Conv-Basis: A New Paradigm for Efficient Attention Inference and Gradient Computation in Transformers

Large Language Models (LLMs) have profoundly changed the world. Their self-attention mechanism is the key to the success of transformers in LLMs. However, the quadratic computational cost O(n²) to the length n input sequence is the notorious obstacle for further improvement and scalability in the longer context. In this work, we leverage the convolution-like structure of attention matrices to develop an efficient approximation method for attention computation using convolution matrices. We propose a conv basis system, "similar" to the rank basis, and show that any lower triangular (attention) matrix can always be decomposed as a sum of k structured convolution matrices in this basis system. We then design an algorithm to quickly decompose the attention matrix into k convolution matrices. Thanks to Fast Fourier Transforms (FFT), the attention { inference} can be computed in O(knd log n) time, where d is the hidden dimension. In practice, we have d n, i.e., $d=3,072$ and $n=1,000,000$ for Gemma. Thus, when kd = nᵒ⁽¹⁾, our algorithm achieve almost linear time, i.e., n¹⁺ᵒ⁽¹⁾. Furthermore, the attention { training forward} and { backward gradient} can be computed in n¹⁺ᵒ⁽¹⁾ as well. Our approach can avoid explicitly computing the n × n attention matrix, which may largely alleviate the quadratic computational complexity. Furthermore, our algorithm works on any input matrices. This work provides a new paradigm for accelerating attention computation in transformers to enable their application to longer contexts.