{
  "abstract": "We study first-order optimization with inexact gradients whose errors combine a component proportional to the true gradient norm and a uniformly bounded absolute component. We develop an adaptive gradient descent scheme that selects estimates of the smoothness constant and both error parameters through a computable backtracking condition using exact function values. Its analysis yields convergence for smooth objectives and functions satisfying Polyak-Lojasiewicz condition. For strongly convex objectives, we propose a composite-error intermediate gradient method, that adapts to unknown smoothness while using known error bounds and a strong convexity parameter $μ$. With fixed smoothness $L$, acceleration is retained when $α=\\mathcal{O}\\left(\\left(μ/ L \\right)^{1/4}\\right)$, larger admissible errors yield intermediate rates. Throughout the range $0 \\leqslant α\\leqslant 1/3$, the absolute error bound $δ$ contributes at most $\\mathcal{O} \\left(δ^2/μ\\right)$ to the objective residual, without additional amplification by the condition number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24863",
  "author_total": 3,
  "authors": [
    {
      "name": "Artem Vasin"
    },
    {
      "name": "Alexander Tyurin"
    },
    {
      "name": "Alexander Gasnikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4yhfgx9czhk6w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24863",
  "fulltext_url": null,
  "id": "4yhfgx9czhk6w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24863",
  "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": "W7223028312",
  "pdf_url": "https://arxiv.org/pdf/2610.24863",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4yhfgx9czhk6w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24863",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24863"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Adaptive First-Order Methods with Composite Gradient Error",
  "title_text": "Adaptive First-Order Methods with Composite Gradient Error",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/4yhfgx9czhk6w",
  "venue": "arXiv",
  "versions": []
}