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.

翻译：单调性约束是统计建模中强大的正则化方法。它们能够支持计算机辅助决策中的公平性，并提升数据驱动科学模型的合理性。开创性的最小-最大（MM）神经网络架构确保了单调性，但在训练过程中常因梯度消失而陷入不理想的局部最优。我们提出对MM网络进行简单改进，采用严格递增的平滑非线性函数来缓解这一问题。由此产生的平滑最小-最大（SMM）网络模块继承了MM架构的渐近逼近特性，可在端到端训练的更深层深度学习系统中使用。相较于用于单调建模的最新神经网络，SMM模块明显更简单且计算需求更低。尽管如此，在我们的实验中，它在泛化性能方面与替代的神经及非神经方法相比仍具有竞争优势。