Semi-Supervised Algorithms for Approximately Optimal and Accurate Clustering

We study $k$-means clustering in a semi-supervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are sufficient to efficiently recover a clustering that, with probability at least $(1 - \delta)$, simultaneously has a cost of at most $(1 + \epsilon)$ times the optimal cost and an accuracy of at least $(1 - \epsilon)$? We show how to achieve such a clustering on $n$ points with $O{((k^2 \log n) \cdot m{(Q, \epsilon^4, \delta / (k\log n))})}$ oracle queries, when the $k$ clusters can be learned with an $\epsilon'$ error and a failure probability $\delta'$ using $m(Q, \epsilon',\delta')$ labeled samples in the supervised setting, where $Q$ is the set of candidate cluster centers. We show that $m(Q, \epsilon', \delta')$ is small both for $k$-means instances in Euclidean space and for those in finite metric spaces. We further show that, for the Euclidean $k$-means instances, we can avoid the dependency on $n$ in the query complexity at the expense of an increased dependency on $k$: specifically, we give a slightly more involved algorithm that uses $O(k^4/(\epsilon^2 \delta) + (k^{9}/\epsilon^4) \log(1/\delta) + k \cdot m(\mathbb{R}^r, \epsilon^4/k, \delta))$ oracle queries. We also show that the number of queries needed for $(1 - \epsilon)$-accuracy in Euclidean $k$-means must linearly depend on the dimension of the underlying Euclidean space, and for finite metric space $k$-means, we show that it must at least be logarithmic in the number of candidate centers. This shows that our query complexities capture the right dependencies on the respective parameters.