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

Siwei Wang, Yifei Shen, Shi Feng, Haoran Sun, Shang-Hua Teng, Wei Chen

TL;DR

This paper initiates a theoretical investigation into the emergence of planning capabilities in Transformer-based LLMs via their next-word prediction mechanisms, and shows that Transformer architectures can execute path-finding by embedding the adjacency and reachability matrices within their weights.

Abstract

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. When applying our methodology to the real-world planning benchmark Blocksworld, our observations remain consistent. Additionally, our analyses uncover a fundamental limitation of current Transformer architectures in path-finding: these architectures cannot identify reachability relationships through transitivity, which leads to failures in generating paths when concatenation is required. These findings provide new insights into how the internal mechanisms of autoregressive learning facilitate intelligent planning and deepen our understanding of how future LLMs might achieve more advanced and general planning-and-reasoning capabilities across diverse applications.

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

TL;DR

This paper initiates a theoretical investigation into the emergence of planning capabilities in Transformer-based LLMs via their next-word prediction mechanisms, and shows that Transformer architectures can execute path-finding by embedding the adjacency and reachability matrices within their weights.

Abstract

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. When applying our methodology to the real-world planning benchmark Blocksworld, our observations remain consistent. Additionally, our analyses uncover a fundamental limitation of current Transformer architectures in path-finding: these architectures cannot identify reachability relationships through transitivity, which leads to failures in generating paths when concatenation is required. These findings provide new insights into how the internal mechanisms of autoregressive learning facilitate intelligent planning and deepen our understanding of how future LLMs might achieve more advanced and general planning-and-reasoning capabilities across diverse applications.
Paper Structure (28 sections, 4 theorems, 24 equations, 13 figures, 1 table, 1 algorithm)

This paper contains 28 sections, 4 theorems, 24 equations, 13 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Autoregressive Transformer Architecture and Loss Function
  4. Path-Planning Dataset: Syntax and Data Sources
  5. Expressiveness and Learning Dynamics of Transformer Models
  6. Expressiveness
  7. Learning Dynamics
  8. Learning the adjacency matrix.
  9. Learning the reachability matrix.
  10. Predicting the next node on a path.
  11. Empirical Evaluations: Peeking into a Trained Transformer
  12. Accuracy on Test Dataset
  13. Peeking into a Trained Transformer
  14. Attention.
  15. Adjacency Matrix.
  16. ...and 13 more sections

Key Result

Theorem 2

Given a graph $\mathcal{G}$ (with adjacency matrix $\bm{A}^{\text{\rm true}}$ and reachability matrix $\bm{R}^{\text{\rm true}}$), for every $\varepsilon>0$, there exists a $1$-layer, $1$-head, and $O(|\mathcal{V}|)$-embedding-size Transformer model that generates a valid path for every valid source

Figures (13)

  • Figure 1: Empirical verification regarding the learning of the adjacency matrix.
  • Figure 2: Empirical verification regarding the learning of the observed reachability matrix.
  • Figure 3: Accuracy on the test datasets with embedding size $d= 120$.
  • Figure 4: The average attention in the 1-layer and 1-head Transformers.
  • Figure 5: The first 20 rows and columns of $\bm{W}^{M'}$ (the red boxes correspond to $1$'s in the adjacency matrix), and the average weight gap between edge terms and non-edge terms in $\bm{W}^{M'}$.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Theorem 2
  • Theorem 3
  • proof : Proof Sketch.
  • Theorem 3
  • proof
  • Theorem 3
  • proof