  Nearest neighbor is a popular class of classification methods with many
desirable properties. For a large data set which cannot be loaded into the
memory of a single machine due to computation, communication, privacy, or
ownership limitations, we consider the divide and conquer scheme: the entire
data set is divided into small subsamples, on which nearest neighbor
predictions are made, and then a final decision is reached by aggregating the
predictions on subsamples by majority voting. We name this method the big
Nearest Neighbor (bigNN) classifier, and provide its rates of convergence under
minimal assumptions, in terms of both the excess risk and the classification
instability, which are proven to be the same rates as the oracle nearest
neighbor classifier and cannot be improved. To significantly reduce the
prediction time that is required for achieving the optimal rate, we also
consider the pre-training acceleration technique applied to the bigNN method,
with proven convergence rate. We find that in the distributed setting, the
optimal choice of the neighbor $k$ should scale with both the total sample size
and the number of partitions, and there is a theoretical upper limit for the
latter. Numerical studies have verified the theoretical findings.
