# Saddle Networks: Structure-Preserving Architectures for Convex-Concave Functions

arXiv: https://arxiv.org/abs/2605.28894 · HTML: https://mlsanity.com/paper/2605.28894/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 15 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 9 of 9)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Near-Optimal Algorithms for Minimax Optimization](https://mlsanity.com/paper/2002.02417/) (2020)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [The Online Saddle Point Problem and Online Convex Optimization with Knapsacks](https://mlsanity.com/paper/1806.08301/) (2018)
- [Disciplined Saddle Programming](https://mlsanity.com/paper/2301.13427/) (2023)
- [The GroupMax neural network approximation of convex functions](https://mlsanity.com/paper/2206.06622/) (2022)
- [Input Convex Kolmogorov Arnold Networks](https://mlsanity.com/paper/2505.21208/) (2025)
- [On Well-Structured Convex-Concave Saddle Point Problems and Variational Inequalities with Monotone Operators](https://mlsanity.com/paper/2102.01002/) (2021)
- [An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function](https://mlsanity.com/paper/2104.06206/) (2021)

## Abstract (arXiv)

Saddle-point models arise throughout optimization, optimal transport, robust learning, and control. In many applications, the relevant function f(x,y) is convex in x and concave in y, and preserving this geometry is essential for obtaining tractable min--max formulations and reliable certificates. We introduce a structured separable decomposition that preserves the convex-concave geometry and prove a complete one-dimensional approximation theorem under a mixed Monge-type convexity condition. We then describe practical saddle network architectures that preserve convexity in x and concavity in y by construction. The proposed architectures require only convexity-preserving neural networks, together with simple output transformations enforcing sign and concavity constraints. Finally, we report numerical benchmarks in dimension 1 and 5, showing that the proposed saddle networks achieve high accuracy on smooth, nonsmooth, and high-rank convex--concave test functions.