  This paper considers the problem of online clustering with bandit feedback. A
set of arms (or items) can be partitioned into various groups that are unknown.
Within each group, the observations associated to each of the arms follow the
same distribution with the same mean vector. At each time step, the agent
queries or pulls an arm and obtains an independent observation from the
distribution it is associated to. Subsequent pulls depend on previous ones as
well as the previously obtained samples. The agent's task is to uncover the
underlying partition of the arms with the least number of arm pulls and with a
probability of error not exceeding a prescribed constant $\delta$. The problem
proposed finds numerous applications from clustering of variants of viruses to
online market segmentation. We present an instance-dependent
information-theoretic lower bound on the expected sample complexity for this
task, and design a computationally efficient and asymptotically optimal
algorithm, namely Bandit Online Clustering (BOC). The algorithm includes a
novel stopping rule for adaptive sequential testing that circumvents the need
to exactly solve any NP-hard weighted clustering problem as its subroutines. We
show through extensive simulations on synthetic and real-world datasets that
BOC's performance matches the lower bound asymptotically, and significantly
outperforms a non-adaptive baseline algorithm.
