{
  "abstract": "Hard-label classification is usually trained with smooth surrogate losses such as softmax cross-entropy. We isolate a mechanism by which this mismatch between smooth surrogate and discrete labels produces power-law learning curves for a one-layer softmax student trained by online SGD on Gaussian inputs with argmax teacher labels. In the thermodynamic limit, the dynamics reduce to a growing student-teacher alignment and a residual student variance sustained by SGD noise. At late times, only shrinking layers of examples around teacher decision boundaries remain active. This yields an $α^{-1/3}$ law in training time $α$ not only for the test loss but also for the generalization error, one minus test accuracy. The error law is an online-noise effect. The loss law persists without gradient noise. Learning-rate annealing separates the two exponents, improving the error exponent from 1/3 toward 1/2 while slowing the loss. Simulations support the predictions. Correlated Gaussian inputs and whitened pretrained features change transients, not the late-time decay. Experiments with deep neural networks can show compatible error scaling. The mechanism is asymptotic and complementary to spectral explanations and data-statistics explanations of neural scaling laws.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.22341",
  "author_total": 3,
  "authors": [
    {
      "name": "Marcel Kühn"
    },
    {
      "name": "Yoon Thelge"
    },
    {
      "name": "Bernd Rosenow"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9awvcqtjjmqb1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.22341",
  "fulltext_url": null,
  "id": "9awvcqtjjmqb1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.22341",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7162132149",
  "pdf_url": "https://arxiv.org/pdf/2605.22341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9awvcqtjjmqb1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2605.22341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.22341"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "A Boundary-Layer Mechanism for One-Third Scaling in Online Softmax Classification",
  "title_text": "A Boundary-Layer Mechanism for One-Third Scaling in Online Softmax Classification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:04Z",
  "url": "https://research.draconitic.com/paper/9awvcqtjjmqb1",
  "venue": "arXiv",
  "versions": []
}