  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, each involving the
operation of a sequence of attention heads across layers. Using this
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 identified convergence phenomena
on different variants of standard transformer architectures.
