{
  "abstract": "We determine the minimax cumulative log-loss regret of a finite-alphabet, exogenously driven source with genuinely infinite input memory: independent Rademacher inputs $(U_t)$ are observed sequentially and the next binary mark has logit $\\sum_{j\\ge1}θ_jU_{t+1-j}$, the unknown coefficients obeying a summable envelope $|θ_j|\\le r_j$, $\\sum_jr_j\\le B$. At horizon $T$, lag $j$ can move the logit by at most $r_j$ and is exercised in only $n_{T,j}=(T-j+1)_+$ rounds, and the two limitations combine into the sum $Γ_T(r)=\\sum_{j\\le T}\\log(1+n_{T,j}r_j^{2})$. One coordinate-localised Bayesian mixture achieves $R_T(r)\\le CΓ_T(r)$ for \\emph{every} summable envelope with $C$ universal. Our main result is a matching nonasymptotic converse for the canonical exponential and polynomial envelopes; its new ingredients are a modular finite-sample information bound for logistic experiments with an exogenous random design, and a conditioning estimate for the overlapping Toeplitz lag matrix obtained by exhibiting each off-diagonal Gram sum as a sum of independent Rademacher variables indexed by the edges of a forest, needing neither local asymptotic normality nor any spectral theorem for random Toeplitz matrices. So $Γ_T(r)$ is the minimax regret scale here, giving $Θ(α^{-1}\\log^{2}T)$ for $r_j=Ae^{-αj}$ and $Θ(T^{1/(2s)})$ for $r_j=Aj^{-s}$, $s>1$ --- the latter without the extra $(\\log T)^{1-1/(2s)}$ factor any window-truncation analysis pays. We also show memory decay cannot determine regret, and that a profile-scaled online Newton predictor attains $O_B(Γ_T(r))$ in polynomial time per round.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26515",
  "author_total": 1,
  "authors": [
    {
      "name": "Vaneet Aggarwal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0j61w8p1yq1rs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26515",
  "fulltext_url": null,
  "id": "0j61w8p1yq1rs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26515",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7204557208",
  "pdf_url": "https://arxiv.org/pdf/2608.26515",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0j61w8p1yq1rs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.26515",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26515"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Sharp Minimax Regret for Infinite-Memory Logistic Prediction",
  "title_text": "Sharp Minimax Regret for Infinite-Memory Logistic Prediction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/0j61w8p1yq1rs",
  "venue": "arXiv",
  "versions": []
}