A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs

[2607.22567] A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2607.22567 (math) This paper has been withdrawn by Mingyu Li [Submitted on 31 May 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs Authors: Shihao Ji , Mingyu Li , Zihui Song No PDF available, click to view other formats Abstract: Zeroth-order Newton-type methods are useful when gradients and Hessians are unavailable, but they behave quite differently from first-order gradient-free methods. We develop a kinetic framework for algorithms that estimate both gradient and Hessian from black-box function values. The naive random-direction Hessian estimator turns out to be biased even on quadratics; a Gaussian--Stein correction is needed to estimate the Hessian of the Gaussian-smoothed objective. Linearizing the inverse Hessian exposes two noise channels: gradient noise preconditioned by the inverse Hessian, and Hessian noise transmitted through an inverse-Hessian sandwich. Under a noisy oracle the second channel carries the second-difference factor $\mu_H^{-4}$. A small-mass kinetic lift links the finite-step Newton update to an underdamped phase-space model; the overdamped spatial limit yields a Lyapunov bound that exposes the curvature--variance trade-off between step size, batch sizes, smoothing radii, and regularization. Numerical experiments confirm estimator identities, the gradient and Hessian variance laws, dimension scaling, inverse-perturbation accuracy, and optimization behavior under query-budget and regularization ablations. Comments: Withdrawn due to serious concerns regarding the authenticity and accuracy of the listed authorship. The identity of one or more listed authors cannot presently be verified, and the author list may not represent distinct contributors. The manuscript is withdrawn pending institutional review Subjects: Optimization and Control (math.OC) ; Artificial Intelligence (cs.AI) Cite as: arXiv:2607.22567 [math.OC] (or arXiv:2607.22567v2 [math.OC] for this version) https://doi.org/10.48550/arXiv.2607.22567 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mingyu Li [ view email ] [v1] Sun, 31 May 2026 08:34:25 UTC (119 KB) [v2] Sat, 22 Aug 2026 07:09:38 UTC (1 KB) (withdrawn) Full-text links: Access Paper: Withdrawn Current browse context: math.OC < prev | next > new | recent | 2026-07 Change to browse by: cs cs.AI math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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