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

תקציר מקורי באנגליתarXiv: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-ma