[
    {
        "id": "osp-21082",
        "type": "article-journal",
        "title": "Differential Privacy of Gradient Descent on Perturbed Objectives",
        "author": [
            {
                "family": "Watkins",
                "given": "Austin"
            },
            {
                "family": "Arora",
                "given": "Raman"
            }
        ],
        "URL": "https://omanscience.com/en/articles/differential-privacy-of-gradient-descent-on-perturbed-objectives",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "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."
    }
]