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Representational Homomorphism Predicts and Improves Compositional Generalization In Transformer Language Model

Zhiyu An, Wan Du

TL;DR

This work tackles the persistent challenge of compositional generalization in neural models by formalizing compositionality as an approximate homomorphism and introducing Homomorphism Error (HE) as a task-independent diagnostic of internal representations. It demonstrates that HE tracks out-of-distribution compositional generalization in controlled SCAN-style experiments and shows that HE-regularized training causally improves OOD performance under noise, with a significant predictive link between HE and OOD accuracy ($R^2=0.73$). The study provides both a diagnostic tool and a training-time intervention, suggesting that shaping representation structure can enhance generalization to novel expressions. While conducted on synthetic data with decoder-only Transformers, the results point to a promising direction for composition-aware training and potential extensions to established compositional benchmarks and larger scale models.

Abstract

Compositional generalization-the ability to interpret novel combinations of familiar components-remains a persistent challenge for neural networks. Behavioral evaluations reveal when models fail but offer limited insight into why failures arise at the representational level. We introduce Homomorphism Error (HE), a structural metric that quantifies deviations from approximate homomorphisms between the expression algebra and a model's hidden-state space. We instantiate HE for two compositional operators in SCAN-style tasks: modifier HE for unary composition and sequence HE for binary composition, measured by learning representation-level operators that predict composed representations from their parts. Across controlled experiments with small decoder-only Transformers, HE predicts out-of-distribution (OOD) compositional generalization under noise injection, achieving R^2 = 0.73 correlation between modifier HE and OOD accuracy. Ablations show that model depth has minimal effect on either HE or OOD accuracy, training data coverage exhibits threshold effects (insufficient coverage sharply increases HE and degrades OOD performance), and randomly inserted noise tokens systematically increase HE. Finally, we test if HE-regularized training improves OOD accuracy. Experiment shows that explicitly enforcing low modifier HE during training significantly reduces modifier HE (p = 1.1x10-4) and sequence HE (p = 0.001) and yields a statistically significant improvement in OOD accuracy (p = 0.023). Together, these results indicate the potential of HE to be both a diagnostic and an actionable training signal for improving compositional generalization. Code to reproduce our experiments is open-sourced.

Representational Homomorphism Predicts and Improves Compositional Generalization In Transformer Language Model

TL;DR

This work tackles the persistent challenge of compositional generalization in neural models by formalizing compositionality as an approximate homomorphism and introducing Homomorphism Error (HE) as a task-independent diagnostic of internal representations. It demonstrates that HE tracks out-of-distribution compositional generalization in controlled SCAN-style experiments and shows that HE-regularized training causally improves OOD performance under noise, with a significant predictive link between HE and OOD accuracy (). The study provides both a diagnostic tool and a training-time intervention, suggesting that shaping representation structure can enhance generalization to novel expressions. While conducted on synthetic data with decoder-only Transformers, the results point to a promising direction for composition-aware training and potential extensions to established compositional benchmarks and larger scale models.

Abstract

Compositional generalization-the ability to interpret novel combinations of familiar components-remains a persistent challenge for neural networks. Behavioral evaluations reveal when models fail but offer limited insight into why failures arise at the representational level. We introduce Homomorphism Error (HE), a structural metric that quantifies deviations from approximate homomorphisms between the expression algebra and a model's hidden-state space. We instantiate HE for two compositional operators in SCAN-style tasks: modifier HE for unary composition and sequence HE for binary composition, measured by learning representation-level operators that predict composed representations from their parts. Across controlled experiments with small decoder-only Transformers, HE predicts out-of-distribution (OOD) compositional generalization under noise injection, achieving R^2 = 0.73 correlation between modifier HE and OOD accuracy. Ablations show that model depth has minimal effect on either HE or OOD accuracy, training data coverage exhibits threshold effects (insufficient coverage sharply increases HE and degrades OOD performance), and randomly inserted noise tokens systematically increase HE. Finally, we test if HE-regularized training improves OOD accuracy. Experiment shows that explicitly enforcing low modifier HE during training significantly reduces modifier HE (p = 1.1x10-4) and sequence HE (p = 0.001) and yields a statistically significant improvement in OOD accuracy (p = 0.023). Together, these results indicate the potential of HE to be both a diagnostic and an actionable training signal for improving compositional generalization. Code to reproduce our experiments is open-sourced.
Paper Structure (37 sections, 13 equations, 7 figures, 1 algorithm)

This paper contains 37 sections, 13 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Compositional Generalization Benchmarks.
  4. Theoretical Frameworks.
  5. Architectural Solutions.
  6. Internal Representation Analysis.
  7. Homomorphism Error as Structural Metric for Compositionality
  8. Compositionality as homomorphism.
  9. Approximate homomorphism in language models.
  10. HE-Regularized Training for Causal Compositionality
  11. Training Objective
  12. Modifier HE Regularizer
  13. Efficient HE-Regularized Training
  14. Interpretation
  15. Experiments
  16. ...and 22 more sections

Figures (7)

  • Figure 1: Compositionality as approximate homomorphism. Solid arrows form the representation-level square; dashed arrows show semantic interpretation $\llbracket\cdot\rrbracket$ and an evaluation/readout $I$. Homomorphism Error (HE) quantifies how well $\Phi$ preserves composition.
  • Figure 2: Model size ablation results. Lines represent number of transformer layers $L \in \{1, 2, \dots, 10\}$ in the language model.
  • Figure 3: Training data sparsity results. Lines represent number of primitives in the expressions that the training set contains up-to. $4$ meaning the training set contains expressions with up-to $4$ primitives.
  • Figure 4: Noise injection results. Lines represent number of noise tokens inserted in training data.
  • Figure 5: Analysis result of the noise injection experiment. Correlation between mean OOD generalization accuracy and mean modifier HE is shown. Polynomial regression with various degrees is conducted and $R^2$ is reported.
  • ...and 2 more figures