Doubly Regularized Entropic Wasserstein Barycenter

We study a general formulation of regularized Wasserstein barycenters that enjoy 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 the $(\lambda ,\tau )$-barycenter, where $\lambda$ is the inner regularization strength and $\tau$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $\lambda ,\tau \ge 0$ and generalizes them. First, we show that, as $\lambda , \tau \rightarrow 0$, regularizing doubly can decrease the approximation error compared to a single regularization. More specifically, we show that for smooth densities and the quadratic cost, the leading order term of the suboptimality in the (unregularized) Wasserstein barycenter objective cancels when $\tau \sim \frac{\lambda }{2}$. We discuss also this phenomenon for isotropic Gaussian distributions where all $(\lambda ,\tau )$-barycenters have closed-form. Second, we show that for $\lambda ,\tau >0$, 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 $n^{-1/2}$. Finally, this formulation is amenable to a grid-free optimization algorithm: we propose a simple Noisy Particle Gradient Descent method which, in the mean-field limit, converges globally at an exponential rate to the $(\lambda ,\tau )$-barycenter.