{
  "abstract": "In this paper, we establish the first asymptotic convergence guarantees for the Muon algorithm through a more accurate proxy for the Newton-Schultz iteration than the typical matrix sign function. We prove that, for appropriate choices of hyperparameters, the iterates satisfy $\\lim_{k\\to\\infty}\\|\\nabla f(x_k)\\|=0$, and, under a global Polyak-Łojasiewicz condition, that the sequence of function values converges linearly. The key insight is that the regularization, implicit in Muon's Newton-Schulz implementation, induces a bounded preconditioner, exposing Muon as a \\emph{preconditioned Polyak heavy-ball} method and enabling a classical Lyapunov analysis. This observation naturally motivates applying the same preconditioning structure to the Nesterov gradient evaluation. We formalize this idea by introducing \\emph{Muesterov}, a Nesterov-based variant of Muon, and prove that it enjoys the same convergence guarantees, extending the theoretical framework beyond the heavy-ball setting. Numerical experiments on a scalar cross-entropy problem corroborate the theory and illuminate the joint role of the learning rate and the Newton-Schulz regularizer in controlling convergence. Preliminary numerical simulations training the nanoGPT dataset provide intuition regarding the relevance of the observations in this paper to practical applications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30546",
  "author_total": 4,
  "authors": [
    {
      "name": "Arthur C. B. de Oliveira"
    },
    {
      "name": "Dhruv D. Jatkar"
    },
    {
      "name": "Guilherme S. Vicinansa"
    },
    {
      "name": "Eduardo D. Sontag"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1g7rswthkbvx9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30546",
  "fulltext_url": null,
  "id": "1g7rswthkbvx9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30546",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214607794",
  "pdf_url": "https://arxiv.org/pdf/2609.30546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1g7rswthkbvx9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Convergence guarantees for Muon: New parameter regimes and generalizations",
  "title_text": "Convergence guarantees for Muon: New parameter regimes and generalizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/1g7rswthkbvx9",
  "venue": "arXiv",
  "versions": []
}