Understanding the internal representations learned by neural networks is a cornerstone challenge in the science of machine learning. While there have been significant recent strides in some cases towards understanding how neural networks implement specific target functions, this paper explores a complementary question -- why do networks arrive at particular computational strategies? Our inquiry focuses on the algebraic learning tasks of modular addition, sparse parities, and finite group operations. Our primary theoretical findings analytically characterize the features learned by stylized neural networks for these algebraic tasks. Notably, our main technique demonstrates how the principle of margin maximization alone can be used to fully specify the features learned by the network. Specifically, we prove that the trained networks utilize Fourier features to perform modular addition and employ features corresponding to irreducible group-theoretic representations to perform compositions in general groups, aligning closely with the empirical observations of Nanda et al. and Chughtai et al. More generally, we hope our techniques can help to foster a deeper understanding of why neural networks adopt specific computational strategies.

翻译：理解神经网络学习到的内部表征是机器学习科学中的核心挑战。尽管近年来在某些情况下对神经网络如何实现特定目标函数取得了显著进展，但本文探讨了一个互补性问题——为何网络会采用特定的计算策略？我们的研究聚焦于模加法、稀疏奇偶性和有限群运算等代数学习任务。我们的主要理论发现对简化神经网络在这些代数任务中学习到的特征进行了分析性刻画。值得注意的是，我们的核心技术表明，仅凭最大化边界原则即可完全确定网络学习到的特征。具体而言，我们证明训练后的网络利用傅里叶特征执行模加法，并采用与群论不可约表示对应的特征来执行一般群中的复合运算，这与Nanda等人及Chughtai等人的实证观察高度吻合。更广泛而言，我们希望我们的技术能够促进对神经网络为何采用特定计算策略的更深入理解。