{
  "abstract": "Attention mechanisms are central to modern foundation models, yet their training dynamics remain poorly understood, especially when the attention matrices have extensive rank. In this work, we study attention-indexed models, a broad framework that can represent multi-layer and multi-head attention architectures. First, we show that, in a suitable high-dimensional limit, the population-loss landscape is characterized by a finite set of trace order parameters. In contrast, online stochastic gradient descent (SGD) is governed by an infinite hierarchy of matrix moments, which we show can be exponentially well-approximated by a finite truncated system. Second, this framework reveals that attention parameterization itself can act as an architectural implicit bias. Direct optimization of an attention matrix $S\\in\\mathbb{R}^{d\\times d}$ can remain trapped in an uninformative state. Tied attention ($S=WW^\\top$) induces an automatic symmetry-breaking mechanism and yields weak recovery in $Θ(d^2\\log d)$ samples. For untied attention, $S=UV^\\top$, we uncover a fast-slow mechanism: the pre-activation mean first evolves on a fast timescale, while the overlaps evolve on a slower one. Weak recovery on the $Θ(d^2\\log d)$ scale occurs when the state selected by the fast dynamics breaks the initial symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03858",
  "author_total": 4,
  "authors": [
    {
      "name": "Yizhou Xu"
    },
    {
      "name": "Margarita Sagitova"
    },
    {
      "name": "Lenka Zdeborová"
    },
    {
      "name": "Florent Krzakala"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hy7t2d4h6d0y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03858",
  "fulltext_url": null,
  "id": "3hy7t2d4h6d0y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03858",
  "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|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7208771770",
  "pdf_url": "https://arxiv.org/pdf/2609.03858",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hy7t2d4h6d0y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:42Z",
      "native_id": "2609.03858",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03858"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "High-Dimensional Learning Dynamics of Attention-Indexed Models",
  "title_text": "High-Dimensional Learning Dynamics of Attention-Indexed Models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:42Z",
  "url": "https://research.draconitic.com/paper/3hy7t2d4h6d0y",
  "venue": "arXiv",
  "versions": []
}