arXiv：cs.LG· Elias Khalife, Mazen Farhood, Pierre-Loic Garoche·· 13 小时前
神经网络可达性分析的二次特征刻画方法
Quadratic Characterizations for Reachability Analysis of Neural Networks
AI 导读
该论文提出一套构造标量关系已验证二次约束的框架：通过在二维实平面上求解凸二次规划生成候选不等式，并用平方和证书在精确半代数描述或松弛多项式描述上做全局验证。所得约束与基于二次约束及逐点积分二次约束的现有分析框架兼容，可嵌入半正定规划用于前馈神经网络的可达性与安全性分析。
正文
Abstract:Quadratic constraints (QCs) are widely used to characterize nonlinearities and uncertainties, but generic analytical characterizations can be conservative on bounded domains. This paper develops a framework for constructing verified quadratic characterizations of scalar relations in the two-dimensional real plane. Candidate quadratic inequalities are locally generated by solving convex quadratic programs using samples from the relation and exterior sample points. They are then verified globally using sum-of-squares certificates over an exact semialgebraic description or, in the case of nonpolynomial relations, over relaxed polynomial descriptions. The resulting verified constraints define a sound overapproximation of the scalar relations over the considered domains. These constraints are directly compatible with existing analysis frameworks based on QCs and pointwise integral quadratic constraints (IQCs) for static nonlinearities and uncertainties, and they can also be embedded in QC-based semidefinite programs for reachability and safety analysis of feedforward neural networks. For smooth activations such as $\tanh$, the method yields domain-dependent quadratic characterizations that constitute an alternative to generic sector- or slope-based descriptions. For ReLU networks, we give methods to reduce conservatism in QC-based reachability analysis of feedforward networks by exploiting dependencies between neurons and tighter local bounds. Numerical examples demonstrate improved reachability results for smooth activations, reduced conservatism for ReLU networks, and applicability beyond neural networks through an example involving saturation.
| Subjects: | Machine Learning (cs.LG); Systems and Control (eess.SY) |
| Cite as: | arXiv:2605.20482 [cs.LG] |
| (or arXiv:2605.20482v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.20482 arXiv-issued DOI via DataCite |
Submission history
From: Elias Khalife [view email]
[v1]
Tue, 19 May 2026 20:45:31 UTC (735 KB)
[v2]
Fri, 9 Oct 2026 04:57:49 UTC (738 KB)
来源：arXiv：cs.LG · arxiv.org