$O(n)$ Connections are Expressive Enough: Universal Approximability of Sparse Transformers
This addresses the scalability issue for long sequences in NLP, providing theoretical justification for sparse attention models, though it is incremental as it builds on existing sparse Transformer research.
The paper tackles the problem of quadratic computational cost in Transformer attention layers by proving that sparse Transformers with only O(n) connections per layer can universally approximate any sequence-to-sequence function, matching the capability of dense models with n^2 connections.
Recently, Transformer networks have redefined the state of the art in many NLP tasks. However, these models suffer from quadratic computational cost in the input sequence length $n$ to compute pairwise attention in each layer. This has prompted recent research into sparse Transformers that sparsify the connections in the attention layers. While empirically promising for long sequences, fundamental questions remain unanswered: Can sparse Transformers approximate any arbitrary sequence-to-sequence function, similar to their dense counterparts? How does the sparsity pattern and the sparsity level affect their performance? In this paper, we address these questions and provide a unifying framework that captures existing sparse attention models. We propose sufficient conditions under which we prove that a sparse attention model can universally approximate any sequence-to-sequence function. Surprisingly, our results show that sparse Transformers with only $O(n)$ connections per attention layer can approximate the same function class as the dense model with $n^2$ connections. Lastly, we present experiments comparing different patterns/levels of sparsity on standard NLP tasks.