{
  "abstract": "What information is needed to turn a natural-gradient direction into a useful finite update? Under a population Kullback-Leibler (KL) budget, we call a step useful if it is feasible and loses at most a fraction $\\varepsilon$ of the best feasible gain along the direction. We construct a four-state exponential family whose laws share their initial gradient, scalar Fisher information and natural gradient, yet two laws have disjoint useful-step sets. With these quantities supplied exactly and the law otherwise known only through draws, the family's worst-case sample complexity is $Θ(\\log(1/δ)/(p\\varepsilon^2))$ for small $\\varepsilon$, where $p$ scales rare-state probabilities and $δ$ is the failure probability. The budget is fixed and the optimal gain stays bounded away from zero, so the step length, not the direction, carries this cost. For succinctly described event-tilt models, returning a useful step is NP-hard even with the exact natural gradient and efficient exact sampling. Recovering the unit natural gradient to constant error is also NP-hard even in a two-parameter logistic family with Fisher condition number at most 3. We also give matching sample bounds for event tilts, sample bounds for damped Fisher solves and a population-KL certificate for affine classifiers. In frozen-feature classifier heads, stopping at a sampled KL boundary succeeds in about half of the trials, and a 10% KL margin raises joint success above 93% at a KL budget of 0.01. Thus, knowing where to move is not enough: how far to move can carry an update's entire cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33499",
  "author_total": 3,
  "authors": [
    {
      "name": "Subhransu S. Bhattacharjee"
    },
    {
      "name": "Dylan Campbell"
    },
    {
      "name": "Rahul Shome"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5kkqwb9q0gx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33499",
  "fulltext_url": null,
  "id": "c5kkqwb9q0gx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33499",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214732408",
  "pdf_url": "https://arxiv.org/pdf/2609.33499",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5kkqwb9q0gx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33499",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33499"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The cost of useful natural gradient updates",
  "title_text": "The cost of useful natural gradient updates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/c5kkqwb9q0gx6",
  "venue": "arXiv",
  "versions": []
}