mlsanityData to 14 Oct 2026

POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks

arXiv 2211.01340 ↗First posted 2 Nov 2022cs.LG cs.CV stat.MLcited by 19

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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

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

11 of 26 references.

References · 11

  1. 2014Explaining and Harnessing Adversarial Examplescited by 7,364
  2. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  3. 2017Provable defenses against adversarial examples via the convex outer adversarial polytopecited by 567
  4. 2014On the Number of Linear Regions of Deep Neural Networkscited by 381
  5. 2017Adversarial Examples: Attacks and Defenses for Deep Learningcited by 351
  6. 2019Algorithms for Verifying Deep Neural Networkscited by 221
  7. 2018Training and Inference with Integers in Deep Neural Networkscited by 86
  8. 2019The Geometry of Deep Networks: Power Diagram Subdivisioncited by 53
  9. 2018Concolic Testing for Deep Neural Networkscited by 51
  10. 2022Polarity Sampling: Quality and Diversity Control of Pre-Trained Generative Networks via Singular Valuescited by 21
  11. 2018From Hard to Soft: Understanding Deep Network Nonlinearities via Vector Quantization and Statistical Inferencecited by 13

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 8; 2024: 2; 2025: 4; 2026: 5.

19 papers.

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  1. 2023NIFTY: Neural Object Interaction Fields for Guided Human Motion Synthesiscited by 78
  2. 2024Deep Networks Always Grok and Here is Whycited by 51
  3. 2024HardNet: Hard-Constrained Neural Networks with Universal Approximation Guaranteescited by 31
  4. 2023RAYEN: Imposition of Hard Convex Constraints on Neural Networkscited by 30
  5. 2025ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projectioncited by 17
  6. 2023Polyhedral Complex Extraction from ReLU Networks using Edge Subdivisioncited by 14
  7. 2023A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraintscited by 12
  8. 2023Neural Fields with Hard Constraints of Arbitrary Differential Ordercited by 5
  9. 2023Modularity in Deep Learning: A Surveycited by 3
  10. 2025Provably-Safe Neural Network Training Using Hybrid Zonotope Reachability Analysiscited by 3
  11. 2025ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guaranteescited by 2
  12. 2026HardNet++: Nonlinear Constraint Enforcement in Neural Networkscited by 2
7 more
  1. 2023Constrained Empirical Risk Minimization: Theory and Practicecited by 1
  2. 2025mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networkscited by 1
  3. 2023Safety without alignmentcited by 0
  4. 2026CAffNet: Hard Constraint-Affine Neural Networkscited by 0
  5. 2026Embedding Linear Equality Constraints in Probabilistic Neural Networks for Dynamic Modellingcited by 0
  6. 2026ShardNet: Training Neural Controllers with Hard, Non-Convex Constraintscited by 0
  7. 2026CG4AI: A Column Generation Framework for Training AI Models Under Constraintscited by 0