{
  "abstract": "We study Langevin-based methods for non-convex optimization under smoothness and dissipativity assumptions, focusing on non-asymptotic expected excess-risk bounds rather than sampling guarantees for the full target distribution. A main methodological message is that relative-entropy sampling guarantees can be converted directly into expected objective-value guarantees, without passing through Wasserstein distance. This direct KL-to-objective route yields sharper dependence on the Log-Sobolev constant, which may scale exponentially with inverse temperature and dimension in non-convex problems. We first analyze the Unadjusted Langevin Algorithm with exact gradients and derive explicit bounds in terms of inverse temperature, dimension, stepsize, smoothness and dissipativity parameters, and the Log-Sobolev constant. We then extend the result to inexact ULA, allowing biased and stochastic gradient surrogates whose mean-square error grows at most quadratically in the state. This framework covers stochastic gradients and zeroth-order estimators based on function evaluations. We show that Gaussian and spherical finite-difference estimators fit the theory and obtain explicit function-evaluation complexity bounds. To the best of our knowledge, these are the first non-asymptotic global non-convex optimization complexity bounds for zeroth-order ULA. We also provide numerical experiments illustrating the spherical zeroth-order scheme.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.22353",
  "author_total": 4,
  "authors": [
    {
      "name": "Emanuele Naldi"
    },
    {
      "name": "Marco Rando"
    },
    {
      "name": "Lorenzo Rosasco"
    },
    {
      "name": "Silvia Villa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42p1ssey7787t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.22353",
  "fulltext_url": null,
  "id": "42p1ssey7787t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.22353",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7171372264",
  "pdf_url": "https://arxiv.org/pdf/2607.22353",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42p1ssey7787t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2607.22353",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.22353"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Langevin for Nonconvex Optimization: Exact, Inexact and Zeroth-Order",
  "title_text": "Langevin for Nonconvex Optimization: Exact, Inexact and Zeroth-Order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/42p1ssey7787t",
  "venue": "arXiv",
  "versions": []
}