  This paper addresses the problem of online learning in metric spaces using
exponential weights. We extend the analysis of the exponentially weighted
average forecaster, traditionally studied in a Euclidean settings, to a more
abstract framework. Our results rely on the notion of barycenters, a suitable
version of Jensen's inequality and a synthetic notion of lower curvature bound
in metric spaces known as the measure contraction property. We also adapt the
online-to-batch conversion principle to apply our results to a statistical
learning framework.
