{
  "abstract": "Local Gaussian models of constant-step learning predict output variability and expected losses, but weak convergence alone does not justify these moment predictions. We establish moment-accurate Gaussian mixtures by matching stationary energy with local Ornstein--Uhlenbeck limits, ruling out quadratic tail mass invisible to weak convergence. For step size $a$, the second-order Wasserstein error is $o(\\sqrt a)$, uniformly over invariant laws, using each law's actual root weights. The assumptions combine confinement, descent, finitely many hyperbolic equilibria and root continuity with finite-variance innovations. The result yields observable covariances, expected objective gaps and first-order mean shifts, while allowing singular covariances, compatible saddles and weights without a limit. For additive noise given by a fixed invertible transform of independent standardized Student $t_3$ coordinates, symmetry gives an order-sharp $\\sqrt a$ smooth-test bound. Numerical transport calculations demonstrate the value of root-specific covariances; controlled SGD studies assess observable predictions across step sizes, batch sizes and model geometries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05595",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoli Li"
    },
    {
      "name": "Wei Biao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dy51ghhwmp624.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05595",
  "fulltext_url": null,
  "id": "dy51ghhwmp624",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05595",
  "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": "W7220722482",
  "pdf_url": "https://arxiv.org/pdf/2610.05595",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dy51ghhwmp624.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:16Z",
      "native_id": "2610.05595",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05595"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Moment-Accurate Gaussian Mixtures for Constant-Step Stochastic Approximation",
  "title_text": "Moment-Accurate Gaussian Mixtures for Constant-Step Stochastic Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/dy51ghhwmp624",
  "venue": "arXiv",
  "versions": []
}