{
  "abstract": "Windowed Anderson acceleration AA($m$) combines at most $m+1$ recent iterates by least squares. For a real affine contraction $x\\mapsto Wx+b$ with $q=\\|W\\| \\in(0,1)$, we determine exactly when AA($m$) converges from every starting point with an asymptotic factor below $q$. We give a polynomial criterion for this improvement and an equivalent spectral condition: there is no probability distribution on the phases of the eigenvalues of modulus $q$ whose first $m$ cosine moments are $q,q^2,\\ldots,q^m$. If such a distribution exists, we construct a starting point from which AA($m$) reproduces fixed-point iteration at every step. Otherwise, over a fixed number of consecutive steps, every run contracts the residual norm by a common factor strictly smaller than the corresponding power of $q$. The polynomial criterion also gives explicit convergence-rate bounds with constants independent of the dimension. When $q$ equals the spectral radius, this characterizes improvement over the asymptotic convergence factor of fixed-point iteration. For general affine contractions, the asymptotic factor never exceeds the numerical radius of $W$, and a related moment condition gives a uniform upper bound strictly below that radius.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30073",
  "author_total": 2,
  "authors": [
    {
      "name": "Gilad Lerman"
    },
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aemhkzrg8q4wq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30073",
  "fulltext_url": null,
  "id": "aemhkzrg8q4wq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30073",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7223255449",
  "pdf_url": "https://arxiv.org/pdf/2610.30073",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aemhkzrg8q4wq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.30073",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30073"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "When Windowed Anderson Acceleration Converges Faster",
  "title_text": "When Windowed Anderson Acceleration Converges Faster",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/aemhkzrg8q4wq",
  "venue": "arXiv",
  "versions": []
}