Doubly Regularized Entropic Wasserstein Barycenters

Abstract

We study a general formulation of regularized Wasserstein barycenters that enjoys favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it $(λ,τ)$-barycenter, where $λ$ is the inner regularization strength and $τ$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $λ,τ\geq 0$ and generalizes them. First, in spite of -- and in fact owing to -- being \emph{doubly} regularized, we show that our formulation is debiased for $τ=λ/2$: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength $λ^2$ of entropic regularization, instead of $\max\{λ,τ\}$ in general. We discuss this phenomenon for isotropic Gaussians where all $(λ,τ)$-barycenters have closed form. Second, we show that for $λ,τ>0$, this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given $n$ samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate $n^{-1/2}$. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple \emph{noisy particle gradient descent} which, in the mean-field limit, converges globally at an exponential rate to the barycenter.