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In the Attention is all you need paper, on the 4th page, we have equation 1, which describes the self-attention mechanism of the transformer architecture

$$ \text { Attention }(Q, K, V)=\operatorname{softmax}\left(\frac{Q K^{T}}{\sqrt{d_{k}}}\right) V $$

Everything is fine up to here.

Then they introduce the multi-head attention, which is described by the following equation.

$$ \begin{aligned} \text { MultiHead }(Q, K, V) &=\text { Concat}\left(\text {head}_{1}, \ldots, \text {head}_{\mathrm{h}}\right) W^{O} \\ \text { where head}_{\mathrm{i}} &=\text {Attention}\left(Q W_{i}^{Q}, K W_{i}^{K}, V W_{i}^{V}\right) \end{aligned} $$

Once the multi-head attention is motivated at the end of page 4, they state that for a single head (the $i$th head), the query $Q$ and key $K$ inputs are first linearly projected by $W_i^Q$ and $W_i^K$, then dot product is calculated, let's say $Q_i^p = Q W_i^Q$ and $K_i^p = K W_i^K$.

Therefore, the dot product of the projected query and key becomes the following from simple linear algebra.

$$Q_i^p {K_i^p}^\intercal = Q W_i^Q {W_i^K}^T K^T = Q W_i K^T,$$

where

$$W_i = W_i^Q {W_i^K}^T$$

Here, $W$ is the outer product of query projection by the key projection matrix. However, it is a matrix with shape $d_{model} \times d_{model}$. Why did the authors not define only a $W_i$ instead of $W_i^Q$ and $W_i^K$ pair which have $2 \times d_{model} \times d_{k}$ elements? In deep learning applications, I think it would be very inefficient.

Is there something that I am missing, like these 2 matrices $W_i^Q$ and $W_i^K$ should be separate because of this and that?

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2 Answers 2

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I'll use notation from the paper you cited, and any other readers should refer to the paper (widely available) for definitions of notation. The utility of using $W^Q$ and $W^K$, rather than $W$, lies in the fact that they allow us to add fewer parameters to our architecture. $W$ has dimension $d_{model} \times d_{model}$, which means that we are adding $d_{model}^2$ parameters to our architecture. $W^Q$ and $W^K$ each have dimension $d_{model} \times d_k$, and $d_k=\frac{d_{model}}{h}$. If we use these two matrices, we only add $2\frac{d_{model}^2}{h}$ parameters to our architecture, even though their multiplication (with the transpose) allows us to have the correct dimensions for matrix multiplication with $Q$ and $K$.

We do use $h$ attention heads, which then brings our number of parameters back up, but the multiple heads let the model attend to different pieces of information in our data.

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  • $\begingroup$ I understand now, actually the resulting $W$ matrix is rank deficient when defined by inner product of $W^Q$ and $W^K$. This makes parameters fewer :) Thanks. $\endgroup$ Dec 17, 2020 at 5:42
  • $\begingroup$ I just read a paper (arxiv.org/pdf/1906.07510.pdf) in which the authors use multi-head attention from Vaswani et al, but have $W^Q$ and $W^K$ be of dimension $d_{model} \times d_{model}$. I don't know if there is a reason for this, or if it's a typo. I plan to email them, and if there is a reason for it, I'll post it here. $\endgroup$
    – BioBroo
    Dec 17, 2020 at 17:59
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In practice, matrices $W^Q, W^K, W^V$ (each of size $d_{model}$ x $d_{model}$) are completely removed instead, and Transformer implementations just learn a single set of matrices $\{ W_i^{Q*}, W_i^{K*}, W_i^{V*} \}$ (each of size $d_{model}$ x $\frac{d_{model}}{h}$) for each head, where

$W_i^{Q*} = W^Q W_i^Q \\ W_i^{K*} = W^K W_i^K \\ W_i^{V*} = W^V W_i^V $

so that:

$Q_i(x) = x W_i^{Q*} = x W^Q W_i^{Q*} = Q W_i^Q\\ K_i(x) = x W_i^{K*} = x W^K W_i^{K*} = K W_i^K\\ V_i(x) = x W_i^{V*} = x W^V W_i^{V*} = V W_i^V\\ head_i(x) = softmax \left(\frac{Q_i(x) K_i(x)^T}{\sqrt{d_k}} \right) V_i(x)$.

I can confirm this with the original Transformer implementation in Tensor2Tensor, and also the BERT code that uses the encoding part of the Transformer.

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  • $\begingroup$ I missed the part where $d_k=d_{model} / h$ . Thank you. $\endgroup$ Dec 17, 2020 at 5:42
  • $\begingroup$ I just stumbled upon this Question after not understanding the original paper. $$ \text { where head }_{\mathrm{i}}=\text { Attention }\left(Q W_{i}^{Q}, K W_{i}^{K}, V W_{i}^{V}\right) $$I would really like to know if this redundant now and it can be$$ \text { head }_{\mathrm{i}}=\text { Attention }\left(Q_{i}, K_{i}, V_{i}\right)$$ where $$Q_{i}, K_{i}, V_{i} $$are simply generated from the input to the layer and the matrices $$ \left\{W_{i}^{Q }, W_{i}^{K }, W_{i}^{V }\right\}\in \mathbb{R} ^{D\times \dfrac{D}{h}} $$ respectively $\endgroup$
    – JimSi
    Jan 17, 2021 at 11:19

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