Composing Global Solutions to Reasoning Tasks via Algebraic Objects in Neural Nets

We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and $L_2$ loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables analytical construction of global optimal solutions from partial solutions that only satisfy part of the loss, despite its high nonlinearity. We coin the framework as CoGS (Composing Global Solutions). Specifically, we show that the weight space over different numbers of hidden nodes of the 2-layer network is equipped with a semi-ring algebraic structure, and the loss function to be optimized consists of sum potentials, which are ring homomorphisms, allowing partial solutions to be composed into global ones by ring addition and multiplication. Our experiments show that around $95\%$ of the solutions obtained by gradient descent match exactly our theoretical constructions. Although the global solutions constructed only required a small number of hidden nodes, our analysis on gradient dynamics shows that overparameterization asymptotically decouples training dynamics and is beneficial. We further show that training dynamics favors simpler solutions under weight decay, and thus high-order global solutions such as perfect memorization are unfavorable. The code is open sourced at https://github.com/facebookresearch/luckmatters/tree/yuandong3/ssl/real-dataset.