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@ -30,7 +30,14 @@ One example of loss is the absolute loss: absolute difference between numbers\\
\section{Loss} \section{Loss}
\subsection{Absolute Loss} \subsection{Absolute Loss}
$$\ell(y,\hat{y} = | y - \hat{y} | \Rightarrow absolute \quad loss\\ $$ $$\ell(y,\hat{y} = | y - \hat{y} | \Rightarrow absolute \quad loss\\ $$
--- DISEGNO ---\\\\ \\
\begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{../img/lez2-img1.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\\
Some inconvenient properties: Some inconvenient properties:
\begin{itemize} \begin{itemize}
@ -40,8 +47,14 @@ Some inconvenient properties:
\subsection{Square Loss} \subsection{Square Loss}
$$ \ell(y,\hat{y} = ( y - \hat{y} )^2 \Rightarrow \textit{square loss}\\$$ $$ \ell(y,\hat{y} = ( y - \hat{y} )^2 \Rightarrow \textit{square loss}\\$$
-- DISEGNO ---\\ \\
Derivative : \begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{../img/lez2-img2.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\\Derivative :
\begin{itemize} \begin{itemize}
\item more informative \item more informative
\item and differentible \item and differentible

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Chapter 1.
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@ -19,8 +19,14 @@ Example $(x,y)$ \qquad y is the label associated with x\\
Learning with example $(x_1,y_1)...(x_m,y_m) \quad \textit{training set} $\\\\ Learning with example $(x_1,y_1)...(x_m,y_m) \quad \textit{training set} $\\\\
Training set is a set of examples with every algorithm can learn.......\\\\ Training set is a set of examples with every algorithm can learn.......\\\\
Learning algorithm take training set as input and produces a predictor as output.\\\\ Learning algorithm take training set as input and produces a predictor as output.\\\\
......DISEGNO \\\\ \\
With image recognition we use as measurement pixels.\\ \begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{../img/lez3-img1.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\\With image recognition we use as measurement pixels.\\
How do we measure the power of a predictor?\\ How do we measure the power of a predictor?\\
A learning algorithm will look at training set, algorithm and generate the predictor. Now the problem is verify the score. \\ A learning algorithm will look at training set, algorithm and generate the predictor. Now the problem is verify the score. \\
Now we can consider a test set collection of example Now we can consider a test set collection of example
@ -194,20 +200,29 @@ $
S is the traning set $(x_1,y_1)...(x_m,y_m) \\ x_t \in \barra{R}^d \qquad y_t \in \{-1,1\} \\\\ S is the traning set $(x_1,y_1)...(x_m,y_m) \\ x_t \in \barra{R}^d \qquad y_t \in \{-1,1\} \\\\
d = 2 \rightarrow \textit{2-dimensional vector}\\ d = 2 \rightarrow \textit{2-dimensional vector}\\
$\\ $\\
....-- DISEGNO --...
\\
where + and - are labels where + and - are labels
\\\\ \newpage
\textbf{Point of test set} \textbf{Point of test set}
\\ \\
If i want to predict this point? If i want to predict this point?
\begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{../img/lez3-img2.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
Maybe if point is close to point with label i know then. Maybe they have the same label. Maybe if point is close to point with label i know then. Maybe they have the same label.
\\ \\
$\hat{y} = + \quad or \quad \hat{y} = - $ $\hat{y} = + \quad or \quad \hat{y} = - $
\\\\ \\\\
.....-- DISEGNO -- ... \begin{figure}[h]
\\\ \centering
\includegraphics[width=0.4\linewidth]{../img/lez3-img3.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\\
I can came up with some sort of classifier. I can came up with some sort of classifier.
\\\\ \\\\
Given $S$ training set, i can define $\hnn$ $X \rightarrow \{-1,1\}\\ Given $S$ training set, i can define $\hnn$ $X \rightarrow \{-1,1\}\\

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@ -49,8 +49,12 @@ You can vary this algorithm as you want.\\\\
Lets go back to Binary classification.\\ Lets go back to Binary classification.\\
The $k$ parameter is the effect of making the structure of classifier more The $k$ parameter is the effect of making the structure of classifier more
complex and less complex for small value of $k$.\\\\ complex and less complex for small value of $k$.\\\\
--.. DISEGNO ..-- \begin{figure}[h]
\\ \centering
\includegraphics[width=0.4\linewidth]{../img/lez4-img1.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
Fix training set and test set\\ Fix training set and test set\\
Accury as oppose to the error Accury as oppose to the error
\\\\ \\\\
@ -83,6 +87,12 @@ I want to avoid comparing $x_i$ with $x_j$, $i\neq j $\\
so comparing different feature and we want to compare each feature with so comparing different feature and we want to compare each feature with
each self. I dont want to mix them up.\\ each self. I dont want to mix them up.\\
We can use a tree! We can use a tree!
\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{../img/lez4-img2.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\\ \\
I have 3 features: I have 3 features:
\begin{itemize} \begin{itemize}
@ -90,17 +100,21 @@ I have 3 features:
\item humidity $= \{[0,100]\}$ \item humidity $= \{[0,100]\}$
\item windy $ = \{yes,no\}$ \item windy $ = \{yes,no\}$
\end{itemize} \end{itemize}
... -- DISEGNO -- ...\\\\
Tree is a natural way of doing decision and abstraction of decision process of Tree is a natural way of doing decision and abstraction of decision process of
one person. It is a good way to deal with categorical variables.\\ one person. It is a good way to deal with categorical variables.\\
What kind of tree we are talking about?\\ What kind of tree we are talking about?\\
Tree has inner node and leaves. Leaves are associated with labels $(Y)$ and Tree has inner node and leaves. Leaves are associated with labels $(Y)$ and
inner nodes are associated with test. inner nodes are associated with test.
\begin{figure}[h]
\centering
\includegraphics[width=0.3\linewidth]{../img/lez4-img3.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\begin{itemize} \begin{itemize}
\item Inner node $\rightarrow$ test \item Inner node $\rightarrow$ test
\item Leaves $\rightarrow$ label in Y \item Leaves $\rightarrow$ label in Y
\end{itemize} \end{itemize}
%... -- DISEGNO -- ...
Test if a function $f$ (NOT A PREDICTOR!) \\ Test if a function $f$ (NOT A PREDICTOR!) \\
Test $ \qquad f_i \, X_i \rightarrow \{1,...,k\}$ Test $ \qquad f_i \, X_i \rightarrow \{1,...,k\}$
\\ where $k$ is the number of children (inner node) to which test is assigned \\ where $k$ is the number of children (inner node) to which test is assigned
@ -146,19 +160,37 @@ $ Y = \{-1, +1 \}$
\\\\ \\\\
What's the simplest way?\\ What's the simplest way?\\
Initial tree and correspond to a costant classifier Initial tree and correspond to a costant classifier
\\\\ \\
-- DISEGNO -- \begin{figure}[h]
\\\\ \centering
\includegraphics[width=0.2\linewidth]{../img/lez4-img4.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
\textbf{Majority of all example} \textbf{Majority of all example}
\\\\ \\
-- DISEGNO -- \begin{figure}[h]
\\\\ \centering
\includegraphics[width=0.2\linewidth]{../img/lez4-img5.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
$(x_1, y_1) ... (x_m, y_m)$ \\ $(x_1, y_1) ... (x_m, y_m)$ \\
$ x_t \in X$ \qquad $ y_t \in \{-1,+1\}$\\ $ x_t \in X$ \qquad $ y_t \in \{-1,+1\}$\\
Training set $S = \{ (x,y) \in S$, x is routed to $\ell\}$\\ Training set $S = \{ (x,y) \in S$, x is routed to $\ell\}$\\
$S_{\ell}^+$ $S_{\ell}^+$
\\\\ \\
-- DISEGNO -- \begin{figure}[h]
\\\\ \centering
$ S_{\ell}$ and $ S_{\ell}$ are given by the result of the test, not the labels and $\ell$ and $\ell'$. \includegraphics[width=0.8\linewidth]{../img/lez4-img6.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}\\
$ S_{\ell}$ and $ S_{\ell}$ are given by the result of the test, not the labels and $\ell$ and $\ell'$.\\
\begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{../img/lez4-img7.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\end{document} \end{document}

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@ -9,9 +9,14 @@ Supposed we groped a tree up to this point and we are wandering how to
grow it. grow it.
\\ \\
$S$ Training set $(x_1,y_1)...(x_m,y_m)$, $x_1 \in X$ $S$ Training set $(x_1,y_1)...(x_m,y_m)$, $x_1 \in X$
\\\\ \\
-- DISEGNO \begin{figure}[h]
\\\\ \centering
\includegraphics[width=0.4\linewidth]{../img/lez5-img1.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
$$ $$
\sll \equiv \{(x_1,y_1) \, x_t \quad \textit{is router to } \ell \} \sll \equiv \{(x_1,y_1) \, x_t \quad \textit{is router to } \ell \}
$$ $$
@ -57,22 +62,39 @@ where $\psi(a) = min \{a, 1-a \} \qquad a \in [0,1] $
\\ \\
I want to replace inner node with other leaves. I want to replace inner node with other leaves.
\\ \\
-- DISEGNO -- \begin{figure}[h]
\\\\ \centering
\includegraphics[width=0.8\linewidth]{../img/lez5-img2.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
How is traning error going to change? How is traning error going to change?
(when i replace inner nodes with other leaves) (when i replace inner nodes with other leaves)
\\ \\
Im hoping my algorithm is not going to overfit (if training error goes to 0 also Im hoping my algorithm is not going to overfit (if training error goes to 0 also
testing error goes to 0).\\ testing error goes to 0).
\newpage
\section{Jensens inequality} \section{Jensens inequality}
If $\psi$ is a concave function $\longrightarrow $ (like $log$ or $\sqrt[2]{..}$ )\\ If $\psi$ is a concave function $\longrightarrow $ (like $log$ or $\sqrt[2]{..}$ )
\\
\begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{../img/lez5-img3.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
Also $\psi$ is a function that map $0$ to $1$, \quad $\longrightarrow$ \quad $\psi\:[0,1]\rightarrow \barra{R}$\\ Also $\psi$ is a function that map $0$ to $1$, \quad $\longrightarrow$ \quad $\psi\:[0,1]\rightarrow \barra{R}$\\
$$ $$
\psi(\alpha \cdot a + (1-\alpha) \cdot b ) \geq \alpha \cdot \psi(a) + (1-\alpha) \cdot \psi(b) \psi(\alpha \cdot a + (1-\alpha) \cdot b ) \geq \alpha \cdot \psi(a) + (1-\alpha) \cdot \psi(b)
\qquad \textit{Also 2° derivative is negative}$$ \qquad \textit{Also 2° derivative is negative}$$
\\ \begin{figure}[h]
-- DISEGNO -- \centering
\includegraphics[width=0.5\linewidth]{../img/lez5-img4.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
$$ \hat{\ell}(h_T) = \frac{1}{m} \cdot \sum_{\ell}{} \psi (\frac{\nl^+}{\nl}) \cdot \nl $$ \hat{\ell}(h_T) = \frac{1}{m} \cdot \sum_{\ell}{} \psi (\frac{\nl^+}{\nl}) \cdot \nl
$$ $$
@ -89,8 +111,13 @@ $
\\\\ \\\\
I want to check function $min$ concave between 0 and 1.\\ I want to check function $min$ concave between 0 and 1.\\
$$min (0,1) = 0 \qquad \psi(a) = min(\alpha, 1- \alpha) $$ $$min (0,1) = 0 \qquad \psi(a) = min(\alpha, 1- \alpha) $$
\\ -- DISEGNO -- \begin{figure}[h]
\\\\ \centering
\includegraphics[width=0.4\linewidth]{../img/lez5-img5.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
\red{This is a concave function and now I can apply Jensen's inquality} \red{This is a concave function and now I can apply Jensen's inquality}
\\\\ \\\\
$$ $$
@ -139,8 +166,13 @@ the approximately since we are not every time sure.
\\\\ \\\\
--- MANCA PARTE --- --- MANCA PARTE ---
\\ \\
--- IMMAGINE --- \begin{figure}[h]
\\\\ \centering
\includegraphics[width=0.8\linewidth]{../img/lez5-img6.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
$ p = 0.8 \qquad q = 1 \qquad r = 1 \qquad \alpha = 60\%$ $ p = 0.8 \qquad q = 1 \qquad r = 1 \qquad \alpha = 60\%$
\\ \\
Net Change in number of mistakes\\ Net Change in number of mistakes\\
@ -155,9 +187,19 @@ Fraction of example miss classified $\ell -$ error $\ell' +$ error $\ell"$ \\
$$ $$
= 0.2 - ( \frac{1}{2} \cdot 0.4 + \frac{1}{2} \cdot 0 ) = 0 = 0.2 - ( \frac{1}{2} \cdot 0.4 + \frac{1}{2} \cdot 0 ) = 0
$$ $$
\begin{figure}[h]
\centering
\includegraphics[width=0.6\linewidth]{../img/lez5-img7.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{../img/lez5-img8.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
--- DISEGNO ---
\\\\
Idea is to replace minimum function with convex combination. Idea is to replace minimum function with convex combination.
$$ $$
\psi(\alpha) = min\ \{\alpha, 1-\alpha\} \qquad \psi(a) \geq \psi(\alpha) \psi(\alpha) = min\ \{\alpha, 1-\alpha\} \qquad \psi(a) \geq \psi(\alpha)
@ -170,11 +212,15 @@ $$
)} )}
\end{cases} \end{cases}
$$ $$
All this functions has this shape (concave???)\\ All this functions has this shape (concave???)
-- DISEGNO -- \begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{../img/lez5-img9.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
In practise Machine Learning algorithm use GNI or entropy to control the split In practise Machine Learning algorithm use GNI or entropy to control the split
\\\\
\section{Tree Predictor} \section{Tree Predictor}
\begin{itemize} \begin{itemize}
\item Multi class classification $|Y| > 2$ $\longrightarrow$ \red{take majority} \item Multi class classification $|Y| > 2$ $\longrightarrow$ \red{take majority}
@ -189,7 +235,12 @@ Unless leaves are \textit{"pured"}, the training error will be bigger than 0.
\\\\ \\\\
In general, i can always write $\hat{\ell}(h_t)$ to 0 by growing enough the tree unless there are $x_1$ in the Time Series such that $(x_t, y_t)(x_t,y_t)$ with $y_t \neq y_t$ both occur. In general, i can always write $\hat{\ell}(h_t)$ to 0 by growing enough the tree unless there are $x_1$ in the Time Series such that $(x_t, y_t)(x_t,y_t)$ with $y_t \neq y_t$ both occur.
\\ \\
--- DISEGNO ---- \begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{../img/lez5-img10.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
$$ if (x_1 = \alpha) \wedge (x_2 = \geq \alpha) \vee (x_1 = b) \vee (x_1 = c) \wedge (x_3= y) \qquad $$ if (x_1 = \alpha) \wedge (x_2 = \geq \alpha) \vee (x_1 = b) \vee (x_1 = c) \wedge (x_3= y) \qquad
$$ $$

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See the pdftex.def package documentation for explanation.
Type H <return> for immediate help.
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Try typing <return> to proceed.
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l.229 C
onditional risk will be high aroun the half so min between the two is...
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@ -79,7 +79,14 @@ where red{$G(\hat{y})$ is equal to the part between $\left[...\right]$}
$$ $$
\frac{d G(\hat{y})}{d\hat{y}} = 2 \cdot \hat{y}- 2 \cdot \barra{E} \left[ \, y | X= x \, \right] = 0 \quad \longrightarrow \quad \red{\textit{So setting derivative to 0}} \frac{d G(\hat{y})}{d\hat{y}} = 2 \cdot \hat{y}- 2 \cdot \barra{E} \left[ \, y | X= x \, \right] = 0 \quad \longrightarrow \quad \red{\textit{So setting derivative to 0}}
$$ $$
\\ --- DISEGNO OPT CURVE ---\\\\ Suppose we have a learning domain\\
\begin{figure}[h]
\centering
\includegraphics[width=0.1\linewidth]{../img/lez6-img0.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
$G' (\hat{y}) = \hat{y}^2 - 2\cdot b \cdot \hat{y}$ $G' (\hat{y}) = \hat{y}^2 - 2\cdot b \cdot \hat{y}$
\\ \\
$$ $$
@ -130,9 +137,14 @@ $$\
$$ $$
D_{y|x} = \{ \eta(x), 1- \eta(x) \} D_{y|x} = \{ \eta(x), 1- \eta(x) \}
$$ $$
\\ \newpage
Suppose we have a learning domain\\ Suppose we have a learning domain\\
--- DISEGNO -- \begin{figure}[h]
\centering
\includegraphics[width=0.6\linewidth]{../img/lez6-img1.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
where $\eta$ is a function of $x$, so i can plot it\\ where $\eta$ is a function of $x$, so i can plot it\\
$\eta$ will te me $Prob (x) = $ $\eta$ will te me $Prob (x) = $
@ -141,10 +153,15 @@ $\eta$ tells me a lot how hard is learning problem in the domain
\\ \\
$\eta(x)$ is not necessary continous $\eta(x)$ is not necessary continous
\\ \\
--- DISEGNO --- \begin{figure}[h]
\\\\ \centering
$\eta(x) \in \{0,1\} $ \qquad $y$ is always determined by $x$ \includegraphics[width=0.6\linewidth]{../img/lez6-img2.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\ \\
$\eta(x) \in \{0,1\} $ \qquad $y$ is always determined by $x$
\newpage
How to get $f^*$ from the graph? How to get $f^*$ from the graph?
\\ \\
$$ $$
@ -153,7 +170,13 @@ $$
$$ $$
Y = \{-1, +1 \} Y = \{-1, +1 \}
$$ $$
--- DISEGNO ---\\ \begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{../img/lez6-img3.JPG}
\caption{Example of domain of $\knn$}
%\label{fig:}
\end{figure}
\\
===============================\\ ===============================\\
MANCA ROBAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\\ MANCA ROBAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\\
============================== ==============================
@ -217,10 +240,11 @@ $$\
$$ $$
\barra{E} \left[ \, \ell , f^*(x) \, \right] = \barra{E} \left[ \, min \, \{ \eta(x) , 1- \eta(x) \} \, \right] \barra{E} \left[ \, \ell , f^*(x) \, \right] = \barra{E} \left[ \, min \, \{ \eta(x) , 1- \eta(x) \} \, \right]
$$ $$
\\
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\includegraphics[width=1\textwidth]{bayesrisk.jpg} \includegraphics[width=0.7\linewidth]{../img/lez6-img4.JPG}
\caption{Example of Bayes Risk} \caption{Example of domain of $\knn$}
%\label{fig:} %\label{fig:}
\end{figure} \end{figure}
\\ \\

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\contentsline {figure}{\numberline {11.4}{\ignorespaces Parametric and non parametric growing as training set getting larger}}{64}% \contentsline {figure}{\numberline {11.4}{\ignorespaces Parametric and non parametric growing as training set getting larger}}{74}%
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\babel@toc {english}{} \babel@toc {english}{}
\contentsline {chapter}{\numberline {1}Lecture 1 - 09-03-2020}{4}% \contentsline {chapter}{\numberline {1}Lecture 1 - 09-03-2020}{5}%
\contentsline {section}{\numberline {1.1}Introduction of the course}{4}% \contentsline {section}{\numberline {1.1}Introduction of the course}{5}%
\contentsline {section}{\numberline {1.2}Examples}{4}% \contentsline {section}{\numberline {1.2}Examples}{5}%
\contentsline {subsection}{\numberline {1.2.1}Spam filtering}{7}% \contentsline {subsection}{\numberline {1.2.1}Spam filtering}{8}%
\contentsline {chapter}{\numberline {2}Lecture 2 - 07-04-2020}{8}% \contentsline {chapter}{\numberline {2}Lecture 2 - 07-04-2020}{9}%
\contentsline {section}{\numberline {2.1}Argomento}{8}% \contentsline {section}{\numberline {2.1}Argomento}{9}%
\contentsline {section}{\numberline {2.2}Loss}{8}% \contentsline {section}{\numberline {2.2}Loss}{9}%
\contentsline {subsection}{\numberline {2.2.1}Absolute Loss}{8}% \contentsline {subsection}{\numberline {2.2.1}Absolute Loss}{9}%
\contentsline {subsection}{\numberline {2.2.2}Square Loss}{9}% \contentsline {subsection}{\numberline {2.2.2}Square Loss}{10}%
\contentsline {subsection}{\numberline {2.2.3}Example of information of square loss}{9}% \contentsline {subsection}{\numberline {2.2.3}Example of information of square loss}{11}%
\contentsline {subsection}{\numberline {2.2.4}labels and losses}{10}% \contentsline {subsection}{\numberline {2.2.4}labels and losses}{12}%
\contentsline {subsection}{\numberline {2.2.5}Example TF(idf) documents encoding}{12}% \contentsline {subsection}{\numberline {2.2.5}Example TF(idf) documents encoding}{14}%
\contentsline {chapter}{\numberline {3}Lecture 3 - 07-04-2020}{14}% \contentsline {chapter}{\numberline {3}Lecture 3 - 07-04-2020}{16}%
\contentsline {section}{\numberline {3.1}Overfitting}{16}% \contentsline {section}{\numberline {3.1}Overfitting}{18}%
\contentsline {subsection}{\numberline {3.1.1}Noise in the data}{16}% \contentsline {subsection}{\numberline {3.1.1}Noise in the data}{18}%
\contentsline {section}{\numberline {3.2}Underfitting}{17}% \contentsline {section}{\numberline {3.2}Underfitting}{20}%
\contentsline {section}{\numberline {3.3}Nearest neighbour}{18}% \contentsline {section}{\numberline {3.3}Nearest neighbour}{20}%
\contentsline {chapter}{\numberline {4}Lecture 4 - 07-04-2020}{20}% \contentsline {chapter}{\numberline {4}Lecture 4 - 07-04-2020}{23}%
\contentsline {section}{\numberline {4.1}Computing $h_{NN}$}{20}% \contentsline {section}{\numberline {4.1}Computing $h_{NN}$}{23}%
\contentsline {section}{\numberline {4.2}Tree Predictor}{21}% \contentsline {section}{\numberline {4.2}Tree Predictor}{25}%
\contentsline {chapter}{\numberline {5}Lecture 5 - 07-04-2020}{24}% \contentsline {chapter}{\numberline {5}Lecture 5 - 07-04-2020}{29}%
\contentsline {section}{\numberline {5.1}Tree Classifier}{24}% \contentsline {section}{\numberline {5.1}Tree Classifier}{29}%
\contentsline {section}{\numberline {5.2}Jensens inequality}{25}% \contentsline {section}{\numberline {5.2}Jensens inequality}{31}%
\contentsline {section}{\numberline {5.3}Tree Predictor}{27}% \contentsline {section}{\numberline {5.3}Tree Predictor}{35}%
\contentsline {section}{\numberline {5.4}Statistical model for Machine Learning}{28}% \contentsline {section}{\numberline {5.4}Statistical model for Machine Learning}{36}%
\contentsline {chapter}{\numberline {6}Lecture 6 - 07-04-2020}{30}% \contentsline {chapter}{\numberline {6}Lecture 6 - 07-04-2020}{38}%
\contentsline {section}{\numberline {6.1}Bayes Optimal Predictor}{30}% \contentsline {section}{\numberline {6.1}Bayes Optimal Predictor}{38}%
\contentsline {subsection}{\numberline {6.1.1}Square Loss}{31}% \contentsline {subsection}{\numberline {6.1.1}Square Loss}{39}%
\contentsline {subsection}{\numberline {6.1.2}Zero-one loss for binary classification}{32}% \contentsline {subsection}{\numberline {6.1.2}Zero-one loss for binary classification}{40}%
\contentsline {section}{\numberline {6.2}Bayes Risk}{34}% \contentsline {section}{\numberline {6.2}Bayes Risk}{43}%
\contentsline {chapter}{\numberline {7}Lecture 7 - 07-04-2020}{35}% \contentsline {chapter}{\numberline {7}Lecture 7 - 07-04-2020}{45}%
\contentsline {section}{\numberline {7.1}Chernoff-Hoffding bound}{35}% \contentsline {section}{\numberline {7.1}Chernoff-Hoffding bound}{45}%
\contentsline {section}{\numberline {7.2}Union Bound}{36}% \contentsline {section}{\numberline {7.2}Union Bound}{46}%
\contentsline {section}{\numberline {7.3}Studying overfitting of a ERM}{40}% \contentsline {section}{\numberline {7.3}Studying overfitting of a ERM}{50}%
\contentsline {chapter}{\numberline {8}Lecture 8 - 07-04-2020}{42}% \contentsline {chapter}{\numberline {8}Lecture 8 - 07-04-2020}{52}%
\contentsline {section}{\numberline {8.1}The problem of estimating risk in practise}{43}% \contentsline {section}{\numberline {8.1}The problem of estimating risk in practise}{53}%
\contentsline {section}{\numberline {8.2}Cross-validation}{45}% \contentsline {section}{\numberline {8.2}Cross-validation}{55}%
\contentsline {section}{\numberline {8.3}Nested cross validation}{47}% \contentsline {section}{\numberline {8.3}Nested cross validation}{57}%
\contentsline {chapter}{\numberline {9}Lecture 9 - 07-04-2020}{48}% \contentsline {chapter}{\numberline {9}Lecture 9 - 07-04-2020}{58}%
\contentsline {section}{\numberline {9.1}Tree predictors}{48}% \contentsline {section}{\numberline {9.1}Tree predictors}{58}%
\contentsline {subsection}{\numberline {9.1.1}Catalan Number}{50}% \contentsline {subsection}{\numberline {9.1.1}Catalan Number}{60}%
\contentsline {chapter}{\numberline {10}Lecture 10 - 07-04-2020}{54}% \contentsline {chapter}{\numberline {10}Lecture 10 - 07-04-2020}{64}%
\contentsline {section}{\numberline {10.1}TO BE DEFINE}{54}% \contentsline {section}{\numberline {10.1}TO BE DEFINE}{64}%
\contentsline {section}{\numberline {10.2}MANCANO 20 MINUTI DI LEZIONE}{54}% \contentsline {section}{\numberline {10.2}MANCANO 20 MINUTI DI LEZIONE}{64}%
\contentsline {section}{\numberline {10.3}Compare risk for zero-one loss}{56}% \contentsline {section}{\numberline {10.3}Compare risk for zero-one loss}{66}%
\contentsline {chapter}{\numberline {11}Lecture 11 - 20-04-2020}{58}% \contentsline {chapter}{\numberline {11}Lecture 11 - 20-04-2020}{68}%
\contentsline {section}{\numberline {11.1}Analysis of $K_{NN}$}{58}% \contentsline {section}{\numberline {11.1}Analysis of $K_{NN}$}{68}%
\contentsline {subsection}{\numberline {11.1.1}Study of $K_{NN}$}{61}% \contentsline {subsection}{\numberline {11.1.1}Study of $K_{NN}$}{71}%
\contentsline {subsection}{\numberline {11.1.2}study of trees}{62}% \contentsline {subsection}{\numberline {11.1.2}study of trees}{72}%
\contentsline {section}{\numberline {11.2}Non-parametric Algorithms}{63}% \contentsline {section}{\numberline {11.2}Non-parametric Algorithms}{73}%
\contentsline {subsection}{\numberline {11.2.1}Example of parametric algorithms}{64}% \contentsline {subsection}{\numberline {11.2.1}Example of parametric algorithms}{74}%
\contentsline {chapter}{\numberline {12}Lecture 12 - 21-04-2020}{65}% \contentsline {chapter}{\numberline {12}Lecture 12 - 21-04-2020}{75}%
\contentsline {section}{\numberline {12.1}Non parametrics algorithms}{65}% \contentsline {section}{\numberline {12.1}Non parametrics algorithms}{75}%