Attention is not all you need: pure attention loses rank doubly exponentially with depth
Yihe Dong, Jean-Baptiste Cordonnier, Andreas Loukas
Abstract
Attention-based architectures have become ubiquitous in machine learning. Yet, our understanding of the reasons for their effectiveness remains limited. This work proposes a new way to understand self-attention networks: we show that their output can be decomposed into a sum of smaller terms—or paths—each involving the operation of a sequence of attention heads across layers. Using this path decomposition, we prove that self-attention possesses a strong inductive bias towards "token uniformity". Specifically, without skip connections or multi-layer perceptrons (MLPs), the output converges doubly exponentially to a rank-1 matrix. On the other hand, skip connections and MLPs stop the output from degeneration. Our experiments verify the convergence results on standard transformer architectures.
BibTeX
@InProceedings{pmlr-v139-dong21a,
title = {Attention is not all you need: pure attention loses rank doubly exponentially with depth},
author = {Dong, Yihe and Cordonnier, Jean-Baptiste and Loukas, Andreas},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {2793--2803},
year = {2021},
editor = {Meila, Marina and Zhang, Tong},
volume = {139},
series = {Proceedings of Machine Learning Research},
month = {18--24 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v139/dong21a/dong21a.pdf},
url = {https://proceedings.mlr.press/v139/dong21a.html},
abstract = {Attention-based architectures have become ubiquitous in machine learning. Yet, our understanding of the reasons for their effectiveness remains limited. This work proposes a new way to understand self-attention networks: we show that their output can be decomposed into a sum of smaller terms—or paths—each involving the operation of a sequence of attention heads across layers. Using this path decomposition, we prove that self-attention possesses a strong inductive bias towards "token uniformity". Specifically, without skip connections or multi-layer perceptrons (MLPs), the output converges doubly exponentially to a rank-1 matrix. On the other hand, skip connections and MLPs stop the output from degeneration. Our experiments verify the convergence results on standard transformer architectures.}
}