Add NLP transformer + decoding strategies

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

@@ -14,5 +14,6 @@
     \include{./sections/_semantics.tex}
     \include{./sections/_rnn.tex}
     \include{./sections/_attention.tex}
+    \include{./sections/_llm.tex}
 
 \end{document}

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

@@ -121,12 +121,15 @@
 
 
 
-\section{Transformers}
+\section{Transformer decoder (for language modelling)}
+
+
+\subsection{Self-attention}
 
 \begin{description}
     \item[Self-attention] \marginnote{Self-attention}
         Component that allows to compute the representation of a token considering the other ones in the input sequence. 
-        
+
         Given an input embedding $\vec{x}_i \in \mathbb{R}^{1 \times d_\text{model}}$, self-attention relies on the following values:
         \begin{descriptionlist}
             \item[Queries] \marginnote{Queries}
@@ -169,4 +172,75 @@
                 \vec{a}_i = \sum_{t: t \leq i} \vec{\alpha}_{i,t} \vec{v}_t
             \end{gathered}
         \]
+\end{description}
+
+
+\subsection{Components}
+
+\begin{description}
+    \item[Input embedding] \marginnote{Input embedding}
+        The input is tokenized using standard tokenizers (e.g., BPE, SentencePiece, \dots). Each token is encoded using a learned embedding matrix.
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.45\linewidth]{./img/_transformer_embedding.pdf}
+        \end{figure}
+
+    \item[Positional encoding] \marginnote{Positional encoding}
+        Learned position embeddings to encode positional information are added to the input token embeddings.
+
+        \begin{remark}
+            Without positional encoding, transformers are invariant to permutations.
+        \end{remark}
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.4\linewidth]{./img/_positional_encoding.pdf}
+        \end{figure}
+
+    \item[Transformer block] \marginnote{Transformer block}
+        Module with the same input and output dimensionality (i.e., allows stacking multiple blocks) composed of:
+        \begin{descriptionlist}
+            \item[Multi-head attention] \marginnote{Multi-head attention}
+                Uses $h$ different self-attention blocks with different queries, keys, and values. Value vectors are of size $\frac{d_v}{h}$. The final projection $W_O$ is applied on the concatenation of the outputs of each head.
+
+                \begin{figure}[H]
+                    \centering
+                    \includegraphics[width=0.5\linewidth]{./img/_multi_head_attention.pdf}
+                \end{figure}
+
+            \item[Feedforward layer]
+                Fully-connected 2-layer network applied at each position of the attention output:
+                \[ \texttt{FFN}(\vec{x}_i) = \texttt{ReLU}(\vec{x}_i\matr{W}_1 + \vec{b}_1)\matr{W}_2 + \vec{b}_2 \]
+                Where the hidden dimension $d_\text{ff}$ is usually larger than $d_\text{model}$.
+
+            \item[Normalization layer]
+                Applies token-wise normalization (i.e., layer norm) to help training stability.
+
+            \item[Residual connection]
+                Helps to propagate information during training.
+
+                \begin{remark}[Residual stream]
+                    An interpretation of residual connections is the residual stream where the input token in enhanced by the output of multi-head attention and the feedforward network.
+
+                    \begin{figure}[H]
+                        \centering
+                        \includegraphics[width=0.38\linewidth]{./img/_residual_stream.pdf}
+                    \end{figure}
+                \end{remark}
+        \end{descriptionlist}
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.4\linewidth]{./img/_attention_block.pdf}
+            \caption{Overall attention block}
+        \end{figure}
+
+    \item[Language modelling head] \marginnote{Language modelling head}
+        Takes as input the output corresponding to a token of the transformer blocks stack and outputs a distribution over the vocabulary.
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.6\linewidth]{./img/_lm_head.pdf}
+        \end{figure}
 \end{description}

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

@@ -0,0 +1,114 @@
+\chapter{Large language models}
+
+
+\begin{description}
+    \item[Conditional generation] \marginnote{Conditional generation}
+        Generate text conditioned on the input tokens (i.e., prompt).
+
+        \begin{figure}[H]
+            \centering
+            \includegraphics[width=0.45\linewidth]{./img/_conditional_generation.pdf}
+        \end{figure}
+
+        \begin{example}[Sentiment analysis]
+            Given the prompt:
+            \[ p = \texttt{the sentiment of the sentence `I like Jackie Chan' is} \]
+            Determine the probability of the tokens \texttt{positive} and \texttt{negative}:
+            \[
+                \prob{\texttt{positive} \mid p} \qquad \prob{\texttt{negative} \mid p}
+            \]
+        \end{example}
+
+        \begin{example}[Question answering]
+            Given the prompt:
+            \[ p = \texttt{Q: who wrote the book `The origin of Species'? A:} \]
+            Determine the tokens of the answer autoregressively:
+            \[
+                \arg\max_{w_1} \prob{w_1 \mid p}, \arg\max_{w_2} \prob{w_2 \mid pw_1}, \dots
+            \]
+        \end{example}
+\end{description}
+
+
+\section{Decoding strategies}
+
+\begin{description}
+    \item[Greedy decoding] \marginnote{Greedy decoding}
+        Select the next token as the most probable of the output distribution.
+
+        \begin{remark}
+            Greedy decoding risks getting stuck in a local optimum.
+
+            \indenttbox
+            \begin{example}
+                Consider the following search tree of possible generated sequences:
+                \begin{figure}[H]
+                    \centering
+                    \includegraphics[width=0.3\linewidth]{./img/_greedy_decoding_local_minimum.pdf}
+                \end{figure}
+
+                Greedy search would select the sequence \texttt{yes yes} which has probability $0.5 \cdot 0.4 = 0.2$. However, the sequence \texttt{ok ok} has a higher probability of $0.4 \cdot 0.7 = 0.28$.
+            \end{example}
+        \end{remark}
+
+    \item[Beam search] \marginnote{Beam search}
+        Given a beam width $k$, perform a breadth-first search keeping at each branching level the top-$k$ tokens based on the probability of that sequence:
+        \[ \log\left( \prob{y \mid x} \right) = \sum_{i=1}^{t} \log\left( \prob{ y_i \mid x, y_1, \dots, y_{i-1} } \right) \]
+
+        \begin{example}
+            Consider the following tree with beam width $k=2$:
+            \begin{figure}[H]
+                \centering
+                \includegraphics[width=0.7\linewidth]{./img/_beam_search.pdf}
+            \end{figure}
+            The selected sequence is \texttt{[BOS] the green witch arrived [EOS]}.
+        \end{example}
+
+        \begin{remark}
+            As each path might generate sequences of different length, the score is usually normalized by the number of tokens as:
+            \[ \log\left( \prob{y \mid x} \right) = \frac{1}{t} \sum_{i=1}^{t} \log\left( \prob{ y_i \mid x, y_1, \dots, y_{i-1} } \right) \]
+        \end{remark}
+
+        \begin{remark}
+            The likelihood of the sequences generated using beam search is higher than using greedy decoding. However, beam search is still not optimal.
+        \end{remark}
+
+    \item[Sampling] \marginnote{Sampling}
+        Sample the next token based on the output distribution.
+
+        \begin{description}
+            \item[Random sampling]
+                Sample considering the distribution over the whole vocabulary.
+
+                \begin{remark}
+                    By adding-up all the low-probability words (which are most likely unreasonable as the next token), their actual chance of getting selected is relatively high.
+                \end{remark}
+
+            \item[Temperature sampling]
+                Skew the distribution to emphasize the most likely words and decrease the probability of less likely words. Given the logits $\vec{u}$ and the temperature $\tau$, the output distribution $\vec{y}$ is determined as:
+                \[ \vec{y} = \texttt{softmax}\left( \frac{\vec{u}}{\tau} \right) \]
+                where:
+                \begin{itemize}
+                    \item Higher temperatures (i.e., $\tau > 1$) allow for considering low-probability words.
+                    \item Lower temperatures (i.e., $\tau \in (0, 1]$) focus on high-probability words.
+                    \begin{remark}
+                        A temperature of $\tau = 0$ corresponds to greedy decoding.
+                    \end{remark}
+                \end{itemize}
+
+
+            \item[Top-k sampling]
+                Consider the top-$k$ most probable words and apply random sampling on their normalized distribution.
+
+                \begin{remark}
+                    $k=1$ corresponds to greedy decoding.
+                \end{remark}
+
+                \begin{remark}
+                    $k$ is fixed and does not account for the shape of the distribution.
+                \end{remark}
+
+            \item[Top-p 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}