{
  "abstract": "Standard neural network training relies on learning-rate schedules tied to a fixed horizon, leading to strong path dependence and costly re-tuning as data availability changes. Schedule-Free (SF) methods address this by removing explicit schedules, yet SF-AdamW, the current state-of-the-art anytime optimizer, consistently underperforms well-tuned AdamW baselines. We propose SF-NorMuon, a schedule-free spectral optimizer that closes this gap: with a single hyperparameter configuration, SF-NorMuon matches or exceeds tuned AdamW on 125M and 772M parameter language models across $1$--$8\\times$ Chinchilla horizons, and trails horizon-aware, cosine-scheduled NorMuon by only $\\sim 0.03$ nats, substantially narrowing the gap between anytime and scheduled spectral optimizers. On the theoretical side, we prove a stationarity guarantee for schedule-free spectral dynamics and identify weight decay at the fast iterate as essential for long-horizon stability. SF-NorMuon enables practitioners to obtain high-quality checkpoints at any point during training without committing to a horizon in advance. Fully matching horizon-aware spectral optimizers without a schedule remains an open problem, which we highlight as a direction for future work. By closing the performance gap with tuned AdamW baselines, SF-NorMuon makes horizon-free optimization more practical, taking a step towards truly open-ended, continual learning.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.23061",
  "author_total": 5,
  "authors": [
    {
      "name": "Anuj Apte"
    },
    {
      "name": "Pranav Deshpande"
    },
    {
      "name": "Niraj Kumar"
    },
    {
      "name": "Shouvanik Chakrabarti"
    },
    {
      "name": "Junhyung Lyle Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1023v3t52gksa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.23061",
  "fulltext_url": null,
  "id": "1023v3t52gksa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.23061",
  "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",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7162299259",
  "pdf_url": "https://arxiv.org/pdf/2605.23061",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1023v3t52gksa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.23061",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.23061"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Anytime Training with Schedule-Free Spectral Optimization",
  "title_text": "Anytime Training with Schedule-Free Spectral Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/1023v3t52gksa",
  "venue": "arXiv",
  "versions": []
}