{
  "abstract": "Synthetic data has become a promising way to scale model training beyond limited human-generated data but it may also induce strong model collapse (Dohmatob et al., 2024), where any fixed fraction of synthetic data prevents model performance from improving under data scaling, leaving a non-vanishing excess risk floor. In this paper, we study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift. We establish finite-sample risk bounds for mixed and two-stage training, separating standard bias and variance from source-mismatch effects, namely fluctuation and persistent drift under mixing and filtered initialization bias under two-stage. These bounds reveal a sharp contrast: mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage, showing that collapse is not inevitable under a simple data curriculum. Under a random sketch model, we further obtain scaling laws for both protocols, with tight results for mixed training in the optimization-saturated regime. These laws show that larger models may amplify synthetic-induced degradation under mixing, and quantify how high-quality synthetic pretraining may reduce bias in two-stage training. Finally, we establish an exact finite-sample necessary-and-sufficient condition for two-stage training to strictly outperform real-only training under the same real-data budget and identical real-stage updates. Overall, our results highlight that synthetic data is neither inherently harmful nor beneficial; its effect depends critically on both its quality and the training protocol used to incorporate it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09572",
  "author_total": 2,
  "authors": [
    {
      "name": "Jichu li"
    },
    {
      "name": "Difan Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fvkw1yrvh92qd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09572",
  "fulltext_url": null,
  "id": "fvkw1yrvh92qd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09572",
  "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.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7212151291",
  "pdf_url": "https://arxiv.org/pdf/2609.09572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fvkw1yrvh92qd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:22Z",
      "native_id": "2609.09572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Learning with Synthetic Data via SGD in High-Dimensional Linear Regression",
  "title_text": "Learning with Synthetic Data via SGD in High-Dimensional Linear Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:22Z",
  "url": "https://research.draconitic.com/paper/fvkw1yrvh92qd",
  "venue": "arXiv",
  "versions": []
}