A Learnability Analysis on Neuro-Symbolic Learning

This paper analyzes the learnability of neuro-symbolic (NeSy) tasks within hybrid systems. We show that the learnability of NeSy tasks can be characterized by their derived constraint satisfaction problems (DCSPs). Specifically, a task is learnable if the corresponding DCSP has a unique solution; otherwise, it is unlearnable. For learnable tasks, we establish error bounds by exploiting the clustering property of the hypothesis space. Additionally, we analyze the asymptotic error for general NeSy tasks, showing that the expected error scales with the disagreement among solutions. Our results offer a principled approach to determining learnability and provide insights into the design of new algorithms.