# Certified Monotonic Neural Networks

arXiv: https://arxiv.org/abs/2011.10219 · HTML: https://mlsanity.com/paper/2011.10219/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2020.

## Built on by (most cited first, 30 of 51)

- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Logic-Based Explainability in Machine Learning](https://mlsanity.com/paper/2211.00541/) (2022)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Explanations for Monotonic Classifiers](https://mlsanity.com/paper/2106.00154/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Constrained Monotonic Neural Networks](https://mlsanity.com/paper/2205.11775/) (2022)
- [Feature Necessity & Relevancy in ML Classifier Explanations](https://mlsanity.com/paper/2210.15675/) (2022)
- [Model Reconstruction Using Counterfactual Explanations: A Perspective From Polytope Theory](https://mlsanity.com/paper/2405.05369/) (2024)
- [Traversing the Local Polytopes of ReLU Neural Networks: A Unified Approach for Network Verification](https://mlsanity.com/paper/2111.08922/) (2021)
- [Monotonic Neural Additive Models: Pursuing Regulated Machine Learning Models for Credit Scoring](https://mlsanity.com/paper/2209.10070/) (2022)
- [Size and depth of monotone neural networks: interpolation and approximation](https://mlsanity.com/paper/2207.05275/) (2022)
- [Cardinality-Minimal Explanations for Monotonic Neural Networks](https://mlsanity.com/paper/2205.09901/) (2022)
- [Expressive Monotonic Neural Networks](https://mlsanity.com/paper/2307.07512/) (2023)
- [Robust and Provably Monotonic Networks](https://mlsanity.com/paper/2112.00038/) (2021)
- [Monotonicity Regularization: Improved Penalties and Novel Applications to Disentangled Representation Learning and Robust Classification](https://mlsanity.com/paper/2205.08247/) (2022)
- [Monotonicity for AI ethics and society: An empirical study of the monotonic neural additive model in criminology, education, health care, and finance](https://mlsanity.com/paper/2301.07060/) (2023)
- [How to address monotonicity for model risk management?](https://mlsanity.com/paper/2305.00799/) (2023)
- [Shape-constrained Symbolic Regression with NSGA-III](https://mlsanity.com/paper/2209.13851/) (2022)
- [Availability Adversarial Attack and Countermeasures for Deep Learning-based Load Forecasting](https://mlsanity.com/paper/2301.01832/) (2023)
- [From Principle to Practice: Vertical Data Minimization for Machine Learning](https://mlsanity.com/paper/2311.10500/) (2023)
- [Compiling High-Level Neural Network Specifications into VNN-LIB Queries](https://mlsanity.com/paper/2402.01353/) (2024)
- [A Practical Theory of Generalization in Selectivity Learning](https://mlsanity.com/paper/2409.07014/) (2024)
- [An interpretable neural network-based non-proportional odds model for ordinal regression](https://mlsanity.com/paper/2303.17823/) (2023)
- [Monotone Tree-Based GAMI Models by Adapting XGBoost](https://mlsanity.com/paper/2309.02426/) (2023)
- [Can I Trust the Explanations? Investigating Explainable Machine Learning Methods for Monotonic Models](https://mlsanity.com/paper/2309.13246/) (2023)
- [The Pros and Cons of Adversarial Robustness](https://mlsanity.com/paper/2312.10911/) (2023)
- [A mathematical certification for positivity conditions in Neural Networks with applications to partial monotonicity and Trustworthy AI](https://mlsanity.com/paper/2406.08525/) (2024)
- [Enhancing Monotonic Modeling with Spatio-Temporal Adaptive Awareness in Diverse Marketing](https://mlsanity.com/paper/2406.14132/) (2024)
- [FinML-Chain: A Blockchain-Integrated Dataset for Enhanced Financial Machine Learning](https://mlsanity.com/paper/2411.16277/) (2024)
- [A Novel Switch-Type Policy Network for Resource Allocation Problems: Technical Report](https://mlsanity.com/paper/2501.11136/) (2025)

## Abstract (arXiv)

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.