{"ID":6138195,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T10:22:07.798851522Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07249","arxiv_id":"2607.07249","title":"Gradient-free stochastic optimization of derivatives under strong convexity","abstract":"We consider the problem of minimizing the $k$-th order partial derivative $f=\\partial_j^k g$ of an unknown function $g$ along a fixed coordinate direction $j$, based on noisy queries of $g$. Assuming that $g$ has Hölder regularity ${β+k}$ for some $β\\ge 2$, that $f$ is strongly convex on a compact convex set $Θ\\subset\\mathbb{R}^d$ and that $g$ and $f$ satisfy mild boundedness and Lipschitz regularity conditions on $Θ$, we propose a kernel-based estimator of $\\nabla f$ and analyze the projected stochastic gradient algorithm driven by this estimator. We obtain a non-asymptotic upper bound on the optimization error of the order $d^{(2β+k-1)/(β+k)}\\,N^{-(β-1)/(β+k)}$, where $N$ is the total number of queries. We also establish a minimax lower bound of the order $N^{-(β-1)/(β+k)}$ showing that this rate is optimal in $N$ over all sequential algorithms.","short_abstract":"We consider the problem of minimizing the $k$-th order partial derivative $f=\\partial_j^k g$ of an unknown function $g$ along a fixed coordinate direction $j$, based on noisy queries of $g$. Assuming that $g$ has Hölder regularity ${β+k}$ for some $β\\ge 2$, that $f$ is strongly convex on a compact convex set $Θ\\subset\\...","url_abs":"https://arxiv.org/abs/2607.07249","url_pdf":"https://arxiv.org/pdf/2607.07249v1","authors":"[\"Arya Akhavan\",\"Sirine Louati\",\"Alexandre B. Tsybakov\"]","published":"2026-07-08T10:30:57Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}