{
  "id": "2103.08280",
  "version": "v1",
  "published": "2021-03-15T11:20:31.000Z",
  "updated": "2021-03-15T11:20:31.000Z",
  "title": "Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction",
  "authors": [
    "Yuze Han",
    "Guangzeng Xie",
    "Zhihua Zhang"
  ],
  "categories": [
    "math.OC",
    "cs.LG",
    "stat.ML"
  ],
  "abstract": "The contribution of this paper includes two aspects. First, we study the lower bound complexity for the minimax optimization problem whose objective function is the average of $n$ individual smooth component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have access to gradient and proximal oracle for each individual component. We develop a novel approach for constructing adversarial problems, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of incremental gradient and proximal oracle. With this approach, we demonstrate the lower bounds of first-order algorithms for finding an $\\varepsilon$-suboptimal point and an $\\varepsilon$-stationary point in different settings. Second, we also derive the lower bounds of minimization optimization with PIFO algorithms from our approach, which can cover the results in \\citep{woodworth2016tight} and improve the results in \\citep{zhou2019lower}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-15T11:20:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite-sum optimization problems",
      "lower complexity bounds",
      "construction",
      "proximal oracle",
      "individual smooth component functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}