimension; and Noisy Distance Sampling where the oracle returns the true distance corrupted by noise. We propose active algorithms, based on ideas such as UCB, Thompson Sampling and Track-and-Stop developed in the closely related Multi-Armed Bandit problem, which adaptively decide which queries to send to the oracle and are able to solve the $k$ -center problem within an approximation ratio of two with high probability. We analytically characterize instance-dependent query complexity of our algorithms and also demonstrate significant improvements over naive implementations via numerical evaluations on two real-world datasets (Tiny ImageNet and UT Zappos50 K).

中文翻译：

---

来自嘈杂距离样本的贪婪$k$-Center

我们研究规范的变体$k$-在度量空间中的一组顶点上的中心问题，其中基础距离是先验未知的。相反，我们可以查询一个预言机，它提供任何一对顶点之间距离的嘈杂/不完整估计。我们考虑两种预言机模型：维度采样，其中对预言机的每个查询都返回一个维度中一对点之间的距离；和噪声距离采样，预言机返回被噪声破坏的真实距离。我们提出了主动算法，基于在密切相关的 Multi-Armed Bandit 问题中开发的 UCB、Thompson Sampling 和 Track-and-Stop 等思想，它们自适应地决定将哪些查询发送到预言机并能够解决$k$ -中心问题在两个近似比率内的概率很高。我们分析地描述了我们算法的实例相关查询复杂性，并通过对两个真实世界数据集（Tiny ImageNet 和 UT Zappos50 K）的数值评估，展示了对简单实现的显着改进。