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This paper analyzes the exact dual geometry of Second-Order Cone Input Convex Neural Networks (SOC-ICNNs), which have an exact representation as value functions of second-order cone programs (SOCPs). It demonstrates that supporting slopes, subdifferentials, directional derivatives, and local Hessians can be directly recovered from the optimal dual variables of the SOCP. This enables a white-box inference approach for SOC-ICNNs, offering an alternative to black-box automatic differentiation.
Unlock white-box inference for SOC-ICNNs by directly reading out geometric primitives like Hessians from the optimal dual variables, bypassing black-box differentiation.
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/.
