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Chain and Causal Attention for Efficient Entity Tracking

Erwan Fagnou, Paul Caillon, Blaise Delattre, Alexandre Allauzen

TL;DR

This work identifies a fundamental limit for Transformer-based entity tracking: to handle $n$ state changes, a decoder-only Transformer requires at least $L_{\ extmin}$ layers with $L_{\ extmin} = \lceil \log_2(\text{depth}(\mathcal{G}) + 1)\rceil$. It then introduces ChaCAL, an attention variant that treats the attention matrix as an adjacency matrix and uses a fixed-point, closed-form update $\mathbf{Y} = (1-\gamma) \mathbf{A} (I - \gamma \mathbf{A})^{-1} \mathbf{V}$ to capture long-range dependencies within a single layer. Through toy, Boxes, and language-modeling experiments, ChaCAL achieves near-perfect entity-tracking performance with far fewer layers, while maintaining competitive results on standard pre-training tasks. The findings underscore the potential for task-specific attention designs to dramatically reduce computational requirements for structured reasoning while highlighting directions for pre-training and broader-domain evaluation.

Abstract

This paper investigates the limitations of transformers for entity-tracking tasks in large language models. We identify a theoretical constraint, showing that transformers require at least $\log_2 (n+1)$ layers to handle entity tracking with $n$ state changes. To address this issue, we propose an efficient and frugal enhancement to the standard attention mechanism, enabling it to manage long-term dependencies more efficiently. By considering attention as an adjacency matrix, our model can track entity states with a single layer. Empirical results demonstrate significant improvements in entity tracking datasets while keeping competitive performance on standard natural language modeling. Our modified attention allows us to achieve the same performance with drastically fewer layers. Additionally, our enhanced mechanism reveals structured internal representations of attention. Extensive experiments on both toy and complex datasets validate our approach. Our contributions include theoretical insights, an improved attention mechanism, and empirical validation.

Chain and Causal Attention for Efficient Entity Tracking

TL;DR

This work identifies a fundamental limit for Transformer-based entity tracking: to handle state changes, a decoder-only Transformer requires at least layers with . It then introduces ChaCAL, an attention variant that treats the attention matrix as an adjacency matrix and uses a fixed-point, closed-form update to capture long-range dependencies within a single layer. Through toy, Boxes, and language-modeling experiments, ChaCAL achieves near-perfect entity-tracking performance with far fewer layers, while maintaining competitive results on standard pre-training tasks. The findings underscore the potential for task-specific attention designs to dramatically reduce computational requirements for structured reasoning while highlighting directions for pre-training and broader-domain evaluation.

Abstract

This paper investigates the limitations of transformers for entity-tracking tasks in large language models. We identify a theoretical constraint, showing that transformers require at least layers to handle entity tracking with state changes. To address this issue, we propose an efficient and frugal enhancement to the standard attention mechanism, enabling it to manage long-term dependencies more efficiently. By considering attention as an adjacency matrix, our model can track entity states with a single layer. Empirical results demonstrate significant improvements in entity tracking datasets while keeping competitive performance on standard natural language modeling. Our modified attention allows us to achieve the same performance with drastically fewer layers. Additionally, our enhanced mechanism reveals structured internal representations of attention. Extensive experiments on both toy and complex datasets validate our approach. Our contributions include theoretical insights, an improved attention mechanism, and empirical validation.
Paper Structure (48 sections, 1 theorem, 8 equations, 8 figures, 7 tables)

This paper contains 48 sections, 1 theorem, 8 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Self-attention layer
  4. Multi-hop attention
  5. Entity tracking
  6. Entity tracking and interpretability
  7. Emergence of entity tracking abilities
  8. Theoretical limits of transformers for entity tracking
  9. Entity tracking as a graph traversal task
  10. Representation as a computational graph
  11. A theoretical limitation of transformers
  12. Method
  13. Enhanced attention
  14. Adjacency matrix perspective
  15. Chain and Causal Attention Layer (ChaCAL)
  16. ...and 33 more sections

Key Result

Theorem 1

Given an entity tracking task instance, let $\mathcal{G}$ be its corresponding computational graph as defined in Section sec:graph_definition. We assume that each attention layer has a receptive field equal to 1 in the computational graph In other words, a layer cannot make the connection between tw where we define $\text{depth}(\mathcal{G})$ as the length of its longest path.

Figures (8)

  • Figure 2: Illustration of how a standard transformer can process a sequence containing chained dependencies. Red arrows represent the reference to a previous node, and black lines show attention connections. With $8$ nodes (represented as colored circles), $log_2(8) = 3$ attention layers are needed to process and gather all the information.
  • Figure 3: Test accuracy of transformer models during training on our toy dataset. The standard transformer architecture struggles to learn the task and needs 4 to 5 layers to reach 100%, while our enhanced attention consistently solves the task with only one layer. We show the average, min and max values over 4 runs for each model.
  • Figure 4: Exact match rate on the test set of each model during training on the advanced version of the boxes dataset.
  • Figure 5: Impact of $\gamma$ over performance and convergence. Accuracy is shown in blue, and the number of epochs to reach a perfect accuracy is in red.
  • Figure :
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1