arXiv：cs.LG· Kihun Rhee, Hanjoon Byun·· 4 小时前AI 评分28
Gauss-Newton 精度与不定 Hessian：低成本集内的一致共存
Gauss-Newton Accuracy and Indefinite Hessians: Uniform Coexistence in Low-Cost Sets
AI 导读
研究在岭正则化非线性最小二乘中，Gauss-Newton 曲率精度存在两种曲率机制的一致共存：全局极小点存在且其相对 Hessian 误差低于 (1+√2)/8，而同一低成本集内另有点的 Hessian 不定、相对误差至少 15/8。单一正岭上限适用于固定邻域内所有独立的中心与标签扰动，且邻域不随岭权重趋于零而收缩。基于当前预测水平集的逐点证书可控制 Hessian 修正的法向、混合与切向分量。
正文
Abstract:We study the accuracy of Gauss-Newton curvature in ridge-regularized nonlinear least squares. Under local regularity and persistence of level-set curvature magnitude along an exact-fit section, we prove uniform coexistence of two curvature regimes. Global minimizers exist, and every global minimizer has relative Hessian error below $(1+\sqrt2)/8$, while the same low-cost set contains a point with an indefinite Hessian and relative error at least $15/8$. One positive ridge cap works for all independent center and label perturbations in fixed neighborhoods and every positive ridge weight up to the cap. These neighborhoods do not shrink as the ridge weight tends to zero. A pointwise certificate based on the current prediction level set controls the normal, mixed, and tangent parts of the Hessian correction. We prove a sharp relative-error bound over the stated pointwise class when the prediction map and ridge vary. Analytic examples describe the roles of output alignment, curvature orientation, and persistence. A separate structural result gives full Jacobian row rank throughout low-cost sets and exact interpolation near a rank-deficient reference.
| Comments: | 28 pages, 1 table |
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.09675 [cs.LG] |
| (or arXiv:2610.09675v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.09675 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Kihun Rhee [view email]
[v1]
Wed, 7 Oct 2026 08:38:13 UTC (34 KB)
来源：arXiv：cs.LG · arxiv.org