Differentiable Approximations for Multi-resource Spatial Coverage Problems

Resource allocation for coverage of physical spaces is a challenging problem in robotic surveillance, mobile sensor networks and security domains. Recent gradient-based optimization approaches to this problem estimate utilities of actions by using neural networks to learn a differentiable approximation to spatial coverage objectives. In this work, we empirically show that spatial coverage objectives with multiple-resources are combinatorially hard to approximate for neural networks and lead to sub-optimal policies. As our major contribution, we propose a tractable framework to approximate a general class of spatial coverage objectives and their gradients using a combination of Newton-Leibniz theorem, spatial discretization and implicit boundary differentiation. We empirically demonstrate the efficacy of our proposed framework on single and multi-agent spatial coverage problems.