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Constrained Machine Learning Through Hyperspherical Representation

Gaetano Signorelli, Michele Lombardi

TL;DR

This paper tackles the challenge of enforcing hard output constraints in machine learning, especially for convex and bounded feasible regions. It introduces the Hyperspherical Constrained Representation (HCR), which maps outputs to a hyperspherical-like space anchored at a feasible origin, ensuring feasibility by construction and enabling standard supervised training with minimal inference overhead. The authors formalize the transform, propose an acceleration technique, and provide a practical training pipeline, including a generalization to star domains. Empirical results show 100% constraint satisfaction with competitive predictive accuracy and substantially lower inference-time cost than projection-based methods, highlighting HCR's potential for safety-critical and control-oriented applications.

Abstract

The problem of ensuring constraints satisfaction on the output of machine learning models is critical for many applications, especially in safety-critical domains. Modern approaches rely on penalty-based methods at training time, which do not guarantee to avoid constraints violations; or constraint-specific model architectures (e.g., for monotonocity); or on output projection, which requires to solve an optimization problem that might be computationally demanding. We present the Hypersherical Constrained Representation, a novel method to enforce constraints in the output space for convex and bounded feasibility regions (generalizable to star domains). Our method operates on a different representation system, where Euclidean coordinates are converted into hyperspherical coordinates relative to the constrained region, which can only inherently represent feasible points. Experiments on a synthetic and a real-world dataset show that our method has predictive performance comparable to the other approaches, can guarantee 100% constraint satisfaction, and has a minimal computational cost at inference time.

Constrained Machine Learning Through Hyperspherical Representation

TL;DR

This paper tackles the challenge of enforcing hard output constraints in machine learning, especially for convex and bounded feasible regions. It introduces the Hyperspherical Constrained Representation (HCR), which maps outputs to a hyperspherical-like space anchored at a feasible origin, ensuring feasibility by construction and enabling standard supervised training with minimal inference overhead. The authors formalize the transform, propose an acceleration technique, and provide a practical training pipeline, including a generalization to star domains. Empirical results show 100% constraint satisfaction with competitive predictive accuracy and substantially lower inference-time cost than projection-based methods, highlighting HCR's potential for safety-critical and control-oriented applications.

Abstract

The problem of ensuring constraints satisfaction on the output of machine learning models is critical for many applications, especially in safety-critical domains. Modern approaches rely on penalty-based methods at training time, which do not guarantee to avoid constraints violations; or constraint-specific model architectures (e.g., for monotonocity); or on output projection, which requires to solve an optimization problem that might be computationally demanding. We present the Hypersherical Constrained Representation, a novel method to enforce constraints in the output space for convex and bounded feasibility regions (generalizable to star domains). Our method operates on a different representation system, where Euclidean coordinates are converted into hyperspherical coordinates relative to the constrained region, which can only inherently represent feasible points. Experiments on a synthetic and a real-world dataset show that our method has predictive performance comparable to the other approaches, can guarantee 100% constraint satisfaction, and has a minimal computational cost at inference time.

Paper Structure

This paper contains 14 sections, 7 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Related work
  4. Method
  5. Formalization:
  6. A Numerical Example:
  7. Acceleration Technique:
  8. Training Process:
  9. Generalization to Star Domains:
  10. Experiments
  11. Synthetic benchmark:
  12. M4 Forecasting Competition Dataset:
  13. Results:
  14. Conclusions

Figures (1)

  • Figure 1: Example of an instance of conversion. The constrained region is $C$: a circle centered at $O=(0,0)$ and radius $R=10$. The point $y_0=(5, 0)$ is converted to hyperspherical coordinates $(d=(1, 0), r=0.5)$ by finding the intersection point $S$.