# Smooth Min-Max Monotonic Networks

arXiv: https://arxiv.org/abs/2306.01147 · HTML: https://mlsanity.com/paper/2306.01147/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 1 Jun 2023.

## Builds on (most cited first, 11 of 11)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Counterexample-Guided Learning of Monotonic Neural Networks](https://mlsanity.com/paper/2006.08852/) (2020)
- [How to Incorporate Monotonicity in Deep Networks While Preserving Flexibility?](https://mlsanity.com/paper/1909.10662/) (2019)
- [Monotonic classification: an overview on algorithms, performance measures and data sets](https://mlsanity.com/paper/1811.07155/) (2018)
- [Deontological Ethics By Monotonicity Shape Constraints](https://mlsanity.com/paper/2001.11990/) (2020)
- [Size and depth of monotone neural networks: interpolation and approximation](https://mlsanity.com/paper/2207.05275/) (2022)
- [Shifting Mean Activation Towards Zero with Bipolar Activation Functions](https://mlsanity.com/paper/1709.04054/) (2017)
- [Avoiding Resentment Via Monotonic Fairness](https://mlsanity.com/paper/1909.01251/) (2019)

## Built on by (most cited first, 5 of 5)

- [Learning Monotonic Probabilities with a Generative Cost Model](https://mlsanity.com/paper/2506.03542/) (2025)
- [Monotonic Kolmogorov-Arnold Networks: A Theoretical and Empirical Study of Monotonicity as an Inductive Bias](https://mlsanity.com/paper/2606.17886/) (2026)
- [Knowledge-informed Bidding with Dual-process Control for Online Advertising](https://mlsanity.com/paper/2603.04920/) (2026)
- [A CDF-First Framework for Free-Form Density Estimation](https://mlsanity.com/paper/2603.25204/) (2026)
- [SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization](https://mlsanity.com/paper/2609.33515/) (2026)

## Abstract (arXiv)

Monotonicity constraints are powerful regularizers in statistical modelling. They can support fairness in computer-aided 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 partial derivatives of the MM nonlinearities being zero. We propose a simple modification of the MM network using strictly-increasing smooth minimum and maximum functions 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 conceptually simple and computationally less demanding than state-of-the-art neural networks for monotonic modelling. Our experiments show that this does not come with a loss in generalization performance compared to alternative neural and non-neural approaches.