Answers.Body: <p>I have written a blog to answer this question, <strong>please see <a href="https://say-hello2y.github.io/2022-09-07/attention-gradient" rel="nofollow noreferrer">https://say-hello2y.github.io/2022-09-07/attention-gradient</a></strong></p>
  <p>I use Matrix calculus to solve this question, here I put the final result of the gradient of an attention unit.
  <span class="math-container">$$
  \frac{\partial f(X)}{\partial W_h^K }=\gamma K^T\mathbb{P}_h\frac{\partial f(X)}{\partial A  }^TQW_h^Q
  $$</span>
  <span class="math-container">$$
  \frac{\partial f(X)}{\partial W_h^Q }=\gamma Q^T\frac{\partial f(X)}{\partial A  }\mathbb{P}_{h}^{T}KW_h^K
  $$</span>
  <span class="math-container">$$
  \frac{\partial f(X)}{\partial W^v  }=A^T\frac{\partial f(X)}{\partial X  }(W^O)^T
  $$</span>
  <span class="math-container">$$
  \frac{\partial f(X)}{\partial W^O }=\mathrm{Concat}(\mathrm{head_1}, ..., \mathrm{head_H})^T\frac{\partial f(X)}{\partial X }
  $$</span>
  For more detail, <strong>please see <a href="https://say-hello2y.github.io/2022-09-07/attention-gradient" rel="nofollow noreferrer">https://say-hello2y.github.io/2022-09-07/attention-gradient</a></strong>.If you have any questions, feel free to contact me ,my email is longxhe@gmail.com.</p>
  
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Answers.CreationDate: 2024-08-17T18:57:58.297Z
Answers.Text: You didn't define several symbols.
Answers.UserDisplayName: ABIM
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-09-07T03:33:00.227Z
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Answers.UserDisplayName: He LongXiang
Body: <p>The paper <a href="https://papers.nips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf" rel="nofollow noreferrer">Attention Is All You Need</a> describes the Transformer architecture, which describes attention as a function of the queries <span class="math-container">$Q = x W^Q$</span>, keys <span class="math-container">$K = x W^K$</span>, and values <span class="math-container">$V = x W^V$</span>:</p>
  <p><span class="math-container">$\text{Attention(Q, K, V)} = \text{softmax}\left( \frac{QK^T}{\sqrt{d_k}} \right) V \\
  = \text{softmax}\left( \frac{x W^Q (W^K)^T x}{\sqrt{d_k}} \right) x W^V$</span></p>
  <p>In the Transformer, there are 3 different flavors of attention:</p>
  <ol>
  <li><strong>Self-attention in the Encoder</strong>, where the queries, keys, and values all come from the input to the Encoder.</li>
  <li><strong>Encoder-Decoder attention in the Decoder</strong>, where the queries come from the input to the Decoder, and the keys and values come from the output of the Encoder</li>
  <li><strong>Masked self-attention in the Decoder</strong>, where the queries, keys and values all come from the input to the Decoder, and, for each token, the <span class="math-container">$\text{softmax}\left( \frac{QK^T}{\sqrt{d_k}} \right)$</span> operation is masked out (zero'd out) for all tokens to the right of that token (to prevent look-ahead, which is cheating during training).</li>
  </ol>
  <p>What is the gradient (i.e. the partial derivatives of the loss function w.r.t. <span class="math-container">$x$</span>, <span class="math-container">$W^Q$</span>, <span class="math-container">$W^K$</span>, <span class="math-container">$W^V$</span>, and any bias term(s)) of each of these attention units? I am having a difficult time wrapping my head around derivating a gradient equation because I'm not sure how the softmax function interacts with the partial derivatives, and also, for the Encoder-Decoder attention in the Decoder, I'm not clear how to incorporate the encoder output into the equation.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-12-08T00:57:57.227Z
Comments.Text: Let me try to understand what you're really looking for. You're not looking for all the details of the partial derivatives with respect to the parameters, i.e. you're not looking for someone to describe to you how back-prop works, right? If I am right, can you please explain what type of answer you're looking for? Anyway, I suspect that the original authors of the transformer did not have to compute the gradients by hand, but let the library perform automatic differentiation (i.e. back-prop) automatically, as it's usually the case. You just need to make sure your operations are differentiable.
Comments.UserDisplayName: nbro
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Comments.CreationDate: 2020-12-08T01:16:19.630Z
Comments.Text: I understand how backprop works in general for deep networks, networks with convolutions, and networks with residuals. I guess I am asking for what the mathematical equation of a gradient is for a specific layer of the neural network, where the layer is an attention unit. Specifically, deriving the partial derivatives of the loss w.r.t. the input $x$, weights $W^Q$, $W^K$, $W^V$, and the bias terms (if the attention unit has any -- I wasn't clear on that from the paper), assuming we have the partial derivatives of the later layers already. This may be solved by software, but I'm still curious!
Comments.UserDisplayName: user3667125
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-04-17T14:18:55.430Z
Comments.Text: Have you got any solution for this. If so please let me know. I'm also trying the same
Comments.UserDisplayName: SS Varshini
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-08-17T18:58:20.567Z
Comments.Text: https://math.stackexchange.com/questions/4692899/derivation-of-the-attention-block-of-a-transformer
Comments.UserDisplayName: ABIM
Tags: <neural-networks><natural-language-processing><gradient-descent><transformer><attention>
Title: What is the gradient of an attention unit?
UserDisplayName: user3667125
ThreadText: # What is the gradient of an attention unit?
  
  (Asked by: username_0 on 2020-12-08)
  
  The paper [Attention Is All You Need](https://papers.nips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf) describes the Transformer architecture, which describes attention as a function of the queries $Q = x W^Q$, keys $K = x W^K$, and values $V = x W^V$:
  
  $\text{Attention(Q, K, V)} = \text{softmax}\left( \frac{QK^T}{\sqrt{d\_k}} \right) V \\
  = \text{softmax}\left( \frac{x W^Q (W^K)^T x}{\sqrt{d\_k}} \right) x W^V$
  
  In the Transformer, there are 3 different flavors of attention:
  
  1. **Self-attention in the Encoder**, where the queries, keys, and values all come from the input to the Encoder.
  2. **Encoder-Decoder attention in the Decoder**, where the queries come from the input to the Decoder, and the keys and values come from the output of the Encoder
  3. **Masked self-attention in the Decoder**, where the queries, keys and values all come from the input to the Decoder, and, for each token, the $\text{softmax}\left( \frac{QK^T}{\sqrt{d\_k}} \right)$ operation is masked out (zero'd out) for all tokens to the right of that token (to prevent look-ahead, which is cheating during training).
  
  What is the gradient (i.e. the partial derivatives of the loss function w.r.t. $x$, $W^Q$, $W^K$, $W^V$, and any bias term(s)) of each of these attention units? I am having a difficult time wrapping my head around derivating a gradient equation because I'm not sure how the softmax function interacts with the partial derivatives, and also, for the Encoder-Decoder attention in the Decoder, I'm not clear how to incorporate the encoder output into the equation.
  
  **Tags:** neural-networks, natural-language-processing, gradient-descent, transformer, attention
  > **Comments:**
  > **username_1**: Let me try to understand what you're really looking for. You're not looking for all the details of the partial derivatives with respect to the parameters, i.e. you're not looking for someone to describe to you how back-prop works, right? If I am right, can you please explain what type of answer you're looking for? Anyway, I suspect that the original authors of the transformer did not have to compute the gradients by hand, but let the library perform automatic differentiation (i.e. back-prop) automatically, as it's usually the case. You just need to make sure your operations are differentiable.  
  > **username_0**: I understand how backprop works in general for deep networks, networks with convolutions, and networks with residuals. I guess I am asking for what the mathematical equation of a gradient is for a specific layer of the neural network, where the layer is an attention unit. Specifically, deriving the partial derivatives of the loss w.r.t. the input $x$, weights $W^Q$, $W^K$, $W^V$, and the bias terms (if the attention unit has any -- I wasn't clear on that from the paper), assuming we have the partial derivatives of the later layers already. This may be solved by software, but I'm still curious!  
  > **username_2**: Have you got any solution for this. If so please let me know. I'm also trying the same  
  > **username_3**: https://math.stackexchange.com/questions/4692899/derivation-of-the-attention-block-of-a-transformer  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_4 on 2022-09-07. Score: 4
  
  I have written a blog to answer this question, **please see <https://say-hello2y.github.io/2022-09-07/attention-gradient>**
  
  I use Matrix calculus to solve this question, here I put the final result of the gradient of an attention unit.
  $$
  \frac{\partial f(X)}{\partial W\_h^K }=\gamma K^T\mathbb{P}\_h\frac{\partial f(X)}{\partial A }^TQW\_h^Q
  $$
  $$
  \frac{\partial f(X)}{\partial W\_h^Q }=\gamma Q^T\frac{\partial f(X)}{\partial A }\mathbb{P}\_{h}^{T}KW\_h^K
  $$
  $$
  \frac{\partial f(X)}{\partial W^v }=A^T\frac{\partial f(X)}{\partial X }(W^O)^T
  $$
  $$
  \frac{\partial f(X)}{\partial W^O }=\mathrm{Concat}(\mathrm{head\_1}, ..., \mathrm{head\_H})^T\frac{\partial f(X)}{\partial X }
  $$
  For more detail, **please see <https://say-hello2y.github.io/2022-09-07/attention-gradient>**.If you have any questions, feel free to contact me ,my email is longxhe@gmail.com.
  
  > **Comments:**
  > **username_3**: You didn't define several symbols.  
  
  ---
  
  