# T-Norms Driven Loss Functions for Machine Learning

arXiv: https://arxiv.org/abs/1907.11468 · HTML: https://mlsanity.com/paper/1907.11468/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2019.

## Built on by (most cited first, 8 of 8)

- [Predicated Diffusion: Predicate Logic-Based Attention Guidance for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2311.16117/) (2023)
- [Deep Explainable Relational Reinforcement Learning: A Neuro-Symbolic Approach](https://mlsanity.com/paper/2304.08349/) (2023)
- [Self-Supervised Transformers as Iterative Solution Improvers for Constraint Satisfaction](https://mlsanity.com/paper/2502.15794/) (2025)
- [Understanding the Logic of Direct Preference Alignment through Logic](https://mlsanity.com/paper/2412.17696/) (2024)
- [Improving Neural-based Classification with Logical Background Knowledge](https://mlsanity.com/paper/2402.13019/) (2024)
- [Compact Rule-Based Classifier Learning via Gradient Descent](https://mlsanity.com/paper/2502.01375/) (2025)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)
- [A Complexity Map of Probabilistic Reasoning for Neurosymbolic Classification Techniques](https://mlsanity.com/paper/2404.08404/) (2024)

## Abstract (arXiv)

Neural-symbolic approaches have recently gained popularity to inject prior knowledge into a learner without requiring it to induce this knowledge from data. These approaches can potentially learn competitive solutions with a significant reduction of the amount of supervised data. A large class of neural-symbolic approaches is based on First-Order Logic to represent prior knowledge, relaxed to a differentiable form using fuzzy logic. This paper shows that the loss function expressing these neural-symbolic learning tasks can be unambiguously determined given the selection of a t-norm generator. When restricted to supervised learning, the presented theoretical apparatus provides a clean justification to the popular cross-entropy loss, which has been shown to provide faster convergence and to reduce the vanishing gradient problem in very deep structures. However, the proposed learning formulation extends the advantages of the cross-entropy loss to the general knowledge that can be represented by a neural-symbolic method. Therefore, the methodology allows the development of a novel class of loss functions, which are shown in the experimental results to lead to faster convergence rates than the approaches previously proposed in the literature.