{
  "abstract": "We study the last iterate of stochastic gradient descent (SGD) on streaming least squares with discretely distributed features (Slock's model), which decouples the dynamics across eigendirections. Under power-law source and capacity conditions, we derive the exact finite-time risk and asymptotic equivalents of constant and Warmup-Stable-Decay (WSD) schedules. With a tuned base learning-rate, the resulting scaling law decays as $Θ(T^{-q})$, where $T$ is the horizon, and $q$ is independent of the cooldown length $c \\in (0,1]$ but is strictly smaller for the constant schedule: any cooldown of fixed relative length $c > 0$, however short, improves the risk by a polynomial factor, while its length only affects the constant, which is minimized by linear decay ($c = 1$). Finally, we show that the optimal risk depends on the batch size $b$ and horizon $T$ only through $bT$, up to a critical batch size.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16518",
  "author_total": 3,
  "authors": [
    {
      "name": "Noé Colomban"
    },
    {
      "name": "Francis Bach"
    },
    {
      "name": "Fabian Schaipp"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6a5py5hv2xq0b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16518",
  "fulltext_url": null,
  "id": "6a5py5hv2xq0b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16518",
  "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",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221761198",
  "pdf_url": "https://arxiv.org/pdf/2610.16518",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6a5py5hv2xq0b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16518",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16518"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Asymptotic Risk of Learning-Rate Schedules for Least-Squares Regression with Discretely Distributed Features",
  "title_text": "Asymptotic Risk of Learning-Rate Schedules for Least-Squares Regression with Discretely Distributed Features",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/6a5py5hv2xq0b",
  "venue": "arXiv",
  "versions": []
}