{
  "id": "1405.4980",
  "version": "v2",
  "published": "2014-05-20T07:50:56.000Z",
  "updated": "2015-11-16T18:52:04.000Z",
  "title": "Convex Optimization: Algorithms and Complexity",
  "authors": [
    "Sébastien Bubeck"
  ],
  "comment": "A previous version of the manuscript was titled \"Theory of Convex Optimization for Machine Learning\"",
  "journal": "In Foundations and Trends in Machine Learning, Vol. 8: No. 3-4, pp 231-357, 2015",
  "categories": [
    "math.OC",
    "cs.CC",
    "cs.LG",
    "cs.NA",
    "stat.ML"
  ],
  "abstract": "This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-20T07:50:56.000Z",
      "title": "Theory of Convex Optimization for Machine Learning",
      "abstract": "This monograph presents the main mathematical ideas in convex optimization. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by the seminal book of Nesterov, includes the analysis of the Ellipsoid Method, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, Mirror Descent, and Dual Averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), Saddle-Point Mirror Prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of Interior Point Methods. In stochastic optimization we discuss Stochastic Gradient Descent, mini-batches, Random Coordinate Descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-16T18:52:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex optimization",
      "machine learning",
      "stochastic optimization",
      "black-box optimization",
      "structural optimization"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4980B"
    }
  }
}