Smooth Monotonic Networks

Monotonicity constraints are powerful regularizers in statistical modelling. They can support fairness in computer supported decision making and increase plausibility in data-driven scientific models. The seminal min-max (MM) neural network architecture ensures monotonicity, but often gets stuck in undesired local optima during training because of vanishing gradients. We propose a simple modification of the MM network using strictly-increasing smooth non-linearities that alleviates this problem. The resulting smooth min-max (SMM) network module inherits the asymptotic approximation properties from the MM architecture. It can be used within larger deep learning systems trained end-to-end. The SMM module is considerably simpler and less computationally demanding than state-of-the-art neural networks for monotonic modelling. Still, in our experiments, it compared favorably to alternative neural and non-neural approaches in terms of generalization performance.