  In deep learning theory, the covariance matrix of the representations serves
as a proxy to examine the network's trainability. Motivated by the success of
Transformers, we study the covariance matrix of a modified Softmax-based
attention model with skip connections in the proportional limit of
infinite-depth-and-width. We show that at initialization the limiting
distribution can be described by a stochastic differential equation (SDE)
indexed by the depth-to-width ratio. To achieve a well-defined stochastic
limit, the Transformer's attention mechanism is modified by centering the
Softmax output at identity, and scaling the Softmax logits by a width-dependent
temperature parameter. We examine the stability of the network through the
corresponding SDE, showing how the scale of both the drift and diffusion can be
elegantly controlled with the aid of residual connections. The existence of a
stable SDE implies that the covariance structure is well-behaved, even for very
large depth and width, thus preventing the notorious issues of rank degeneracy
in deep attention models. Finally, we show, through simulations, that the SDE
provides a surprisingly good description of the corresponding finite-size
model. We coin the name shaped Transformer for these architectural
modifications.
