Soft-Radial Projection for Constrained End-to-End Learning

Constrained end-to-end learning trains neural networks whose outputs must satisfy feasibility constraints, such as resource, budget, or operational limits. While projection and optimization layers guarantee constraint satisfaction, boundary-based projections can introduce unfavorable optimization geometry: regions of the unconstrained output space are mapped to the same active face of the feasible set. This makes the resulting map locally rank-deficient, suppressing gradients in constrained directions and degrading optimization—a bottleneck known as gradient saturation. We propose Soft-Radial Projection, a differentiable reparameterization layer that maps unconstrained network outputs into the relative interior of a convex feasible set by smoothly rescaling rays from a strictly feasible anchor point. Unlike hard radial or orthogonal projections, the resulting map is one-to-one and has a full-rank Jacobian almost everywhere, while guaranteeing strict feasibility. We prove that constrained networks equipped with Soft-Radial Projection retain universal approximation guarantees. For standard convex sets such as simplices and balls, the layer admits a closed-form forward pass, avoiding the iterative solvers required by optimization-based layers. Experiments on decision-focused learning benchmarks show improved optimization stability and solution quality compared with optimization- and projection-based baselines.