  Optimal transport is now a standard tool for solving many problems in
statistics and machine learning. The optimal "transport of probability
measures" is also a recurring theme in stochastic control and distributed
control, where in the latter application the probability measure corresponds to
an empirical distribution associated with a large collection of distributed
agents, subject to local and global control. The goal of this paper is to make
precise these connections, which inspires new relaxations of optimal transport
for application in new and traditional domains. The proposed relaxation
replaces a target measure with a "moment class": a set of probability measures
defined by generalized moment constraints. This is motivated by applications to
control, outlier detection, and to address computational complexity. The main
conclusions are (i) A characterization of the solution is obtained, similar to
Kantorovich duality, in which one of the dual functions in the classical theory
is replaced by a linear combination of the features defining the generalized
moments. Hence the dimension of the optimization problem coincides with the
number of constraints, even with an uncountable state space; (ii) By
introducing regularization in the form of relative entropy, the solution can be
interpreted as replacing a maximum with a softmax in the dual; (iii) In
applications such as control for which it is not known a-priori if the moment
class is non-empty, a relaxation is proposed whose solution admits a similar
characterization; (iv) The gradient of the dual function can be expressed in
terms of the expectation of the features under a tilted probability measure,
which motivates Monte-Carlo techniques for computation.
