{
  "abstract": "This paper studies convex optimization when the gradient cannot be evaluated exactly, but only approximated by a hierarchy of algorithms whose compute grows like $δ^{-γ}$ in the accuracy $δ$. When $γ>2$, falling into the Harder-Than-Monte-Carlo (HTMC) regime, the price of accuracy outruns the variance reduction that Monte Carlo would buy and we show that minimizing a loss function costs no more, up to a factor depending only on $γ$, than a single evaluation of its gradient at the accuracy the problem demands. A randomized multilevel oracle replaces the deterministic approximation of accuracy $δ$ by an unbiased estimator of it, whose variance $σ^2$ becomes a second, independently priced dial: the cost of one call drops from $δ^{-γ}$ to $δ^{2-γ}σ^{-2}$. Plain inexact gradient descent driven by that oracle reaches loss $\\varepsilon$ at expected compute $Θ(\\varepsilon^{-γ})$ in the convex case, against $Θ(\\varepsilon^{-(γ+1)})$ for the same method run at a fixed accuracy: randomization buys a full power of $\\varepsilon$. Under $μ$-strong convexity the exponent halves, to $\\varepsilon^{-γ/2}$, because the iterates settle at a noise floor and the bias budget relaxes accordingly. Both bounds are independent of the step size, and hence of the smoothness constant, and we show that the cost is a functional of the underlying gradient flow rather than of any discretization of it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35418",
  "author_total": 2,
  "authors": [
    {
      "name": "Arthur Paing"
    },
    {
      "name": "Arthur Jacot"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23bfzefwxw70e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35418",
  "fulltext_url": null,
  "id": "23bfzefwxw70e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35418",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/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": "W7214720691",
  "pdf_url": "https://arxiv.org/pdf/2609.35418",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23bfzefwxw70e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35418",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35418"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Convex Optimization Is Free When Accuracy Is Expensive",
  "title_text": "Convex Optimization Is Free When Accuracy Is Expensive",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/23bfzefwxw70e",
  "venue": "arXiv",
  "versions": []
}