{
  "id": "2203.17226",
  "version": "v1",
  "published": "2022-03-29T19:26:20.000Z",
  "updated": "2022-03-29T19:26:20.000Z",
  "title": "A Derivation of Nesterov's Accelerated Gradient Algorithm from Optimal Control Theory",
  "authors": [
    "I. M. Ross"
  ],
  "comment": "7 pages. arXiv admin note: text overlap with arXiv:1902.09004",
  "categories": [
    "math.OC",
    "cs.LG"
  ],
  "abstract": "Nesterov's accelerated gradient algorithm is derived from first principles. The first principles are founded on the recently-developed optimal control theory for optimization. This theory frames an optimization problem as an optimal control problem whose trajectories generate various continuous-time algorithms. The algorithmic trajectories satisfy the necessary conditions for optimal control. The necessary conditions produce a controllable dynamical system for accelerated optimization. Stabilizing this system via a quadratic control Lyapunov function generates an ordinary differential equation. An Euler discretization of the resulting differential equation produces Nesterov's algorithm. In this context, this result solves the purported mystery surrounding the algorithm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-29T19:26:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C25",
      "49K15",
      "93D05",
      "68T07",
      "G.1.6",
      "I.2.8"
    ],
    "keywords": [
      "nesterovs accelerated gradient algorithm",
      "optimal control theory",
      "control lyapunov function generates",
      "differential equation produces nesterovs algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}