We study a general clustering setting in which we have $n$ elements to be clustered, and we aim to perform as few queries as possible to an oracle that returns a noisy sample of the weighted similarity between two elements. Our setting encompasses many application domains in which the similarity function is costly to compute and inherently noisy. We introduce two novel formulations of online learning problems rooted in the paradigm of Pure Exploration in Combinatorial Multi-Armed Bandits (PE-CMAB): fixed confidence and fixed budget settings. For both settings, we design algorithms that combine a sampling strategy with a classic approximation algorithm for correlation clustering and study their theoretical guarantees. Our results are the first examples of polynomial-time algorithms that work for the case of PE-CMAB in which the underlying offline optimization problem is NP-hard.

翻译：我们研究一种通用的聚类场景，其中包含待聚类的 $n$ 个元素，目标是尽可能少地向预言机进行查询，该预言机返回两个元素之间加权相似度的带噪声样本。我们的设定涵盖了许多应用领域，其中相似度函数的计算成本高昂且本质上是带噪声的。我们基于组合多臂老虎机纯探索范式，提出了两种新颖的在线学习问题形式化表述：固定置信度与固定预算设定。针对这两种设定，我们设计了将采样策略与经典相关聚类近似算法相结合的算法，并研究了其理论保证。我们的结果是首批适用于组合多臂老虎机纯探索场景的多项式时间算法，其中底层离线优化问题是NP难的。