On the Complexity of Approximating Wasserstein Barycenter

Abstract

We study the complexity of approximating Wassertein barycenter of \(m\) discrete measures, or histograms of size \(n\) by contrasting two alternative approaches, both using entropic regularization. The first approach is based on the Iterative Bregman Projections (IBP) algorithm for which our novel analysis gives a complexity bound proportional to \(\frac{mn^2}{\varepsilon^2}\) to approximate the original non-regularized barycenter. Using an alternative accelerated-gradient-descent-based approach, we obtain a complexity proportional to \(\frac{mn^{2.5}}{\varepsilon} \). As a byproduct, we show that the regularization parameter in both approaches has to be proportional to \(\varepsilon\), which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology.