Online Clustering of Data Sequences with Bandit Information
We study the problem of online clustering of data sequences in the multi-armed bandit (MAB) framework under the fixed-confidence setting. There are $M$ arms, each providing i.i.d. samples from a parametric distribution whose parameters are unknown. The $M$ arms form $K$ clusters based on the distance between the true parameters. In the MAB setting, one arm can be sampled at each time. The objective is to estimate the clusters of the arms using as few samples as possible from the arms, subject to an upper bound on the error probability. Our setting allows for: arms within a cluster to have non-identical distributions, vector parameter arms, vector observations, and $K \le M$ clusters. We propose and analyze the Average Tracking Bandit Online Clustering (ATBOC) algorithm. ATBOC is asymptotically order-optimal for multivariate Gaussian arms, with expected sample complexity grows at most twice as fast as the lower bound as $δ\rightarrow 0$, and this guarantee extends to multivariate sub-Gaussian arms. For single-parameter exponential family arms, ATBOC is asymptotically optimal, matching the lower bound. We also propose a computationally more efficient alternatives Lower and Upper Confidence Bound based Bandit Online Clustering Algorithm (LUCBBOC), and Bandit Online Clustering-Elimination (BOC-ELIM). We derive the computational complexity of the proposed algorithms and compare their per-sample runtime through simulations. LUCBBOC and BOC-ELIM require lower per-sample runtime than ATBOC while achieving comparable performance. All the proposed algorithms are $δ$-Probably correct, i.e., the error probability of cluster estimate at the stopping time is atmost $δ$. We validate the asymptotic optimality guarantees through simulations, and present the comparison of our proposed algorithms with other related work through simulations on both synthetic and real-world datasets.