{
  "abstract": "We study finite-horizon one-gradient realizations with the Chebyshev minimax terminal residual on $[μ,L]$. Under time-dependent Hessian perturbations, the terminal first variation is governed by a time-ordered spectral kernel $K_s(λ,ν)$; its sharp $\\ell_2$ gain is $A_N$. For the prefix-exact Chebyshev recurrence, $$A_N^{pref}=\\frac4{\\sqrt3}\\frac{N^{3/2}}{L-μ}ε_N^\\star(1+o(1)),$$ and this is sharp in the causal two-term class with Chebyshev exactness at every prefix. For terminal-only exactness, Jacobi coordinates give $P_N=2^{1-N}T_N$: the spectrum is fixed at the midpoint Chebyshev nodes, while the spectral weights parametrize the realizations. The sine weights give a final-exact Jacobi method with the same terminal residual and $$A_N(J_N^{\\sin})=2\\sqrt{c_{\\sin}}\\frac{N^{3/2}}{L-μ}ε_N^\\star(1+o(1)), 2\\sqrt{c_{\\sin}}\\approx2.1379<4/\\sqrt3.$$ Thus the Chebyshev terminal polynomial does not determine the first-order Hessian-drift gain. Momentum is necessary for this order: every Chebyshev-exact stepsize schedule has $A_N\\ge2\\sqrt{2/3} N^2ε_N^\\star/(L-μ)$. Beyond first order, the response to drift $E_s=H_s-H$ of size $δ=\\max_s\\|E_s\\|$ is a finite polynomial expansion, and for both realizations the terminal error is at most $ε_N^\\star T_N(1+2δ/(L-μ))$, the Chebyshev bound of the interval widened by $δ$. This bound is attained, holds for objectives with Lipschitz Hessian, and shows that drift reduces the accelerated rate exponent by the factor $1-\\sqrt{δ/μ}$. The two realizations thus share the worst-case response and differ in the $\\ell_2$-in-time response measured by $A_N$. The experiments show the finite-horizon effect: lower stochastic curvature overhead, larger admissible-block frontiers, accurate time-varying quadratic predictions, and lower restart cost on an endpoint-coupled smooth strongly convex GLM.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.16671",
  "author_total": 2,
  "authors": [
    {
      "name": "Dmitry Pasechnyuk-Vilensky"
    },
    {
      "name": "Martin Takáč"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1gzactmb3qdx2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.16671",
  "fulltext_url": null,
  "id": "1gzactmb3qdx2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.16671",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7164902966",
  "pdf_url": "https://arxiv.org/pdf/2606.16671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1gzactmb3qdx2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2606.16671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.16671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Chebyshev-Exact Acceleration under Hessian Variation, I: Sine-Jacobi Method",
  "title_text": "Chebyshev-Exact Acceleration under Hessian Variation, I: Sine-Jacobi Method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/1gzactmb3qdx2",
  "venue": "arXiv",
  "versions": []
}