# POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks

arXiv: https://arxiv.org/abs/2211.01340 · HTML: https://mlsanity.com/paper/2211.01340/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 2 Nov 2022.

## Builds on (most cited first, 11 of 11)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Adversarial Examples: Attacks and Defenses for Deep Learning](https://mlsanity.com/paper/1712.07107/) (2017)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Training and Inference with Integers in Deep Neural Networks](https://mlsanity.com/paper/1802.04680/) (2018)
- [The Geometry of Deep Networks: Power Diagram Subdivision](https://mlsanity.com/paper/1905.08443/) (2019)
- [Concolic Testing for Deep Neural Networks](https://mlsanity.com/paper/1805.00089/) (2018)
- [Polarity Sampling: Quality and Diversity Control of Pre-Trained Generative Networks via Singular Values](https://mlsanity.com/paper/2203.01993/) (2022)
- [From Hard to Soft: Understanding Deep Network Nonlinearities via Vector Quantization and Statistical Inference](https://mlsanity.com/paper/1810.09274/) (2018)

## Built on by (most cited first, 19 of 19)

- [NIFTY: Neural Object Interaction Fields for Guided Human Motion Synthesis](https://mlsanity.com/paper/2307.07511/) (2023)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Neural Fields with Hard Constraints of Arbitrary Differential Order](https://mlsanity.com/paper/2306.08943/) (2023)
- [Modularity in Deep Learning: A Survey](https://mlsanity.com/paper/2310.01154/) (2023)
- [Provably-Safe Neural Network Training Using Hybrid Zonotope Reachability Analysis](https://mlsanity.com/paper/2501.13023/) (2025)
- [ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2512.01984/) (2025)
- [HardNet++: Nonlinear Constraint Enforcement in Neural Networks](https://mlsanity.com/paper/2604.19669/) (2026)
- [Constrained Empirical Risk Minimization: Theory and Practice](https://mlsanity.com/paper/2302.04729/) (2023)
- [mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks](https://mlsanity.com/paper/2502.02434/) (2025)
- [Safety without alignment](https://mlsanity.com/paper/2303.00752/) (2023)
- [CAffNet: Hard Constraint-Affine Neural Networks](https://mlsanity.com/paper/2605.24437/) (2026)
- [Embedding Linear Equality Constraints in Probabilistic Neural Networks for Dynamic Modelling](https://mlsanity.com/paper/2606.21728/) (2026)
- [ShardNet: Training Neural Controllers with Hard, Non-Convex Constraints](https://mlsanity.com/paper/2606.30935/) (2026)
- [CG4AI: A Column Generation Framework for Training AI Models Under Constraints](https://mlsanity.com/paper/2608.26375/) (2026)

## Abstract (arXiv)

Deep Neural Networks (DNNs) outshine alternative function approximators in many settings thanks to their modularity in composing any desired differentiable operator. The formed parametrized functional is then tuned to solve a task at hand from simple gradient descent. This modularity comes at the cost of making strict enforcement of constraints on DNNs, e.g. from a priori knowledge of the task, or from desired physical properties, an open challenge. In this paper we propose the first provable affine constraint enforcement method for DNNs that only requires minimal changes into a given DNN's forward-pass, that is computationally friendly, and that leaves the optimization of the DNN's parameter to be unconstrained, i.e. standard gradient-based method can be employed. Our method does not require any sampling and provably ensures that the DNN fulfills the affine constraint on a given input space's region at any point during training, and testing. We coin this method POLICE, standing for Provably Optimal LInear Constraint Enforcement. Github: https://github.com/RandallBalestriero/POLICE