Transformer Models and the Attention Mechanism

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January 28, 2025

Introduction

This document delves into the Transformer architecture, a pioneering model introduced by Google in the seminal paper “Attention is All You Need” [@vaswani2017attention]. The Transformer has revolutionized the field of natural language processing (NLP) and serves as a foundational component of numerous state-of-the-art systems, including ChatGPT, BERT, and other large language models. The primary objectives of this lecture are to:

Provide a comprehensive understanding of the overall architecture of Transformer models.
Explore the concept of Multi-Head Attention, elucidating its significance in capturing intricate relationships within sequential data.
Detail how input sequences are processed, tokenized, numericalized, and transformed into meaningful embeddings.
Analyze the role of positional encoding in maintaining sequence information and discuss its evolution.
Offer insights into the functioning of BERT and its diverse applications in various NLP tasks.

The Transformer model, particularly its encoder component as used in BERT, has significantly advanced the field of natural language processing by introducing the Multi-Head Attention mechanism and maintaining input dimensionality throughout the encoding process.

Background and Motivation

The Transformer model, a revolutionary architecture in the field of NLP, was introduced in the seminal paper “Attention is All You Need” by Vaswani et al. (2017) [@vaswani2017attention]. This groundbreaking work has garnered an exceptionally high number of citations, as depicted in Figure 1, underscoring its profound impact on the research community and its widespread adoption.

Transformer Model and its Impact

Initially, the Transformer was conceived for machine translation tasks. The core objective was to develop a model capable of accurately translating text from a source language to a target language. The original Transformer architecture, as illustrated in Figure 2, comprises two primary components:

Encoder: This component is responsible for processing the input sequence in the source language and encoding it into a rich, contextualized series of vectors. These vectors capture the semantic essence of the input.
Decoder: This component takes the encoded information generated by the encoder and utilizes it to generate the output sequence in the target language, effectively performing the translation.

Basic Structure of Encoder-Decoder Model for Machine Translation

This encoder-decoder architecture mirrors the structure commonly employed in LSTM-based models for sequence-to-sequence tasks. However, the Transformer distinguishes itself through its innovative use of the attention mechanism, which we will explore in subsequent sections.

The development of the Transformer model marked a significant departure from traditional recurrent and convolutional neural networks for sequence processing. Its reliance on the attention mechanism allowed for more efficient parallelization and improved handling of long-range dependencies.

Evolution to BERT

Following the groundbreaking introduction of the Transformer architecture, a significant advancement emerged in the form of BERT (Bidirectional Encoder Representations from Transformers) [@devlin2018bert]. BERT represents a pivotal evolution in the application of Transformer models, focusing solely on the encoder component of the original architecture. This strategic simplification, depicted in Figure 3, proved to be a key factor in BERT’s ability to achieve remarkable performance across a wide range of natural language processing tasks.

BERT Model: A Simplified Transformer Encoder

By concentrating exclusively on the encoder, BERT leverages the power of bidirectional context understanding inherent in the Transformer’s design. Unlike traditional language models that process text in a single direction (left-to-right or right-to-left), BERT’s encoder can simultaneously consider the entire context of a word, leading to richer and more nuanced representations.

The decision to focus solely on the encoder in BERT was motivated by the observation that the encoder’s ability to capture bidirectional context was sufficient for many NLP tasks, and removing the decoder simplified the architecture without sacrificing performance.

This architectural choice enabled BERT to excel in tasks such as:

Question Answering: Providing accurate answers to questions based on a given passage.
Text Classification: Categorizing text into predefined classes (e.g., sentiment analysis, topic classification).
Named Entity Recognition: Identifying and classifying named entities in text (e.g., persons, organizations, locations).

The success of BERT has spurred further research and development in Transformer-based models, leading to a proliferation of powerful language models that continue to push the boundaries of NLP.

Core Concepts of Transformer Encoders

Transformer encoders employ a unique approach to processing input sequences, differing significantly from traditional recurrent models like LSTMs. This section elucidates the fundamental concepts underlying their operation.

Input Representation

A key distinction of Transformer encoders is their ability to maintain the dimensionality of the input throughout the encoding process. Unlike some architectures that progressively reduce dimensionality, Transformers preserve the richness of the input representation. Each word in the input sequence is represented by a numerical vector known as a word embedding.

Word Embeddings: A word embedding is a vector representation of a word, designed to capture its semantic meaning and relationships with other words. For instance, a word like "dog" might be represented by a 300-dimensional vector, where each dimension encodes a specific aspect of the word’s meaning.

The process of transforming an input sequence into a format suitable for the Transformer encoder involves the following steps:

Input: The raw input is a sequence of words, such as a sentence or a paragraph (e.g., "Hello, how are you?").
Tokenization: The input sequence is divided into individual tokens. Tokens can be words, subwords, or even individual characters, depending on the specific tokenization scheme used. In our example, the sentence "Hello, how are you?" would be tokenized into: "Hello", ",", "how", "are", "you", "?".
Numericalization: Each token is mapped to a unique numerical index based on a predefined vocabulary or dictionary. This dictionary is typically constructed during the training phase and contains all the unique tokens encountered in the training data. For example, "Hello" might be mapped to index 34, "," to 90, and "how" to 15.
Embedding: Each numerical index is then converted into its corresponding word embedding vector. These embeddings can be pre-trained (learned from large text corpora using methods like Word2Vec or GloVe) or learned from scratch during the training of the Transformer model.

Consider the sentence "Hello, how are you?". The tokenization and numericalization process would yield the following:

Tokens: "Hello", ",", "how", "are", "you", "?"
Numerical Indices: 34, 90, 15, 20, 25, 40 (assuming a hypothetical vocabulary)

Each of these numerical indices would then be replaced by its corresponding embedding vector, resulting in a sequence of vectors that represent the input sentence.

Maintaining Dimensionality

A crucial aspect of Transformer encoders is their ability to preserve the dimensionality of the input throughout the encoding process. This is illustrated in Figure 4. If the input word embeddings have a dimensionality of 300, the output vectors produced by each encoder block will also maintain that same dimensionality.

Maintaining Dimensionality in a Transformer Encoder Block

Maintaining dimensionality allows for the stacking of multiple encoder blocks without information loss due to dimensionality reduction, enabling the construction of deep and powerful Transformer models.

Encoder Blocks

The modular design of Transformer encoders allows for the stacking of multiple encoder blocks to create deeper and more powerful models. Each encoder block receives an input matrix and produces an output matrix of the same dimensions. This property enables the seamless addition of blocks without a reduction in the size of the representation.

Small BERT: Employs 12 encoder blocks stacked sequentially.
Large BERT: Employs a more substantial stack of 24 encoder blocks.
Other Models: Models like ChatGPT can have even more blocks, such as 49 or 94.

Stacked Encoder Blocks in a Transformer Model

As depicted in Figure 5, the output of one encoder block serves as the input to the next, allowing for the gradual refinement of the encoded representation.

Fixed Input Size

Transformer encoders are designed to handle input sequences of a fixed length. This constraint necessitates mechanisms for handling sequences that are shorter or longer than the predetermined input size.

Input Size: The original BERT model, for instance, is designed to process a maximum of 512 tokens. Other models may have different input size limitations.
Padding: Input sequences shorter than the fixed size are padded with special "padding" tokens to fill the remaining slots. These padding tokens are typically ignored during the attention computations.
Truncation: Sequences longer than the fixed size are truncated to fit within the limit. This typically involves discarding tokens from either the beginning or the end of the sequence.

Handling Input Sequences of Varying Lengths via Padding and Truncation

Figure 6 illustrates how input sequences are adjusted to meet the fixed input size requirement.

Input Matrix

The input to a Transformer encoder is not just a sequence of vectors but is organized as a matrix. Each row of this matrix represents the embedding of a single token in the input sequence. The dimensions of this matrix are determined by two factors:

The number of tokens in the input sequence (which is fixed, as discussed earlier).
The dimensionality of the word embeddings.

Matrix Dimensions: The input matrix has dimensions of (Number of tokens) $\times$ (Embedding dimensionality).
Example: For the original BERT model, which processes 512 tokens and uses 300-dimensional embeddings, the input matrix has dimensions of $512 \times 300$.

\[\text{Input Matrix} = \begin{bmatrix} \text{Token}_1 \\ \text{Token}_2 \\ \vdots \\ \text{Token}_{512} \end{bmatrix} = \begin{bmatrix} e_{1,1} & e_{1,2} & \cdots & e_{1,300} \\ e_{2,1} & e_{2,2} & \cdots & e_{2,300} \\ \vdots & \vdots & \ddots & \vdots \\ e_{512,1} & e_{512,2} & \cdots & e_{512,300} \end{bmatrix} \label{eq:input_matrix}\]

The matrix representation allows for efficient parallel computation of attention scores across all tokens in the sequence, a key advantage of the Transformer architecture.

Tokenization and Numericalization

Before an input sequence can be processed by a Transformer encoder, it must undergo two crucial preprocessing steps: tokenization and numericalization.

Tokenization: Tokenization is the process of splitting a sequence of text into individual tokens. These tokens can be words, subwords, or even individual characters, depending on the specific tokenization scheme employed.

Numericalization: Numericalization is the process of mapping each unique token in the vocabulary to a unique numerical index. This allows the model to process the tokens as numerical inputs.

Consider the sentence "Hello, how are you?". The tokenization process would break it down into the following tokens:

"Hello"
","
"how"
"are"
"you"
"?"

Each of these tokens would then be mapped to a numerical index based on a predefined vocabulary. For instance:

"Hello" $\rightarrow$ 34
"," $\rightarrow$ 90
"how" $\rightarrow$ 15
"are" $\rightarrow$ 20
"you" $\rightarrow$ 25
"?" $\rightarrow$ 40

(These indices are for illustrative purposes only and would depend on the actual vocabulary used.)

Word Embeddings

As defined earlier in Definition [def:word_embeddings], each token is represented by a word embedding, a vector that captures the semantic meaning of the word. These embeddings can be:

Pre-trained Embeddings: These embeddings are learned from vast text corpora using unsupervised methods like Word2Vec or GloVe. They capture general semantic relationships between words based on their co-occurrence patterns in the training data.
Learned Embeddings: Alternatively, the embeddings can be learned from scratch during the training of the Transformer model itself. This allows the embeddings to be fine-tuned to the specific task and dataset.

The choice between pre-trained and learned embeddings depends on factors such as the availability of large pre-trained models, the size of the training dataset, and the specific requirements of the task.

Positional Encoding

Unlike recurrent models that process sequences sequentially, Transformer encoders process all tokens in parallel. This parallelism, while computationally efficient, means that the model inherently lacks information about the order of tokens in the input sequence. To address this, positional encoding is introduced.

Positional Encoding: Positional encoding is a vector that is added to the word embedding of each token to provide information about the token’s position within the sequence.

The purpose of positional encoding is to inject information about the order of tokens into the model. The original Transformer paper proposed a specific method for generating these encodings:

Method: The positional encoding vector is typically of the same dimension as the word embedding. Each element of the positional encoding vector is calculated using sine and cosine functions with varying frequencies. This allows the model to learn to attend to relative positions.
Original Implementation: In the initial Transformer model, the positional encoding was based on a sequence of numbers (1, 2, 3, ...) representing the position of each token. However, this approach has been refined over time.
Current Status: Many modern implementations of Transformers use learned positional embeddings or alternative encoding schemes. The effectiveness of the original sinusoidal positional encoding has been debated, and it is often removed or modified in practice due to its limited impact on performance in some cases.

Mathematically, the positional encoding for a token at position $pos$ and dimension $i$ can be represented as:

\[PE(pos, 2i) = \sin\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \label{eq:pe_sin}\]

\[PE(pos, 2i+1) = \cos\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \label{eq:pe_cos}\]

where $d_{\text{model}}$ is the dimensionality of the embeddings.

The final input representation for each token is obtained by adding the positional encoding to the word embedding:

\[\text{Input with PE} = \text{Word Embedding} + \text{Positional Encoding} \label{eq:input_with_pe}\]

While positional encoding is a core concept in the original Transformer, its necessity and optimal implementation remain active areas of research. Some studies suggest that it may not always be crucial for good performance, particularly in models with learned positional embeddings.

Multi-Head Attention

Multi-Head Attention is arguably the most crucial component of the Transformer architecture. It empowers the model to focus on different segments of the input sequence and discern intricate relationships between tokens, even those that are far apart.

Overview

The Multi-Head Attention mechanism allows the model to jointly attend to information from different representation subspaces at different positions within the sequence. This capability is central to the Transformer’s ability to capture complex dependencies and patterns in the data.

Multi-Head Attention: Multi-Head Attention is a mechanism that enables the model to simultaneously attend to information from diverse representation subspaces at various positions in the input sequence. It computes multiple attention scores, each focusing on a different aspect of the relationships between tokens.

Figure 7 provides a high-level overview of the Multi-Head Attention mechanism. The input matrix, representing the sequence of token embeddings, is fed into the Multi-Head Attention block, which produces an output matrix of the same dimensions.

Multi-Head Attention Mechanism

Input Splitting

The first step in the Multi-Head Attention mechanism is to split the input matrix into three identical copies. These copies are then transformed into three distinct representations: Query (Q), Key (K), and Value (V).

Query (Q): Represents the current focus of attention. It is used to query the other tokens to determine their relevance to the current token.
Key (K): Represents the relevance or "label" of each token. It is compared with the query to determine the attention scores.
Value (V): Represents the information content of each token. The attention scores are used to weight the values, and the weighted values are summed to produce the output.

This process can be summarized by the following equation:

\[\text{Input Matrix} \xrightarrow{\text{Split}} \text{Q, K, V} \label{eq:input_split}\]

Imagine you have a set of documents (tokens). The Query is like a search query you are using to find relevant information. The Key is like the title or keywords of each document, and the Value is the actual content of the document. The attention mechanism determines which documents are most relevant to your query based on their titles and then extracts the relevant information from those documents.

Linear Projections

Each of the Q, K, and V matrices undergoes a linear transformation, projecting the information into different subspaces. These linear transformations are performed by multiplying each matrix with a separate weight matrix. These weight matrices are learnable parameters that are adjusted during the training process.

Linear Projection: A linear projection is a transformation that maps a vector from one vector space to another using a matrix multiplication. It is defined by a weight matrix that determines the mapping.

The linear projections for the $i$-th attention head can be expressed as:

\[Q_i = \text{Input} \times W^Q_i \label{eq:query_projection}\] \[K_i = \text{Input} \times W^K_i \label{eq:key_projection}\] \[V_i = \text{Input} \times W^V_i \label{eq:value_projection}\]

where $W^Q_i$, $W^K_i$, and $W^V_i$ are the weight matrices for the Query, Key, and Value projections of the $i$-th head, respectively.

Projecting the Q, K, and V matrices into different subspaces allows each attention head to focus on different aspects of the relationships between tokens. It’s like looking at the input from multiple perspectives, each highlighting a different type of information.

Scaled Dot-Product Attention

The core of the attention mechanism is the scaled dot-product attention, which calculates the attention scores between the Query and Key matrices.

Scaled Dot-Product Attention: Scaled Dot-Product Attention computes the attention scores between the Query and Key matrices by taking their dot product, scaling the result, and then applying a softmax function to obtain a probability distribution over the tokens. This distribution represents the attention weights, indicating the relevance of each token to the query.

The scaled dot-product attention is computed as follows:

\[\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V \label{eq:scaled_dot_product_attention}\]

where $d_k$ is the dimensionality of the Key vectors. The scaling factor $\frac{1}{\sqrt{d_k}}$ is used to prevent the dot products from becoming too large, which can lead to issues with the gradients during training.

Scaling by $\frac{1}{\sqrt{d_k}}$ is crucial for stable training. Without it, the dot products can grow large, pushing the softmax function into regions with extremely small gradients, hindering the learning process.

Multiple Heads

The "Multi-Head" aspect of the attention mechanism refers to the use of multiple attention heads, each with its own set of Q, K, and V matrices and its own set of learnable weight matrices. Each head computes the scaled dot-product attention independently, allowing the model to capture diverse relationships between tokens.

Multiple Heads: Each attention head operates independently, computing its own attention scores and producing its own output.
Purpose: The use of multiple heads allows the model to attend to different types of relationships between tokens simultaneously. Each head can learn to focus on a different aspect of the input, such as syntactic dependencies, semantic relationships, or long-range dependencies.

Figure 8 illustrates the concept of multiple attention heads operating in parallel.

Multiple Attention Heads Operating in Parallel

Concatenation and Linear Projection

After each attention head has computed its output, the outputs are concatenated and then passed through a final linear layer. This step combines the information captured by the different heads and produces the final output of the Multi-Head Attention mechanism.

The concatenation and linear projection can be expressed as:

\[\text{MultiHead}(Q, K, V) = \text{Concat}(\text{head}_1, \text{head}_2, \ldots, \text{head}_h)W\label{eq:multi_head_output}\]

where $\text{head}_i = \text{Attention}(Q_i, K_i, V_i)$ is the output of the $i$-th attention head, and $W^O$ is the output weight matrix, another learnable parameter.

The final linear layer serves to integrate the information from all the attention heads and project it back into the original embedding space, ensuring that the output of the Multi-Head Attention block has the same dimensions as the input.

Conclusion

The Transformer architecture, particularly its encoder component as exemplified by BERT, represents a monumental leap forward in the field of natural language processing. Its innovative use of attention mechanisms and its ability to maintain input dimensionality have significantly improved the ability of models to understand and process text.

Key takeaways from this lecture include:

Dimensionality Preservation: Transformers maintain the dimensionality of the input throughout the encoding process, allowing for the construction of deep models without information loss due to dimensionality reduction.
Multi-Head Attention: The Multi-Head Attention mechanism is the cornerstone of the Transformer’s success. It enables the model to capture complex relationships between tokens by attending to different representation subspaces, effectively looking at the input from multiple perspectives.
Positional Encoding: While initially considered crucial for injecting positional information into the model, positional encoding has undergone modifications and its necessity is sometimes debated in recent implementations. Its role is often replaced or augmented by learned positional embeddings.
BERT’s Impact: BERT, with its focused use of the Transformer’s encoder, has demonstrated state-of-the-art performance on a wide array of NLP tasks, highlighting the power of bidirectional context understanding.

The Transformer architecture has revolutionized NLP by introducing the powerful Multi-Head Attention mechanism and demonstrating the effectiveness of maintaining input dimensionality. BERT’s success further underscores the importance of bidirectional context in language understanding.

Follow-up Questions for the Next Lecture:

To further our understanding of Transformer models and their broader implications, the following questions will be addressed in the next lecture:

Transformer Decoder:
- How does the decoder part of the Transformer work, and how is it employed in sequence-to-sequence tasks like machine translation? What are the intricacies of its architecture and how does it interact with the encoder?
Limitations and Solutions: What are the inherent limitations of the Transformer architecture, such as its computational cost for very long sequences or its potential biases? How can these limitations be addressed through architectural modifications or training strategies?
Evolution of Transformers: How do recent advancements, such as GPT and other large language models, build upon the foundational Transformer architecture? What are the key innovations and improvements introduced by these models?
Beyond NLP: What are some practical applications of Transformer models beyond the realm of natural language processing? How are they being adapted for use in computer vision, time series analysis, or other domains, and what are the challenges and opportunities in these areas?

--- title: "Transformer Models and the Attention Mechanism" author: "Your Name" date: "2025-01-28" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This document delves into the Transformer architecture, a pioneering model introduced by Google in the seminal paper "*Attention is All You Need*" [@vaswani2017attention]. The Transformer has revolutionized the field of natural language processing (NLP) and serves as a foundational component of numerous state-of-the-art systems, including ChatGPT, BERT, and other large language models. The primary objectives of this lecture are to: - Provide a comprehensive understanding of the overall architecture of Transformer models. - Explore the concept of Multi-Head Attention, elucidating its significance in capturing intricate relationships within sequential data. - Detail how input sequences are processed, tokenized, numericalized, and transformed into meaningful embeddings. - Analyze the role of positional encoding in maintaining sequence information and discuss its evolution. - Offer insights into the functioning of BERT and its diverse applications in various NLP tasks. ::: tcolorbox The Transformer model, particularly its encoder component as used in BERT, has significantly advanced the field of natural language processing by introducing the Multi-Head Attention mechanism and maintaining input dimensionality throughout the encoding process. ::: # Background and Motivation The Transformer model, a revolutionary architecture in the field of NLP, was introduced in the seminal paper "*Attention is All You Need*" by Vaswani et al. (2017) [@vaswani2017attention]. This groundbreaking work has garnered an exceptionally high number of citations, as depicted in Figure [1](#fig:transformer_impact){reference-type="ref" reference="fig:transformer_impact"}, underscoring its profound impact on the research community and its widespread adoption. <figure id="fig:transformer_impact"> <figcaption>Transformer Model and its Impact</figcaption> </figure> Initially, the Transformer was conceived for machine translation tasks. The core objective was to develop a model capable of accurately translating text from a source language to a target language. The original Transformer architecture, as illustrated in Figure [2](#fig:encoder_decoder){reference-type="ref" reference="fig:encoder_decoder"}, comprises two primary components: - **Encoder**: This component is responsible for processing the input sequence in the source language and encoding it into a rich, contextualized series of vectors. These vectors capture the semantic essence of the input. - **Decoder**: This component takes the encoded information generated by the encoder and utilizes it to generate the output sequence in the target language, effectively performing the translation. <figure id="fig:encoder_decoder"> <figcaption>Basic Structure of Encoder-Decoder Model for Machine Translation</figcaption> </figure> This encoder-decoder architecture mirrors the structure commonly employed in LSTM-based models for sequence-to-sequence tasks. However, the Transformer distinguishes itself through its innovative use of the attention mechanism, which we will explore in subsequent sections. ::: tcolorbox The development of the Transformer model marked a significant departure from traditional recurrent and convolutional neural networks for sequence processing. Its reliance on the attention mechanism allowed for more efficient parallelization and improved handling of long-range dependencies. ::: # Evolution to BERT Following the groundbreaking introduction of the Transformer architecture, a significant advancement emerged in the form of BERT (Bidirectional Encoder Representations from Transformers) [@devlin2018bert]. BERT represents a pivotal evolution in the application of Transformer models, focusing solely on the encoder component of the original architecture. This strategic simplification, depicted in Figure [3](#fig:bert){reference-type="ref" reference="fig:bert"}, proved to be a key factor in BERT's ability to achieve remarkable performance across a wide range of natural language processing tasks. <figure id="fig:bert"> <figcaption>BERT Model: A Simplified Transformer Encoder</figcaption> </figure> By concentrating exclusively on the encoder, BERT leverages the power of bidirectional context understanding inherent in the Transformer's design. Unlike traditional language models that process text in a single direction (left-to-right or right-to-left), BERT's encoder can simultaneously consider the entire context of a word, leading to richer and more nuanced representations. ::: tcolorbox The decision to focus solely on the encoder in BERT was motivated by the observation that the encoder's ability to capture bidirectional context was sufficient for many NLP tasks, and removing the decoder simplified the architecture without sacrificing performance. ::: This architectural choice enabled BERT to excel in tasks such as: - **Question Answering**: Providing accurate answers to questions based on a given passage. - **Text Classification**: Categorizing text into predefined classes (e.g., sentiment analysis, topic classification). - **Named Entity Recognition**: Identifying and classifying named entities in text (e.g., persons, organizations, locations). The success of BERT has spurred further research and development in Transformer-based models, leading to a proliferation of powerful language models that continue to push the boundaries of NLP. # Core Concepts of Transformer Encoders Transformer encoders employ a unique approach to processing input sequences, differing significantly from traditional recurrent models like LSTMs. This section elucidates the fundamental concepts underlying their operation. ## Input Representation A key distinction of Transformer encoders is their ability to maintain the dimensionality of the input throughout the encoding process. Unlike some architectures that progressively reduce dimensionality, Transformers preserve the richness of the input representation. Each word in the input sequence is represented by a numerical vector known as a word embedding. ::: tcolorbox **Word Embeddings:** A word embedding is a vector representation of a word, designed to capture its semantic meaning and relationships with other words. For instance, a word like \"dog\" might be represented by a 300-dimensional vector, where each dimension encodes a specific aspect of the word's meaning. ::: The process of transforming an input sequence into a format suitable for the Transformer encoder involves the following steps: 1. **Input**: The raw input is a sequence of words, such as a sentence or a paragraph (e.g., \"Hello, how are you?\"). 2. **Tokenization**: The input sequence is divided into individual tokens. Tokens can be words, subwords, or even individual characters, depending on the specific tokenization scheme used. In our example, the sentence \"Hello, how are you?\" would be tokenized into: \"Hello\", \",\", \"how\", \"are\", \"you\", \"?\". 3. **Numericalization**: Each token is mapped to a unique numerical index based on a predefined vocabulary or dictionary. This dictionary is typically constructed during the training phase and contains all the unique tokens encountered in the training data. For example, \"Hello\" might be mapped to index 34, \",\" to 90, and \"how\" to 15. 4. **Embedding**: Each numerical index is then converted into its corresponding word embedding vector. These embeddings can be pre-trained (learned from large text corpora using methods like Word2Vec or GloVe) or learned from scratch during the training of the Transformer model. ::: tcolorbox Consider the sentence \"Hello, how are you?\". The tokenization and numericalization process would yield the following: - **Tokens**: \"Hello\", \",\", \"how\", \"are\", \"you\", \"?\" - **Numerical Indices**: 34, 90, 15, 20, 25, 40 (assuming a hypothetical vocabulary) Each of these numerical indices would then be replaced by its corresponding embedding vector, resulting in a sequence of vectors that represent the input sentence. ::: ## Maintaining Dimensionality A crucial aspect of Transformer encoders is their ability to preserve the dimensionality of the input throughout the encoding process. This is illustrated in Figure [4](#fig:dimensionality){reference-type="ref" reference="fig:dimensionality"}. If the input word embeddings have a dimensionality of 300, the output vectors produced by each encoder block will also maintain that same dimensionality. <figure id="fig:dimensionality"> <figcaption>Maintaining Dimensionality in a Transformer Encoder Block</figcaption> </figure> ::: tcolorbox Maintaining dimensionality allows for the stacking of multiple encoder blocks without information loss due to dimensionality reduction, enabling the construction of deep and powerful Transformer models. ::: ## Encoder Blocks The modular design of Transformer encoders allows for the stacking of multiple encoder blocks to create deeper and more powerful models. Each encoder block receives an input matrix and produces an output matrix of the same dimensions. This property enables the seamless addition of blocks without a reduction in the size of the representation. - **Small BERT**: Employs 12 encoder blocks stacked sequentially. - **Large BERT**: Employs a more substantial stack of 24 encoder blocks. - **Other Models**: Models like ChatGPT can have even more blocks, such as 49 or 94. <figure id="fig:stacked_encoders"> <figcaption>Stacked Encoder Blocks in a Transformer Model</figcaption> </figure> As depicted in Figure [5](#fig:stacked_encoders){reference-type="ref" reference="fig:stacked_encoders"}, the output of one encoder block serves as the input to the next, allowing for the gradual refinement of the encoded representation. ## Fixed Input Size Transformer encoders are designed to handle input sequences of a fixed length. This constraint necessitates mechanisms for handling sequences that are shorter or longer than the predetermined input size. - **Input Size**: The original BERT model, for instance, is designed to process a maximum of 512 tokens. Other models may have different input size limitations. - **Padding**: Input sequences shorter than the fixed size are padded with special \"padding\" tokens to fill the remaining slots. These padding tokens are typically ignored during the attention computations. - **Truncation**: Sequences longer than the fixed size are truncated to fit within the limit. This typically involves discarding tokens from either the beginning or the end of the sequence. <figure id="fig:input_handling"> <figcaption>Handling Input Sequences of Varying Lengths via Padding and Truncation</figcaption> </figure> Figure [6](#fig:input_handling){reference-type="ref" reference="fig:input_handling"} illustrates how input sequences are adjusted to meet the fixed input size requirement. ## Input Matrix The input to a Transformer encoder is not just a sequence of vectors but is organized as a matrix. Each row of this matrix represents the embedding of a single token in the input sequence. The dimensions of this matrix are determined by two factors: 1. The number of tokens in the input sequence (which is fixed, as discussed earlier). 2. The dimensionality of the word embeddings. - **Matrix Dimensions**: The input matrix has dimensions of (Number of tokens) $\times$ (Embedding dimensionality). - **Example**: For the original BERT model, which processes 512 tokens and uses 300-dimensional embeddings, the input matrix has dimensions of $512 \times 300$. $$\text{Input Matrix} = \begin{bmatrix} \text{Token}_1 \\ \text{Token}_2 \\ \vdots \\ \text{Token}_{512} \end{bmatrix} = \begin{bmatrix} e_{1,1} & e_{1,2} & \cdots & e_{1,300} \\ e_{2,1} & e_{2,2} & \cdots & e_{2,300} \\ \vdots & \vdots & \ddots & \vdots \\ e_{512,1} & e_{512,2} & \cdots & e_{512,300} \end{bmatrix} \label{eq:input_matrix}$$ ::: tcolorbox The matrix representation allows for efficient parallel computation of attention scores across all tokens in the sequence, a key advantage of the Transformer architecture. ::: ## Tokenization and Numericalization Before an input sequence can be processed by a Transformer encoder, it must undergo two crucial preprocessing steps: tokenization and numericalization. ::: tcolorbox **Tokenization:** Tokenization is the process of splitting a sequence of text into individual tokens. These tokens can be words, subwords, or even individual characters, depending on the specific tokenization scheme employed. ::: ::: tcolorbox **Numericalization:** Numericalization is the process of mapping each unique token in the vocabulary to a unique numerical index. This allows the model to process the tokens as numerical inputs. ::: ::: tcolorbox Consider the sentence \"Hello, how are you?\". The tokenization process would break it down into the following tokens: - \"Hello\" - \",\" - \"how\" - \"are\" - \"you\" - \"?\" Each of these tokens would then be mapped to a numerical index based on a predefined vocabulary. For instance: - \"Hello\" $\rightarrow$ 34 - \",\" $\rightarrow$ 90 - \"how\" $\rightarrow$ 15 - \"are\" $\rightarrow$ 20 - \"you\" $\rightarrow$ 25 - \"?\" $\rightarrow$ 40 (These indices are for illustrative purposes only and would depend on the actual vocabulary used.) ::: ## Word Embeddings As defined earlier in Definition [\[def:word_embeddings\]](#def:word_embeddings){reference-type="ref" reference="def:word_embeddings"}, each token is represented by a word embedding, a vector that captures the semantic meaning of the word. These embeddings can be: - **Pre-trained Embeddings**: These embeddings are learned from vast text corpora using unsupervised methods like Word2Vec or GloVe. They capture general semantic relationships between words based on their co-occurrence patterns in the training data. - **Learned Embeddings**: Alternatively, the embeddings can be learned from scratch during the training of the Transformer model itself. This allows the embeddings to be fine-tuned to the specific task and dataset. ::: tcolorbox The choice between pre-trained and learned embeddings depends on factors such as the availability of large pre-trained models, the size of the training dataset, and the specific requirements of the task. ::: ## Positional Encoding Unlike recurrent models that process sequences sequentially, Transformer encoders process all tokens in parallel. This parallelism, while computationally efficient, means that the model inherently lacks information about the order of tokens in the input sequence. To address this, positional encoding is introduced. ::: tcolorbox **Positional Encoding:** Positional encoding is a vector that is added to the word embedding of each token to provide information about the token's position within the sequence. ::: The purpose of positional encoding is to inject information about the order of tokens into the model. The original Transformer paper proposed a specific method for generating these encodings: - **Method**: The positional encoding vector is typically of the same dimension as the word embedding. Each element of the positional encoding vector is calculated using sine and cosine functions with varying frequencies. This allows the model to learn to attend to relative positions. - **Original Implementation**: In the initial Transformer model, the positional encoding was based on a sequence of numbers (1, 2, 3, \...) representing the position of each token. However, this approach has been refined over time. - **Current Status**: Many modern implementations of Transformers use learned positional embeddings or alternative encoding schemes. The effectiveness of the original sinusoidal positional encoding has been debated, and it is often removed or modified in practice due to its limited impact on performance in some cases. Mathematically, the positional encoding for a token at position $pos$ and dimension $i$ can be represented as: $$PE(pos, 2i) = \sin\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \label{eq:pe_sin}$$ $$PE(pos, 2i+1) = \cos\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \label{eq:pe_cos}$$ where $d_{\text{model}}$ is the dimensionality of the embeddings. The final input representation for each token is obtained by adding the positional encoding to the word embedding: $$\text{Input with PE} = \text{Word Embedding} + \text{Positional Encoding} \label{eq:input_with_pe}$$ ::: tcolorbox While positional encoding is a core concept in the original Transformer, its necessity and optimal implementation remain active areas of research. Some studies suggest that it may not always be crucial for good performance, particularly in models with learned positional embeddings. ::: # Multi-Head Attention Multi-Head Attention is arguably the most crucial component of the Transformer architecture. It empowers the model to focus on different segments of the input sequence and discern intricate relationships between tokens, even those that are far apart. ## Overview The Multi-Head Attention mechanism allows the model to jointly attend to information from different representation subspaces at different positions within the sequence. This capability is central to the Transformer's ability to capture complex dependencies and patterns in the data. ::: tcolorbox **Multi-Head Attention:** Multi-Head Attention is a mechanism that enables the model to simultaneously attend to information from diverse representation subspaces at various positions in the input sequence. It computes multiple attention scores, each focusing on a different aspect of the relationships between tokens. ::: Figure [7](#fig:mha){reference-type="ref" reference="fig:mha"} provides a high-level overview of the Multi-Head Attention mechanism. The input matrix, representing the sequence of token embeddings, is fed into the Multi-Head Attention block, which produces an output matrix of the same dimensions. <figure id="fig:mha"> <figcaption>Multi-Head Attention Mechanism</figcaption> </figure> ## Input Splitting The first step in the Multi-Head Attention mechanism is to split the input matrix into three identical copies. These copies are then transformed into three distinct representations: Query (Q), Key (K), and Value (V). - **Query (Q)**: Represents the current focus of attention. It is used to query the other tokens to determine their relevance to the current token. - **Key (K)**: Represents the relevance or \"label\" of each token. It is compared with the query to determine the attention scores. - **Value (V)**: Represents the information content of each token. The attention scores are used to weight the values, and the weighted values are summed to produce the output. This process can be summarized by the following equation: $$\text{Input Matrix} \xrightarrow{\text{Split}} \text{Q, K, V} \label{eq:input_split}$$ ::: tcolorbox Imagine you have a set of documents (tokens). The Query is like a search query you are using to find relevant information. The Key is like the title or keywords of each document, and the Value is the actual content of the document. The attention mechanism determines which documents are most relevant to your query based on their titles and then extracts the relevant information from those documents. ::: ## Linear Projections Each of the Q, K, and V matrices undergoes a linear transformation, projecting the information into different subspaces. These linear transformations are performed by multiplying each matrix with a separate weight matrix. These weight matrices are learnable parameters that are adjusted during the training process. ::: tcolorbox **Linear Projection:** A linear projection is a transformation that maps a vector from one vector space to another using a matrix multiplication. It is defined by a weight matrix that determines the mapping. ::: The linear projections for the $i$-th attention head can be expressed as: $$Q_i = \text{Input} \times W^Q_i \label{eq:query_projection}$$ $$K_i = \text{Input} \times W^K_i \label{eq:key_projection}$$ $$V_i = \text{Input} \times W^V_i \label{eq:value_projection}$$ where $W^Q_i$, $W^K_i$, and $W^V_i$ are the weight matrices for the Query, Key, and Value projections of the $i$-th head, respectively. ::: tcolorbox Projecting the Q, K, and V matrices into different subspaces allows each attention head to focus on different aspects of the relationships between tokens. It's like looking at the input from multiple perspectives, each highlighting a different type of information. ::: ## Scaled Dot-Product Attention The core of the attention mechanism is the scaled dot-product attention, which calculates the attention scores between the Query and Key matrices. ::: tcolorbox **Scaled Dot-Product Attention:** Scaled Dot-Product Attention computes the attention scores between the Query and Key matrices by taking their dot product, scaling the result, and then applying a softmax function to obtain a probability distribution over the tokens. This distribution represents the attention weights, indicating the relevance of each token to the query. ::: The scaled dot-product attention is computed as follows: $$\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V \label{eq:scaled_dot_product_attention}$$ where $d_k$ is the dimensionality of the Key vectors. The scaling factor $\frac{1}{\sqrt{d_k}}$ is used to prevent the dot products from becoming too large, which can lead to issues with the gradients during training. ::: tcolorbox Scaling by $\frac{1}{\sqrt{d_k}}$ is crucial for stable training. Without it, the dot products can grow large, pushing the softmax function into regions with extremely small gradients, hindering the learning process. ::: ## Multiple Heads The \"Multi-Head\" aspect of the attention mechanism refers to the use of multiple attention heads, each with its own set of Q, K, and V matrices and its own set of learnable weight matrices. Each head computes the scaled dot-product attention independently, allowing the model to capture diverse relationships between tokens. - **Multiple Heads**: Each attention head operates independently, computing its own attention scores and producing its own output. - **Purpose**: The use of multiple heads allows the model to attend to different types of relationships between tokens simultaneously. Each head can learn to focus on a different aspect of the input, such as syntactic dependencies, semantic relationships, or long-range dependencies. Figure [8](#fig:multiple_heads){reference-type="ref" reference="fig:multiple_heads"} illustrates the concept of multiple attention heads operating in parallel. <figure id="fig:multiple_heads"> <figcaption>Multiple Attention Heads Operating in Parallel</figcaption> </figure> ## Concatenation and Linear Projection After each attention head has computed its output, the outputs are concatenated and then passed through a final linear layer. This step combines the information captured by the different heads and produces the final output of the Multi-Head Attention mechanism. The concatenation and linear projection can be expressed as: $$\text{MultiHead}(Q, K, V) = \text{Concat}(\text{head}_1, \text{head}_2, \ldots, \text{head}_h)W\label{eq:multi_head_output}$$ where $\text{head}_i = \text{Attention}(Q_i, K_i, V_i)$ is the output of the $i$-th attention head, and $W^O$ is the output weight matrix, another learnable parameter. ::: tcolorbox The final linear layer serves to integrate the information from all the attention heads and project it back into the original embedding space, ensuring that the output of the Multi-Head Attention block has the same dimensions as the input. ::: # Conclusion The Transformer architecture, particularly its encoder component as exemplified by BERT, represents a monumental leap forward in the field of natural language processing. Its innovative use of attention mechanisms and its ability to maintain input dimensionality have significantly improved the ability of models to understand and process text. Key takeaways from this lecture include: - **Dimensionality Preservation**: Transformers maintain the dimensionality of the input throughout the encoding process, allowing for the construction of deep models without information loss due to dimensionality reduction. - **Multi-Head Attention**: The Multi-Head Attention mechanism is the cornerstone of the Transformer's success. It enables the model to capture complex relationships between tokens by attending to different representation subspaces, effectively looking at the input from multiple perspectives. - **Positional Encoding**: While initially considered crucial for injecting positional information into the model, positional encoding has undergone modifications and its necessity is sometimes debated in recent implementations. Its role is often replaced or augmented by learned positional embeddings. - **BERT's Impact**: BERT, with its focused use of the Transformer's encoder, has demonstrated state-of-the-art performance on a wide array of NLP tasks, highlighting the power of bidirectional context understanding. ::: tcolorbox The Transformer architecture has revolutionized NLP by introducing the powerful Multi-Head Attention mechanism and demonstrating the effectiveness of maintaining input dimensionality. BERT's success further underscores the importance of bidirectional context in language understanding. ::: **Follow-up Questions for the Next Lecture:** To further our understanding of Transformer models and their broader implications, the following questions will be addressed in the next lecture: - **Transformer Decoder**: - How does the decoder part of the Transformer work, and how is it employed in sequence-to-sequence tasks like machine translation? What are the intricacies of its architecture and how does it interact with the encoder? - **Limitations and Solutions**: What are the inherent limitations of the Transformer architecture, such as its computational cost for very long sequences or its potential biases? How can these limitations be addressed through architectural modifications or training strategies? - **Evolution of Transformers**: How do recent advancements, such as GPT and other large language models, build upon the foundational Transformer architecture? What are the key innovations and improvements introduced by these models? - **Beyond NLP**: What are some practical applications of Transformer models beyond the realm of natural language processing? How are they being adapted for use in computer vision, time series analysis, or other domains, and what are the challenges and opportunities in these areas?