  We present and study a novel algorithm for the computation of 2-Wasserstein
population barycenters of absolutely continuous probability measures on
Euclidean space. The proposed method can be seen as a stochastic gradient
descent procedure in the 2-Wasserstein space, as well as a manifestation of a
Law of Large Numbers therein. The algorithm aims at finding a Karcher mean or
critical point in this setting, and can be implemented ``online", sequentially
using i.i.d. random measures sampled from the population law. We provide
natural sufficient conditions for this algorithm to a.s. converge in the
Wasserstein space towards the population barycenter, and we introduce a novel,
general condition which ensures uniqueness of Karcher means and moreover allows
us to obtain explicit, parametric convergence rates for the expected optimality
gap. We furthermore study the mini-batch version of this algorithm, and discuss
examples of families of population laws to which our method and results can be
applied. This work expands and deepens ideas and results introduced in an early
version of \cite{backhoff2018bayesian}, in which a statistical application (and
numerical implementation) of this method is developed in the context of
Bayesian learning.
