A Formal Kinetic Theory for Zeroth-Order Newton Dynamics:Stein-Corrected Hessian Estimation and Curvature--Variance Trade-offs
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 ...
| Authors | Shihao Ji, Mingyu Li, Zihui Song |
|---|---|
| Primary category | math.OC |
| Published | 31 May 2026 |
| Last updated at source | 22 August 2026 |
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