  Iterative clustering algorithms help us to learn the insights behind the
data. Unfortunately, this may allow adversaries to infer the privacy of
individuals with some background knowledge. In the worst case, the adversaries
know the centroids of an arbitrary iteration and the information of n-1 out of
n items. To protect individual privacy against such an inference attack,
preserving differential privacy (DP) for the iterative clustering algorithms
has been extensively studied in the interactive settings. However, existing
interactive differentially private clustering algorithms suffer from a
non-convergence problem, i.e., these algorithms may not terminate without a
predefined number of iterations. This problem severely impacts the clustering
quality and the efficiency of a differentially private algorithm. To resolve
this problem, in this paper, we propose a novel differentially private
clustering framework in the interactive settings which controls the orientation
of the movement of the centroids over the iterations to ensure the convergence
by injecting DP noise in a selected area. We prove that, in the expected case,
algorithm under our framework converges in at most twice the iterations of
Lloyd's algorithm. We perform experimental evaluations on real-world datasets
to show that our algorithm outperforms the state-of-the-art of the interactive
differentially private clustering algorithms with guaranteed convergence and
better clustering quality to meet the same DP requirement.
