{
  "abstract": "For smooth nonconvex stochastic multi-objective optimization, stochastic multi-gradient descent (SMG) computes an approximate steepest common descent direction of the objectives from stochastic gradients. Under standard assumptions, this note establishes a new convergence rate for SMG in terms of the squared Pareto-stationarity (PS) measure. With a constant stepsize and linearly growing mini-batches, the expected squared empirical PS measure at the algorithm's output is $\\tilde{O}(T^{-1})$ after $T$ iterations. This improves on the $\\tilde{O}(T^{-1/4})$ bound obtained by Chen et al. (JMLR, 2024) for linearly growing batches, without requiring bounded gradients. Here, $\\tilde{O}(\\cdot)$ suppresses logarithmic factors. The key to the rate improvement is to exploit the Lipschitz continuity of the PS measure, defined by the norm of the multi-gradient descent algorithm (MGDA) direction, rather than the $(1/2)$-Hölder continuity of the MGDA direction used by Chen et al. (2024). For variants of MoCo and MoCo+ with exact MGDA updates and constant batch sizes, the analysis gives expected squared empirical PS rates of $O(T^{-1/2})$ and $\\tilde{O}(T^{-2/3})$, respectively. These bounds match the corresponding single-objective momentum rates up to logarithmic factors. The proof was accidentally discovered while the author was preparing homework for a graduate course: ChatGPT 5.4 Thinking Extended generated the initial proof strategy for the main SMG result in response to an author-written homework-solution prompt; the author then verified and reorganized the resulting argument. The appendices document the prompt, summarize the student submissions with LLM disclosures, and present additional proofs and results on momentum-based stochastic MGDA.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18174",
  "author_total": 1,
  "authors": [
    {
      "name": "Lisha Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b6205yz7mnh9z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18174",
  "fulltext_url": null,
  "id": "b6205yz7mnh9z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18174",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7169888281",
  "pdf_url": "https://arxiv.org/pdf/2607.18174",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b6205yz7mnh9z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2607.18174",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18174"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Improved Convergence Rates for Stochastic Multi-Gradient Descent which Close the Gap: A Proof by AI",
  "title_text": "Improved Convergence Rates for Stochastic Multi-Gradient Descent which Close the Gap: A Proof by AI",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/b6205yz7mnh9z",
  "venue": "arXiv",
  "versions": []
}