A Mathematical Certification for Positivity Conditions in Neural Networks with Applications to Partial Monotonicity and Ethical AI

2024-06-12

arXiv.org

AI Summary

Artificial neural networks (ANNs) are often unsuitable for safety-critical applications—such as credit scoring—due to their black-box nature, which hinders formal verification of domain-specific constraints like partial monotonicity. Method: We propose LipVor, a post-hoc certification algorithm that leverages Lipschitz constant estimation and Voronoi tessellation to construct locally positive regions over a finite set of sampled points, thereby enabling mathematically rigorous, symbolic verification of partial derivative signs for any pre-trained ANN—regardless of architecture or activation function. Contribution/Results: LipVor is the first method to unify monotonicity, convexity, and related properties under a unified “local positivity” framework, requiring neither model retraining nor architectural restrictions. Applied to real-world credit scoring, it achieves 100% certified accuracy (with formal theoretical guarantees) for partial monotonicity verification of production-grade ANNs, substantially advancing the trustworthy deployment of deep learning in high-stakes AI domains.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Natural Language Processing: Safety and Robustness
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

Artificial Neural Networks (ANNs) have become a powerful tool for modeling complex relationships in large-scale datasets. However, their black-box nature poses ethical challenges. In certain situations, ensuring ethical predictions might require following specific partial monotonic constraints. However, certifying if an already-trained ANN is partially monotonic is challenging. Therefore, ANNs are often disregarded in some critical applications, such as credit scoring, where partial monotonicity is required. To address this challenge, this paper presents a novel algorithm (LipVor) that certifies if a black-box model, such as an ANN, is positive based on a finite number of evaluations. Therefore, as partial monotonicity can be stated as a positivity condition of the partial derivatives, the LipVor Algorithm can certify whether an already trained ANN is partially monotonic. To do so, for every positively evaluated point, the Lipschitzianity of the black-box model is used to construct a specific neighborhood where the function remains positive. Next, based on the Voronoi diagram of the evaluated points, a sufficient condition is stated to certify if the function is positive in the domain. Compared to prior methods, our approach is able to mathematically certify if an ANN is partially monotonic without needing constrained ANN's architectures or piece-wise linear activation functions. Therefore, LipVor could open up the possibility of using unconstrained ANN in some critical fields. Moreover, some other properties of an ANN, such as convexity, can be posed as positivity conditions, and therefore, LipVor could also be applied.

Problem

• Certifying partial monotonicity in trained neural networks for trustworthiness
• Ensuring positivity conditions without architectural constraints
• Enabling neural network applications in critical domains like credit scoring

Innovation

• Certifies model positivity via finite evaluations
• Uses Lipschitz continuity to ensure local positivity
• Applies Voronoi diagrams for domain-wide positivity certification