{
  "abstract": "Adaptive algorithms increasingly make decisions while reshaping the dynamics that generate their future data. We establish a shrinking-tube concentration bound for projected stochastic approximation driven by an adaptive Markov chain. The bound guarantees, with high probability, that every iterate after a chosen time remains within a tolerance around the target that tightens over time. The probability of any exit after the chosen time admits a polynomially decaying upper bound, and a matching lower bound shows that its polynomial exponent cannot be improved in general under finite second moments. The result therefore identifies a sharp tradeoff between how quickly the tolerance shrinks and how rapidly the probability of any future exit decreases. We also extend the analysis to recursions with additional martingale-difference noise and predictable bias, showing how growth in the martingale-difference noise scale slows the decay of the exit-probability bound while predictable bias restricts the admissible tube shrinkage. The proof combines backward kernel replacement, a finite-time mean-squared-error bound, and a blockwise maximal first-exit argument. We apply the theory to inventory learning with stockout-dependent demand and fixed stockout costs, and quantify how numerical gradient accuracy affects the all-future reliability of the resulting policies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29833",
  "author_total": 3,
  "authors": [
    {
      "name": "Jin Li"
    },
    {
      "name": "Ye Luo"
    },
    {
      "name": "Xiaowei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpmtk0bfhk902.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29833",
  "fulltext_url": null,
  "id": "fpmtk0bfhk902",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29833",
  "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"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1803|Simulation Techniques and Applications"
    ]
  ],
  "openalex": "W7214300741",
  "pdf_url": "https://arxiv.org/pdf/2609.29833",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpmtk0bfhk902.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29833",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29833"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Shrinking-Tube Concentration for Adaptive Markovian Stochastic Approximation",
  "title_text": "Shrinking-Tube Concentration for Adaptive Markovian Stochastic Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/fpmtk0bfhk902",
  "venue": "arXiv",
  "versions": []
}