{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-ml-finite-class-bound",
    "area": "machine-learning",
    "topic": "theory",
    "origin": "original",
    "title": "Read a finite-class generalization bound",
    "skills": [
      "generalization",
      "model-complexity"
    ],
    "priority": "stretch",
    "difficulty": "medium",
    "estimated_minutes": 7,
    "prerequisites": [
      "logarithms",
      "probability"
    ],
    "prompt": "A finite hypothesis class has $\\lvert\\mathcal{H}\\rvert=100$ and $n=1000$ independent examples. Using\n\n$$\\varepsilon=\\sqrt{\\frac{\\log\\left(2\\lvert\\mathcal{H}\\rvert/\\delta\\right)}{2n}}$$\n\nwith $\\delta=0.05$, compute $\\varepsilon$ and explain what the simultaneous bound says.\n",
    "answer": "$$\\varepsilon=\\sqrt{\\frac{\\log(4000)}{2000}}\\approx 0.0644.$$\n\nWith probability at least $0.95$, every hypothesis in $\\mathcal{H}$ has empirical and population error differing by at most this amount under the bound's assumptions.\n",
    "check": {
      "kind": "numeric",
      "id": "original-ml-finite-class-bound",
      "values": [
        0.06439739761862287
      ]
    }
  }
}