{
  "abstract": "We study episodic continuous-time control with an unknown vector control gain, scalar state, quadratic action cost, and smooth convex terminal cost. In the scalar Gaussian experiment, let $Δ(H)$ denote the minimax improvement over zero control and set $δ=\\sqrt2H^2-1$. We prove the critical law $$ Δ(H)\\asymp δ^4\\sqrt{\\log(1/δ)} \\qquad (δ\\downarrow0), $$ with a matching lower and upper bound. The lower bound follows from a uniform deficit--energy inequality valid for fully adaptive controls with unbounded amplitudes, while a moving soft-threshold feedback attains the rate. For local parameters $θ=N^{-1/4}h$, $|h|\\le H$, we show that the normalized minimax regret over $N$ episodes is within $O(N^{-1/2})$ of a fixed-horizon Gaussian sequential control problem, without an additional dimension factor. The terminal task enters the limit only through $c_g=\\operatorname{Var}(g(Z))$. The Gaussian problem exhibits an exact no-learning phase transition: $$ C_d^T(H)=TH^2/2 \\iff H^4T\\le d^2/2, $$ yielding the asymptotic task boundary $c_gH^4=d^2/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33154",
  "author_total": 1,
  "authors": [
    {
      "name": "Chen Jia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cg0v2b022rvs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33154",
  "fulltext_url": null,
  "id": "1cg0v2b022rvs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33154",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7214806498",
  "pdf_url": "https://arxiv.org/pdf/2609.33154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cg0v2b022rvs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33154"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control",
  "title_text": "Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1cg0v2b022rvs",
  "venue": "arXiv",
  "versions": []
}