Differentially Private Wasserstein Barycenters

The Wasserstein barycenter is defined as the mean of a set of probability measures under the optimal transport metric, and has numerous applications in machine learning. In practice these input measures are empirical distributions, often built from sensitive datasets. We present, to our knowledge, the first algorithms for computing Wasserstein barycenters under differential privacy. Empirically, on large-scale US population datasets, our methods produce high-quality private barycenters with strong accuracy-privacy tradeoffs. As a byproduct of our analysis we give a simple reduction that can significantly accelerate the runtime for non-private barycenters, which may be of broader interest.