{
  "abstract": "We develop a comprehensive theoretical analysis of Matrix Shampoo for tracking time-varying optima in nonstationary stochastic optimization. We derive finite-horizon high-probability bounds for average tracking error that disentangle initialization, optimizer drift, momentum-induced drift penalty, stochastic noise, and adaptive preconditioning, with explicit dependence on momentum parameter \\(β_1\\) and stabilization parameter \\(\\varepsilon\\). These bounds reveal an explicit drift--noise and stability--geometry tradeoffs: noise-dominated regimes favor larger $β_1$, which provides stronger first-moment averaging and greater suppression of gradient noise, whereas drift-dominated regimes favor smaller $β_1$, allowing faster adaptation to moving optimizer. Larger $\\varepsilon$ reduces the preconditioner-perturbation term while weakening Shampoo's ability to exploit matrix geometry, thereby reducing adaptivity. We establish a matching minimax lower bound under squared path-variation constraint, proving the information-theoretic optimality of Shampoo over the class of causal optimization procedures. Under a constant stepsize, we establish high-probability bounds for the average projected stationarity gap along general nonstationary nonconvex objectives. We validate our theoretical findings on both synthetic and multiple real-world problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16923",
  "author_total": 3,
  "authors": [
    {
      "name": "Abir Sarkar"
    },
    {
      "name": "Nirmalya Mandal"
    },
    {
      "name": "Martin T. Wells"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5prt4v4mn86h7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16923",
  "fulltext_url": null,
  "id": "5prt4v4mn86h7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16923",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221766923",
  "pdf_url": "https://arxiv.org/pdf/2610.16923",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5prt4v4mn86h7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:26Z",
      "native_id": "2610.16923",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16923"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization",
  "title_text": "Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/5prt4v4mn86h7",
  "venue": "arXiv",
  "versions": []
}