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mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks

Mohammadmehdi Ataei, Hyunmin Cheong, Adrian Butscher

Abstract

Deep neural networks are increasingly used in safety-critical domains such as robotics and scientific modeling, where strict adherence to output constraints is essential. Methods like POLICE, which are tailored for single convex regions, face challenges when extended to multiple disjoint regions, often leading to constraint violations or unwanted affine behavior across regions. This paper proposes mPOLICE, a new approach that generalizes POLICE to provably enforce affine constraints over multiple disjoint convex regions. At its core, mPOLICE assigns distinct neuron activation patterns to each constrained region, enabling localized affine behavior and avoiding unintended generalization. This is implemented through a layer-wise optimization of the network parameters. Additionally, we introduce a training algorithm that incorporates mPOLICE into conventional deep learning pipelines, balancing task-specific performance with constraint enforcement using periodic sign pattern enforcement. We validate the flexibility and effectiveness of mPOLICE through experiments across various applications, including safety-critical reinforcement learning, implicit 3D shape representation with geometric constraints, and fluid dynamics simulations with boundary condition enforcement. Importantly, mPOLICE incurs no runtime overhead during inference, making it a practical and reliable solution for constraint handling in deep neural networks.

mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks

Abstract

Deep neural networks are increasingly used in safety-critical domains such as robotics and scientific modeling, where strict adherence to output constraints is essential. Methods like POLICE, which are tailored for single convex regions, face challenges when extended to multiple disjoint regions, often leading to constraint violations or unwanted affine behavior across regions. This paper proposes mPOLICE, a new approach that generalizes POLICE to provably enforce affine constraints over multiple disjoint convex regions. At its core, mPOLICE assigns distinct neuron activation patterns to each constrained region, enabling localized affine behavior and avoiding unintended generalization. This is implemented through a layer-wise optimization of the network parameters. Additionally, we introduce a training algorithm that incorporates mPOLICE into conventional deep learning pipelines, balancing task-specific performance with constraint enforcement using periodic sign pattern enforcement. We validate the flexibility and effectiveness of mPOLICE through experiments across various applications, including safety-critical reinforcement learning, implicit 3D shape representation with geometric constraints, and fluid dynamics simulations with boundary condition enforcement. Importantly, mPOLICE incurs no runtime overhead during inference, making it a practical and reliable solution for constraint handling in deep neural networks.

Paper Structure

This paper contains 40 sections, 1 theorem, 13 equations, 9 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Piecewise Affine Structure of ReLU Networks
  5. Problem Setup and Preliminaries
  6. From Single to Multiple Regions and the Convex Hull Problem
  7. Problem Formulation: Multi-Region Sign Assignment and Parameter Adjustments
  8. Strategies for Sign Assignment
  9. Majority Voting.
  10. Pre-Activation Mean-based.
  11. Enforcing Sign Patterns
  12. Algorithm for Imposing Affine Constraints during Training
  13. Experiments
  14. Reinforcement Learning
  15. Implicit Occupancy-Field Learning with Geometric Constraints
  16. ...and 25 more sections

Key Result

Theorem 1

Let $f_{\boldsymbol{\theta}}$ be a feedforward ReLU network of depth $L$. Consider a convex region $R = \mathrm{conv}\{\boldsymbol{v}_1,\dots,\boldsymbol{v}_P\}$ in the input space, and a set of $N$ disjoint convex regions $\{R_i\}_{i=1}^N$, where each $R_i = \mathrm{conv}\{\boldsymbol{v}_{i,1},\dot

Figures (9)

  • Figure 1: POLICE (single-region) vs. mPOLICE (multi-region) enforcement. POLICE uses one activation pattern, making the network affine over the combined convex hull of all regions (middle). mPOLICE assigns unique patterns to each region, localizing affine behavior and preventing unintended affine behavior across regions (right).
  • Figure 2: Reinforcement learning results. Left: Policy network affine partitions with learned policy (blue arrows), target (green circle), obstacles (red), buffer regions (light pink), and affine polytope boundaries from ReLU activations (thin black lines). Right: Training curves showing average evaluation reward, critic losses, and actor loss.
  • Figure 3: 3D shape representations: Ground Truth, Baseline Prediction, and mPOLICE Prediction. The baseline's voxelized output incorrectly fills holes (magenta), while mPOLICE accurately depicts empty cylindrical holes, satisfying constraints.
  • Figure 4: Two affine regions approximating a saddle background field. On the left, the large gap between squares spans several polytopes, yielding no affine guarantees between them. On the right, placing the squares very close turns the gap into a shared boundary of two polytopes, effectively approximating a non-convex shape with two convex pieces.
  • Figure 5: Comparison of fluid flow velocity magnitude fields: (Top) Ground Truth, (Middle) Baseline Prediction, and (Bottom) mPOLICE Constrained Prediction. The red dashed squares indicate the regions where zero velocity is enforced by mPOLICE. The color bar indicates velocity magnitude. Data shown is for the interior of the domain.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1: Localized Affine Behavior with Unique Activation Patterns
  • proof