# Tailored max-out networks for learning convex PWQ functions

arXiv: https://arxiv.org/abs/2206.06826 · HTML: https://mlsanity.com/paper/2206.06826/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 13 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 7 of 7)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [Efficient representation and approximation of model predictive control laws via deep learning](https://mlsanity.com/paper/1806.10644/) (2018)
- [AReN: Assured ReLU NN Architecture for Model Predictive Control of LTI Systems](https://mlsanity.com/paper/1911.01608/) (2019)
- [Polytopic Input Constraints in Learning-Based Optimal Control Using Neural Networks](https://mlsanity.com/paper/2105.03376/) (2021)
- [Tailored neural networks for learning optimal value functions in MPC](https://mlsanity.com/paper/2112.03975/) (2021)

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

- [Mapping back and forth between model predictive control and neural networks](https://mlsanity.com/paper/2404.12030/) (2024)

## Abstract (arXiv)

Convex piecewise quadratic (PWQ) functions frequently appear in control and elsewhere. For instance, it is well-known that the optimal value function (OVF) as well as Q-functions for linear MPC are convex PWQ functions. Now, in learning-based control, these functions are often represented with the help of artificial neural networks (NN). In this context, a recurring question is how to choose the topology of the NN in terms of depth, width, and activations in order to enable efficient learning. An elegant answer to that question could be a topology that, in principle, allows to exactly describe the function to be learned. Such solutions are already available for related problems. In fact, suitable topologies are known for piecewise affine (PWA) functions that can, for example, reflect the optimal control law in linear MPC. Following this direction, we show in this paper that convex PWQ functions can be exactly described by max-out-NN with only one hidden layer and two neurons.