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.

本文提出了一种名为 LipVor 的新算法，用于通过有限数量的评估来验证黑盒模型，例如人工神经网络（ANN），是否在部分区域上是单调增的。与先前的方法相比，我们的方法能够在不需要受限的 ANN 架构或分段线性激活函数的情况下，数学上验证 ANN 是否部分单调增，从而扩展了在某些关键领域中使用不受限制的 ANN 的可能性。此外，ANN 的其他属性，例如凸性，也可以被形式化为正性条件，因此 LipVor 也可以应用于此。