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Attention Bias

Here're some resources about Attention Bias

Intro

As alternative mechanisms to explicitly encoding positional information, attention bias have been explored to capture the sequentiality and temporality of natural language incorporated into attention kernel.

Table of Contents

Definition

As shown in equation below, the attention bias is depicted as a matrix, denoted as $B$, which is added to the unnormalized attention weights matrix $P$ before applying the softmax operation. Each element of this matrix, indexed by $(i,j)$, carries positional information encoded by a function $\mathcal{B}$. Thus, it is reasonable to regard the attention bias as a form of relative PEs.

$$ \begin{align} & \widetilde{P} := P + B, \quad B \in \mathbb{R}^{L\times L},\quad where\quad B_{ij} := \mathcal{B}(i,j), \quad\forall i,j \in {0,1,..,L-1} \end{align} $$

Super Baseline

Transformer Upgrade Roadmap: 7. Length Extrapolation and Local Attention

$$ \begin{align} & \mathcal{B}_{super\text{-}baseline}(i,j) := \begin{cases} 0, & |i-j| \in [0, max\text{-}length]\\ -\infty, & otherwise\\ \end{cases} \end{align} $$

blog link: here

citation:

@misc{transformer-upgrade-7,
    author = "Su, Jianlin",
    title = "Transformer Upgrade Roadmap: 7. Length Extrapolation and Local Attention",
    year = "2023",
    month = "Jan",
    howpublished = "\url{https://spaces.ac.cn/archives/9431}"
}

illustration: Su introduced a simple method in his blog where he utilizes a super-baseline approach during inference, as illustrated in the equation. This method relies on a local causal attention mask, where each query attends to keys whose distances have not exceeded $L_{max}$ while still applying RoPE. According to Su's experiments, this approach proves to be simple, low-cost and performs sufficiently well compared to the more elaborate designs below, thus referred as a super-baseline.

More Elaborate Designs

Dissecting transformer length extrapolation via the lens of receptive field analysis (Sandwitch)

$$ \begin{align} & \mathcal{B}_{Sandwitch}(i,j) := \lambda\cdot \langle \mathrm{SinPE}(i), \mathrm{SinPE}(j) \rangle \end{align} $$

paper link: here

citation:

@inproceedings{chi2023dissecting,
  title={Dissecting transformer length extrapolation via the lens of receptive field analysis},
  author={Chi, Ta-Chung and Fan, Ting-Han and Rudnicky, Alexander and Ramadge, Peter},
  booktitle={Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)},
  pages={13522--13537},
  year={2023}
}

Kerple: Kernelized relative positional embedding for length extrapolation

$$ \begin{align} & \mathcal{B}_{\textit{KERPLE}}(i,j) := \begin{cases} -r_1\log(1 + r_2|i-j|), \quad r_1, r_2 > 0\\ -r_1 |i-j|^{r_2}, \quad r_1 > 0, r_2 \in (0,2]\\ \end{cases} \end{align} $$

paper link: here

citation:

@article{chi2022kerple,
  title={Kerple: Kernelized relative positional embedding for length extrapolation},
  author={Chi, Ta-Chung and Fan, Ting-Han and Ramadge, Peter J and Rudnicky, Alexander},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={8386--8399},
  year={2022}
}

Train short, test long: Attention with linear biases enables input length extrapolation (Alibi)

$$ \begin{align} & \mathcal{B}_{ALiBi}^{(h)}(i,j) := -\lambda^{(h)}\cdot |i-j|, \quad\lambda^{(h)} := \cfrac{1}{2^h} \space or\space \cfrac{1}{2^{h/2}} \end{align} $$

paper link: here

citation:

@article{press2021train,
  title={Train short, test long: Attention with linear biases enables input length extrapolation},
  author={Press, Ofir and Smith, Noah A and Lewis, Mike},
  journal={arXiv preprint arXiv:2108.12409},
  year={2021}
}