  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 $(\lambda,\tau)$-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
$\lambda,\tau \geq 0$ and generalizes them. First, in spite of -- and in fact
owing to -- being \emph{doubly} regularized, we show that our formulation is
debiased for $\tau=\lambda/2$: the suboptimality in the (unregularized)
Wasserstein barycenter objective is, for smooth densities, of the order of the
strength $\lambda^2$ of entropic regularization, instead of
$\max\{\lambda,\tau\}$ in general. We discuss this phenomenon for isotropic
Gaussians 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
$n^{-1/2}$. 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.
