{
  "abstract": "Understanding how gradient descent learns features remains a central challenge in neural network theory. We study this question in a minimal multi-index setting that already exhibits rich multi-phase gradient dynamics: a well-specified two-neuron ReLU teacher-student model under isotropic Gaussian inputs, trained by population gradient flow from small initialization. We show that the dynamics are organized by two directions: the \"easy\" bisector direction, which carries the leading signal, and the \"hard\" splitting direction, which governs specialization. To characterize the loss structure and learning behavior, we analyze the population squared-loss landscape and show that every critical point with both student weights nonzero is either a global minimum or a saddle. We then track the gradient-flow trajectory, showing that student neurons first collapse toward a bisector saddle before escaping and specializing to teacher neurons. Our results provide both a landscape and dynamics account of the multi-phase symmetry-breaking behavior that arises in a simple multi-index model and under standard algorithmic and architectural choices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.42118",
  "author_total": 4,
  "authors": [
    {
      "name": "Binghua Li"
    },
    {
      "name": "Mengzhe Li"
    },
    {
      "name": "Denny Wu"
    },
    {
      "name": "Tianhao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1kg5b5zsh5nd2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.42118",
  "fulltext_url": null,
  "id": "1kg5b5zsh5nd2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.42118",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.42118",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1kg5b5zsh5nd2.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:25Z",
      "native_id": "2610.42118",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.42118"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics",
  "title_text": "Gradient Descent on Two ReLU Neurons: Global Landscape and Bifurcation Dynamics",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/1kg5b5zsh5nd2",
  "venue": "arXiv",
  "versions": []
}