Add missing corollary and sections reorder

src/languages-and-algorithms-for-ai/module3/sections/_computational_learning.tex

@@ -79,36 +79,39 @@
         \end{description}
 \end{description}
 
-\begin{example}[Axes-aligned rectangles in $\mathbb{R}^2_{[0, 1]}$]
-    Consider the instance space $X = \mathbb{R}^2_{[0, 1]}$
-    and the concept class $\mathcal{C}$ of concepts represented by all the points contained within a rectangle parallel to the axes of arbitrary size.
 
-    \begin{figure}[H]
+
+\section{Axes-aligned rectangles over $\mathbb{R}^2_{[0, 1]}$}
+
+Consider the instance space $X = \mathbb{R}^2_{[0, 1]}$
+and the concept class $\mathcal{C}$ of concepts represented by all the points contained within a rectangle parallel to the axes of arbitrary size.
+
+\begin{figure}[H]
     \centering
     \includegraphics[width=0.2\linewidth]{./img/_learning_rectangle.pdf}
     \caption{Example of problem instance. The gray rectangle is the target concept, red dots are positive data points and blue dots are negative data points.}
-    \end{figure}
+\end{figure}
 
-    An algorithm has to guess a classifier (i.e. a rectangle) without knowing the target concept and the distribution of its training data.
-    Let an algorithm $\mathcal{A}_\text{BFP}$ be defined as follows:
-    \begin{itemize}
+An algorithm has to guess a classifier (i.e. a rectangle) without knowing the target concept and the distribution of its training data.
+Let an algorithm $\mathcal{A}_\text{BFP}$ be defined as follows:
+\begin{itemize}
     \item Take as input some data $\{ ((x_1, y_1), p_1), \dots, ((x_n, y_n), p_n) \}$ where 
         $(x_i, y_i)$ are the coordinates of the point and $p_i$ indicates if the point is within the target rectangle.
     \item Return the smallest rectangle that includes all the positive instances.
-    \end{itemize}
+\end{itemize}
 
-    Given the rectangle $R$ predicted by $\mathcal{A}_\text{BFP}$ and the target rectangle $T$,
-    the probability of error in using $R$ in place of $T$ is:
-    \[ \text{error}_{\mathcal{D}, T}(R) = \mathcal{P}_{x \sim \mathcal{D}} [ x \in (R \smallsetminus T) \cup (T \smallsetminus R) ] \]
-    In other words, a point is misclassified if it is in $R$ but not in $T$ or vice versa.
-    \begin{remark}
+Given the rectangle $R$ predicted by $\mathcal{A}_\text{BFP}$ and the target rectangle $T$,
+the probability of error in using $R$ in place of $T$ is:
+\[ \text{error}_{\mathcal{D}, T}(R) = \mathcal{P}_{x \sim \mathcal{D}} [ x \in (R \smallsetminus T) \cup (T \smallsetminus R) ] \]
+In other words, a point is misclassified if it is in $R$ but not in $T$ or vice versa.
+\begin{remark}
     By definition of $\mathcal{A}_\text{BFP}$, it always holds that $R \subseteq T$. 
     Therefore, $(R \smallsetminus T) = \varnothing$ and the error can be rewritten as:
     \[ \text{error}_{\mathcal{D}, T}(R) = \mathcal{P}_{x \sim \mathcal{D}} [ x \in (T \smallsetminus R) ] \]
-    \end{remark}
+\end{remark}
 
 
-    \begin{theorem}[Axes-aligned rectangles in $\mathbb{R}^2_{[0, 1]}$ PAC learnability]
+\begin{theorem}[Axes-aligned rectangles over $\mathbb{R}^2_{[0, 1]}$ PAC learnability]
     It holds that:
     \begin{itemize}
         \item For every distribution $\mathcal{D}$,
@@ -155,5 +158,9 @@
         
         \textit{To be continued\dots}
     \end{proof}
-    \end{theorem}
-\end{example}
+\end{theorem}
+
+
+\begin{corollary}
+    The concept class of axis-aligned rectangles over $\mathbb{R}^2_{[0, 1]}$ is efficiently PAC learnable.
+\end{corollary}