Greedy k-Center from Noisy Distance Samples

TL;DR

This paper introduces active algorithms for the k-center problem in metric spaces with unknown distances, utilizing noisy or incomplete oracle queries, achieving near-optimal approximation ratios with proven query complexity bounds.

Contribution

It develops novel active algorithms inspired by bandit techniques to solve the k-center problem with noisy distance samples, providing theoretical analysis and empirical validation.

Findings

Algorithms achieve approximation ratio of 2 with high probability.

Significant reduction in query complexity over naive methods.

Effective performance demonstrated on real-world datasets.

Abstract

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 Zappos50K).