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Optimal Clustering with Noisy Queries via Multi-Armed Bandit

Motivated by many applications, we study clustering with a faulty oracle. In this problem, there are $n$ items belonging to $k$ unknown clusters, and the algorithm is allowed to ask the oracle whether two items belong to the same cluster or not. However, the answer from the oracle is correct only with probability $\frac{1}{2}+\fracδ{2}$. The goal is to recover the hidden clusters with minimum number of noisy queries. Previous works have shown that the problem can be solved with $O(\frac{nk\log n}{δ^2} + \text{poly}(k,\frac{1}δ, \log n))$ queries, while $Ω(\frac{nk}{δ^2})$ queries is known to be necessary. So, for any values of $k$ and $δ$, there is still a non-trivial gap between upper and lower bounds. In this work, we obtain the first matching upper and lower bounds for a wide range of parameters. In particular, a new polynomial time algorithm with $O(\frac{n(k+\log n)}{δ^2} + \text{poly}(k,\frac{1}δ, \log n))$ queries is proposed. Moreover, we prove a new lower bound of $Ω(\frac{n\log n}{δ^2})$, which, combined with the existing $Ω(\frac{nk}{δ^2})$ bound, matches our upper bound up to an additive $\text{poly}(k,\frac{1}δ,\log n)$ term. To obtain the new results, our main ingredient is an interesting connection between our problem and multi-armed bandit, which might provide useful insights for other similar problems.