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Beyond Parallelism: Synergistic Computational Graph Effects in Multi-Head Attention

Haitz Sáez de Ocáriz Borde

TL;DR

The paper tackles whether multi-head attention provides benefits beyond simple parallelism by reframing each head as a synergistic feedforward DAG with a common sink. It develops a graph-theoretic framework to analyze information propagation via mixing time, showing $T_{\mathrm{mix}}(\overline{W},\epsilon) \lesssim \frac{2N}{p}$ with $N = n-1$ and $p = \sum_h \alpha_h p_h$, and studies minimax fidelity through diffusion matrices $\Delta^{(h)}$ and the multi-head operator $\overline{\Delta}$. Theoretical results indicate adaptive head weighting can match the fastest head's mixing time and that cross-head interactions can amplify fidelity beyond any single head, a claim supported by empirical results on toy sequence tasks with the same parameter budget. Overall, the work provides interpretable metrics for attention dynamics, informing adaptive weighting and pruning strategies while offering code to reproduce the experiments.

Abstract

Multi-head attention powers Transformer networks, the primary deep learning architecture behind the success of large language models (LLMs). Yet, the theoretical advantages of multi-head versus single-head attention, beyond mere parallel processing, remain underexplored. In this paper, we reframe multi-head attention as a system of potentially synergistic computational graphs, where each head functions as a feedforward directed acyclic graph (DAG) with a common sink state. We provide intuition and preliminary theoretical analysis of mixing time and minimax fidelity in this framework. Our results show that multi-head attention can synergistically enhance information propagation, yielding faster mixing times and minimax fidelity amplification under specific head-diversity conditions. Finally, we train single-head and multi-head Transformers, each with the same total number of parameters, on sequence manipulation tasks and empirically verify the predicted effects. The code is available at https://github.com/haitzsaezdeocariz/beyondparallelism.

Beyond Parallelism: Synergistic Computational Graph Effects in Multi-Head Attention

TL;DR

The paper tackles whether multi-head attention provides benefits beyond simple parallelism by reframing each head as a synergistic feedforward DAG with a common sink. It develops a graph-theoretic framework to analyze information propagation via mixing time, showing with and , and studies minimax fidelity through diffusion matrices and the multi-head operator . Theoretical results indicate adaptive head weighting can match the fastest head's mixing time and that cross-head interactions can amplify fidelity beyond any single head, a claim supported by empirical results on toy sequence tasks with the same parameter budget. Overall, the work provides interpretable metrics for attention dynamics, informing adaptive weighting and pruning strategies while offering code to reproduce the experiments.

Abstract

Multi-head attention powers Transformer networks, the primary deep learning architecture behind the success of large language models (LLMs). Yet, the theoretical advantages of multi-head versus single-head attention, beyond mere parallel processing, remain underexplored. In this paper, we reframe multi-head attention as a system of potentially synergistic computational graphs, where each head functions as a feedforward directed acyclic graph (DAG) with a common sink state. We provide intuition and preliminary theoretical analysis of mixing time and minimax fidelity in this framework. Our results show that multi-head attention can synergistically enhance information propagation, yielding faster mixing times and minimax fidelity amplification under specific head-diversity conditions. Finally, we train single-head and multi-head Transformers, each with the same total number of parameters, on sequence manipulation tasks and empirically verify the predicted effects. The code is available at https://github.com/haitzsaezdeocariz/beyondparallelism.

Paper Structure

This paper contains 17 sections, 3 theorems, 25 equations, 4 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multi-Head Mixing Time
  3. Multi-Head Stationary Distributions
  4. Having Multiple Heads can Improve Mixing Time
  5. Multi-Head Minimax Fidelity
  6. Having Multiple Heads can Amplify Fidelity
  7. Experimental Validation
  8. Data Generation
  9. Model Architecture and Training
  10. Mixing Time Proxy Estimation
  11. Minimax Fidelity Proxy Estimation
  12. Conclusion
  13. Proofs
  14. Example \ref{['ex:example']} temporal dynamics discussion
  15. Compute Resources
  16. ...and 2 more sections

Key Result

lemma 1

Let $G$ be a feedforward graph on $n$ vertices with a unique sink $\tau$. Then the only stationary distribution for the random walk matrix $W$ is $1_{\tau}$, the distribution taking value $1$ at $\tau$ and $0$ elsewhere what.

Figures (4)

  • Figure 1: Example feedforward DAG with $n=5$ nodes.
  • Figure 2: Multi-head sink visualization.
  • Figure 3: Diffusion of signal from nodes $u,v,w$ to the sink $\tau$ under single-head and multi-head diffusion kernels. Solid lines show signal arrival percentages over diffusion steps, while the dashed line $\phi_{\min}$ indicates the cumulative fidelity.
  • Figure 4: Mixing time and fidelity for Transformer model trained on synthetic sequence manipulation tasks.

Theorems & Definitions (24)

  • definition 1: Feedforward graph
  • definition 2: Unique sink
  • definition 3: Random walk matrix
  • definition 4: Stationary distribution
  • lemma 1: Stationary Distribution for a Single-Head Unique Sink
  • lemma 2: Stationary Distribution for a Multi-Head Unique Sink
  • proof
  • definition 5: Mixing Time
  • theorem 1: Multi-Head Mixing Time Bound via Forward Moves
  • proof
  • ...and 14 more