  Optimal transport distances are powerful tools to compare probability
distributions and have found many applications in machine learning. Yet their
algorithmic complexity prevents their direct use on large scale datasets. To
overcome this challenge, practitioners compute these distances on minibatches
{\em i.e.} they average the outcome of several smaller optimal transport
problems. We propose in this paper an analysis of this practice, which effects
are not well understood so far. We notably argue that it is equivalent to an
implicit regularization of the original problem, with appealing properties such
as unbiased estimators, gradients and a concentration bound around the
expectation, but also with defects such as loss of distance property. Along
with this theoretical analysis, we also conduct empirical experiments on
gradient flows, GANs or color transfer that highlight the practical interest of
this strategy.
