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Linear attention is (maybe) all you need (to understand transformer optimization)

Kwangjun Ahn, Xiang Cheng, Minhak Song, Chulhee Yun, Ali Jadbabaie, Suvrit Sra

TL;DR

The paper investigates why Transformer training is difficult and proposes a minimal linear attention model as an abstraction that reproduces known optimization phenomena. By training a shallow linear Transformer on random linear regression tasks, the authors demonstrate the model captures heavy-tailed gradient noise, ill-conditioned landscapes, directional and generalized smoothness, and the advantage of adaptive optimizers over SGD. They further analyze the impact of data distribution and network depth, showing these features intensify with heavier-tailed data and more layers. The work provides a practical, tractable framework for theory development and optimization algorithm design for Transformers.

Abstract

Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet canonical linearized shallow Transformer model. Specifically, we train linear Transformers to solve regression tasks, inspired by J.~von Oswald et al.~(ICML 2023), and K.~Ahn et al.~(NeurIPS 2023). Most importantly, we observe that our proposed linearized models can reproduce several prominent aspects of Transformer training dynamics. Consequently, the results obtained in this paper suggest that a simple linearized Transformer model could actually be a valuable, realistic abstraction for understanding Transformer optimization.

Linear attention is (maybe) all you need (to understand transformer optimization)

TL;DR

The paper investigates why Transformer training is difficult and proposes a minimal linear attention model as an abstraction that reproduces known optimization phenomena. By training a shallow linear Transformer on random linear regression tasks, the authors demonstrate the model captures heavy-tailed gradient noise, ill-conditioned landscapes, directional and generalized smoothness, and the advantage of adaptive optimizers over SGD. They further analyze the impact of data distribution and network depth, showing these features intensify with heavier-tailed data and more layers. The work provides a practical, tractable framework for theory development and optimization algorithm design for Transformers.

Abstract

Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet canonical linearized shallow Transformer model. Specifically, we train linear Transformers to solve regression tasks, inspired by J.~von Oswald et al.~(ICML 2023), and K.~Ahn et al.~(NeurIPS 2023). Most importantly, we observe that our proposed linearized models can reproduce several prominent aspects of Transformer training dynamics. Consequently, the results obtained in this paper suggest that a simple linearized Transformer model could actually be a valuable, realistic abstraction for understanding Transformer optimization.
Paper Structure (17 sections, 6 equations, 16 figures, 2 tables)

This paper contains 17 sections, 6 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Distinctive features of Transformer optimization
  3. A whirlwind tour of (convex) optimization theory
  4. Heavy-tailed gradient noise zhang2020adaptivekunstner2023noise
  5. Ill-conditioned landscape jiang2022does
  6. Directional Smoothness pan2023toward
  7. Generalized smoothness zhang2019gradient
  8. Linear shallow Transformers have the same loss landscape as practical deep Transformers
  9. Linear Transformer on linear regression
  10. Linear Transformers as a fruitful abstraction
  11. Understanding features based on linear Transformers
  12. Effect of data distribution
  13. Effect of more layers
  14. Conclusion
  15. Hyperparameters for the experiments
  16. ...and 2 more sections

Figures (16)

  • Figure 1: Adaptive optimization methods like Adam are much more effective than SGD for training Transformers. This experimental result is taken from kunstner2023noise. (+m) denotes "with momentum".
  • Figure 2: For Transformer optimization, adaptive methods like Adam are strictly better than SGD. (+m) denotes "with momentum" and (-m) denotes without momentum. Our plots only show the momentum variants of SGD and Adam as they perform better in all cases. Left 3 plots: Full Transformers, from kunstner2023noise. Right 3 plots: Shallow linear Transformers (see Settings 1, 2, and 3 from Table \ref{['table:setting']}).
  • Figure 3: The stochastic gradient noise is heavy-tailed for Transformer optimization. The top-right corner of each plot is the quantile-quantile (q-q) plot between the histogram ($y$-axis) and its best fit Gaussian ($x$-axis). The q-q plot is above the $y=x$ line toward the right, showing its heavy-tailedness. Left 3 plots: Full Transformers, from kunstner2023noise. Right 3 plots: Shallow linear Transformers (see Settings 1, 2, and 3 from \ref{['table:setting']}).
  • Figure 4: The comparison of the robust condition number (see \ref{['jiang2022does']}) between SGD and Adam for Transformer optimization. Numbers in parentheses show standard deviation. Left table: Full Transformers, from jiang2022does. Right table: Shallow linear Transformers, see \ref{['table:setting']}.
  • Figure 5: $\log$(directional smoothness) against iteration (see \ref{['pan2023toward']}) for shallow linear Transformers (see Settings 1, 2, 3 from \ref{['table:setting']}).
  • ...and 11 more figures

Theorems & Definitions (1)

  • Definition 1