Are Transformers with One Layer Self-Attention Using Low-Rank Weight Matrices Universal Approximators?
This addresses a theoretical gap in understanding the expressive capacity of Transformers for researchers, providing a more practical alignment with models used in practice, though it is incremental in clarifying existing mathematical interpretations.
The paper tackles the problem of Transformer models requiring excessively deep layers for data memorization, and proves that a single layer of self-attention with low-rank weight matrices can perfectly capture the context of an entire input sequence, showing that one-layer Transformers are universal approximators for continuous permutation equivariant functions.
Existing analyses of the expressive capacity of Transformer models have required excessively deep layers for data memorization, leading to a discrepancy with the Transformers actually used in practice. This is primarily due to the interpretation of the softmax function as an approximation of the hardmax function. By clarifying the connection between the softmax function and the Boltzmann operator, we prove that a single layer of self-attention with low-rank weight matrices possesses the capability to perfectly capture the context of an entire input sequence. As a consequence, we show that one-layer and single-head Transformers have a memorization capacity for finite samples, and that Transformers consisting of one self-attention layer with two feed-forward neural networks are universal approximators for continuous permutation equivariant functions on a compact domain.