# A mathematical theory of balancing relational generalization and memorization

arXiv: https://arxiv.org/abs/2605.22972 · HTML: https://mlsanity.com/paper/2605.22972/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 51 of its 116 references that are in the map.
- First posted on arXiv 21 May 2026.

## Builds on (most cited first, 30 of 51)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Measuring and Narrowing the Compositionality Gap in Language Models](https://mlsanity.com/paper/2210.03350/) (2022)
- [Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution](https://mlsanity.com/paper/2202.10054/) (2022)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [The Reversal Curse: LLMs trained on "A is B" fail to learn "B is A"](https://mlsanity.com/paper/2309.12288/) (2023)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Measuring Compositional Generalization: A Comprehensive Method on Realistic Data](https://mlsanity.com/paper/1912.09713/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Compositionality decomposed: how do neural networks generalise?](https://mlsanity.com/paper/1908.08351/) (2019)
- [Language models show human-like content effects on reasoning tasks](https://mlsanity.com/paper/2207.07051/) (2022)
- [Do Large Language Models Latently Perform Multi-Hop Reasoning?](https://mlsanity.com/paper/2402.16837/) (2024)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Benign overfitting in ridge regression](https://mlsanity.com/paper/2009.14286/) (2020)
- [A Kernel-Based View of Language Model Fine-Tuning](https://mlsanity.com/paper/2210.05643/) (2022)
- [Optimal Regularization Can Mitigate Double Descent](https://mlsanity.com/paper/2003.01897/) (2020)
- [Revisiting Catastrophic Forgetting in Large Language Model Tuning](https://mlsanity.com/paper/2406.04836/) (2024)
- [An analytic theory of generalization dynamics and transfer learning in deep linear networks](https://mlsanity.com/paper/1809.10374/) (2018)
- [Generalization on the Unseen, Logic Reasoning and Degree Curriculum](https://mlsanity.com/paper/2301.13105/) (2023)
- [Compositional Generalization from First Principles](https://mlsanity.com/paper/2307.05596/) (2023)
- [Gradient descent follows the regularization path for general losses](https://mlsanity.com/paper/2006.11226/) (2020)

## Abstract (arXiv)

Humans, animals, and modern machine learning models exhibit impressive abilities to learn complex behaviors and generalize these behaviors to unseen situations. This ability requires us to learn rules and regularities that allow for such generalizations. At the same time, in most complex environments, any rule will have its exceptions. How do learning systems balance between learning general regularities and memorizing exceptions? We argue that a lack of task paradigms has hindered the study of this essential ability. To address this gap, we introduce a novel task, transitive inference with exceptions, that tests for relational generalization and memorization of an exception to the relational rule. We then analytically characterize the behavior of a simple, theoretically tractable model of neural network learning (kernel ridge regression) across a broad family of representations and task parameters. We find that these models can balance between relational generalization and memorization, but unlike for transitive inference without an exception, successful generalization is sensitive to the specific representational geometry. We explain why this task is more challenging mechanistically by drawing on our analytical theory. Finally, we validate our theoretical insights in pretrained language models that are finetuned on ordered relations, finding that these models successfully generalize according to the transitive rule, but also make the kinds of systematic mistakes predicted by our theory. Overall, our theory shows how learning systems can balance between relational generalization and memorization, explains how this can go wrong, and emphasizes the need for new task paradigms designed to probe this ability.