  Semidiscrete optimal transport is a challenging generalization of the
classical transportation problem in linear programming. The goal is to design a
joint distribution for two random variables (one continuous, one discrete) with
fixed marginals, in a way that minimizes expected cost. We formulate a novel
variant of this problem in which the cost functions are unknown, but can be
learned through noisy observations; however, only one function can be sampled
at a time. We develop a semi-myopic algorithm that couples online learning with
stochastic approximation, and prove that it achieves optimal convergence rates,
despite the non-smoothness of the stochastic gradient and the lack of strong
concavity in the objective function.
