{
  "abstract": "Zeroth-order optimization methods are essential for solving black-box problems where gradient information is unavailable or expensive to compute. This paper presents POEM-ES, a novel parameter-free stochastic zeroth-order algorithm that extends the recent POEM method by integrating subspace preconditioning with ellipsoidal randomized sampling. In contrast to traditional zeroth-order approaches that rely on isotropic random directions, POEM-ES performs anisotropic sampling guided by a fixed structural symmetric positive semi-definite (SPSD) preconditioner $\\hatΣ$ that encodes the underlying low-dimensional geometry. Under a standard structural spectral normalization where $λ_{\\max}(\\hatΣ) = 1$, we introduce the use of the empirical effective dimension $d^* = \\operatorname{tr}(\\hatΣ)$, which reflects the intrinsic dimensionality of the problem and guides both the sampling and randomized smoothing parameter schedules. In practice, such a preconditioner can be effectively obtained via pilot sampling, historical trajectories, or domain-specific expert knowledge. We prove that POEM-ES achieves a dimension-reduced convergence rate under low-rank structural assumptions, requiring only $\\tilde{\\mathcal{O}}\\left( \\frac{\\left( r^2 κ(\\hatΣ) + d^* \\right) L^2 D_{\\mathcal{X}}^2}{\\varepsilon^2} \\right)$ stochastic zeroth-order oracle queries. The method remains fully parameter-free and demonstrates significant improvements over the original POEM in problems with low-rank structure where $d^* \\ll d$. Numerical experiments on hinge-loss binary classification tasks using LibSVM datasets confirm the practical superiority of the proposed approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38561",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexander Sholokhov"
    },
    {
      "name": "Alexander Rogozin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38t0j5rtjtsnd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38561",
  "fulltext_url": null,
  "id": "38t0j5rtjtsnd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38561",
  "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"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7215116740",
  "pdf_url": "https://arxiv.org/pdf/2609.38561",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38t0j5rtjtsnd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.38561",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38561"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Parameter-Free Zeroth-Order Optimization with Ellipsoidal Sampling",
  "title_text": "Parameter-Free Zeroth-Order Optimization with Ellipsoidal Sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/38t0j5rtjtsnd",
  "venue": "arXiv",
  "versions": []
}