Fix typos <noupdate>

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

@@ -46,7 +46,7 @@
                             \vec{s}^{(t)} \cdot \vec{h}^{(N)} 
                         \end{bmatrix} \in \mathbb{R}^{N} 
                 \]
-                $\vec{e}^{(t)}$ is used to determine the attention distribution $\vec{\alpha}^{(t)}$ that is required to obtain the attention output $\vec{a}^{(t)}$ as the weighted encoder hidden states :
+                $\vec{e}^{(t)}$ is used to determine the attention distribution $\vec{\alpha}^{(t)}$ that is required to obtain the attention output $\vec{a}^{(t)}$ as the weighted sum of the encoder hidden states:
                 \[
                     \begin{gathered}
                         \mathbb{R}^{N} \ni \vec{\alpha}^{(t)} = \texttt{softmax}(\vec{e}^{(t)}) \\

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

@@ -71,7 +71,7 @@
                 \hat{c} &= \arg\max_{c \in C} \prob{c | d} \\
                 &= \arg\max_{c \in C} \underbrace{\prob{d | c}}_{\text{likelihood}} \underbrace{\prob{c}}_{\text{prior}} \\
                 &= \arg\max_{c \in C} \prob{w_1, \dots, w_n | c} \prob{c} \\
-                &= \arg\max_{c \in C} \prod_{i} \prob{w_i | c} \prob{c} \\
+                &\approx \arg\max_{c \in C} \prod_{i} \prob{w_i | c} \prob{c} \\
                 &= \arg\max_{c \in C} \sum_{i} \log\prob{w_i | c} \log\prob{c} \\
             \end{split}
         \]
@@ -82,7 +82,7 @@
             \qquad
             \prob{w_i | c} = \frac{\texttt{count}(w_i, c)}{\sum_{v \in V} \texttt{count}(v, c)}
         \]
-        where $N_c$ is the number of documents with class $c$ and $\texttt{count}(w, c)$ counts the occurrences of the word $w$ in the training samples with class $c$.
+        where $N_c$ is the number of documents of class $c$ and $\texttt{count}(w, c)$ counts the occurrences of the word $w$ in the training samples of class $c$.
 
         \begin{remark}
             Laplace smoothing is used to avoid zero probabilities.
@@ -226,7 +226,7 @@ Naive Bayes has the following properties:
                     \end{cases} 
                 \]
                 By applying a log-transformation and inverting the sign, this corresponds to the cross-entropy loss in the binary case:
-                \[ \mathcal{L}_{\text{BCE}}(\hat{y}, y) = -\log \prob{y | x} = -[y \log(\hat{y}) + (1-y)\log(1-\hat{y})] \]
+                \[ \mathcal{L}_{\text{BCE}}(\hat{y}, y) = -\log(\prob{y | x}) = -[y \log(\hat{y}) + (1-y)\log(1-\hat{y})] \]
 
             \item[Optimization]
                 As cross-entropy is convex, SGD is well suited to find the parameters $\vec{\theta}$ of a logistic regressor $f$ over batches of $m$ examples by solving:
@@ -255,7 +255,7 @@ Logistic regression has the following properties:
 \end{itemize}
 
 \begin{remark}
-    Logistic regression is also a good baseline when experimenting with text classifications.
+    Logistic regression is also a good baseline when experimenting with text classification. 
     
     As they are lightweight to train, it is a good idea to test both naive Bayes and logistic regression to determine the best baseline for other experiments.
 \end{remark}

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

@@ -60,7 +60,7 @@
         \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 computes as:
+        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 computed as:
         \[ \log\left( \prob{y \mid x} \right) = \sum_{i=1}^{t} \log\left( \prob{ y_i \mid x, y_1, \dots, y_{i-1} } \right) \]
 
         \begin{example}
@@ -192,7 +192,7 @@
         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.
+            This architecture performs feature extraction and is more suited for classification tasks.
         \end{remark}
 
         \begin{description}
@@ -206,7 +206,7 @@
         \end{description}
 
     \item[Contextual embedding] \marginnote{Contextual embedding}
-        Represent the meaning of word instances (i.e., dynamically depending on the surroundings).
+        Represent the meaning of word instances (i.e., in a dynamic manner depending on the surroundings).
 
         \begin{remark}[Sequence embedding]
             Encoders usually have a classifier token (e.g., \texttt{[CLS]}) to model the whole sentence.

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

@@ -61,7 +61,7 @@
 
             \item Fine-tune the language model (i.e., train the policy) using an RL algorithm (e.g., PPO) and the learned reward model. 
 
-            Given a prompt $x$ and an answer $y$, the reward $r$ used for RL update is computed as:
+            Given a prompt $x$ and an answer $y$, the reward $r$ used for the RL update is computed as:
             \[ r = r_\theta(y \mid x) - \lambda_\text{KL} D_\text{KL}(\pi_{\text{PPO}}(y \mid x) \Vert \pi_{\text{base}}(y \mid x)) \]
             where:
             \begin{itemize}

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

@@ -207,7 +207,7 @@
                 Provide an initial hidden state to the RNN (e.g., speech-to-text).
         \end{description}
 
-    \item[Sequence labelling] \marginnote{Sequence labelling}
+    \item[Sequence labeling] \marginnote{Sequence labeling}
         Assign a class to each input token (e.g., POS-tagging, named-entity recognition, structure prediction, \dots).
 
     \item[Sequence classification] \marginnote{Sequence classification}

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

@@ -406,7 +406,7 @@
     \item[Neural language modeling] \marginnote{Neural language modeling}
         Use a neural network to predict the next word $w_{n+1}$ given an input sequence $w_{1..n}$. The general flow is the following:
         \begin{enumerate}
-            \item Encode the input words into one-hot vectors ($\mathbb{R}^{|V| \times n}$). 
+            \item Encode the input words into one-hot vectors ($\mathbb{N}^{|V| \times n}$). 
             \item Project the input vectors with an embedding matrix $\matr{E} \in \mathbb{R}^{d \times |V|}$ that encodes them into $d$-dimensional vectors.
             \item Pass the embedding into the hidden layers.
             \item The final layer is a probability distribution over the vocabulary ($\mathbb{R}^{|V| \times 1}$).
@@ -478,7 +478,7 @@
                         Use a binary logistic regressor as classifier. The two classes are:
                         \begin{itemize}
                             \item Context words within the context window (positive label).
-                            \item Words randomly sampled (negative label).
+                            \item Randomly sampled words (negative label).
                         \end{itemize} 
                         The probabilities can be computed as:
                         \[