# Universal Representation of Generalized Convex Functions and their Gradients

arXiv: https://arxiv.org/abs/2509.04477 · HTML: https://mlsanity.com/paper/2509.04477/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 61 references that are in the map.
- First posted on arXiv 30 Aug 2025.

## Builds on (most cited first, 30 of 32)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Deep Declarative Networks: A New Hope](https://mlsanity.com/paper/1909.04866/) (2019)
- [Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks](https://mlsanity.com/paper/2106.00774/) (2021)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Automated Mechanism Design via Neural Networks](https://mlsanity.com/paper/1805.03382/) (2018)
- [Kernel Neural Optimal Transport](https://mlsanity.com/paper/2205.15269/) (2022)
- [Auction learning as a two-player game](https://mlsanity.com/paper/2006.05684/) (2020)
- [Optimal-er Auctions through Attention](https://mlsanity.com/paper/2202.13110/) (2022)
- [A Scalable Neural Network for DSIC Affine Maximizer Auction Design](https://mlsanity.com/paper/2305.12162/) (2023)
- [Differentiable Economics for Randomized Affine Maximizer Auctions](https://mlsanity.com/paper/2202.02872/) (2022)
- [On approximating nabla f with neural networks](https://mlsanity.com/paper/1910.12744/) (2019)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)
- [Learning Revenue-Maximizing Auctions With Differentiable Matching](https://mlsanity.com/paper/2106.07877/) (2021)
- [Duality and Optimality of Auctions for Uniform Distributions](https://mlsanity.com/paper/1404.2329/) (2014)
- [Reinforcement Mechanism Design, with Applications to Dynamic Pricing in Sponsored Search Auctions](https://mlsanity.com/paper/1711.10279/) (2017)
- [JacNet: Learning Functions with Structured Jacobians](https://mlsanity.com/paper/2408.13237/) (2024)
- [The GroupMax neural network approximation of convex functions](https://mlsanity.com/paper/2206.06622/) (2022)
- [Adversarial learning for revenue-maximizing auctions](https://mlsanity.com/paper/1909.06806/) (2019)

## Abstract (arXiv)

A wide range of optimization problems can often be written in terms of generalized convex functions (GCFs). When this structure is present, it can convert certain nested bilevel objectives into single-level problems amenable to standard first-order optimization methods. We provide a new differentiable layer with a convex parameter space and show (Theorems 5.1 and 5.2) that it and its gradient are universal approximators for GCFs and their gradients. We demonstrate how this parameterization can be leveraged in practice by (i) learning optimal transport maps with general cost functions and (ii) learning optimal auctions of multiple goods. In both these cases, we show how our layer can be used to convert the existing bilevel or min-max formulations into single-level problems that can be solved efficiently with first-order methods.