揭示SOC-ICNN值函数的精确对偶几何，实现白盒推理
Exact Dual Geometry of SOC-ICNN Value Functions
- 从对偶视角推导出支持斜率、次梯度等几何特征
- 最优对偶变量可直接读出精确的局部黑塞矩阵
- 适用于需可解释性推理的优化与控制场景
输入凸神经网络（ICNN）通常以两阶段方式使用：先训练一个凸网络，再在下游任务中对其输入进行最小化。近期的二阶锥ICNN（SOC-ICNN）通过引入二次和锥模块，丰富了基于ReLU的ICNN，并可被精确表示为二阶锥规划（SOCP）的值函数。该值函数结构使SOC-ICNN推理具备显式的凸分析性质。本文从对偶角度研究了SOC-ICNN的一阶及局部二阶几何。我们证明支持斜率、次微分、方向导数和局部黑塞矩阵均可由最优对偶变量直接恢复。这些结果为白盒推理提供了几何基础，超越了黑盒自动微分。数值实验验证了对偶乘子读出的精确性、局部黑塞公式的正确性以及在结构性退化输入下的集值行为。我们还提供了一个逐步教程，展示读出机制如何构建完整的白盒推理流程。代码见https://anonymous.4open.science/r/SOC-ICNN-Theory-BEFC/。
原文摘要 · Abstract (English)
Input Convex Neural Networks (ICNNs) are commonly used in a two-stage manner: one first trains a convex network and then minimizes it over its input in a downstream inference problem. Recent second-order-cone ICNNs (SOC-ICNNs) enrich ReLU-based ICNNs with quadratic and conic modules and admit an exact representation as value functions of second-order cone programs (SOCPs). This value-function structure enables an explicit convex-analytic treatment of SOC-ICNN inference. In this paper, we study the exact first-order and local second-order geometry of SOC-ICNNs from the dual viewpoint. We show that supporting slopes, subdifferentials, directional derivatives, and local Hessians can be recovered directly from optimal dual variables. These results provide the geometric primitives for white-box SOC-ICNN inference, going beyond black-box automatic differentiation. Numerical experiments validate the exact multiplier readout, the local Hessian formula, and the set-valued behavior at structurally degenerate inputs. We also provide a step-by-step tutorial showing how the readout mechanism instantiates a complete white-box inference loop. The code is available at https://anonymous.4open.science/r/SOC-ICNN-Theory-BEFC/.
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