# Semantic Loss Functions for Neuro-Symbolic Structured Prediction

arXiv: https://arxiv.org/abs/2405.07387 · HTML: https://mlsanity.com/paper/2405.07387/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 51 references that are in the map.
- First posted on arXiv 12 May 2024.

## Builds on (most cited first, 30 of 30)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [StackGAN: Text to Photo-realistic Image Synthesis with Stacked Generative Adversarial Networks](https://mlsanity.com/paper/1612.03242/) (2016)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [MolGAN: An implicit generative model for small molecular graphs](https://mlsanity.com/paper/1805.11973/) (2018)
- [Multi-Task Identification of Entities, Relations, and Coreference for Scientific Knowledge Graph Construction](https://mlsanity.com/paper/1808.09602/) (2018)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Evolving Mario Levels in the Latent Space of a Deep Convolutional Generative Adversarial Network](https://mlsanity.com/paper/1805.00728/) (2018)
- [Coherent Hierarchical Multi-Label Classification Networks](https://mlsanity.com/paper/2010.10151/) (2020)
- [Distributional Smoothing with Virtual Adversarial Training](https://mlsanity.com/paper/1507.00677/) (2015)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [The VGLC: The Video Game Level Corpus](https://mlsanity.com/paper/1606.07487/) (2016)
- [A Frustratingly Easy Approach for Entity and Relation Extraction](https://mlsanity.com/paper/2010.12812/) (2020)
- [Multi-Label Classification Neural Networks with Hard Logical Constraints](https://mlsanity.com/paper/2103.13427/) (2021)
- [SIMPLE: A Gradient Estimator for k-Subset Sampling](https://mlsanity.com/paper/2210.01941/) (2022)
- [Programming with a Differentiable Forth Interpreter](https://mlsanity.com/paper/1605.06640/) (2016)
- [Neuro-Symbolic Entropy Regularization](https://mlsanity.com/paper/2201.11250/) (2022)
- [Semantic Strengthening of Neuro-Symbolic Learning](https://mlsanity.com/paper/2302.14207/) (2023)
- [Tunneling Neural Perception and Logic Reasoning through Abductive Learning](https://mlsanity.com/paper/1802.01173/) (2018)
- [On a Convex Logic Fragment for Learning and Reasoning](https://mlsanity.com/paper/1809.06778/) (2018)

## Built on by (most cited first, 3 of 3)

- [Understanding the Logic of Direct Preference Alignment through Logic](https://mlsanity.com/paper/2412.17696/) (2024)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)
- [NASP-T: A Fuzzy Neuro-Symbolic Transformer for Logic-Constrained Aviation Safety Report Classification](https://mlsanity.com/paper/2510.05451/) (2025)

## Abstract (arXiv)

Structured output prediction problems are ubiquitous in machine learning. The prominent approach leverages neural networks as powerful feature extractors, otherwise assuming the independence of the outputs. These outputs, however, jointly encode an object, e.g. a path in a graph, and are therefore related through the structure underlying the output space. We discuss the semantic loss, which injects knowledge about such structure, defined symbolically, into training by minimizing the network's violation of such dependencies, steering the network towards predicting distributions satisfying the underlying structure. At the same time, it is agnostic to the arrangement of the symbols, and depends only on the semantics expressed thereby, while also enabling efficient end-to-end training and inference. We also discuss key improvements and applications of the semantic loss. One limitations of the semantic loss is that it does not exploit the association of every data point with certain features certifying its membership in a target class. We should therefore prefer minimum-entropy distributions over valid structures, which we obtain by additionally minimizing the neuro-symbolic entropy. We empirically demonstrate the benefits of this more refined formulation. Moreover, the semantic loss is designed to be modular and can be combined with both discriminative and generative neural models. This is illustrated by integrating it into generative adversarial networks, yielding constrained adversarial networks, a novel class of deep generative models able to efficiently synthesize complex objects obeying the structure of the underlying domain.