{
  "abstract": "Pretraining followed by fine-tuning introduces a compute-allocation problem: under a fixed training budget, compute spent improving the upstream objective reduces the compute available for downstream adaptation. Despite its practical importance, this trade-off is not yet well understood theoretically, even in simple models. In this paper, we cast this allocation as a compute-split problem under a two-stage pretrain--fine-tune procedure with fixed total optimisation budget, using regularised least squares trained by gradient descent as a tractable setting. We characterise the optimal split under data-dependent evaluation geometries induced by the fine-tuning problem. Our results show that the allocation depends on how pretraining directions affect fine-tuning predictions and how fine-tuning shifts are seen through downstream data geometry. In particular, the relevant quantities are determined by prediction-relevant spectral components of the pretraining and fine-tuning empirical covariances. Technically, the analysis relies on a basis-invariant, eigenspace-level spectral decomposition, together with perturbative control of the non-commuting pretraining and fine-tuning dynamics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16262",
  "author_total": 3,
  "authors": [
    {
      "name": "Alex Buna"
    },
    {
      "name": "Fanghui Liu"
    },
    {
      "name": "Patrick Rebeschini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0ncdxdryapaj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16262",
  "fulltext_url": null,
  "id": "a0ncdxdryapaj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16262",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213428178",
  "pdf_url": "https://arxiv.org/pdf/2609.16262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0ncdxdryapaj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:51Z",
      "native_id": "2609.16262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Compute-Optimal Pretrain--Fine-tune in Ridge Gradient Descent",
  "title_text": "Compute-Optimal Pretrain--Fine-tune in Ridge Gradient Descent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/a0ncdxdryapaj",
  "venue": "arXiv",
  "versions": []
}