# Learning Monotonic Probabilities with a Generative Cost Model

arXiv: https://arxiv.org/abs/2506.03542 · HTML: https://mlsanity.com/paper/2506.03542/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Monotonic Value Function Factorisation for Deep Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2003.08839/) (2020)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [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)
- [Improving Deep Learning For Airbnb Search](https://mlsanity.com/paper/2002.05515/) (2020)
- [Smooth Min-Max Monotonic Networks](https://mlsanity.com/paper/2306.01147/) (2023)

## Built on by (most cited first, 1 of 1)

- [Knowledge-informed Bidding with Dual-process Control for Online Advertising](https://mlsanity.com/paper/2603.04920/) (2026)

## Abstract (arXiv)

In many machine learning tasks, it is often necessary for the relationship between input and output variables to be monotonic, including both strictly monotonic and implicitly monotonic relationships. Traditional methods for maintaining monotonicity mainly rely on construction or regularization techniques, whereas this paper shows that the issue of strict monotonic probability can be viewed as a partial order between an observable revenue variable and a latent cost variable. This perspective enables us to reformulate the monotonicity challenge into modeling the latent cost variable. To tackle this, we introduce a generative network for the latent cost variable, termed the Generative Cost Model (GCM), which inherently addresses the strict monotonic problem, and propose the Implicit Generative Cost Model (IGCM) to address the implicit monotonic problem. We further validate our approach with a numerical simulation of quantile regression and conduct multiple experiments on public datasets, showing that our method significantly outperforms existing monotonic modeling techniques. The code for our experiments can be found at https://github.com/tyxaaron/GCM.