{
  "abstract": "Natural gradient descent (NGD) underlies common methods in ML. For dually flat families, idealized NGD on the forward Kullback--Leibler objective follows the mixture geodesic which is often longer than the shortest Fisher--Rao path. We quantify this overhead by the inefficiency ratio \\(R \\ge 1\\), the Fisher length of the mixture geodesic divided by the Fisher--Rao distance, and bound its supremum over endpoint pairs as a function of the parameter dimension \\(d\\). A tensor criterion identifies the regime (I) families, with \\(R=1\\) everywhere: exactly those with quadratic potential or dimension one, such as fixed-covariance Gaussians. For non-quadratic families, we prove two further regimes: (II) bounded third-order skewness plus finite Fisher--Rao diameter yields a dimension-independent bound; and (III) for products of scale families---including Gaussian covariances and Gamma rates---\\(R\\) grows as \\(Θ(\\sqrt{\\log d})\\), unbounded in \\(d\\). Under a per-step Fisher-chord budget, \\(R\\) translates to a practical computational cost: NGD requires asymptotically at least \\(R\\) times as many steps as an optimizer following the Fisher--Rao geodesic. Experiments confirm all three regimes: \\(R=1\\) to machine precision for quadratic-potential families (I), the categorical bound \\(π/(2\\sqrt{2})\\) is approached but not attained (II), and sampled scale-product \\(R\\) grows with \\(d\\), reaching \\(R \\approx 1.5\\) for long, high-dimensional moves (III).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07228",
  "author_total": 2,
  "authors": [
    {
      "name": "Guni Sharon"
    },
    {
      "name": "Alan Kuhnle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4jgtn1d722jq2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07228",
  "fulltext_url": null,
  "id": "4jgtn1d722jq2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07228",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7220990030",
  "pdf_url": "https://arxiv.org/pdf/2610.07228",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4jgtn1d722jq2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2610.07228",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07228"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "How Inefficient Is Natural Gradient Descent? From Exact Optimality to Θ( \\sqrt{ \\log d } ) Divergence",
  "title_text": "How Inefficient Is Natural Gradient Descent? From Exact Optimality to Θ( \\sqrt{ \\log d } ) Divergence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/4jgtn1d722jq2",
  "venue": "arXiv",
  "versions": []
}