Generalization on the Unseen, Logic Reasoning and Degree Curriculum
It addresses out-of-distribution generalization for logical reasoning tasks, offering theoretical insights and a practical algorithm, though it is incremental in extending existing concepts to specific architectures.
The paper investigates how neural networks generalize to unseen logical functions, showing that models like Transformers learn a minimal-degree interpolator on out-of-distribution data, which explains length generalization in Boolean tasks and leads to a curriculum learning method for more efficient monomial learning.
This paper considers the learning of logical (Boolean) functions with a focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for sparse functions and a class of network models including instances of Transformers, random features models, and linear networks, a min-degree-interpolator is learned on the unseen. More specifically, this means an interpolator of the training data that has minimal Fourier mass on the higher degree basis elements. These findings lead to two implications: (1) we provide an explanation to the length generalization problem for Boolean functions (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports. Finally, we discuss extensions to other models or non-sparse regimes where the min-degree bias may still occur or fade, as well as how it can be potentially corrected when undesirable.