{
  "id": "2403.11787",
  "version": "v2",
  "published": "2024-03-18T13:46:28.000Z",
  "updated": "2024-09-27T15:27:37.000Z",
  "title": "On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems",
  "authors": [
    "Zehui Zhou"
  ],
  "comment": "45 pages, 3 figures",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.OC"
  ],
  "abstract": "Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. In this work, we analyze a new data-driven regularized stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in infinite dimensional Hilbert spaces. At each step of the iteration, the method randomly selects one equation from the nonlinear system combined with a corresponding equation from the learned system based on training data to obtain a stochastic estimate of the gradient and then performs a descent step with the estimated gradient. We prove the regularizing property of this method under the tangential cone condition and a priori parameter choice and then derive the convergence rates under the additional source condition and range invariance conditions. Several numerical experiments are provided to complement the analysis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-27T15:27:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65J20",
      "65J22",
      "47J06"
    ],
    "keywords": [
      "data-driven regularized stochastic gradient descent",
      "nonlinear ill-posed problems",
      "convergence",
      "infinite dimensional hilbert spaces",
      "additional source condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}