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  • 整体思路以及计算方式
  • 时间复杂度
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  1. Pe

Translational Equivariance in Kernelizable Attention

PreviousTransformer-XL Attentive Language Models Beyond a Fixed-Length ContextNextTransformer Language Models without Positional Encodings Still Learn Positional Information

Last updated 2 years ago

论文地址：

整体思路以及计算方式

论文讨论的是在线性Attention中添加相对位置编码信息，整体思路如下。

首先回顾Attention计算方式，其中ei\mathbf e_iei​是word embedding，ui\mathbf u_iui​是position embedding：

A~i,j=⟨(ei+ui)Wq,(ej+uj)Wk⟩=(ei+ui)WqWk⊤(ej+uj)⊤=eiWqWk⊤ej⊤+eiWqWk⊤uj⊤+uiWqWk⊤ej⊤+uiWqWk⊤uj⊤\begin{aligned} \tilde{\mathbf{A}}_{i, j} &=\left\langle\left(\mathbf e_{i}+\mathbf u_{i}\right) \mathbf {W}_{q},\left(\mathbf e_{j}+\mathbf u_{j}\right) \mathbf {W}_{k}\right\rangle=\left(\mathbf e_{i}+\mathbf u_{i}\right) \mathbf {W}_{q} \mathbf {W}_{k}^{\top}\left(\mathbf e_{j}+\mathbf u_{j}\right)^{\top} \\ &=\mathbf e_{i} \mathbf {W}_{q} \mathbf {W}_{k}^{\top}\mathbf e_{j}^{\top}+\mathbf e_{i} \mathbf {W}_{q} \mathbf {W}_{k}^{\top} \mathbf u_{j}^{\top}+\mathbf u_{i} \mathbf {W}_{q} \mathbf {W}_{k}^{\top} \mathbf e_{j}^{\top}+\mathbf u_{i} \mathbf {W}_{q} \mathbf {W}_{k}^{\top} \mathbf u_{j}^{\top} \end{aligned}A~i,j​​=⟨(ei​+ui​)Wq​,(ej​+uj​)Wk​⟩=(ei​+ui​)Wq​Wk⊤​(ej​+uj​)⊤=ei​Wq​Wk⊤​ej⊤​+ei​Wq​Wk⊤​uj⊤​+ui​Wq​Wk⊤​ej⊤​+ui​Wq​Wk⊤​uj⊤​​

论文的思路是对该计算方式进行重构，并且仍然能保持线性Attention的性质。

方案1：

A~i,j=eiWqWk⊤ej⊤+uiWqWk⊤uj⊤\begin{aligned} \tilde{\mathbf{A}}_{i, j} &=\mathbf e_{i} \mathbf {W}_{q} \mathbf {W}_{k}^{\top}\mathbf e_{j}^{\top} +\mathbf u_{i} \mathbf {W}_{q} \mathbf {W}_{k}^{\top}\mathbf u_{j}^{\top} \end{aligned}A~i,j​​=ei​Wq​Wk⊤​ej⊤​+ui​Wq​Wk⊤​uj⊤​​

其中：

Wq∗=blockdiag⁡([α1β1−β1α1],…,[αmβm−βmαm])Wk∗=I2m\begin{aligned} \mathbf {W}_{q}^{*} &=\operatorname{blockdiag}\left(\left[\begin{array}{cc} \alpha_{1} & \beta_{1} \\ -\beta_{1} & \alpha_{1} \end{array}\right], \ldots,\left[\begin{array}{cc} \alpha_{m} & \beta_{m} \\ -\beta_{m} & \alpha_{m} \end{array}\right]\right)\\ \mathbf {W}_{k}^{*}&=\mathbb{I}_{2 m} \end{aligned}Wq∗​Wk∗​​=blockdiag([α1​−β1​​β1​α1​​],…,[αm​−βm​​βm​αm​​])=I2m​​

该方案的特点是，如果位置编码的形式为：

ux=ϕ(x)=[sin⁡(ω1x),cos⁡(ω1x),…,sin⁡(ωmx),cos⁡(ωmx)]u_x =\phi(x)=\left[\sin \left(\omega_{1} x\right), \cos \left(\omega_{1} x\right), \ldots, \sin \left(\omega_{m} x\right), \cos \left(\omega_{m} x\right)\right]ux​=ϕ(x)=[sin(ω1​x),cos(ω1​x),…,sin(ωm​x),cos(ωm​x)]

那么满足如下性质：

⟨ui−uWq,uj−uWk⟩=⟨uiWq,ujWk⟩\left\langle \mathbf u_{i-u} \mathbf {W}_{q},\mathbf u_{j-u} \mathbf {W}_{k}\right\rangle =\left\langle \mathbf u_{i} \mathbf {W}_{q},\mathbf u_{j} \mathbf {W}_{k}\right\rangle⟨ui−u​Wq​,uj−u​Wk​⟩=⟨ui​Wq​,uj​Wk​⟩

方案2：

A~ij=eiWq(ejWk+aij)⊤=eiWqWk⊤ej⊤+eiWqaij⊤\tilde{\mathbf{A}}_{i j}=\mathbf e_{i} \mathbf{W}_{q}\left(\mathbf e_{j} \mathbf{W}_{k}+{a}_{i j}\right)^{\top}=\mathbf e_{i} \mathbf{W}_{q} \mathbf{W}_{k}^{\top} \mathbf e_{j}^{\top}+\mathbf e_{i} \mathbf{W}_{q} \mathbf {a}_{i j}^{\top}A~ij​=ei​Wq​(ej​Wk​+aij​)⊤=ei​Wq​Wk⊤​ej⊤​+ei​Wq​aij⊤​

其中

aij=wclip⁡(j−i,k)clip⁡(x,k)=max⁡(−k,min⁡(k,x))\begin{aligned} \mathbf {a}_{i j} &=\mathbf{w}_{\operatorname{clip}(j-i, k)} \\ \operatorname{clip}(x, k) &=\max (-k, \min (k, x)) \end{aligned}aij​clip(x,k)​=wclip(j−i,k)​=max(−k,min(k,x))​

clip函数的含义是将输入截断至[−k,k][-k, k][−k,k]之间。

如果使用普通的实现方式，那么时间复杂度为O(L2d)O(L^2 d)O(L2d)，但是注意到：

∑j=1L⟨qi′,aij′⟩vj=⟨qi′,wk′⟩∑j=1Lvj+∑m⟨qi′,aim′⟩\sum_{j=1}^{L}\left\langle\mathbf{q}_{i}^{\prime}, \mathbf{a}_{i j}^{\prime}\right\rangle \mathbf{v}_{j}=\left\langle\mathbf{q}_{i}^{\prime}, \mathbf{w}_{k}^{\prime}\right\rangle \sum_{j=1}^{L} \mathbf{v}_{j}+\sum_{m}\left\langle\mathbf{q}_{i}^{\prime}, \mathbf{a}_{i m}^{\prime} \right\ranglej=1∑L​⟨qi′​,aij′​⟩vj​=⟨qi′​,wk′​⟩j=1∑L​vj​+m∑​⟨qi′​,aim′​⟩

可以将时间复杂度降低为O(Ld2)O(Ld^2)O(Ld2)，其中ddd为embedding维度。

时间复杂度

和线性Attention一致，依然为O(Ld2)O(Ld^2)O(Ld2)

训练以及loss

不变。

代码

实验以及适用场景

适用于所有场景，实验只测试了图像任务，但是效果一般。

细节

暂无。

简评

总体感觉新意尚可，但效果一般。

https://arxiv.org/abs/2102.07680
https://github.com/ExpectationMax/Translational-Equivariant-Performers