Solving Attention Kernel Regression Problem via Pre-conditioner

The attention mechanism is the key to large language models, and the attention matrix serves as an algorithmic and computational bottleneck for such a scheme. In this paper, we define two problems, motivated by designing fast algorithms for proxy of attention matrix and solving regressions against them. Given an input matrix $A\in \mathbb{R}^{n\times d}$ with $n\gg d$ and a response vector $b$, we first consider the matrix exponential of the matrix $A^\top A$ as a proxy, and we in turn design algorithms for two types of regression problems: $\min_{x\in \mathbb{R}^d}\|(A^\top A)^jx-b\|_2$ and $\min_{x\in \mathbb{R}^d}\|A(A^\top A)^jx-b\|_2$ for any positive integer $j$. Studying algorithms for these regressions is essential, as matrix exponential can be approximated term-by-term via these smaller problems. The second proxy is applying exponential entrywise to the Gram matrix, denoted by $\exp(AA^\top)$ and solving the regression $\min_{x\in \mathbb{R}^n}\|\exp(AA^\top)x-b \|_2$. We call this problem the attention kernel regression problem, as the matrix $\exp(AA^\top)$ could be viewed as a kernel function with respect to $A$. We design fast algorithms for these regression problems, based on sketching and preconditioning. We hope these efforts will provide an alternative perspective of studying efficient approximation of attention matrices.