Doubly Regularized Entropic Wasserstein Barycenters

arXiv2303.11844MaRDI QIDQ6430335

Lénaïc Chizat

Publication date: 21 March 2023

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 $(lambda,au)$-barycenter, where $lambda$ is the inner regularization strength and $au$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $lambda,augeq0$ and generalizes them. First, in spite of -- and in fact owing to -- being emph{doubly} regularized, we show that our formulation is debiased for $au=lambda/2$: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength $lambd{a}^2$ of entropic regularization, instead of $maxlambda,au$ in general. We discuss this phenomenon for isotropic Gaussians where all $(lambda,au)$-barycenters have closed form. Second, we show that for $lambda,au>0$, 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.

Has companion code repository: https://github.com/lchizat/2023-doubly-entropic-barycenter

Mathematics Subject Classification ID

Nonlinear programming (90C30) Miscellaneous topics in calculus of variations and optimal control (49N99)

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