{
  "abstract": "We study convergence rates of Nesterov's Accelerated Gradient Descent (NAG) method for convex optimization in both deterministic and stochastic settings. We focus on a more general smoothness condition raised from several machine learning problems empirically and theoretically. We show the accelerated convergence rate of order $\\mathcal{O}\\left(1/T^2\\right)$ in terms of the function value gap, given access to exact gradients of objective functions, matching the optimal rate for standard smooth convex optimization in \\citep{nesterov1983method}. Under the relaxed affine-variance noise assumption for stochastic optimization, we establish the high-probability convergence rate of order $\\tilde{\\mathcal{O}}\\left(\\sqrt{\\log\\left(1/δ\\right)/T}\\right)$ and this rate could improve to $\\tilde{\\mathcal{O}}\\left(\\log\\left(1/δ\\right)/T^2\\right)$ when the noise parameters are sufficiently small. Here, $T$ denotes the total number of iterations and $δ$ is the probability margin. Up to logarithm factors, our probabilistic convergence rate reaches the same order of the expected rate obtained in \\citep{ghadimi2016accelerated} where the assumptions of bounded variance noise and Lipschitz smoothness are required. We additionally establish the convergence rates of the studied algorithms in the non-convex setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33761",
  "author_total": 2,
  "authors": [
    {
      "name": "Chenhao Yu"
    },
    {
      "name": "Junhong Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7eq88a7k38vhq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33761",
  "fulltext_url": null,
  "id": "7eq88a7k38vhq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33761",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7223841166",
  "pdf_url": "https://arxiv.org/pdf/2610.33761",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7eq88a7k38vhq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.33761",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33761"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Convergence Analysis of Nesterov's Accelerated Gradient Descent under Relaxed Assumptions",
  "title_text": "Convergence Analysis of Nesterov's Accelerated Gradient Descent under Relaxed Assumptions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7eq88a7k38vhq",
  "venue": "arXiv",
  "versions": []
}