aiqbench analysis
This paper advances zeroth-order optimization by introducing a kinetic theory to correct Hessian estimation biases using Gaussian-Stein methods. It rigorously analyzes noise propagation in black-box settings and establishes theoretical bounds linking curvature, variance, and regularization parameters for gradient-free algorithms. These findings provide essential mathematical foundations for improving derivative-free machine learning techniques where explicit gradients are inaccessible.
A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs
arXiv:2607.22567v1 Announce Type: cross 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.
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.
Optimization and Control (math.OC); Artificial Intelligence (cs.AI)
arXiv:2607.22567 [math.OC]
(or arXiv:2607.22567v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2607.22567 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 253 words
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