Certified Robustness via Sparse Second-Order Cone Relaxations

Certified robustness is critical for deploying neural networks in safety-sensitive settings, but computing exact certificates typically requires solving nonconvex optimization problems. Practical verifiers use convex relaxations: linear programming (LP) relaxations are scalable but often too conservative, whereas semidefinite programming (SDP) relaxations can be substantially tighter but are computationally expensive. We introduce a hierarchy of second-order cone programming (SOCP) relaxations that provide stronger certificates than LP while avoiding the cost of global positive-semidefinite constraints. Our construction combines convex-hull relaxations of ReLU activations, selective McCormick lifting over certified variable bounds, and second-order cone constraints induced by 2-by-2 principal minors. Adding these constraints progressively yields a monotone hierarchy of increasingly tight bounds with a controllable computation-tightness trade-off. On fully connected and convolutional networks on MNIST and CIFAR-10, SOCP-based verification improves certified robust accuracy over LP-based verification and is competitive with alpha,beta-CROWN, while remaining orders of magnitude faster than SDP-based verification. These results position selective second-order constraints as a practical middle ground between scalable LP-based and tight SDP-based certification. Finally, we introduce class-optimized robust evaluation (CORE), which applies tighter relaxations only to target classes not certified by cheaper ones. On the fully connected MNIST benchmark, CORE variants match SDP-cert's certified robust accuracy while reducing average verification time by up to 86.30%.