# Chapter 6: Self-Attention and Multi-Head Attention in Transformers

## 6.2 Self-Attention in Transformers

Attention mechanisms have been around for a while, but it wasn't until the advent of Transformers that the concept of self-attention or intra-attention was introduced. Self-attention takes attention a step further by deriving queries, keys, and values from the same input, as opposed to using separate inputs for each, which was the case for previous attention mechanisms.

This approach has proven to be highly effective in natural language processing and has led to significant improvements in tasks such as machine translation, sentiment analysis, and text summarization. As such, self-attention is widely regarded as a key innovation in the field of deep learning.

### 6.2.1 Definition and Intuition of Self-Attention

Self-attention, a mechanism that allows each word in the input sequence to associate with every other word, is a powerful tool for understanding the context and resolving the meaning of words in a sentence. By generating a weight matrix that defines the contribution of each word in deciding the context of every other word, self-attention enables a more holistic understanding of the sentence.

Moreover, the intuition behind self-attention is that it allows each word to look at every other word in the sequence. This means that instead of only paying attention to its position and the target, like in traditional seq2seq models, self-attention allows each word to "see" the bigger picture of the sentence. By doing so, self-attention provides a more comprehensive understanding of the relationships between the words in a sentence, which is essential in natural language processing tasks such as machine translation, question answering, and text summarization.

### 6.2.2 Calculating Self-Attention

Self-attention is a fundamental mechanism in natural language processing that helps models understand the relationships between different words in a sentence. To better understand how it works, let's take a closer look at the mechanics of self-attention.

Consider an input sequence represented by a matrix X, where each row is the embedding of a word. Here, the self-attention mechanism first projects X into three different spaces: a Query space (Q), a Key space (K), and a Value space (V). These spaces are computed by multiplying X by three weight matrices that are learned during training: Wq, Wk, and Wv. Essentially, the queries, keys, and values are just different ways of representing the same input sequence, but they each play a different role in the self-attention mechanism.

Once we have these three spaces, we can compute the self-attention scores between each pair of words in the input sequence. The self-attention score for a pair of words is computed by taking the dot product of their respective queries and keys, and then normalizing the result using a softmax function. This gives us a probability distribution over the values, which we can use to compute a weighted sum of the values for each word. The result of this computation is a new set of embeddings for the input sequence that takes into account the relationships between the different words.

Self-attention is a powerful mechanism that allows models to capture complex dependencies between words in a sentence. By projecting the input sequence into different spaces and computing attention scores between each pair of words, self-attention gives models a way to reason about the relationships between different parts of the input, even when those relationships are not easily captured by a fixed set of rules.

Once we have Q, K, and V, we can compute the self-attention of the input sequence as follows:

**Dot product of Q and K**

This results in a matrix that represents the scores of the words. Each element in this matrix corresponds to the score of an interaction between two words.

The dot product of Q and K is a fundamental operation in the field of natural language processing. It is used to compute the similarity between two sets of vectors, which in turn can be used to identify relevant information in large datasets. The resulting matrix represents the scores of the words, providing us with valuable insights into the relationships between different words.

Each element in this matrix corresponds to the score of an interaction between two words, which can be used to identify patterns and trends in the data. By analyzing these scores, we can gain a deeper understanding of the underlying structure of the data and make more informed decisions based on this knowledge.

**Scale the scores**

We scale the scores by dividing them by the square root of the dimension of the keys. This scaling is done to prevent the scores from growing too large.

We scale the scores by dividing them by the square root of the dimension of the keys, which is a technique used to prevent the scores from growing too large. This is an important step in the process as it helps us to ensure that the scores are accurate and reliable.

Additionally, by scaling the scores, we are able to better understand the data and identify any patterns or trends that may be present. This allows us to make more informed decisions and take appropriate actions based on the results of our analysis. In summary, scaling the scores is a crucial step in the process of analyzing data and is essential for obtaining accurate and reliable results.

**Apply softmax**

In order to convert the scores into weights, we apply the softmax function. The softmax function is used to transform the scores into a probability distribution that assigns probabilities to each possible output. The softmax function is commonly used in machine learning and deep learning models. It is a popular choice for the output layer of classification models, where the goal is to assign each input to one of several possible output categories.

By applying the softmax function, we can ensure that the scores are normalized and that they sum to one. This makes it easier to interpret the results and to make comparisons between different inputs and outputs. Overall, the softmax function is an essential tool for any machine learning practitioner who wants to build accurate and reliable models.

**Multiply by V**

The final step in the process involves multiplying the weights by V, which is a crucial step in computing the output sequence. This step is essential as it allows for the transformation of the input sequence into the output sequence, which is used in a variety of applications across different fields. The multiplication by V is an important step as it helps to ensure that the output sequence is accurate and reliable, which is critical in many industries.

Without this step, the output sequence would not be able to effectively capture the information contained in the input sequence, which would lead to inaccurate and unreliable results. As such, it is important to ensure that this step is performed correctly and accurately to ensure that the output sequence is of the highest quality and reliability possible.

**Example:**

Here's a simple Python function that demonstrates the computation of self-attention:

`import numpy as np`

def self_attention(X, Wq, Wk, Wv):

# Project to Query, Key and Value spaces

Q = np.dot(X, Wq)

K = np.dot(X, Wk)

V = np.dot(X, Wv)

# Compute scores

scores = np.dot(Q, K.T) / np.sqrt(K.shape[-1])

# Compute weights

weights = np.exp(scores) / np.sum(np.exp(scores), axis=-1, keepdims=True)

# Compute output

output = np.dot(weights, V)

return output, weights

# Test the function with random data

X = np.random.rand(2, 3) # Input sequence

Wq = np.random.rand(3, 3) # Weight matrix for Query

Wk = np.random.rand(3, 3) # Weight matrix for Key

Wv = np.random.rand(3, 3) # Weight matrix for Value

output, weights = self_attention(X, Wq, Wk, Wv)

print("Output:", output)

print("Self-attention weights:", weights)

In this code, `X`

is the input sequence, and `Wq`

, `Wk`

, and `Wv`

are the weight matrices for the Query, Key, and Value projections, respectively. The `self_attention`

function computes the Query, Key, and Value projections, computes the scores,

applies softmax to get the weights, and finally computes the output sequence as a weighted sum of the Value vectors.

## 6.2 Self-Attention in Transformers

Attention mechanisms have been around for a while, but it wasn't until the advent of Transformers that the concept of self-attention or intra-attention was introduced. Self-attention takes attention a step further by deriving queries, keys, and values from the same input, as opposed to using separate inputs for each, which was the case for previous attention mechanisms.

This approach has proven to be highly effective in natural language processing and has led to significant improvements in tasks such as machine translation, sentiment analysis, and text summarization. As such, self-attention is widely regarded as a key innovation in the field of deep learning.

### 6.2.1 Definition and Intuition of Self-Attention

Self-attention, a mechanism that allows each word in the input sequence to associate with every other word, is a powerful tool for understanding the context and resolving the meaning of words in a sentence. By generating a weight matrix that defines the contribution of each word in deciding the context of every other word, self-attention enables a more holistic understanding of the sentence.

Moreover, the intuition behind self-attention is that it allows each word to look at every other word in the sequence. This means that instead of only paying attention to its position and the target, like in traditional seq2seq models, self-attention allows each word to "see" the bigger picture of the sentence. By doing so, self-attention provides a more comprehensive understanding of the relationships between the words in a sentence, which is essential in natural language processing tasks such as machine translation, question answering, and text summarization.

### 6.2.2 Calculating Self-Attention

Self-attention is a fundamental mechanism in natural language processing that helps models understand the relationships between different words in a sentence. To better understand how it works, let's take a closer look at the mechanics of self-attention.

Consider an input sequence represented by a matrix X, where each row is the embedding of a word. Here, the self-attention mechanism first projects X into three different spaces: a Query space (Q), a Key space (K), and a Value space (V). These spaces are computed by multiplying X by three weight matrices that are learned during training: Wq, Wk, and Wv. Essentially, the queries, keys, and values are just different ways of representing the same input sequence, but they each play a different role in the self-attention mechanism.

Once we have these three spaces, we can compute the self-attention scores between each pair of words in the input sequence. The self-attention score for a pair of words is computed by taking the dot product of their respective queries and keys, and then normalizing the result using a softmax function. This gives us a probability distribution over the values, which we can use to compute a weighted sum of the values for each word. The result of this computation is a new set of embeddings for the input sequence that takes into account the relationships between the different words.

Self-attention is a powerful mechanism that allows models to capture complex dependencies between words in a sentence. By projecting the input sequence into different spaces and computing attention scores between each pair of words, self-attention gives models a way to reason about the relationships between different parts of the input, even when those relationships are not easily captured by a fixed set of rules.

Once we have Q, K, and V, we can compute the self-attention of the input sequence as follows:

**Dot product of Q and K**

This results in a matrix that represents the scores of the words. Each element in this matrix corresponds to the score of an interaction between two words.

The dot product of Q and K is a fundamental operation in the field of natural language processing. It is used to compute the similarity between two sets of vectors, which in turn can be used to identify relevant information in large datasets. The resulting matrix represents the scores of the words, providing us with valuable insights into the relationships between different words.

Each element in this matrix corresponds to the score of an interaction between two words, which can be used to identify patterns and trends in the data. By analyzing these scores, we can gain a deeper understanding of the underlying structure of the data and make more informed decisions based on this knowledge.

**Scale the scores**

We scale the scores by dividing them by the square root of the dimension of the keys. This scaling is done to prevent the scores from growing too large.

We scale the scores by dividing them by the square root of the dimension of the keys, which is a technique used to prevent the scores from growing too large. This is an important step in the process as it helps us to ensure that the scores are accurate and reliable.

Additionally, by scaling the scores, we are able to better understand the data and identify any patterns or trends that may be present. This allows us to make more informed decisions and take appropriate actions based on the results of our analysis. In summary, scaling the scores is a crucial step in the process of analyzing data and is essential for obtaining accurate and reliable results.

**Apply softmax**

In order to convert the scores into weights, we apply the softmax function. The softmax function is used to transform the scores into a probability distribution that assigns probabilities to each possible output. The softmax function is commonly used in machine learning and deep learning models. It is a popular choice for the output layer of classification models, where the goal is to assign each input to one of several possible output categories.

By applying the softmax function, we can ensure that the scores are normalized and that they sum to one. This makes it easier to interpret the results and to make comparisons between different inputs and outputs. Overall, the softmax function is an essential tool for any machine learning practitioner who wants to build accurate and reliable models.

**Multiply by V**

The final step in the process involves multiplying the weights by V, which is a crucial step in computing the output sequence. This step is essential as it allows for the transformation of the input sequence into the output sequence, which is used in a variety of applications across different fields. The multiplication by V is an important step as it helps to ensure that the output sequence is accurate and reliable, which is critical in many industries.

Without this step, the output sequence would not be able to effectively capture the information contained in the input sequence, which would lead to inaccurate and unreliable results. As such, it is important to ensure that this step is performed correctly and accurately to ensure that the output sequence is of the highest quality and reliability possible.

**Example:**

Here's a simple Python function that demonstrates the computation of self-attention:

`import numpy as np`

def self_attention(X, Wq, Wk, Wv):

# Project to Query, Key and Value spaces

Q = np.dot(X, Wq)

K = np.dot(X, Wk)

V = np.dot(X, Wv)

# Compute scores

scores = np.dot(Q, K.T) / np.sqrt(K.shape[-1])

# Compute weights

weights = np.exp(scores) / np.sum(np.exp(scores), axis=-1, keepdims=True)

# Compute output

output = np.dot(weights, V)

return output, weights

# Test the function with random data

X = np.random.rand(2, 3) # Input sequence

Wq = np.random.rand(3, 3) # Weight matrix for Query

Wk = np.random.rand(3, 3) # Weight matrix for Key

Wv = np.random.rand(3, 3) # Weight matrix for Value

output, weights = self_attention(X, Wq, Wk, Wv)

print("Output:", output)

print("Self-attention weights:", weights)

In this code, `X`

is the input sequence, and `Wq`

, `Wk`

, and `Wv`

are the weight matrices for the Query, Key, and Value projections, respectively. The `self_attention`

function computes the Query, Key, and Value projections, computes the scores,

applies softmax to get the weights, and finally computes the output sequence as a weighted sum of the Value vectors.

## 6.2 Self-Attention in Transformers

Attention mechanisms have been around for a while, but it wasn't until the advent of Transformers that the concept of self-attention or intra-attention was introduced. Self-attention takes attention a step further by deriving queries, keys, and values from the same input, as opposed to using separate inputs for each, which was the case for previous attention mechanisms.

This approach has proven to be highly effective in natural language processing and has led to significant improvements in tasks such as machine translation, sentiment analysis, and text summarization. As such, self-attention is widely regarded as a key innovation in the field of deep learning.

### 6.2.1 Definition and Intuition of Self-Attention

Self-attention, a mechanism that allows each word in the input sequence to associate with every other word, is a powerful tool for understanding the context and resolving the meaning of words in a sentence. By generating a weight matrix that defines the contribution of each word in deciding the context of every other word, self-attention enables a more holistic understanding of the sentence.

Moreover, the intuition behind self-attention is that it allows each word to look at every other word in the sequence. This means that instead of only paying attention to its position and the target, like in traditional seq2seq models, self-attention allows each word to "see" the bigger picture of the sentence. By doing so, self-attention provides a more comprehensive understanding of the relationships between the words in a sentence, which is essential in natural language processing tasks such as machine translation, question answering, and text summarization.

### 6.2.2 Calculating Self-Attention

Self-attention is a fundamental mechanism in natural language processing that helps models understand the relationships between different words in a sentence. To better understand how it works, let's take a closer look at the mechanics of self-attention.

Consider an input sequence represented by a matrix X, where each row is the embedding of a word. Here, the self-attention mechanism first projects X into three different spaces: a Query space (Q), a Key space (K), and a Value space (V). These spaces are computed by multiplying X by three weight matrices that are learned during training: Wq, Wk, and Wv. Essentially, the queries, keys, and values are just different ways of representing the same input sequence, but they each play a different role in the self-attention mechanism.

Once we have these three spaces, we can compute the self-attention scores between each pair of words in the input sequence. The self-attention score for a pair of words is computed by taking the dot product of their respective queries and keys, and then normalizing the result using a softmax function. This gives us a probability distribution over the values, which we can use to compute a weighted sum of the values for each word. The result of this computation is a new set of embeddings for the input sequence that takes into account the relationships between the different words.

Self-attention is a powerful mechanism that allows models to capture complex dependencies between words in a sentence. By projecting the input sequence into different spaces and computing attention scores between each pair of words, self-attention gives models a way to reason about the relationships between different parts of the input, even when those relationships are not easily captured by a fixed set of rules.

Once we have Q, K, and V, we can compute the self-attention of the input sequence as follows:

**Dot product of Q and K**

This results in a matrix that represents the scores of the words. Each element in this matrix corresponds to the score of an interaction between two words.

The dot product of Q and K is a fundamental operation in the field of natural language processing. It is used to compute the similarity between two sets of vectors, which in turn can be used to identify relevant information in large datasets. The resulting matrix represents the scores of the words, providing us with valuable insights into the relationships between different words.

Each element in this matrix corresponds to the score of an interaction between two words, which can be used to identify patterns and trends in the data. By analyzing these scores, we can gain a deeper understanding of the underlying structure of the data and make more informed decisions based on this knowledge.

**Scale the scores**

We scale the scores by dividing them by the square root of the dimension of the keys. This scaling is done to prevent the scores from growing too large.

We scale the scores by dividing them by the square root of the dimension of the keys, which is a technique used to prevent the scores from growing too large. This is an important step in the process as it helps us to ensure that the scores are accurate and reliable.

Additionally, by scaling the scores, we are able to better understand the data and identify any patterns or trends that may be present. This allows us to make more informed decisions and take appropriate actions based on the results of our analysis. In summary, scaling the scores is a crucial step in the process of analyzing data and is essential for obtaining accurate and reliable results.

**Apply softmax**

In order to convert the scores into weights, we apply the softmax function. The softmax function is used to transform the scores into a probability distribution that assigns probabilities to each possible output. The softmax function is commonly used in machine learning and deep learning models. It is a popular choice for the output layer of classification models, where the goal is to assign each input to one of several possible output categories.

By applying the softmax function, we can ensure that the scores are normalized and that they sum to one. This makes it easier to interpret the results and to make comparisons between different inputs and outputs. Overall, the softmax function is an essential tool for any machine learning practitioner who wants to build accurate and reliable models.

**Multiply by V**

The final step in the process involves multiplying the weights by V, which is a crucial step in computing the output sequence. This step is essential as it allows for the transformation of the input sequence into the output sequence, which is used in a variety of applications across different fields. The multiplication by V is an important step as it helps to ensure that the output sequence is accurate and reliable, which is critical in many industries.

Without this step, the output sequence would not be able to effectively capture the information contained in the input sequence, which would lead to inaccurate and unreliable results. As such, it is important to ensure that this step is performed correctly and accurately to ensure that the output sequence is of the highest quality and reliability possible.

**Example:**

Here's a simple Python function that demonstrates the computation of self-attention:

`import numpy as np`

def self_attention(X, Wq, Wk, Wv):

# Project to Query, Key and Value spaces

Q = np.dot(X, Wq)

K = np.dot(X, Wk)

V = np.dot(X, Wv)

# Compute scores

scores = np.dot(Q, K.T) / np.sqrt(K.shape[-1])

# Compute weights

weights = np.exp(scores) / np.sum(np.exp(scores), axis=-1, keepdims=True)

# Compute output

output = np.dot(weights, V)

return output, weights

# Test the function with random data

X = np.random.rand(2, 3) # Input sequence

Wq = np.random.rand(3, 3) # Weight matrix for Query

Wk = np.random.rand(3, 3) # Weight matrix for Key

Wv = np.random.rand(3, 3) # Weight matrix for Value

output, weights = self_attention(X, Wq, Wk, Wv)

print("Output:", output)

print("Self-attention weights:", weights)

In this code, `X`

is the input sequence, and `Wq`

, `Wk`

, and `Wv`

are the weight matrices for the Query, Key, and Value projections, respectively. The `self_attention`

function computes the Query, Key, and Value projections, computes the scores,

applies softmax to get the weights, and finally computes the output sequence as a weighted sum of the Value vectors.

## 6.2 Self-Attention in Transformers

### 6.2.1 Definition and Intuition of Self-Attention

### 6.2.2 Calculating Self-Attention

Once we have Q, K, and V, we can compute the self-attention of the input sequence as follows:

**Dot product of Q and K**

**Scale the scores**

**Apply softmax**

**Multiply by V**

**Example:**

Here's a simple Python function that demonstrates the computation of self-attention:

`import numpy as np`

def self_attention(X, Wq, Wk, Wv):

# Project to Query, Key and Value spaces

Q = np.dot(X, Wq)

K = np.dot(X, Wk)

V = np.dot(X, Wv)

# Compute scores

scores = np.dot(Q, K.T) / np.sqrt(K.shape[-1])

# Compute weights

weights = np.exp(scores) / np.sum(np.exp(scores), axis=-1, keepdims=True)

# Compute output

output = np.dot(weights, V)

return output, weights

# Test the function with random data

X = np.random.rand(2, 3) # Input sequence

Wq = np.random.rand(3, 3) # Weight matrix for Query

Wk = np.random.rand(3, 3) # Weight matrix for Key

Wv = np.random.rand(3, 3) # Weight matrix for Value

output, weights = self_attention(X, Wq, Wk, Wv)

print("Output:", output)

print("Self-attention weights:", weights)

`X`

is the input sequence, and `Wq`

, `Wk`

, and `Wv`

are the weight matrices for the Query, Key, and Value projections, respectively. The `self_attention`

function computes the Query, Key, and Value projections, computes the scores,