  We study first order methods to compute the barycenter of a probability
distribution $P$ over the space of probability measures with finite second
moment. We develop a framework to derive global rates of convergence for both
gradient descent and stochastic gradient descent despite the fact that the
barycenter functional is not geodesically convex. Our analysis overcomes this
technical hurdle by employing a Polyak-Lojasiewicz (PL) inequality and relies
on tools from optimal transport and metric geometry. In turn, we establish a PL
inequality when $P$ is supported on the Bures-Wasserstein manifold of Gaussian
probability measures. It leads to the first global rates of convergence for
first order methods in this context.
