{
  "abstract": "We study stochastic fixed-point iterations $x_{n+1}=θ_nx_0+(1-θ_n)\\widehat T(x_n)$ for nonexpansive and contractive operators on Hilbert spaces, with a single-point unbiased oracle of bounded variance. Deterministically, every anchor schedule $θ_n=c_n/(n+2)$ whose density $c_n\\in(0,1]$ does not oscillate between scales is either polynomially suboptimal on contractions or super-polynomially slow on rotations; for constant densities $θ_n=c/(n+2)$ the tradeoff is exact, with $\\norm{x_n}\\asympΓ(c+1)(nφ)^{-c}$ on rotations by angle $φ$ ($φ\\to0$, $nφ\\to\\infty$) and, for the classical schedule, $\\norm{x_n}=\\frac{2|\\sin(nφ/2)|}{nφ}+O(\\frac1n)$. The dichotomy fails exactly under lacunary anchor concentration: an explicit $γ$-oblivious schedule, one per target accuracy, is simultaneously contraction-optimal and rotation-polynomial with exponent $1/α_0$, $α_0=\\log_2(3/2)$, and, for each fixed target accuracy, the matching lower bound holds for Lebesgue-a.e.\\ angle. For stochastic contractions, the minimax rate is $Θ(σ^2\\eps^{-2}(1-γ)^{-2})$ on affine maps, and within the anchored class $Θ(σ^2\\eps^{-2}(1-γ)^{-2}+(1-γ)^{-1}\\ln(D/\\eps))$ for known modulus via geometric batching; without a certified modulus bound no sound distance certificate exists, and a certified ceiling is the exact boundary. For nonexpansive maps we prove $O(Kσ^2D^2\\eps^{-4})$ in every $2$-uniformly smooth Banach space and the Hilbert rate $\\tildeΘ(σ^2\\eps^{-2}+D\\eps^{-1})$ by reduction to monotone inclusions. Numerical experiments are consistent with every scaling prediction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00295",
  "author_total": 1,
  "authors": [
    {
      "name": "Yekini Shehu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fx533812p7551.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00295",
  "fulltext_url": null,
  "id": "fx533812p7551",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00295",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7216197545",
  "pdf_url": "https://arxiv.org/pdf/2610.00295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fx533812p7551.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Adaptivity, Anchoring, and the Exact Oracle Complexity of Stochastic Fixed-Point Iterations",
  "title_text": "Adaptivity, Anchoring, and the Exact Oracle Complexity of Stochastic Fixed-Point Iterations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/fx533812p7551",
  "venue": "arXiv",
  "versions": []
}