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.
