Federated Wasserstein Distance
Alain Rakotomamonjy Kimia Nadjahi Liva Ralaivola
We introduce a principled way of computing the Wasserstein distance between
two distributions in a federated manner. Namely, we show how to estimate the
Wasserstein distance between two samples stored and kept on different
devices/clients whilst a central entity/server orchestrates the computations
(again, without having access to the samples). To achieve this feat, we take
advantage of the geometric properties of the Wasserstein distance -- in
particular, the triangle inequality -- and that of the associated {\em
geodesics}: our algorithm, FedWad (for Federated Wasserstein Distance),
iteratively approximates the Wasserstein distance by manipulating and
exchanging distributions from the space of geodesics in lieu of the input
samples. In addition to establishing the convergence properties of FedWad, we
provide empirical results on federated coresets and federate optimal transport
dataset distance, that we respectively exploit for building a novel federated
model and for boosting performance of popular federated learning algorithms.