  We consider the optimization problem of minimizing a functional defined over
a family of probability distributions, where the objective functional is
assumed to possess a variational form. Such a distributional optimization
problem arises widely in machine learning and statistics, with Monte-Carlo
sampling, variational inference, policy optimization, and generative
adversarial network as examples. For this problem, we propose a novel
particle-based algorithm, dubbed as variational transport, which approximately
performs Wasserstein gradient descent over the manifold of probability
distributions via iteratively pushing a set of particles. Specifically, we
prove that moving along the geodesic in the direction of functional gradient
with respect to the second-order Wasserstein distance is equivalent to applying
a pushforward mapping to a probability distribution, which can be approximated
accurately by pushing a set of particles. Specifically, in each iteration of
variational transport, we first solve the variational problem associated with
the objective functional using the particles, whose solution yields the
Wasserstein gradient direction. Then we update the current distribution by
pushing each particle along the direction specified by such a solution. By
characterizing both the statistical error incurred in estimating the
Wasserstein gradient and the progress of the optimization algorithm, we prove
that when the objective function satisfies a functional version of the
Polyak-\L{}ojasiewicz (PL) (Polyak, 1963) and smoothness conditions,
variational transport converges linearly to the global minimum of the objective
functional up to a certain statistical error, which decays to zero sublinearly
as the number of particles goes to infinity.
