{
  "abstract": "Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17759",
  "author_total": 1,
  "authors": [
    {
      "name": "Pengcheng Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eahzxqgqkch5h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17759",
  "fulltext_url": null,
  "id": "eahzxqgqkch5h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17759",
  "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": "W7213530534",
  "pdf_url": "https://arxiv.org/pdf/2609.17759",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eahzxqgqkch5h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17759",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17759"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Derivative-Free Structured Updates for Muon",
  "title_text": "Derivative-Free Structured Updates for Muon",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/eahzxqgqkch5h",
  "venue": "arXiv",
  "versions": []
}