# Homogeneous Linear Inequality Constraints for Neural Network Activations

## TL;DR

This paper introduces a method to incorporate homogeneous linear inequality constraints directly into neural network architectures, enabling faster inference and consistent constraint satisfaction during training, demonstrated on constrained generative modeling.

## Contribution

The authors present a novel architecture-based approach to enforce linear inequality constraints in neural networks, avoiding costly projection steps and hyperparameter tuning.

## Key findings

- Significant speed-up in inference time, up to two orders of magnitude.
- Constraints are satisfied throughout training without additional optimization.
- Effective application demonstrated on constrained variational autoencoders.

## Abstract

We propose a method to impose homogeneous linear inequality constraints of the form $Ax\leq 0$ on neural network activations. The proposed method allows a data-driven training approach to be combined with modeling prior knowledge about the task. One way to achieve this task is by means of a projection step at test time after unconstrained training. However, this is an expensive operation. By directly incorporating the constraints into the architecture, we can significantly speed-up inference at test time; for instance, our experiments show a speed-up of up to two orders of magnitude over a projection method. Our algorithm computes a suitable parameterization of the feasible set at initialization and uses standard variants of stochastic gradient descent to find solutions to the constrained network. Thus, the modeling constraints are always satisfied during training. Crucially, our approach avoids to solve an optimization problem at each training step or to manually trade-off data and constraint fidelity with additional hyperparameters. We consider constrained generative modeling as an important application domain and experimentally demonstrate the proposed method by constraining a variational autoencoder.