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A mathematical certification for positivity conditions in Neural Networks with applications to partial monotonicity and Trustworthy AI

Alejandro Polo-Molina, David Alfaya, Jose Portela

TL;DR

The paper tackles the challenge of certifying partial monotonicity and related properties for black-box neural networks without constraining architectures. It introduces LipVor, a positivity-certification algorithm leveraging Lipschitz continuity to extend pointwise positivity to neighborhoods and a Voronoi-based global coverage test, providing a finite-step certificate or counterexamples. It additionally derives an upper bound for the Lipschitz constant of the partial derivatives of an ANN, enabling derivative-based monotonicity certification, and extends the framework to training unconstrained monotonic networks via an $oldsymbol{\varepsilon}$-monotonic penalty with iterative certification. Through heat-equation, ESL, and AutoMPG case studies, LipVor demonstrates the ability to certify partial monotonicity, locate counterexamples, and guide model refinement, highlighting potential for trustworthy AI in regulated domains. The work suggests broader applicability to convexity and other properties, and discusses computational strategies to scale Voronoi-based certification in practice.

Abstract

Artificial Neural Networks (ANNs) have become a powerful tool for modeling complex relationships in large-scale datasets. However, their black-box nature poses trustworthiness challenges. In certain situations, ensuring trust in predictions might require following specific partial monotonicity 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. Consequently, since partial monotonicity can be expressed as a positivity condition on partial derivatives, LipVor can certify whether an 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. Unlike prior methods, our approach certifies partial monotonicity without constrained architectures or piece-wise linear activations. 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.

A mathematical certification for positivity conditions in Neural Networks with applications to partial monotonicity and Trustworthy AI

TL;DR

The paper tackles the challenge of certifying partial monotonicity and related properties for black-box neural networks without constraining architectures. It introduces LipVor, a positivity-certification algorithm leveraging Lipschitz continuity to extend pointwise positivity to neighborhoods and a Voronoi-based global coverage test, providing a finite-step certificate or counterexamples. It additionally derives an upper bound for the Lipschitz constant of the partial derivatives of an ANN, enabling derivative-based monotonicity certification, and extends the framework to training unconstrained monotonic networks via an -monotonic penalty with iterative certification. Through heat-equation, ESL, and AutoMPG case studies, LipVor demonstrates the ability to certify partial monotonicity, locate counterexamples, and guide model refinement, highlighting potential for trustworthy AI in regulated domains. The work suggests broader applicability to convexity and other properties, and discusses computational strategies to scale Voronoi-based certification in practice.

Abstract

Artificial Neural Networks (ANNs) have become a powerful tool for modeling complex relationships in large-scale datasets. However, their black-box nature poses trustworthiness challenges. In certain situations, ensuring trust in predictions might require following specific partial monotonicity 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. Consequently, since partial monotonicity can be expressed as a positivity condition on partial derivatives, LipVor can certify whether an 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. Unlike prior methods, our approach certifies partial monotonicity without constrained architectures or piece-wise linear activations. 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.
Paper Structure (18 sections, 10 theorems, 30 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 18 sections, 10 theorems, 30 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Positivity Certification: The LipVor Algorithm
  3. Local positivity Certification
  4. Global Positivity Certification in a Compact Domain
  5. The LipVor Algorithm
  6. Certification of Partial Monotonicity for ANNs
  7. Partial Monotonicity of an ANN
  8. Lipschitz Constant Estimate of the Partial Derivative of an ANN
  9. Extension: Training Certified Monotonic Neural Networks
  10. Computational Implementation Details
  11. Case Studies
  12. Case Study: Heat Equation
  13. Case Study: Trustworthy Monotonic Predictions for the ESL dataset
  14. Case Study: AutoMPG Dataset
  15. Conclusions
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $f: \Omega \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$ with $f \in C^0(\Omega)$ and $x_0 \in \Omega$. If $f$ is L-Lipschitz and $f(x_0)>0$, then there exists a radius $\delta_0 = \frac{f(x_0)}{L}$ such that $f(x)> 0$, $\forall x \in B(x_0,\delta_0)$.

Figures (4)

  • Figure 1: Voronoi Diagram $V(\mathcal{P})$ for a set of 10 randomly allocated points $\mathcal{P} = \{p_0,p_1,\dots,p_{9}\}$ in a 2D space. Each red point represents the furthest vertex to each of the points in $\mathcal{P}$ and each circle is the ball of certified positivity given by Proposition \ref{['prop:rad_mono']}.
  • Figure 2: Surface of the solution of the heat equation \ref{['eq:heat_eq']} and the training (blue), validation (red) and test (green) datasets obtained from the solution with added noise.
  • Figure 3: Evolution of the Voronoi diagram generated by the LipVor Algorithm. (a) Surface plot of the partial derivative of the ANN output w.r.t the input $t$ across the domain. A horizontal plane at $z=0$ is included to help identify regions where the partial derivative is positive or negative.(b) Initialization of the Voronoi diagram using the training set. (c) Voronoi diagram expansion when the first counter-examples is detected by the LipVor Algorithm. (d) Final iteration, after LipVor has reached the maximum number of steps, showing the partial monotonic subdomain and the found counter-examples.
  • Figure 4: Partial monotonicity verification of the ANN using the LipVor Algorithm. (a) Surface plot of the partial derivative of the ANN output w.r.t. input $t$. (b) Voronoi diagram after certifying partial monotonicity.

Theorems & Definitions (17)

  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Theorem 3.2
  • Lemma A.1
  • proof
  • Theorem A.2
  • ...and 7 more