{
  "abstract": "This paper examines how row-wise renormalization affects Muon, focusing on the gap between NorMuon's worst-case guarantees and its practical performance (Li et al.). Despite its growing adoption and promising performance in large language model (LLM) pretraining, NorMuon's worst-case guarantees remain poorly understood. One fundamental question is: Does row normalization yield provable convergence gains, potentially through its interaction with approximate polar computation and exponential moving-average momentum? Our results show that row normalization introduces a dimension-dependent factor in the worst-case iteration complexity under the operator-norm geometry, which persists even with exact polar computation and any fixed momentum parameters. Indeed, we establish an algorithm-dependent lower bound and a matching upper bound in deterministic settings, and extend our upper bound analysis to stochastic settings. Both upper-bound analyses allow approximate polar computation. Experiments show that NorMuon is slower than Muon on synthetic problems inspired by our worst-case construction, yet outperforms Muon in LLM pretraining. These findings sharpen the puzzle of why row normalization helps in practice and complement the recent findings of Dewulf et al.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39114",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiayu Zhang"
    },
    {
      "name": "Tianyi Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bdfnt5vhxwf7m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39114",
  "fulltext_url": null,
  "id": "bdfnt5vhxwf7m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39114",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7215055460",
  "pdf_url": "https://arxiv.org/pdf/2609.39114",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bdfnt5vhxwf7m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.39114",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39114"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Row Normalization Puzzle in Muon",
  "title_text": "The Row Normalization Puzzle in Muon",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/bdfnt5vhxwf7m",
  "venue": "arXiv",
  "versions": []
}