{
  "id": "2205.09647",
  "version": "v1",
  "published": "2022-05-19T16:04:40.000Z",
  "updated": "2022-05-19T16:04:40.000Z",
  "title": "The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization",
  "authors": [
    "Dmitry Kovalev",
    "Alexander Gasnikov"
  ],
  "categories": [
    "math.OC",
    "cs.LG"
  ],
  "abstract": "In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\\Omega\\left(\\epsilon^{-2/(3p+1)}\\right)$ on the number of the $p$-th order oracle calls required by an algorithm to find an $\\epsilon$-accurate solution to the problem, where the $p$-th order oracle stands for the computation of the objective function value and the derivatives up to the order $p$. However, the existing state-of-the-art high-order methods of Gasnikov et al. (2019b); Bubeck et al. (2019); Jiang et al. (2019) achieve the oracle complexity $\\mathcal{O}\\left(\\epsilon^{-2/(3p+1)} \\log (1/\\epsilon)\\right)$, which does not match the lower bound. The reason for this is that these algorithms require performing a complex binary search procedure, which makes them neither optimal nor practical. We fix this fundamental issue by providing the first algorithm with $\\mathcal{O}\\left(\\epsilon^{-2/(3p+1)}\\right)$ $p$-th order oracle complexity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T16:04:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smooth convex optimization",
      "first optimal acceleration",
      "high-order methods",
      "smooth convex minimization problems",
      "th order oracle calls"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}