🤖 AI Summary
This work investigates the data conditions and theoretical limits under which structured representations—particularly disentangled representations—enable compositional generalization. We introduce “conjunctive additivity,” a novel statistical constraint that formally characterizes learnability for compositional tasks: kernel methods achieve unbiased compositional generalization if and only if training data satisfy this condition. Our theoretical analysis identifies two fundamental failure modes—memory leakage and shortcut bias—both arising from structural deficiencies in the data distribution. We extend the theory to deep neural networks and empirically validate it across CNNs, ResNets, and Vision Transformers, demonstrating that compositional generalization success strictly hinges on whether training data meet conjunctive additivity. The results provide a unified explanation for diverse compositional generalization phenomena and establish a principled foundation for constructing generalization-friendly training datasets.
📝 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.