{
  "id": "2303.11844",
  "version": "v1",
  "published": "2023-03-21T13:42:43.000Z",
  "updated": "2023-03-21T13:42:43.000Z",
  "title": "Doubly Regularized Entropic Wasserstein Barycenters",
  "authors": [
    "Lénaïc Chizat"
  ],
  "categories": [
    "math.OC",
    "cs.LG",
    "stat.ML"
  ],
  "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 $(\\lambda,\\tau)$-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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-21T13:42:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49N99",
      "62G05",
      "90C30"
    ],
    "keywords": [
      "doubly regularized entropic wasserstein barycenters",
      "formulation",
      "smooth density",
      "grid-free optimization algorithm",
      "entropic optimal transport"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}