{
  "id": "2407.17063",
  "version": "v1",
  "published": "2024-07-24T07:45:13.000Z",
  "updated": "2024-07-24T07:45:13.000Z",
  "title": "Strong Convergence of FISTA Iterates under H{ö}lderian and Quadratic Growth Conditions",
  "authors": [
    "Jean-François Aujol",
    "Charles Dossal",
    "Hippolyte Labarrière",
    "Aude Rondepierre"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "Introduced by Beck and Teboulle, FISTA (for Fast Iterative Shrinkage-Thresholding Algorithm) is a first-order method widely used in convex optimization. Adapted from Nesterov's accelerated gradient method for convex functions, the generated sequence guarantees a decay of the function values of $\\mathcal{O}\\left(n^{-2}\\right)$ in the convex setting. We show that for coercive functions satisfying some local growth condition (namely a H\\''olderian or quadratic growth condition), this sequence strongly converges to a minimizer. This property, which has never been proved without assuming the uniqueness of the minimizer, is associated with improved convergence rates for the function values. The proposed analysis is based on a preliminary study of the Asymptotic Vanishing Damping system introduced by Su et al. in to modelNesterov's accelerated gradient method in a continuous setting. Novel improved convergence results are also shown for the solutions of this dynamical system, including the finite length of the trajectory under the aforementioned geometry conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-24T07:45:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic growth condition",
      "fista iterates",
      "strong convergence",
      "function values",
      "modelnesterovs accelerated gradient method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}