  Clustering is a fundamental problem in statistics and machine learning.
Lloyd's algorithm, proposed in 1957, is still possibly the most widely used
clustering algorithm in practice due to its simplicity and empirical
performance. However, there has been little theoretical investigation on the
statistical and computational guarantees of Lloyd's algorithm. This paper is an
attempt to bridge this gap between practice and theory. We investigate the
performance of Lloyd's algorithm on clustering sub-Gaussian mixtures. Under an
appropriate initialization for labels or centers, we show that Lloyd's
algorithm converges to an exponentially small clustering error after an order
of $\log n$ iterations, where $n$ is the sample size. The error rate is shown
to be minimax optimal. For the two-mixture case, we only require the
initializer to be slightly better than random guess.
  In addition, we extend the Lloyd's algorithm and its analysis to community
detection and crowdsourcing, two problems that have received a lot of attention
recently in statistics and machine learning. Two variants of Lloyd's algorithm
are proposed respectively for community detection and crowdsourcing. On the
theoretical side, we provide statistical and computational guarantees of the
two algorithms, and the results improve upon some previous signal-to-noise
ratio conditions in literature for both problems. Experimental results on
simulated and real data sets demonstrate competitive performance of our
algorithms to the state-of-the-art methods.
