Stochastic Approximation versus Sample Average Approximation for population Wasserstein barycenters

In machine learning and optimization community there are two main approaches for convex risk minimization problem, namely, the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on the specific problem, however, starting from work \cite{nemirovski2009robust} it was generally accepted that the SA is better than the SAA. Nevertheless, in case of large-scale problems SA may run out of memory as storing all data on one machine and organizing online access to it can be impossible without communications with other machines. SAA in contradistinction to SA allows parallel/distributed calculations. In this paper, we shed new light on the comparison of SA and SAA for particular problem of calculating the population (regularized) Wasserstein barycenter of discrete measures. The conclusion is valid even for non-parallel (non-decentralized) setup.