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

A. Polo Molina, D. Alfaya, J. Portela

Artificial Neural Networks (ANNs) are powerful for modeling complex relationships but face trust challenges due to their black-box nature. Ensuring trust may require partial monotonicity constraints, yet certifying if a trained ANN meets these constraints is challenging. Therefore, this paper introduces LipVor, a novel algorithm certifying if a black-box model, like an ANN, is positive based on finite evaluations. Partial monotonicity can be expressed as a positivity condition of partial derivatives, enabling LipVor to certify if an ANN is partially monotonic. LipVor leverages Lipschitzianity to construct neighborhoods where the function remains positive for each positively evaluated point. Using the Voronoi diagram of evaluated points, LipVor provides a sufficient condition to certify positivity across the domain. Unlike prior methods, LipVor doesn’t require constrained architectures or piece-wise linear activation functions, enabling unconstrained ANNs in critical fields.

Palabras clave: Artificial Neural Networks, Partial Monotonicity, Mathematical Certification, Trustworthy AI

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