Add NLP LLM + MLM

src/year2/natural-language-processing/nlp.tex

@@ -15,5 +15,6 @@
     \include{./sections/_rnn.tex}
     \include{./sections/_attention.tex}
     \include{./sections/_llm.tex}
+    \include{./sections/_mlm.tex}
 
 \end{document}

src/year2/natural-language-processing/sections/_attention.tex

@@ -144,15 +144,15 @@
             \end{descriptionlist}
         where $\matr{W}_Q \in \mathbb{R}^{d_\text{model} \times d_k}$, $\matr{W}_K \in \mathbb{R}^{d_\text{model} \times d_k}$, and $\matr{W}_V \in \mathbb{R}^{d_\text{model} \times d_v}$ are parameters.
 
-        Then, the attention weights $\vec{\alpha}_{i,j}$ between two embeddings $\vec{x}_i$ and $\vec{x}_j$ are computed as:
+        Then, the attention weights $\alpha_{i,j}$ between two embeddings $\vec{x}_i$ and $\vec{x}_j$ are computed as:
         \[
             \begin{gathered}
-                \texttt{scores}(\vec{x}_i, \vec{x}_j) = \frac{\vec{q}_i \cdot \vec{k}_j}{\sqrt{d_k}} \\
-                \vec{\alpha}_{i,j} = \texttt{softmax}_j\left( \texttt{scores}(\vec{x}_i, \vec{x}_j) \right)
+                \texttt{scores}(\vec{x}_i, \vec{x}_j) = \frac{\vec{q}_i \vec{k}_j}{\sqrt{d_k}} \\
+                \alpha_{i,j} = \texttt{softmax}_j\left( \left[\texttt{scores}(\vec{x}_i, \vec{x}_1), \dots, \texttt{scores}(\vec{x}_i, \vec{x}_T)\right] \right) \\
             \end{gathered}
         \]
         The output $\vec{a}_i \in \mathbb{R}^{1 \times d_v}$ is a weighted sum of the values of each token:
-        \[ \vec{a}_i = \sum_{t} \vec{\alpha}_{i,t} \vec{v}_t \]
+        \[ \vec{a}_i = \sum_{t} \alpha_{i,t} \vec{v}_t \]
 
         To maintain the input dimension, a final projection $\matr{W}_O \in \mathbb{R}^{d_v \times d_\text{model}}$ is applied.
 
@@ -166,12 +166,18 @@
         Self-attention mechanism where only past tokens can be used to determine the representation of a token at a specific position. It is computed by modifying the standard self-attention as:
         \[
             \begin{gathered}
-                \forall j \leq i: \texttt{scores}(\vec{x}_i, \vec{x}_j) = \frac{\vec{q}_i \cdot \vec{k}_j}{\sqrt{d_k}} \qquad
-                \forall j > i: \texttt{scores}(\vec{x}_i, \vec{x}_j) = \nullvec \\
-                \vec{\alpha}_{i,j} = \texttt{softmax}_j\left( \texttt{scores}(\vec{x}_i, \vec{x}_j) \right) \\
-                \vec{a}_i = \sum_{t: t \leq i} \vec{\alpha}_{i,t} \vec{v}_t
+                \forall j \leq i: \texttt{scores}(\vec{x}_i, \vec{x}_j) = \frac{\vec{q}_i \vec{k}_j}{\sqrt{d_k}} \qquad
+                \forall j > i: \texttt{scores}(\vec{x}_i, \vec{x}_j) = -\infty \\
+                \alpha_{i,j} = \texttt{softmax}_j\left( \left[\texttt{scores}(\vec{x}_i, \vec{x}_1), \dots, \texttt{scores}(\vec{x}_i, \vec{x}_T)\right] \right) \\
+                \vec{a}_i = \sum_{t: t \leq i} \alpha_{i,t} \vec{v}_t
             \end{gathered}
         \]
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.2\linewidth]{./img/_masked_attention.pdf}
+            \caption{Score matrix with causal attention}
+        \end{figure}
 \end{description}
 
 
@@ -195,7 +201,7 @@
 
         \begin{figure}[H]
             \centering
-            \includegraphics[width=0.4\linewidth]{./img/_positional_encoding.pdf}
+            \includegraphics[width=0.45\linewidth]{./img/_positional_encoding.pdf}
         \end{figure}
 
     \item[Transformer block] \marginnote{Transformer block}
@@ -206,7 +212,7 @@
 
                 \begin{figure}[H]
                     \centering
-                    \includegraphics[width=0.5\linewidth]{./img/_multi_head_attention.pdf}
+                    \includegraphics[width=0.6\linewidth]{./img/_multi_head_attention.pdf}
                 \end{figure}
 
             \item[Feedforward layer]

src/year2/natural-language-processing/sections/_llm.tex

@@ -108,7 +108,85 @@
                     $k$ is fixed and does not account for the shape of the distribution.
                 \end{remark}
 
-            \item[Top-p sampling]
+            \item[Top-p/nucleus sampling]
                 Consider the most likely words such that their probability mass adds up to $p$. Then, apply random sampling on their normalized distribution.
         \end{description}
+\end{description}
+
+
+
+\section{Training}
+
+
+\subsection{Pre-training}
+
+\begin{description}
+    \item[Pre-training] \marginnote{Pre-training}
+        Use self-supervision and teacher forcing to train the whole context window in parallel on a large text corpus.
+
+        \begin{remark}
+            Results are highly dependent on the training corpora. Important aspects to consider are:
+            \begin{descriptionlist}
+                \item[Language] Most of the available data is in English.
+                \item[Data quality] Prefer high-quality sources such as Wikipedia or books. Boilerplate removal and deduplication might be needed. 
+                \item[Safety filtering] Toxicity removal might be needed
+                \item[Ethical and legal issues] Use of copyrighted material, permission from data owners, use of private information, \dots
+            \end{descriptionlist}
+        \end{remark}
+
+    \item[Scaling laws] \marginnote{Scaling laws}
+        Empirical laws that put in relationship:
+        \begin{itemize}
+            \item Non-embedding parameters $N$ ($N \approx 2 d_\text{model} n_\text{layer} (2 d_\text{attention} + d_\text{ff})$),
+            \item Training data size $D$,
+            \item Compute budget $C$ (i.e., training iterations).
+        \end{itemize}
+        By keeping two of the three factors constant, the loss $\mathcal{L}$ of an LLM can be estimated as a function of the third variable:
+        \[ 
+            \mathcal{L}(N) = \left( \frac{N_c}{N} \right)^{\alpha N} 
+            \qquad
+            \mathcal{L}(D) = \left( \frac{D_c}{D} \right)^{\alpha D} 
+            \qquad
+            \mathcal{L}(C) = \left( \frac{C_c}{C} \right)^{\alpha C} 
+        \]
+\end{description}
+
+
+\subsection{Fine-tuning}
+
+\begin{description}
+    \item[Fine-tuning] \marginnote{Fine-tuning}
+        Specialize an LLM to a specific domain or task.
+
+        \begin{description}
+            \item[Continued pre-training] \marginnote{Continued pre-training}
+                Continue pre-training with a domain-specific corpus.
+
+            \item[Model adaptation]
+                Specialize a model by adding new learnable parameters.
+
+                \begin{description}
+                    \item[Parameter-efficient fine-tuning (PEFT)] \marginnote{Parameter-efficient fine-tuning (PEFT)}
+                        Continue training a selected subset of parameters.
+
+                        \begin{description}
+                            \item[Low-rank adaptation (LoRA)] \marginnote{Low-rank adaptation (LoRA)}
+                                Method to update weights by learning an offset that uses fewer parameters.
+
+                                Consider a weight matrix $\matr{W} \in \mathbb{R}^{d \times k}$, LoRA decomposes the update into two learnable matrices $\matr{A} \in \mathbb{R}^{d \times r}$ and $\matr{B} \in \mathbb{R}^{r \times k}$ (with $r \ll d, k$). Weights update is performed as:
+                                \[ \matr{W}_{\text{fine-tuned}} = \matr{W}_{\text{pre-trained}} + \matr{AB} \]
+
+                                \begin{figure}[H]
+                                    \centering
+                                    \includegraphics[width=0.35\linewidth]{./img/_lora.pdf}
+                                \end{figure}
+                        \end{description}
+
+                    \item[Task-specific fine-tuning] \marginnote{Task-specific fine-tuning}
+                        Add a new trainable head on top of the model.
+                \end{description}
+
+            \item[Supervised fine-tuning] \marginnote{Supervised fine-tuning}
+                Continue training using a supervised dataset to align the model to human's expectation.
+        \end{description}
 \end{description}

src/year2/natural-language-processing/sections/_mlm.tex

@@ -0,0 +1,55 @@
+\chapter{Masked language models}
+
+
+
+\section{Bidirectional transformer encoder}
+
+\begin{description}
+    \item[Transformer encoder] \marginnote{Transformer encoder}
+        Architecture that produces contextual embeddings by considering both left-to-right and right-to-left context.
+
+        \begin{remark}
+            This architecture does feature extraction and is more suited for classification tasks.
+        \end{remark}
+
+        \begin{description}
+            \item[Architecture]
+                Similar to a transformer decoder, but self-attention is not causal.
+
+                \begin{figure}[H]
+                    \centering
+                    \includegraphics[width=0.75\linewidth]{./img/_decoder_vs_encoder.pdf}
+                \end{figure}
+        \end{description}
+\end{description}
+
+
+\subsection{Masked language modelling}
+
+\begin{description}
+    \item[Masked language modelling] \marginnote{Masked language modelling}
+        Main training task of transformer encoders. It consists of predicting missing or corrupted tokens in a sequence.
+
+        \begin{remark}
+            Transformer encoders output embeddings. For training purposes, a head to output a distribution over the vocabulary is added.
+        \end{remark}
+
+        \begin{example}
+            Given a training corpus, BERT is trained by randomly sampling $15\%$ of the tokens in the training data and either:
+            \begin{itemize}
+                \item Mask it with a special \texttt{[MASK]} token ($80\%$ of the time).
+                \item Replace it with a different token ($10\%$ of the time).
+                \item Do nothing ($10\%$ of the time).
+            \end{itemize}
+
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=0.6\linewidth]{./img/_bert_training.pdf}
+            \end{figure}
+
+            \indenttbox
+            \begin{remark}
+                BERT's training approach is inefficient as masks are determined before training and only $15\%$ of the corpus tokens are actually used for training. Other models (e.g., RoBERTa), dynamically determine the mask at training time, allowing for more variety.
+            \end{remark}
+        \end{example}
+\end{description}