The Shaped Transformer: Attention Models in the Infinite Depth-and-Width
  Limit
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.
64a22d6d8f953945fa4b4f69
Lorenzo Noci
64a22d6d8f953945fa4b4f6a
Chuning Li
64a22d6d8f953945fa4b4f6b
Mufan Bill Li
64a22d6d8f953945fa4b4f6c
Bobby He
64a22d6d8f953945fa4b4f6d
Thomas Hofmann
64a22d6d8f953945fa4b4f6e
Chris Maddison
64a22d6d8f953945fa4b4f6f
Daniel M. Roy
The Shaped Transformer: Attention Models in the Infinite Depth-and-Width
  Limit
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.
A modified Softmax-based attention model with skip connections exhibits well-behaved covariance structure and stability in the proportional limit of infinite-depth-and-width, preventing rank degeneracy issues.
covariance matrix representations trainability Transformers Softmax-based attention model skip connections proportional limit infinite-depth-and-width stochastic differential equation SDE depth-to-width ratio centering width-dependent temperature parameter residual connections stability rank degeneracy finite-size model shaped Transformer