{"ID":2823565,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.24527","arxiv_id":"2512.24527","title":"Dimension-free estimators of gradients of functions with(out) non-independent variables","abstract":"This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using $\\ell_p$-spherical distributions on $\\R^d$ with $d, p\\geq 1$. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any $p\\geq 1$. Also, the mean squared errors (MSEs) of the gradient estimators are bounded by $K_0 N^{-1} d$ for any $p \\in [1, 2]$, and by $K_1 N^{-1} d^{2/p}$ when $2 \\leq p \\ll d$ with $N$ the sample size and $K_0, K_1$ some constants. Taking $\\max\\left\\{2, \\log(d) \\right\\} \u003c p \\ll d$ allows for achieving dimension-free upper-bounds of MSEs. In the case where $d\\ll p\u003c +\\infty$, the upper-bound $K_2 N^{-1} d^{2-2/p}/ (d+2)^2$ is reached with $K_2$ a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach.","short_abstract":"This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using $\\ell_p$-spherical distributions on $\\R^d$ with $d, p\\geq 1$. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dim...","url_abs":"https://arxiv.org/abs/2512.24527","url_pdf":"https://arxiv.org/pdf/2512.24527v1","authors":"[\"Matieyendou Lamboni\"]","published":"2025-12-31T00:22:47Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.OC\",\"math.PR\"]","methods":"[]","has_code":false}