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.

翻译：我们提出了一种以联邦方式计算两个分布之间Wasserstein距离的原理性方法。具体而言，我们展示了如何在中央实体/服务器协调计算（同样无需访问样本）的情况下，估算存储并保留在不同设备/客户端上的两个样本之间的Wasserstein距离。为实现这一目标，我们利用了Wasserstein距离的几何性质——尤其是三角不等式——以及相关测地线的性质：我们的算法FedWad（联邦Wasserstein距离）通过操纵和交换来自测地线空间的分布（而非输入样本）来迭代逼近Wasserstein距离。除了确立FedWad的收敛性质外，我们还提供了关于联邦核心集和联邦最优传输数据集距离的实证结果，分别用于构建新型联邦模型以及提升主流联邦学习算法的性能。