{
  "id": "2406.18789",
  "version": "v1",
  "published": "2024-06-26T23:17:33.000Z",
  "updated": "2024-06-26T23:17:33.000Z",
  "title": "Fast Convergence of Frank-Wolfe algorithms on polytopes",
  "authors": [
    "Elias Wirth",
    "Javier Pena",
    "Sebastian Pokutta"
  ],
  "comment": "28 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "We provide a template to derive convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-26T23:17:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C25",
      "90C52"
    ],
    "keywords": [
      "frank-wolfe algorithm",
      "fast convergence",
      "error bound",
      "popular versions",
      "derive convergence rates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}