  Motivated by the mode estimation problem of an unknown multivariate
probability density function, we study the problem of identifying the point
with the minimum k-th nearest neighbor distance for a given dataset of n
points. We study the case where the pairwise distances are apriori unknown, but
we have access to an oracle which we can query to get noisy information about
the distance between any pair of points. For two natural oracle models, we
design a sequential learning algorithm, based on the idea of confidence
intervals, which adaptively decides which queries to send to the oracle and is
able to correctly solve the problem with high probability. We derive
instance-dependent upper bounds on the query complexity of our proposed scheme
and also demonstrate significant improvement over the performance of other
baselines via extensive numerical evaluations.
