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Add LAAI3 computational learning
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\input{sections/_intro.tex}
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\input{sections/_turing.tex}
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\input{sections/_complexity.tex}
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\input{sections/_computational_learning.tex}
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\end{document}
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@ -0,0 +1,159 @@
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\chapter{Computational learning theory}
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\begin{description}
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\item[Instance space] \marginnote{Instance space}
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Set $X$ of (encoded) instances of objects that a learner wants to classify.
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Data from the instance space is drawn from a distribution $\mathcal{D}$ unknown to the learner.
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\item[Concept] \marginnote{Concept}
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Subset $c \subseteq X$ of the instance space which can be intended as properties of objects (i.e. a way to classify the instance space).
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\item[Concept class] \marginnote{Concept class}
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Collection $\mathcal{C} \subseteq \mathbb{P}(X)$ of concepts.
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It represents the concepts that are sufficiently simple for the algorithm to handle (i.e. the space of learnable concepts).
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\begin{description}
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\item[Target concept]
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Concept $c \in \mathcal{C}$ that the learner wants to learn.
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\end{description}
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\begin{remark}
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A learning algorithm is designed to learn concepts from a concept class
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neither knowing the target concept nor its data distribution.
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\end{remark}
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\item[Learning algorithm] \marginnote{Learning algorithm}
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Given a concept class $\mathcal{C}$ and a target concept $c \in \mathcal{C}$ with unknown distribution $\mathcal{D}$,
|
||||
a learning algorithm $\mathcal{A}$ takes as input:
|
||||
\begin{itemize}
|
||||
\item $\varepsilon$, the error parameter (or accuracy if seen as $(1-\varepsilon)$),
|
||||
\item $\delta$, the confidence parameter,
|
||||
\item $EX(c, \mathcal{D})$, an oracle that $\mathcal{A}$ can call to retrieve a data point $x \sim \mathcal{D}$
|
||||
with a label to indicate whether it is in the target concept $c$ or not (i.e. training data),
|
||||
\end{itemize}
|
||||
and outputs a concept $h \in \mathcal{C}$.
|
||||
\begin{center}
|
||||
\includegraphics[width=0.3\linewidth]{./img/_learning_algorithm.pdf}
|
||||
\end{center}
|
||||
|
||||
\begin{description}
|
||||
\item[Probability of error] \marginnote{Probability of error}
|
||||
Given a concept class $\mathcal{C}$,
|
||||
a target concept $c \in \mathcal{C}$ with unknown distribution $\mathcal{D}$ and
|
||||
a learning algorithm $\mathcal{A}$,
|
||||
the probability of error (i.e. misclassifications) for any output $h \in \mathcal{C}$ of $\mathcal{A}$ is defined as:
|
||||
\[ \text{error}_{\mathcal{D}, c} = \mathcal{P}_{x \sim \mathcal{D}}[ h(x) \neq c(x) ] \]
|
||||
\end{description}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.35\linewidth]{./img/_learning_model.pdf}
|
||||
\caption{General idea of a learning algorithm $\mathcal{A}$ computed as a function $f_\mathcal{A}$}
|
||||
\end{figure}
|
||||
|
||||
\item[PAC learnability] \marginnote{PAC learnability}
|
||||
A concept class $\mathcal{C}$ over the instance space $X$ is probably approximately correct (PAC) learnable iff there is an algorithm $\mathcal{A}$ such that:
|
||||
\begin{itemize}
|
||||
\item For each target concept $c \in \mathcal{C}$,
|
||||
\item For each distribution $\mathcal{D}$,
|
||||
\item For each error $0 < \varepsilon < \frac{1}{2}$,
|
||||
\item For each confidence $0 < \delta < \frac{1}{2}$,
|
||||
\end{itemize}
|
||||
it holds that:
|
||||
\[ \mathcal{P}\left[ \text{error}_{\mathcal{D}, c}\Big( \mathcal{A}\big( EX(c, \mathcal{D}), \varepsilon, \delta \big) \Big) < \varepsilon \right] > 1-\delta \]
|
||||
where the probability is computed by sampling data points from $EX(c, \mathcal{D})$.
|
||||
|
||||
In other words, the probability that $\mathcal{A}$ has an error rate lower than $\varepsilon$ (or an accuracy higher than $(1-\varepsilon)$) is greater than $(1-\delta)$.
|
||||
|
||||
\begin{description}
|
||||
\item[Efficient PAC learnability] \marginnote{Efficient PAC learnability}
|
||||
A concept class $\mathcal{C}$ is efficiently PAC learnable iff
|
||||
it is PAC learnable and the algorithm $\mathcal{A}$ that learns it has
|
||||
a time complexity bound to a polynomial in $\frac{1}{\varepsilon}$ and $\frac{1}{\delta}$.
|
||||
|
||||
\begin{remark}
|
||||
The complexity of $\mathcal{A}$ is measured taking into account the number of calls to $EX(c, \mathcal{D})$.
|
||||
\end{remark}
|
||||
\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]
|
||||
\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}
|
||||
|
||||
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}
|
||||
|
||||
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}
|
||||
|
||||
|
||||
\begin{theorem}[Axes-aligned rectangles in $\mathbb{R}^2_{[0, 1]}$ PAC learnability]
|
||||
It holds that:
|
||||
\begin{itemize}
|
||||
\item For every distribution $\mathcal{D}$,
|
||||
\item For every error $0 < \varepsilon < \frac{1}{2}$,
|
||||
\item For every confidence $0 < \delta < \frac{1}{2}$,
|
||||
\end{itemize}
|
||||
if $m \geq \frac{4}{\varepsilon}\ln\left( \frac{4}{\delta} \right)$, then:
|
||||
\[
|
||||
\mathcal{P}_{D \sim \mathcal{D}^m}
|
||||
\left[ \text{error}_{\mathcal{D}, T}\Big( \mathcal{A}_\text{BFP}\big(T(D)\big) \Big) < \varepsilon \right] > 1 - \delta
|
||||
\]
|
||||
where $D \sim \mathcal{D}^m$ is a sample of $m$ data points (i.e. training data)
|
||||
and $T(\cdot)$ labels the input data wrt to the target rectangle $T$.
|
||||
|
||||
\begin{proof}
|
||||
By definition, the error of $\mathcal{A}_\text{BFP}$ is defined as:
|
||||
\[ \text{error}_{\mathcal{D}, T}(R) = \mathcal{P}_{x \sim \mathcal{D}} [ x \in (T \smallsetminus R) ] \]
|
||||
|
||||
Consider the space defined by $(T \smallsetminus R)$ divided in four sections $E_1 \cup \dots \cup E_4 = (T \smallsetminus R)$:
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\linewidth]{./img/_rectangle_space.pdf}
|
||||
\end{figure}
|
||||
|
||||
Consider the probabilistic event "$x \in E_i$".
|
||||
For the training data $x \sim \mathcal{D}$ this holds iff none of those points
|
||||
end up in $E_i$ as, if a training point is in $E_i$, $R$ would be bigger to include it and $E_i$ would be smaller.
|
||||
|
||||
Now consider four other regions $F_1, \dots, F_4$ of the plane related to $E_i$ but defined differently
|
||||
in such a way that $\mathcal{P}_{x \sim D}[x \in F_i] = \frac{\varepsilon}{4}$.
|
||||
This can be achieved by expanding the $E_i$ regions to take some area of the rectangle $R$.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\linewidth]{./img/_rectangle_space2.pdf}
|
||||
\end{figure}
|
||||
|
||||
Then, as $E_i$ are smaller than $F_i$, it holds that:
|
||||
\[
|
||||
\begin{split}
|
||||
\mathcal{P}_{x \sim D}[x \in E_i] < \frac{\varepsilon}{4} &\Rightarrow \mathcal{P}_{x \sim D}[x \in (T \smallsetminus R)] < \varepsilon \\
|
||||
& \Rightarrow \text{error}_{\mathcal{D}, T}(R) < \varepsilon
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\textit{To be continued\dots}
|
||||
\end{proof}
|
||||
\end{theorem}
|
||||
\end{example}
|
||||
Reference in New Issue
Block a user