When does compositional structure yield compositional generalization? A kernel theory

TL;DR

This paper develops a kernel theory to understand when compositional structure leads to generalization, revealing fundamental limitations and failure modes in models like neural networks, with empirical validation on deep learning architectures.

Contribution

It introduces a theoretical framework for compositional generalization in kernel models, identifying key limitations and failure modes, and validates these findings empirically on neural networks.

Findings

Kernel models are limited to conjunction-wise additive functions.

They cannot transitively generalize equivalence relations.

Training data biases cause memorization leak and shortcut bias.

Abstract

Compositional generalization (the ability to respond correctly to novel combinations of familiar components) is thought to be a cornerstone of intelligent behavior. Compositionally structured (e.g. disentangled) representations support this ability; however, the conditions under which they are sufficient for the emergence of compositional generalization remain unclear. To address this gap, we present a theory of compositional generalization in kernel models with fixed, compositionally structured representations. This provides a tractable framework for characterizing the impact of training data statistics on generalization. We find that these models are limited to functions that assign values to each combination of components seen during training, and then sum up these values ("conjunction-wise additivity"). This imposes fundamental restrictions on the set of tasks compositionally structured kernel models can learn, in particular preventing them from transitively generalizing equivalence relations. Even for compositional tasks that they can learn in principle, we identify novel failure modes in compositional generalization (memorization leak and shortcut bias) that arise from biases in the training data. Finally, we empirically validate our theory, showing that it captures the behavior of deep neural networks (convolutional networks, residual networks, and Vision Transformers) trained on a set of compositional tasks with similarly structured data. Ultimately, this work examines how statistical structure in the training data can affect compositional generalization, with implications for how to identify and remedy failure modes in deep learning models.