  Learning monotonic models with respect to a subset of the inputs is a
desirable feature to effectively address the fairness, interpretability, and
generalization issues in practice. Existing methods for learning monotonic
neural networks either require specifically designed model structures to ensure
monotonicity, which can be too restrictive/complicated, or enforce monotonicity
by adjusting the learning process, which cannot provably guarantee the learned
model is monotonic on selected features. In this work, we propose to certify
the monotonicity of the general piece-wise linear neural networks by solving a
mixed integer linear programming problem.This provides a new general approach
for learning monotonic neural networks with arbitrary model structures. Our
method allows us to train neural networks with heuristic monotonicity
regularizations, and we can gradually increase the regularization magnitude
until the learned network is certified monotonic. Compared to prior works, our
approach does not require human-designed constraints on the weight space and
also yields more accurate approximation. Empirical studies on various datasets
demonstrate the efficiency of our approach over the state-of-the-art methods,
such as Deep Lattice Networks.
