  The development of modern technology has enabled data collection of
unprecedented size, which poses new challenges to many statistical estimation
and inference problems. This paper studies the maximum score estimator of a
semi-parametric binary choice model under a distributed computing environment
without pre-specifying the noise distribution. An intuitive divide-and-conquer
estimator is computationally expensive and restricted by a non-regular
constraint on the number of machines, due to the highly non-smooth nature of
the objective function. We propose (1) a one-shot divide-and-conquer estimator
after smoothing the objective to relax the constraint, and (2) a multi-round
estimator to completely remove the constraint via iterative smoothing. We
specify an adaptive choice of kernel smoother with a sequentially shrinking
bandwidth to achieve the superlinear improvement of the optimization error over
the multiple iterations. The improved statistical accuracy per iteration is
derived, and a quadratic convergence up to the optimal statistical error rate
is established. We further provide two generalizations to handle the
heterogeneity of datasets and high-dimensional problems where the parameter of
interest is sparse.
