Solving Attention Kernel Regression Problem via Pre-conditioner

Zhao Song, Junze Yin, Lichen Zhang
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:208-216, 2024.

Abstract

Attention mechanism is the key to large language models, and 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 \emph{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 \emph{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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-song24a, title = {Solving Attention Kernel Regression Problem via Pre-conditioner}, author = {Song, Zhao and Yin, Junze and Zhang, Lichen}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {208--216}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/song24a/song24a.pdf}, url = {https://proceedings.mlr.press/v238/song24a.html}, abstract = {Attention mechanism is the key to large language models, and 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 \emph{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 \emph{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.} }
Endnote
%0 Conference Paper %T Solving Attention Kernel Regression Problem via Pre-conditioner %A Zhao Song %A Junze Yin %A Lichen Zhang %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-song24a %I PMLR %P 208--216 %U https://proceedings.mlr.press/v238/song24a.html %V 238 %X Attention mechanism is the key to large language models, and 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 \emph{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 \emph{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.
APA
Song, Z., Yin, J. & Zhang, L.. (2024). Solving Attention Kernel Regression Problem via Pre-conditioner. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:208-216 Available from https://proceedings.mlr.press/v238/song24a.html.

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