mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks

Deep neural networks are increasingly employed in fields such as climate modeling, robotics, and industrial control, where strict output constraints must be upheld. Although prior methods like the POLICE algorithm can enforce affine constraints in a single convex region by adjusting network parameters, they struggle with multiple disjoint regions, often leading to conflicts or unintended affine extensions. We present mPOLICE, a new method that extends POLICE to handle constraints imposed on multiple regions. mPOLICE assigns a distinct activation pattern to each constrained region, preserving exact affine behavior locally while avoiding overreach into other parts of the input domain. We formulate a layer-wise optimization problem that adjusts both the weights and biases to assign unique activation patterns to each convex region, ensuring that constraints are met without conflicts, while maintaining the continuity and smoothness of the learned function. Our experiments show the enforcement of multi-region constraints for multiple scenarios, including regression and classification, function approximation, and non-convex regions through approximation. Notably, mPOLICE adds zero inference overhead and minimal training overhead.