{
  "abstract": "This work addresses decentralized online Riemannian optimization on Hadamard manifolds. Prior work under geodesic convexity (g-convexity) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature. Curvature may also enter the step size or contraction factor of tangent-space Riemannian consensus schemes. In this work, we relax the curvature dependence for a narrower class of horospherical convex (h-convex) functions. We study Distributed Riemannian Online Gradient Descent (D-ROGD), which combines local Riemannian h-subgradient updates with an implicit Fréchet-mean consensus. For h-convex and strongly h-convex local objectives, we establish $O(\\sqrt{T})$ and $O(\\log T)$ static regret, respectively, matching the corresponding Euclidean rates with respect to $T$, with network dependence governed solely by the spectral gap. To our knowledge, these are the first curvature-independent regret guarantees for decentralized online optimization on Hadamard manifolds. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13646",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhanyuan Cai"
    },
    {
      "name": "Emre Sahinoglu"
    },
    {
      "name": "Shahin Shahrampour"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s1jbk7jdt127.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13646",
  "fulltext_url": null,
  "id": "0s1jbk7jdt127",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13646",
  "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"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1705|Distributed Control Multi-Agent Systems"
    ]
  ],
  "openalex": "W7213225151",
  "pdf_url": "https://arxiv.org/pdf/2609.13646",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s1jbk7jdt127.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13646",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13646"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds",
  "title_text": "Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:36Z",
  "url": "https://research.draconitic.com/paper/0s1jbk7jdt127",
  "venue": "arXiv",
  "versions": []
}