Automatic Cost Function Learning with Interpretable Compositional Networks

Cost Function Networks (CFN) are a formalism in Constraint Programming to model combinatorial satisfaction or optimization problems. By associating a function to each constraint type to evaluate the quality of an assignment, it extends the expressivity of regular CSP/COP formalisms but at a price of making harder the problem modeling. Indeed, in addition to regular variables/domains/constraints sets, one must provide a set of cost functions that are not always easy to define. Here we propose a method to automatically learn a cost function of a constraint, given a function deciding if assignments are valid or not. This is to the best of our knowledge the first attempt to automatically learn cost functions. Our method aims to learn cost functions in a supervised fashion, trying to reproduce the Hamming distance, by using a variation of neural networks we named Interpretable Compositional Networks, allowing us to get explainable results, unlike regular artificial neural networks. We experiment it on 5 different constraints to show its versatility. Experiments show that functions learned on small dimensions scale on high dimensions, outputting a perfect or near-perfect Hamming distance for most constraints. Our system can be used to automatically generate cost functions and then having the expressivity of CFN with the same modeling effort than for CSP/COP.