The Shaped Transformer: Attention Models in the Infinite Depth-and-Width Limit
This work addresses stability and trainability issues in deep Transformers for the deep learning theory community, though it is incremental as it modifies existing architectures.
The authors tackled the problem of rank degeneracy in deep attention models by studying a modified Transformer in the infinite depth-and-width limit, showing that its covariance matrix can be described by a stable stochastic differential equation (SDE) that prevents these issues, with simulations confirming the SDE's accuracy for finite models.
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.