{
  "abstract": "Can a network keep learning while its training cross-entropy stays nearly flat at a positive level? We show that it can. On modular addition, a standard testbed for grokking, ReLU networks trained with supported-Polar hidden updates, normalized output-layer updates and weight decay can keep 100% training accuracy while their hidden features change and held-out accuracy rises from zero to above 90%, at relatively flat training cross-entropy. Correct classification need not minimize cross-entropy, which still depends on prediction confidence. We prove this coexistence for one-hidden-layer networks in two finite-window theorems from constructed initializations: one gives a constant gain in the Fourier fraction of the hidden features, and the other holds at arbitrarily low positive loss, where a readout refitted on the training data generalizes before the network. Experiments from random initialization show the same coexistence on a window fixed in advance, and in deeper networks held-out accuracy also rises over post-hoc windows with a net increase in training cross-entropy. With a last-layer rate set per training fraction and head decay scaled with the prime, deeper networks at p=47 and 61 meet a criterion on post-hoc windows, fixed in advance, in nearly all fresh runs. A one-block transformer trained with the same updates and a slowly decaying output gain keeps every training pair correct at every logged checkpoint while its training cross-entropy rises and its held-out accuracy approaches one; with two blocks this holds through the held-out rise, after which training pairs begin to flip. A plateau in training loss need not be a plateau in learning. Code: https://github.com/akashkumar-d/flat-loss-evolving-features.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22593",
  "author_total": 2,
  "authors": [
    {
      "name": "Akash Kumar"
    },
    {
      "name": "Karthik Sankaranarayanan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2pcpgwpapexk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22593",
  "fulltext_url": null,
  "id": "c2pcpgwpapexk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22593",
  "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|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7222940776",
  "pdf_url": "https://arxiv.org/pdf/2610.22593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2pcpgwpapexk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:43Z",
      "native_id": "2610.22593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Flat Loss, Evolving Features: Grokking in Modular Addition",
  "title_text": "Flat Loss, Evolving Features: Grokking in Modular Addition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/c2pcpgwpapexk",
  "venue": "arXiv",
  "versions": []
}