{
  "abstract": "The notion of s-composability was introduced to compose optimized stepsize schedules while preserving sharp guarantees. We show that the same joint potential has a second use: it can serve directly as a certificate of sharp last-iterate performance. We develop this viewpoint for constant and silver schedules. For the constant schedule, we prove the s-composability statement posed as an open question by Grimmer et al. (2025), at every finite horizon, through an explicit nonnegative smooth-convex interpolation certificate. The result identifies the unique constant stepsize minimizing the worst-case final gradient norm of smooth convex gradient descent under an initial-distance bound, thereby proving the optimal stepsize, value, and uniqueness predicted by Conjecture 3 of Taylor et al. (2017) without resolving its full worst-case curve. Through the Moreau envelope, the same constant is also the unique constant relaxation minimizing the worst-case final residual of relaxed proximal point. More generally, for every positive s-composable schedule, we determine the exact worst-case final proximal objective-gap constant and give a matching one-dimensional example. Applying this result to the already s-composable original silver schedule yields its exact last-iterate objective-gap constant, closing a question left open by Wang et al. (2025).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13765",
  "author_total": 1,
  "authors": [
    {
      "name": "Salah Chikhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7prwmqbj2jpjd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13765",
  "fulltext_url": null,
  "id": "7prwmqbj2jpjd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13765",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213372745",
  "pdf_url": "https://arxiv.org/pdf/2609.13765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7prwmqbj2jpjd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.13765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Last-Iterate Performance of Gradient Descent and Relaxed Proximal Point via $s$-Composability",
  "title_text": "Last-Iterate Performance of Gradient Descent and Relaxed Proximal Point via s-Composability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7prwmqbj2jpjd",
  "venue": "arXiv",
  "versions": []
}