ALPINE: Unveiling The Planning Capability of Autoregressive Learning in Language Models

Planning is a crucial element of both human intelligence and contemporary large language models (LLMs). In this paper, we initiate a theoretical investigation into the emergence of planning capabilities in Transformer-based LLMs via their next-word prediction mechanisms. We model planning as a network path-finding task, where the objective is to generate a valid path from a specified source node to a designated target node. Our mathematical characterization shows that Transformer architectures can execute path-finding by embedding the adjacency and reachability matrices within their weights. Furthermore, our theoretical analysis of gradient-based learning dynamics reveals that LLMs can learn both the adjacency and a limited form of the reachability matrices. These theoretical insights are then validated through experiments, which demonstrate that Transformer architectures indeed learn the adjacency and an incomplete reachability matrices, consistent with our theoretical predictions.