{
  "abstract": "Objective perturbation adds a random linear term to a regularized empirical risk and releases the exact perturbed minimizer. We study the finite computation obtained by releasing the $N$-th iterate of deterministic gradient descent on $w\\mapsto F(w;S)+\\langle z,w\\rangle$, where $z\\sim\\mathcal N(0,σ^2I_d)$ is drawn once before optimization. For strongly convex and smooth objectives with Lipschitz Hessian, we prove an explicit condition under which the map $z\\mapsto w_N$ is a $C^1$-diffeomorphism on the bounded domains used in the privacy argument, with a quantitative lower bound on the smallest singular value of its Jacobian. This permits a direct change-of-variables analysis of the finite iterate. For generalized linear models, the resulting privacy-profile bound has no explicit ambient-dimension factor once the iteration condition holds, and its finite-iteration correction decreases geometrically. By letting the free truncation parameter grow slowly with $N$, we recover the corresponding exact-minimizer certificate in the limit. We also bound the expected excess empirical risk by $dσ^2/(2μ)$ plus a geometrically decreasing optimization term, and transfer the result to population risk without an additional multiplicative condition-number factor in the leading statistical terms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02716",
  "author_total": 2,
  "authors": [
    {
      "name": "Austin Watkins"
    },
    {
      "name": "Raman Arora"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4348q71nd4stq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02716",
  "fulltext_url": null,
  "id": "4348q71nd4stq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02716",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7220464799",
  "pdf_url": "https://arxiv.org/pdf/2610.02716",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4348q71nd4stq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:56Z",
      "native_id": "2610.02716",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02716"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Differential Privacy of Gradient Descent on Perturbed Objectives",
  "title_text": "Differential Privacy of Gradient Descent on Perturbed Objectives",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:56Z",
  "url": "https://research.draconitic.com/paper/4348q71nd4stq",
  "venue": "arXiv",
  "versions": []
}