When does compositional structure yield compositional generalization? A kernel theory

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 are essential for this; however, the conditions under which they yield compositional generalization remain unclear. To address this gap, we present a general theory of compositional generalization in kernel models with fixed, potentially nonlinear representations (which also applies to neural networks in the "lazy regime"). We prove that these models are functionally limited to adding up values assigned to conjunctions/combinations of components that have been seen during training ("conjunction-wise additivity"), and identify novel compositionality failure modes that arise from the data and model structure, even for disentangled inputs. For models in the representation learning (or "rich") regime, we show that networks can generalize on an important non-additive task (associative inference), and give a mechanistic explanation for why. Finally, we validate our theory empirically, showing that it captures the behavior of deep neural networks trained on a set of compositional tasks. In sum, our theory characterizes the principles giving rise to compositional generalization in kernel models and shows how representation learning can overcome their limitations. We further provide a formally grounded, novel generalization class for compositional tasks that highlights fundamental differences in the required learning mechanisms (conjunction-wise additivity).

Implicit regularization of multi-task learning and finetuning in overparameterized neural networks

It is common in deep learning to train networks on auxiliary tasks with the expectation that the learning will transfer, at least partially, to another task of interest. In this work, we investigate the inductive biases that result from learning auxiliary tasks, either simultaneously (multi-task learning, MTL) or sequentially (pretraining and subsequent finetuning, PT+FT). In the simplified setting of two-layer diagonal linear networks trained with gradient descent, we identify implicit regularization penalties associated with MTL and PT+FT, both of which incentivize feature sharing between tasks and sparsity in learned task-specific features. Notably, our results imply that during finetuning, networks operate in a hybrid of the kernel (or "lazy") regime and the feature learning ("rich") regime identified in prior work. Moreover, PT+FT can exhibit a novel "nested feature learning" behavior not captured by either regime, which biases it to extract a sparse subset of the features learned during pretraining. In ReLU networks, we reproduce all of these qualitative behaviors. We also observe that PT+FT (but not MTL) is biased to learn features that are correlated with (but distinct from) those needed for the auxiliary task, while MTL is biased toward using identical features for both tasks. As a result, we find that in realistic settings, MTL generalizes better when comparatively little data is available for the task of interest, while PT+FT outperforms it with more data available. We show that our findings hold qualitatively for a deep architecture trained on image classification tasks. Our characterization of the nested feature learning regime also motivates a modification to PT+FT that we find empirically improves performance. Overall, our results shed light on the impact of auxiliary task learning and suggest ways to leverage it more effectively.

The Implicit Bias of Gradient Descent on Generalized Gated Linear Networks

Understanding the asymptotic behavior of gradient-descent training of deep neural networks is essential for revealing inductive biases and improving network performance. We derive the infinite-time training limit of a mathematically tractable class of deep nonlinear neural networks, gated linear networks (GLNs), and generalize these results to gated networks described by general homogeneous polynomials. We study the implications of our results, focusing first on two-layer GLNs. We then apply our theoretical predictions to GLNs trained on MNIST and show how architectural constraints and the implicit bias of gradient descent affect performance. Finally, we show that our theory captures a substantial portion of the inductive bias of ReLU networks. By making the inductive bias explicit, our framework is poised to inform the development of more efficient, biologically plausible, and robust learning algorithms.