{
  "abstract": "We analyze a simple stochastic inertial Krasnosel'skii--Mann (iKM) method for finding a fixed point of a nonexpansive operator in a real Hilbert space. Our method is obtained simply by adding two inertial extrapolations to stochastic KM [Bravo and Cominetti, 2024], and it retains one call to a possibly biased stochastic oracle per update and achieves sharp rates in both the stochastic and deterministic regimes. Specifically, with our proposed parameter schedule, we prove the following last-iterate fixed-point residual bound: \\[ {O}\\!\\left(\\frac{1}{K} +\\frac{σ\\log K}{\\sqrt K} +\\frac{B_K\\log K}{K}\\right), \\] where $K$ is the horizon, $σ$ is the noise level and $B_K$ is the accumulated root-mean-square bias. When $B_K=O(\\sqrt K)$, this yields $\\widetilde O(ε^{-2})$ sample complexity that matches, up to a logarithmic factor, the stochastic-oracle lower bound given under the unbiased subclass of our model [Foster et al., 2019, Theorem 2]. It also improves the best-known $O(ε^{-4})$ random-iterate guarantee for stochastic KM [Bravo and Cominetti, 2024, Corollary 5.4]. To our knowledge, this is the first single-loop method for general nonexpansive fixed-point problems to attain this near-optimal sample complexity without variance reduction or batching. When the oracle is exact, the same method attains the worst-case-optimal $O(K^{-1})$ last-iterate residual rate [Park and Ryu, 2022, Theorem 4.6], improving the $O(K^{-1/2})$ rate of classical KM [Cominetti et al., 2014; Bravo and Cominetti, 2018].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28543",
  "author_total": 5,
  "authors": [
    {
      "name": "Tong Yang"
    },
    {
      "name": "Tao Jiang"
    },
    {
      "name": "Yuejie Chi"
    },
    {
      "name": "Ashok Cutkosky"
    },
    {
      "name": "Lin Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afc4h1acw3b8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28543",
  "fulltext_url": null,
  "id": "afc4h1acw3b8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28543",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214382303",
  "pdf_url": "https://arxiv.org/pdf/2609.28543",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afc4h1acw3b8f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.28543",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28543"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Stochastic Inertial Krasnosel'skii-Mann Iteration Achieves Near-Optimal Sample Complexity",
  "title_text": "Stochastic Inertial Krasnosel'skii-Mann Iteration Achieves Near-Optimal Sample Complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:03Z",
  "url": "https://research.draconitic.com/paper/afc4h1acw3b8f",
  "venue": "arXiv",
  "versions": []
}