{
  "abstract": "Scaling deep learning models efficiently requires predictable performance transfer across compute regimes. Establishing reliable hyperparameter transfer allows optimal training dynamics and structural choices to be discovered on low-cost proxy models, avoiding catastrophic failures during massive-scale training. Existing depth-transfer analyses are architecture-specific and heuristic: hyperparameter transfer is only justified through toy blocks, and both rely on infinite-depth dynamics, a specific initialization, and feature-learning criteria. Building on $L$-smoothness framework of Xu et al. (2026), we understand hyperparameter transfer as how do the Lipschitz and $L$-smoothness constants of the loss scale with the depth $K$? We show that inverse-depth residual branch scaling $α=Θ(1/K)$ is sufficient to make both constants depth-independent at fixed width, uniformly over the bounded parameter domain and for arbitrary $p\\to q$ geometries. Because the bounds depend on each residual branch only through its Lipschitz and smoothness constants, the argument applies to general residual networks (including branches of block depth $\\geq 2$ such as transformer and ResNet blocks) without assumptions on initialization or correlations between layer weights. Since the smoothness constant controls the curvature penalty in the descent inequality, our bounds motivate a learning-rate scale without an additional depth-dependent curvature correction for updates bounded in the prescribed norm. This provides a geometric explanation of inverse-depth residual scaling and its application to the operator-norm geometries associated with AdamW, Muon, and MOGA. Learning-rate sweeps on GPT-2 (depths 2--128) and CIFAR-100 ResNets (8--32 residual blocks) confirm that Muon and MOGA exhibit stable learning-rate transfer under these prescriptions, with a common optimal rate across a 64-fold increase in depth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14344",
  "author_total": 4,
  "authors": [
    {
      "name": "Maximilian Wong"
    },
    {
      "name": "Chenyu Xu"
    },
    {
      "name": "Jiajin Li"
    },
    {
      "name": "Yiping Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cg6jc2ce4cvap.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14344",
  "fulltext_url": null,
  "id": "cg6jc2ce4cvap",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14344",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221809970",
  "pdf_url": "https://arxiv.org/pdf/2610.14344",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cg6jc2ce4cvap.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14344",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14344"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Depth Transfer of Neural Optimizers via L-Smoothness",
  "title_text": "On the Depth Transfer of Neural Optimizers via L-Smoothness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/cg6jc2ce4cvap",
  "venue": "arXiv",
  "versions": []
}