{
  "abstract": "The statistical accuracy of neural networks depends on both their approximation power and the complexity of the class fitted from data. While increasing network size is a natural way to improve approximation, parameter magnitude provides another resource whose role must be quantified in both respects. We establish a sharp width--magnitude tradeoff at fixed depth using one elementary bounded $1$-Lipschitz Dyadic--Triangular Activation. For the unit $β$-Hölder ball on $[0,1]^d$ with $0<β\\leq1$, the optimal $L^p$ approximation error for $0<p<\\infty$ is of order $[N^2\\log(eNT)]^{-β/d}$ when the network width satisfies $N\\geq2d+3$ and the parameter magnitudes are bounded by $T\\geq1$. Matching lower bounds hold for every fixed globally Hölder activation; its Hölder exponent affects the constants but not the rate. Under bounded design densities and independent centered sub-Gaussian noise, approximate least squares over the full clipped class at depth $23$ attains the classical Hölder minimax risk $\\mathcal{O}(M^{-\\frac{2β}{2β+d}})$ without logarithmic loss whenever $N^2\\log(eNT)\\asymp M^{\\frac{d}{2β+d}}$, where $M$ is the sample size. This yields a continuum of statistically optimal choices, ranging from unit parameter radius to fixed network size. At fixed size, four hidden layers with at most $8d+7$ nonzero parameters give a near-optimal radius, while six layers with at most $8d+27$ attain the optimal order $\\log T=\\mathcal{O}(η^{-d/β})$ at approximation error $η$. The same decoding method also yields fixed-size Transformer approximation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25710",
  "author_total": 4,
  "authors": [
    {
      "name": "Baicheng Li"
    },
    {
      "name": "Zuowei Shen"
    },
    {
      "name": "Haizhao Yang"
    },
    {
      "name": "Shijun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s5yk5atvcbvt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25710",
  "fulltext_url": null,
  "id": "0s5yk5atvcbvt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25710",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7214118081",
  "pdf_url": "https://arxiv.org/pdf/2609.25710",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s5yk5atvcbvt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:49Z",
      "native_id": "2609.25710",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25710"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Optimal Tradeoffs Between Network Size and Parameter Magnitude in Neural Approximation and Minimax Regression",
  "title_text": "Optimal Tradeoffs Between Network Size and Parameter Magnitude in Neural Approximation and Minimax Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:49Z",
  "url": "https://research.draconitic.com/paper/0s5yk5atvcbvt",
  "venue": "arXiv",
  "versions": []
}