you are given the following input for transformer encoder: {flying, arrows}. The input embeddings for these two words are [0,1,1,1,1,0] and [1,1,0,-1,-1,1]. 
you are trying to represent the first word 'flying' with the help of self-attention in the first encoder.  
For the first attention head, the query, key and value matrices just take the 2 dimensions from the input each. 
so, first 2 dimensions define the query vector, and so on. What will be the self-attention output for the word 'flying' corresponding to this attention head.  
You are using the scaled dot vector.

Let's break down the problem step-by-step to find the self-attention output for the word "flying" using the given information:

### Given Data
- **Words**: `{flying, arrows}`
- **Embeddings**:
  - **flying**: `[0, 1, 1, 1, 1, 0]`
  - **arrows**: `[1, 1, 0, -1, -1, 1]`
- **Query, Key, Value Extraction**:
  - Each matrix uses 2 dimensions from the embeddings:
    - **Query**: First 2 dimensions
    - **Key**: Next 2 dimensions
    - **Value**: Last 2 dimensions

### Step-by-Step Computation

1. **Extract Query, Key, and Value Vectors**:
   - For the word **"flying"**:
     - Query: `[0, 1]` (first 2 dimensions of the embedding `[0, 1, 1, 1, 1, 0]`)
     - Key: `[1, 1]` (next 2 dimensions of the embedding `[0, 1, 1, 1, 1, 0]`)
     - Value: `[1, 0]` (last 2 dimensions of the embedding `[0, 1, 1, 1, 1, 0]`)
   - For the word **"arrows"**:
     - Query: `[1, 1]` (first 2 dimensions of the embedding `[1, 1, 0, -1, -1, 1]`)
     - Key: `[0, -1]` (next 2 dimensions of the embedding `[1, 1, 0, -1, -1, 1]`)
     - Value: `[-1, 1]` (last 2 dimensions of the embedding `[1, 1, 0, -1, -1, 1]`)

2. **Calculate Attention Scores**:
   The attention scores are calculated using the scaled dot product of the query and key vectors. The formula for the score between a query \( q \) and a key \( k \) is:
   \[
   \text{score} = \frac{q \cdot k^T}{\sqrt{d_k}}
   \]
   where \( d_k \) is the dimension of the key vectors (here, \( d_k = 2 \)).

   - **Score of "flying" with "flying"**:
     \[
     \text{score} = \frac{[0, 1] \cdot [1, 1]}{\sqrt{2}} = \frac{(0 \times 1 + 1 \times 1)}{\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{1}{1.414} \approx 0.707
     \]

   - **Score of "flying" with "arrows"**:
     \[
     \text{score} = \frac{[0, 1] \cdot [0, -1]}{\sqrt{2}} = \frac{(0 \times 0 + 1 \times (-1))}{\sqrt{2}} = \frac{-1}{\sqrt{2}} = \frac{-1}{1.414} \approx -0.707
     \]

3. **Apply Softmax to the Scores**:
   The next step is to apply the softmax function to these scores to get the attention weights.
   \[
   \text{softmax}(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}}
   \]
   Let's denote:
   - \( s_1 = 0.707 \)
   - \( s_2 = -0.707 \)

   \[
   \text{softmax}(s_1) = \frac{e^{0.707}}{e^{0.707} + e^{-0.707}} = \frac{e^{0.707}}{e^{0.707} + e^{-0.707}}
   \]
   Similarly,
   \[
   \text{softmax}(s_2) = \frac{e^{-0.707}}{e^{0.707} + e^{-0.707}}
   \]

   - Compute the values:
     \[
     e^{0.707} \approx 2.028
     \]
     \[
     e^{-0.707} \approx 0.493
     \]

     - Softmax outputs:
     \[
     \text{softmax}(s_1) = \frac{2.028}{2.028 + 0.493} = \frac{2.028}{2.521} \approx 0.805
     \]
     \[
     \text{softmax}(s_2) = \frac{0.493}{2.521} \approx 0.195
     \]

4. **Compute the Self-Attention Output**:
   The self-attention output is a weighted sum of the value vectors:
   \[
   \text{output} = \text{softmax}(s_1) \times \text{Value}_\text{flying} + \text{softmax}(s_2) \times \text{Value}_\text{arrows}
   \]
   Substituting the values:
   \[
   \text{output} = 0.805 \times [1, 0] + 0.195 \times [-1, 1]
   \]

   Calculate each component:
   - First component:
     \[
     0.805 \times 1 + 0.195 \times (-1) = 0.805 - 0.195 = 0.61
     \]
   - Second component:
     \[
     0.805 \times 0 + 0.195 \times 1 = 0 + 0.195 = 0.195
     \]

   - So, the final self-attention output for the word "flying" is:
     \[
     [0.61, 0.195]
     \]

Would you like more details or have any questions about this process?

### Related Questions
1. How does the self-attention mechanism differ in multi-head attention?
2. What is the purpose of scaling the dot product in self-attention?
3. How are the query, key, and value matrices learned during training?
4. What role do positional encodings play in transformer models?
5. How does the transformer handle long-range dependencies compared to RNNs?

### Tip
Always apply softmax to the attention scores to convert them into a probability distribution before using them to compute the weighted sum of values.

You are given the following input for transformer encoder: {flying, arrows}. The input embeddings for these two words are [0,1,1,1,1,0] and [1,1,0,-1,-1,1]. You are trying to represent the first word 'flying' with the help of self-attention in the first encoder. For the first attention head, the query, key and value matrices just take the 2 dimensions from the input each. So, first 2 dimensions define the query vector, and so on. What will be the self-attention output for the word 'flying' corresponding to this attention head? You are using the scaled dot vector.

Discover how to calculate the self-attention output for the word 'flying' in a transformer encoder using the scaled dot-product attention mechanism. This detailed step-by-step solution covers linear algebra concepts such as dot products, scaling, and softmax functions. Ideal for students and professionals in graduate-level machine learning courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation