{
  "id": "2212.06000",
  "version": "v1",
  "published": "2022-12-12T16:00:57.000Z",
  "updated": "2022-12-12T16:00:57.000Z",
  "title": "On the Convergence Rate of Sinkhorn's Algorithm",
  "authors": [
    "Promit Ghosal",
    "Marcel Nutz"
  ],
  "categories": [
    "math.OC",
    "math.AP",
    "math.PR"
  ],
  "abstract": "We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $\\pi_{t}$ is shown to satisfy $H(\\pi_{t}|\\pi_{*})+H(\\pi_{*}|\\pi_{t})=O(t^{-1})$ where $H$ denotes relative entropy and $\\pi_{*}$ the optimal coupling. This holds for a large class of cost functions and marginals, including quadratic cost with subgaussian marginals. We also obtain the rate $O(t^{-1})$ for the dual suboptimality and $O(t^{-2})$ for the marginal entropies. More precisely, we derive non-asymptotic bounds, and in contrast to previous results on linear convergence that are limited to bounded costs, our estimates do not deteriorate exponentially with the regularization parameter. We also obtain a stability result for $\\pi_{*}$ as a function of the marginals, quantified in relative entropy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T16:00:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C25",
      "49N05"
    ],
    "keywords": [
      "convergence rate",
      "entropically regularized optimal transport problem",
      "study sinkhorns algorithm",
      "regularization parameter",
      "linear convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}