arXiv：cs.LG· Hazar Yueksel (Google)·· 4 小时前AI 评分29
一步式追索的带符号几何：路径有效性与符号曲率准则
The Signed Geometry of One-Shot Recourse: On-Path Validity and the Signed-Curvature Criterion
AI 导读
研究对分类器得分做闭式追索时，一步沿单位梯度移动的成败取决于路径曲率 κ 是否非负：在 80 个浅层模型上，步后落在有利一侧的比例与 κ≥0 的比例相关系数达 r=0.985，但 Fashion-MNIST 上前者平均低 8.2 个百分点。
正文
Abstract:Closed-form recourse moves a rejected user along the unit gradient $\hat g$ of the classifier score $f$ by the promised distance $d_p=|f(x)|/\|\nabla f(x)\|$, at which the linearized score reaches zero. We ask when this one-shot step succeeds and what additional model queries change. To leading order the step ends on the favorable side exactly when the path curvature $\kappa=\hat g^\top\nabla^2 f(x)\,\hat g$ is nonnegative. Across 80 shallow models, the fraction of rejected users whose step ends there and the fraction with $\kappa\ge0$ correlate at $r=0.985$, although on Fashion-MNIST the first falls below the second by 8.2 points on average. No rule that uses only the score value and gradient can be valid for every score with path curvature bounded by $K$ without overshooting some by order $Kd_p^2/\|\nabla f(x)\|$. When the curvature is also Lipschitz and the step is short, one evaluation of $f$ at the promised point attains the minimax rate among deterministic one-query rules that know the curvature bound and its Lipschitz constant, and split-conformal calibration makes such a rule reach the first crossing or abstain with probability at least $1-\delta$. Training with an asymmetric curvature penalty lets 99-100% of paths cross within the promised step on undershoot-prone shallow data, at about 4-22 times the overshoot of symmetric penalties (Fashion-MNIST, COMPAS). Because $\kappa$ and $d_p$ depend on how the score is scaled, part of this gain can be a longer promised step, and at matched validity a smaller audit of briefly trained models finds no uniform advantage over tuned inflation. Where a per-user line search along the ray is affordable, it is exact to grid resolution and preferable.
| Comments: | v2: minor corrections and clarifications. Main text 8 pages; 58 pages including appendix and checklist |
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2609.36252 [cs.LG] |
| (or arXiv:2609.36252v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2609.36252 arXiv-issued DOI via DataCite |
Submission history
From: Hazar Yueksel [view email]
[v1]
Mon, 28 Sep 2026 20:44:23 UTC (210 KB)
[v2]
Tue, 6 Oct 2026 20:43:06 UTC (211 KB)
来源：arXiv：cs.LG · arxiv.org