{
  "abstract": "We study Polyak-type step-size selection for extragradient methods for solving deterministic and stochastic monotone root-finding problems. We show that the known projection-type correction for deterministic extragradient arises from minimizing an upper bound on the distance to a solution, paralleling the classical Polyak step-size construction. Using this viewpoint, we provide a unified deterministic analysis of the Polyak-type Extragradient Method (PolyakEG), based on a local critical condition controlling the variation of operator $F$ along the extrapolation direction. This analysis does not require global Lipschitz continuity, and covers sublinear convergence under broader conditions such as Hölder continuity or $(L_0, L_1)$-Lipschitzness and linear convergence under additional strong monotonicity, all through a single framework. We then study the stochastic extensions of this approach. We first prove convergence of a direct stochastic variant, PolyakSEG, when all stochastic component operators share a common solution. We also show that, without this condition, PolyakSEG with nonvanishing step-sizes may fail to converge to a zero of the mean operator. To address this limitation, we propose DecPolyakSEG, which combines decreasing step-sizes with Polyak-type updates, and establish a sublinear residual convergence result without requiring a common solution across the component operators. These results parallel recent developments in stochastic Polyak step-sizes from the convex minimization literature and establish an analogous research avenue in the broader root-finding regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26581",
  "author_total": 4,
  "authors": [
    {
      "name": "TaeHo Yoon"
    },
    {
      "name": "Sayantan Choudhury"
    },
    {
      "name": "Ezra Greenberg"
    },
    {
      "name": "Nicolas Loizou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ma8hzef3hejy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26581",
  "fulltext_url": null,
  "id": "0ma8hzef3hejy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26581",
  "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"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214102371",
  "pdf_url": "https://arxiv.org/pdf/2609.26581",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ma8hzef3hejy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26581",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26581"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Polyak-Type Extragradient Methods for Monotone Root-Finding Problems",
  "title_text": "Polyak-Type Extragradient Methods for Monotone Root-Finding Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/0ma8hzef3hejy",
  "venue": "arXiv",
  "versions": []
}