{
  "id": "1908.08394",
  "version": "v1",
  "published": "2019-08-22T14:02:46.000Z",
  "updated": "2019-08-22T14:02:46.000Z",
  "title": "A General Analysis Framework of Lower Complexity Bounds for Finite-Sum Optimization",
  "authors": [
    "Guangzeng Xie",
    "Luo Luo",
    "Zhihua Zhang"
  ],
  "categories": [
    "math.OC",
    "cs.LG",
    "stat.ML"
  ],
  "abstract": "This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for each individual component. For the strongly-convex case, we prove such an algorithm can not reach an $\\varepsilon$-suboptimal point in fewer than $\\Omega((n+\\sqrt{\\kappa n})\\log(1/\\varepsilon))$ iterations, where $\\kappa$ is the condition number of the objective function. This lower bound is tighter than previous results and perfectly matches the upper bound of the existing proximal incremental first-order oracle algorithm Point-SAGA. We develop a novel construction to show the above result, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of proximal oracle and also could be used to general convex and average smooth cases naturally.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-22T14:02:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower complexity bounds",
      "general analysis framework",
      "finite-sum optimization",
      "first-order oracle algorithm point-saga",
      "proximal incremental first-order oracle algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}