Doubly Regularized Entropic Wasserstein Barycenters

Code

• github.com/lchizat/2023-doubly-entropic-barycenter
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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 (,)-barycenter, where is the inner regularization strength and the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of , 0 and generalizes them. First, in spite of -- and in fact owing to -- being doubly regularized, we show that our formulation is debiased for =/2: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength ^2 of entropic regularization, instead of \,\ in general. We discuss this phenomenon for isotropic Gaussians where all (,)-barycenters have closed form. Second, we show that for ,>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 noisy particle gradient descent which, in the mean-field limit, converges globally at an exponential rate to the barycenter.

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