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.

翻译：深度神经网络日益广泛应用于气候建模、机器人学和工业控制等领域，这些领域要求严格遵守输出约束。尽管现有方法（如POLICE算法）能够通过调整网络参数在单个凸区域内实施仿射约束，但面对多个不相交区域时往往会产生冲突或产生非预期的仿射扩展。本文提出mPOLICE方法，将POLICE扩展至处理多区域约束。mPOLICE为每个约束区域分配独特的激活模式，在局部保持精确仿射行为的同时避免扩展到输入域的其他部分。我们构建了分层优化问题，通过调整权重和偏置为每个凸区域分配唯一激活模式，确保约束在无冲突条件下得到满足，同时保持学习函数的连续性与平滑性。实验展示了多区域约束在多种场景下的实施效果，包括回归与分类、函数逼近以及通过逼近处理非凸区域。值得注意的是，mPOLICE在推理阶段不产生额外开销，在训练阶段仅引入极小开销。