用锥优化提升神经网络曲率表达能力，突破传统线性分段限制
SOC-ICNN: From Polyhedral to Conic Geometry for Learning Convex Surrogate Functions
- 将ICNN的线性规划基础升级为二阶锥规划，引入光滑曲率结构
- 在保持前向计算复杂度不变的前提下，显著增强函数逼近能力
- 适合需要高精度非线性建模的优化与控制场景
基于ReLU的输入凸神经网络（ICNN）等价于线性规划（LP）的最优值函数，其表示能力受限于分段线性的多面体函数。为突破这一瓶颈，本文提出SOC-ICNN，将底层优化问题从线性规划推广至二阶锥规划（SOCP）。通过显式注入半正定曲率和基于欧几里得范数的锥形基元，该架构在保持严格优化理论解释的同时，天然引入光滑曲率。我们证明了SOC-ICNN严格扩展了ReLU-ICNN的表示空间，且不增加前向传播的渐近复杂度。大量实验表明，SOC-ICNN在函数逼近性能上显著优于基线方法，同时在下游决策任务中表现竞争力。代码已公开于https://anonymous.4open.science/r/SOC-ICNN-4B18/。
原文摘要 · Abstract (English)
Classical ReLU-based Input Convex Neural Networks (ICNNs) are equivalent to the optimal value functions of Linear Programming (LP). This intrinsic structural equivalence restricts their representational capacity to piecewise-linear polyhedral functions. To overcome this representational bottleneck, we propose the SOC-ICNN, an architecture that generalizes the underlying optimization class from LP to Second-Order Cone Programming (SOCP). By explicitly injecting positive semi-definite curvature and Euclidean norm-based conic primitives, our formulation introduces native smooth curvature into the representation while preserving a rigorous optimization-theoretic interpretation. We formally prove that SOC-ICNNs strictly expand the representational space of ReLU-ICNNs without increasing the asymptotic order of forward-pass complexity. Extensive experiments demonstrate that SOC-ICNN substantially improves function approximation, while delivering competitive downstream decision quality. The code is available at https://anonymous.4open.science/r/SOC-ICNN-4B18/.
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