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# Multi-Head Attention Collaborate Instead of Concatenate

### Multi-Head Attention: Collaborate Instead of Concatenate

论文地址：

* <https://arxiv.org/abs/2006.16362>

### 整体思路以及计算方式

对$$\mathbf Q,\mathbf K$$降维，然后通过对角阵增加模型表达力，最后达到相当的效果。

计算方式：

* 压缩率$$p$$，$$h$$为头数；
* 输入：$$\mathbf X\in \mathbb R^{n\times d},\mathbf Y\in \mathbb R^{m\times d},m\_i\in \mathbb R,i=1,\ldots, h$$，记$$d\_1=\lfloor pd\rfloor$$
* for $$i=1,\ldots, h$$
  * 计算$$\mathbf Q= \mathbf X\mathbf W\_Q^{(i)} \in \mathbb R^{n\times (d\_1 /h)},\mathbf K= \mathbf X\mathbf W\_K^{(i)} \in \mathbb R^{m\times (d\_1 /h)}, \mathbf V =\mathbf X\mathbf W\_V^{(i)}\in \mathbb R^{m\times (d /h)}$$
  * $$\mathbf H^{(i)}=\mathrm{MHA}(\mathbf Q\mathrm{diag}(m\_i), \mathbf K, \mathbf V)$$
* 输出：$$\mathrm{Concat}\[\mathbf H^{(i)}]$$

说明：

* 尽管原文中不同头算$$\mathbf Q,\mathbf K$$的$$\mathbf W\_Q,\mathbf W\_K$$是共享的，但实际实现的时候并不是；

### 时间复杂度

对于每个头，时间复杂度为：

$$
O(nmd\_1/h + mnd/h)=O(nmpd/h + mnd/h)
$$

所以$$h$$个头的时间复杂度为：

$$
O(mn(p+1)d )
$$

### 训练以及loss

略过。

### 代码

* <https://github.com/lucidrains/x-transformers/blob/main/x_transformers/x_transformers.py#L514>
* <https://github.com/epfml/collaborative-attention/blob/master/src/collaborative_attention/collaborative_attention.py>
* <https://github.com/epfml/collaborative-attention>

### 实验以及适用场景

因为只改了Head部分，所以适用于所有场景；作者进行了大量实验，效果均不错。

### 细节

降维比例为30%的时候也能达到相当效果。

### 简评

总结：

* 很简洁的思路，通过降维减少参数量，然后再通过少量参数恢复性能；
* 非常简洁，值得复现；


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