# EM-NeSy: Expectation Maximization for Neurosymbolic Learning

arXiv: https://arxiv.org/abs/2606.14463 · HTML: https://mlsanity.com/paper/2606.14463/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 25 of its 51 references that are in the map.
- First posted on arXiv 12 Jun 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Neural Logic Machines](https://mlsanity.com/paper/1904.11694/) (2019)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference](https://mlsanity.com/paper/2212.12393/) (2022)
- [NLProlog: Reasoning with Weak Unification for Question Answering in Natural Language](https://mlsanity.com/paper/1906.06187/) (2019)
- [Neural Markov Logic Networks](https://mlsanity.com/paper/1905.13462/) (2019)
- [Lifted Rule Injection for Relation Embeddings](https://mlsanity.com/paper/1606.08359/) (2016)
- [Relational Neural Machines](https://mlsanity.com/paper/2002.02193/) (2020)
- [Adversarial Sets for Regularising Neural Link Predictors](https://mlsanity.com/paper/1707.07596/) (2017)
- [Integrating Deep Learning with Logic Fusion for Information Extraction](https://mlsanity.com/paper/1912.03041/) (2019)
- [Differentiable Sampling of Categorical Distributions Using the CatLog-Derivative Trick](https://mlsanity.com/paper/2311.12569/) (2023)
- [On the Hardness of Probabilistic Neurosymbolic Learning](https://mlsanity.com/paper/2406.04472/) (2024)
- [Efficient Rectification of Neuro-Symbolic Reasoning Inconsistencies by Abductive Reflection](https://mlsanity.com/paper/2412.08457/) (2024)
- [EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic](https://mlsanity.com/paper/2408.08133/) (2024)
- [Hard and Soft EM in Bayesian Network Learning from Incomplete Data](https://mlsanity.com/paper/2012.05269/) (2020)
- [A Smooth Transition Between Induction and Deduction: Fast Abductive Learning Based on Probabilistic Symbol Perception](https://mlsanity.com/paper/2502.12919/) (2025)

## Abstract (arXiv)

Neurosymbolic (NeSy) models integrate neural networks and symbolic reasoning for robust and interpretable AI. State-of-the-art NeSy models require that the symbolic component is expressed in a differentiable way, often complicating the use of approximate inference. We propose EM-NeSy which casts probabilistic NeSy learning as an instance of the Expectation-Maximization (EM) algorithm. In the expectation step, we compute the posterior over the neurally predicted symbols conditioned on the label via probabilistic inference. In the maximization step, we update the neural parameters based on this posterior using gradient descent only through the neural component. This formulation unlocks the full potential of the EM algorithm for NeSy learning. It allows NeSy to extend naturally to approximate reasoning without any additional modifications or differentiability requirements of the symbolic component. Furthermore, it recovers the standard end-to-end gradient-based NeSy setting under exact inference. Our experimental results demonstrate the scalability and computational efficiency of EM-NeSy.