Active-Set Projection Layers for Hard-Constrained Neural Networks

Differentiable projection layers make a network's outputs satisfy hard constraints exactly, but the projection can dominate both training and inference cost, and with the fixed iterative budget used in practice its gradient loses accuracy as more constraints become tight. We show that, for any solver of the projection, the implicit-differentiation system is block-triangular with respect to the active manifold cut out by the tight constraints: the backward pass reduces to one solve on its normal and curved tangent directions, where the curvature enters. Building on this structure, we propose ASPL (Active-Set Projection Layer), which identifies the active manifold in the forward pass and differentiates through the reduced system. We prove finite-time identification and an $\mathcal{O}(1/T)$ end-to-end training rate. We evaluate ASPL on parametric quadratic programs, multi-robot motion planning, inverse kinematics on real teleoperation logs, maze trajectory planning, and constrained flow matching. Its exact gradient costs less than ambient implicit differentiation of equal accuracy, and on quadratic programs its accuracy holds as more constraints bind.