{
  "abstract": "We study how the spectral geometry of the mixing matrix enters the convergence of decentralized stochastic gradient tracking (GT) for smooth strongly convex problems. General bounds contain the factor $Δ_λ^{-3}$ in the higher-order stochastic term, where $Δ_λ$ is the absolute spectral gap. We show that, even for two agents with identical quadratic objectives, the exact steady-state mean-square disagreement scales as $α^2σ^2Δ_λ^{-3}$ as the nontrivial eigenvalue approaches $-1$, provided the stepsize $α$ is sufficiently small relative to the gap. Here $σ^2$ is the gradient-noise variance. By unrolling the disagreement recursion, we establish a stochastic bound that averages all nontrivial eigenvalue contributions for diagonalizable mixing matrices, up to an eigenvector conditioning factor. As a real or complex eigenvalue approaches the unit circle away from $1$, its contribution grows as the inverse cube of its distance to the circle; growth can be slower near $1$. For symmetric mixing with gaps $p$ at $1$ and $ν$ at $-1$, the weighted sum $θ_1/p+θ_2/ν^3$ replaces the prior factor $(Δ_λν^2)^{-1}$. For $N$ agents, $Nθ_1$ and $Nθ_2$ count the nonnegative and negative nontrivial eigenvalues, respectively. On a symmetric family, a finite-horizon lower bound for the average iterate matches the worst-case factor $p^{-1}+ν^{-3}$ up to universal constants. Numerical experiments confirm the predicted scaling.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07958",
  "author_total": 3,
  "authors": [
    {
      "name": "Sulaiman A. Alghunaim"
    },
    {
      "name": "Haoyuan Cai"
    },
    {
      "name": "Ali H. Sayed"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fx2jfvb05zez.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07958",
  "fulltext_url": null,
  "id": "9fx2jfvb05zez",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07958",
  "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.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07958",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fx2jfvb05zez.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07958",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07958"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Spectral Dependence of Decentralized Stochastic Gradient Tracking",
  "title_text": "On the Spectral Dependence of Decentralized Stochastic Gradient Tracking",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/9fx2jfvb05zez",
  "venue": "arXiv",
  "versions": []
}