  We consider the population Wasserstein barycenter problem for random
probability measures supported on a finite set of points and generated by an
online stream of data. This leads to a complicated stochastic optimization
problem where the objective is given as an expectation of a function given as a
solution to a random optimization problem. We employ the structure of the
problem and obtain a convex-concave stochastic saddle-point reformulation of
this problem. In the setting when the distribution of random probability
measures is discrete, we propose a stochastic optimization algorithm and
estimate its complexity. The second result, based on kernel methods, extends
the previous one to the arbitrary distribution of random probability measures.
Moreover, this new algorithm has a total complexity better than the Stochastic
Approximation approach combined with the Sinkhorn algorithm in many cases. We
also illustrate our developments by a series of numerical experiments.
