diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture10.tex b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture10.tex index d0b40d437..f6e26420a 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture10.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture10.tex @@ -1,3 +1,6 @@ +\documentclass[../main.tex]{subfiles} +\begin{document} + \section{Lecture 10 - 07-04-2020} \subsection{TO BE DEFINE} @@ -89,4 +92,6 @@ Depends on dimension: curse of dimensionality $ \ell_d(f*) = 0 \iff min\{ \eta(x), 1-\eta(x)\} =0 \quad$ with probability = 1 \\ -to be true $\eta(x) \in \{0,1\}$ \ No newline at end of file +to be true $\eta(x) \in \{0,1\}$ + +\end{document} \ No newline at end of file diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.log b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.log index 90305c222..54e2c51ff 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.log +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.12) 12 APR 2020 16:28 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.12) 12 APR 2020 16:33 entering extended mode **./lecture4.tex (lecture4.tex @@ -161,43 +161,33 @@ Underfull \hbox (badness 10000) in paragraph at lines 15--17 [] -Underfull \hbox (badness 10000) in paragraph at lines 19--20 - - [] - - Underfull \hbox (badness 10000) in paragraph at lines 21--39 [] -Underfull \hbox (badness 10000) in paragraph at lines 39--40 +Underfull \hbox (badness 10000) in paragraph at lines 42--66 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 42--66 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 42--66 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 42--66 [] [1 {C:/Users/AndreDany/AppData/Local/MiKTeX/2.9/pdftex/config/pdftex.map}] -Underfull \hbox (badness 10000) in paragraph at lines 42--66 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 42--66 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 42--66 - - [] - - -Underfull \hbox (badness 10000) in paragraph at lines 42--66 - - [] - - Underfull \hbox (badness 10000) in paragraph at lines 68--70 [] @@ -284,7 +274,7 @@ s/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr8.pfb> -Output written on lecture4.pdf (4 pages, 125496 bytes). +Output written on lecture4.pdf (4 pages, 125307 bytes). PDF statistics: 64 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.pdf b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.pdf index 2dc366297..8c177ac0c 100644 Binary files a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.pdf and b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.pdf differ diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.synctex.gz b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.synctex.gz index 39e8f6644..32a971cfc 100644 Binary files a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.synctex.gz and b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.synctex.gz differ diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.tex b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.tex index c9973179a..533096124 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture4.tex @@ -6,7 +6,7 @@ We spoke about Knn classifier with voronoi diagram $$ -\hat{\ell}(\hnn) = 0 \qquad \forall Traning set +\hat{\ell}(\hnn) = 0 \qquad \forall \, \textit{Traning set} $$ \\ $\hnn$ predictor needs to store entire dataset. @@ -16,7 +16,7 @@ Computing $\hnn(x)$ requires computing distances between x and points in the tra \\ $$ \Theta(d) \quad \textit{time for each distance} -$$\\ +$$ NN $\rightarrow$ 1-NN\\ We can generalise NN in K-NN with $k = 1,3,5,7$ so odd $K$ \\ @@ -36,7 +36,7 @@ $K = 1$ error is 0.\\ In the second line we switch to $k =3$. Second point doesn’t switch and third will be classify to positive and we have training mistake.\\ Switches corresponds to border of voronoi partition. -$$\knn \qquad \textit{For multiclass classification}$$\\ +$$\knn \qquad \textit{For multiclass classification}$$ $$ (|Y| > 2 ) \qquad \textit{for regression } Y\equiv \barra{R} $$ diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.aux b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.aux new file mode 100644 index 000000000..63c173768 --- /dev/null +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.aux @@ -0,0 +1,8 @@ +\relax +\@nameuse{bbl@beforestart} +\babel@aux{english}{} +\@writefile{toc}{\contentsline {section}{\numberline {1}Lecture 5 - 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a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.tex b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.tex index d28e94a5b..bb439cc7e 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture5.tex @@ -1 +1,248 @@ +\documentclass[../main.tex]{subfiles} +\begin{document} \section{Lecture 5 - 07-04-2020} + +\subsection{Tree Classifier} + +Supposed we groped a tree up to this point and we are wandering how to +grow it. +\\ +$S$ Training set $(x_1,y_1)...(x_m,y_m)$, $x_1 \in X$ +\\\\ +-- DISEGNO +\\\\ +$$ +\sll \equiv \{(x_1,y_1) \, x_t \quad \textit{is router to } \ell \} +$$ +\\ +$y_1 \in \{-1,1\}$ +\\ +$$ +\sll ^+ \equiv \{(x_1,y_1) \in \sll : \quad y_t = +1 \} +$$ +$$ +\sll^- \equiv \{(x_1,y_1) \in \sll : \quad y_t = -1 \} +\qquad +\sll^+ \cap \sll^- \equiv 0 \qquad \sll \equiv \sll^+ \cup \sll^- +$$ + +$$ +\qquad \nl = | \sll| +\qquad \nl^+ = |\sll^+| +\qquad \nl^- = |\sll^-| $$ +$$ +\nl = \nl^- + \nl^+ +$$ +leaf $\ell$ classifies all traning example ($\sll$) +\\ +$$ +Y_{\ell} = +\begin{cases} ++1, & \mbox{If } \nl^+ \geq \nl^- +\\ +-1, & \mbox{If } otherwise +\end{cases} +$$ +\\ +$\ell$ makes a mistake on $min \{\nl^+, \nl^- \}$ example in $\sll$ +$$ +\hat{\ell}(h_T) = \frac{1}{m} +\cdot \sum_{\ell}{} min \{ \frac{\nl^+}{\nl} , \frac{\nl^-}{\nl} \} \cdot \nl = +$$ +$$ += \frac{1}{m} \cdot \sum_{\ell}{}\psi \cdot (\frac{\nl+}{\nl}) \cdot \nl \quad \longrightarrow \qquad \frac{\nl^+}{\nl} = 1 - \frac{\nl}{\nl ??} +$$ +where $\psi(a) = min \{a, 1-a \} \qquad a \in [0,1] $ +\\ +I want to replace inner node with other leaves. +\\ +-- DISEGNO -- +\\\\ +How is traning error going to change? +(when i replace inner nodes with other leaves) +\\ +I’m hoping my algorithm is not going to overfit (if training error goes to 0 also +testing error goes to 0).\\ + +\subsection{Jensen’s inequality} +If $\psi$ is a concave function $\longrightarrow $ (like $log$ or $\sqrt[2]{..}$ )\\ +Also $\psi$ is a function that map $0$ to $1$, \quad $\longrightarrow$ \quad $\psi\:[0,1]\rightarrow \barra{R}$\\ +$$ +\psi(\alpha \cdot a + (1-\alpha) \cdot b ) \geq \alpha \cdot \psi(a) + (1-\alpha) \cdot \psi(b) +\qquad \textit{Also 2° derivative is negative}$$ +\\ +-- DISEGNO -- +\\ +$$ \hat{\ell}(h_T) = \frac{1}{m} \cdot \sum_{\ell}{} \psi (\frac{\nl^+}{\nl}) \cdot \nl +$$ +\\ +Look a single contribution fo a leaf $\ell$ to training error +\\ +$$ +\psi(\frac{\nl^+}{\nl}) \cdot \nl = \psi(\frac{\nl'^+}{\nl'} \cdot \red{\frac{\nl'}{\nl}} + \frac{\nl"^+}{\nl"} \cdot \red{\frac{\nl"}{\nl}}) \cdot \nl +$$ +where $\red{\frac{\nl'}{\nl}} = \alpha$ and $\red{\frac{\nl"}{\nl}} = 1-\alpha$ \qquad so \quad $\red{\frac{\nl'}{\nl}} + \red{\frac{\nl"}{\nl}} = 1$ \qquad $\longrightarrow$ $\alpha + 1 -\alpha = 1$ +\\\\ +$N_{\ell'}^+ + N_{\ell"}^+ = \nl +$ +\\\\ +I want to check function $min$ concave between 0 and 1.\\ +$$min (0,1) = 0 \qquad \psi(a) = min(\alpha, 1- \alpha) $$ +\\ -- DISEGNO -- +\\\\ +\red{This is a concave function and now I can apply Jensen's inquality} +\\\\ +$$ +\psi(\frac{\nl^+}{\nl}) \cdot \nl +\geq +(\frac{\nl'}{\nl} \cdot \psi (\frac{\nl'^+}{\nl'}) ++ +\frac{\nl"}{\nl}\cdot \psi(\frac{\nl"^+}{\nl"})) \cdot \nl = +$$ + +$\qquad \qquad\qquad \qquad \quad= $ +\framebox[1.1\width] +{ +$ +\psi(\frac{\nl'^+}{\nl'})\cdot \nl' ++ +\psi(\frac{\nl"^+}{\nl"})\cdot \nl" +$ +} +\\\\ +\red{This are the contribuion of $\ell'$ and $\ell"$ to the training error} +\\\\ +Every time i split my tree my training error is never going to increase since we +have a concave function.\\ +Whenever I’m growing my tree training error is going to be smaller.\\\\ +\textbf{Every time a leaf is expanded the training error never goes up. +(Hopelly will go down)} +\\ +I’ll should always grow the tree by expanding leave that decrease the training +error as much as possible.\\ +If i take the effort of growing the tree i should get benefits. I can imaging that if +i grow the tree at random my training error is going to drop down error (but +maybe will derive overfitting).\\ +For now is just an intuition since we will introduced statistical learning model.\\\\ +Could be complicated and tree big may have 100 leave and there could be +many way of associating a test with that leaves.\\ +I can spent a lot of time to select which leave is the best promising to split.\\ +\begin{itemize} +\item Grow the tree by expanding leave that decrease the training error as much +as possible +\item In general we can assume:\\greedy algorithm at each step pick the pair leaf and test that cause +(approximative) the largest decrease in training error\\ +\end{itemize} +In practise we want optimise this all the way since it’s time expensive. That’s +the approximately since we are not every time sure. +\\\\ +--- MANCA PARTE --- +\\ +--- IMMAGINE --- +\\\\ +$ p = 0.8 \qquad q = 1 \qquad r = 1 \qquad \alpha = 60\%$ +\\ +Net Change in number of mistakes\\ +$$ +\psi(p) - (\alpha \cdot \psi(q) + (1- \alpha ) \cdot \psi (r) ) = +$$ +$$ +\red{ +\qquad \qquad \ell \quad- \qquad \ell' \quad+ \qquad \ell" \qquad \qquad \qquad \qquad +}$$ +Fraction of example miss classified $\ell -$ error $\ell' +$ error $\ell"$ \\ +$$ += 0.2 - ( \frac{1}{2} \cdot 0.4 + \frac{1}{2} \cdot 0 ) = 0 +$$ +\\ +--- DISEGNO --- +\\\\ +Idea is to replace minimum function with convex combination. +$$ +\psi(\alpha) = min\ \{\alpha, 1-\alpha\} \qquad \psi(a) \geq \psi(\alpha) +$$ +$$ +\begin{cases} +\psi_1(\alpha) = 2\cdot\alpha \cdot (1-\alpha) \longrightarrow \red{GNI} \\ +\psi_2(\alpha)= -\frac{\alpha}{2}\cdot \ln \alpha - \frac{1-\alpha}{2} \cdot \ln (1-\alpha)\longrightarrow \red{ENTROPY} \\ +\psi_3(\alpha) = \sqrt[]{\alpha \cdot (1-\alpha +)} +\end{cases} +$$ +All this functions has this shape (concave???)\\ +-- DISEGNO -- +\\ +In practise Machine Learning algorithm use GNI or entropy to control the split +\\\\ +\subsection{Tree Predictor} +\begin{itemize} +\item Multi class classification $|Y| > 2$ $\longrightarrow$ \red{take majority} +\item Regression $Y = \barra{R} $ $\longrightarrow$ \red{take average of labels in $\sll$} +\end{itemize} +I still take majority among different classes.\\ +Take average of labels in $\sll$ +\\ +Unless $\frac{\nl^+}{\nl} \in {0,1} $ \qquad $\forall$ leaves $\ell$, $\hat{\ell}(h_T) > 0$ +\\ +Unless leaves are \textit{"pured"}, the training error will be bigger than 0. +\\\\ +In general, i can always write $\hat{\ell}(h_t)$ to 0 by growing enough the tree unless there are $x_1$ in the Time Series such that $(x_t, y_t)(x_t,y_t’)$ with $y_t \neq y_t’$ both occur. +\\ +--- DISEGNO ---- +\\ +$$ if (x_1 = \alpha) \wedge (x_2 = \geq \alpha) \vee (x_1 = b) \vee (x_1 = c) \wedge (x_3= y) \qquad +$$ +$$ +\textit{then predict 1} \qquad \qquad +$$ +$ +\qquad \quad \,\,else +$ +$$ +\textit{then predict -1} \qquad \qquad +$$ +\\ +--- Picture of tree classifier of iris dataset. ---\\ +I’m using due attribute at the time.\\ +Each data point is a flower and i can measure how petal and sepal are long. +I can use two attribute and i test this two. I can see the plot of the tree +classifier (second one) making test splitting data space into region that has +this sort of “blackish” shape ( like boxes: blue box, red box, yellow box)\\ +A good exercise in which I want to reconstruct the tree given this picture. +\\\\ +\subsection{Statistical model for Machine Learning} + +To understand Tree classifier, nearest neighbour and other algorithm...\\ +It’s important to understand that the only way to have a guideline in which +model to choose.\\\\ +\textbf{This mathematical model are developed to learning and choose learning +algorithm.}\\\\ +Now let start with theoretical model. +\begin{itemize} +\item How example $(x,y)$ are generated to create test set and training set?\\ +We get the dataset but we need to have a mathematical model for this +process. +$(x,y)$ are drawn from a fixed but unknown probability distribution on the pairs $X$ +and $Y$ ($X$ data space, $Y$ label set o label space) +\item Why $X$ should be random? \\ +In general we assumed that not all the $x$ in $X$ are equally likely to be observed. +I have some distribution over my data point and this said that I’m most like to +get a datapoint to another. +\item How much label?\\ +Often labels are not determined uniquely by their datapoints because labels +are given by human that have their subjective thoughts and also natural +phenomena. Labels are stochastic phenomena given a datapoint: i will have a +distribution. +\end{itemize} +We’re going to write (in capital) $(X, Y)$ since they are random variable drawn +from $D$ on $X \cdot Y$ +A dataset $(X_1, Y_1) ... (X_m, Y_m)$ they are drawn independently from $D$ +(distribution on examples)\\ +When I get a training the abstraction of process collecting a training set\\ +$D$ is a joint probability distribution over $X\cdot Y$\\ +where $D_x$ is the marginal over $X \rightarrow D_y|x$ (conditional of $Y$ given $X$).\\ +I can divided my draw in two part. +I draw sample and label from conditional.??\\ +Any dataset ( training or test ) is a random sample (campione casuale) in the +statistical sense $\longrightarrow$ so we can use all stastical tools to make inference. +\end{document} \ No newline at end of file diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture6.tex b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture6.tex index 163cbcd2e..7abd10e09 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture6.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture6.tex @@ -1 +1,7 @@ +\documentclass[../main.tex]{subfiles} +\begin{document} \section{Lecture 6 - 07-04-2020} + + + +\end{document} diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture7.tex b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture7.tex index df7958cb6..68d20c4dd 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture7.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture7.tex @@ -1 +1,6 @@ +\documentclass[../main.tex]{subfiles} +\begin{document} \section{Lecture 7 - 07-04-2020} + + +\end{document} \ No newline at end of file diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture8.tex b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture8.tex index a3457995b..96f8beb35 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture8.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture8.tex @@ -1 +1,5 @@ +\documentclass[../main.tex]{subfiles} +\begin{document} + \section{Lecture 8 - 07-04-2020} +\end{document} \ No newline at end of file diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial 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09-03-2020}{3}% -\contentsline {subsection}{\numberline {1.1}Introduction}{3}% -\contentsline {section}{\numberline {2}Lecture 2 - 07-04-2020}{6}% -\contentsline {subsection}{\numberline {2.1}Argomento}{6}% -\contentsline {subsection}{\numberline {2.2}Loss}{6}% -\contentsline {subsubsection}{\numberline {2.2.1}Absolute Loss}{6}% -\contentsline {subsubsection}{\numberline {2.2.2}Square Loss}{7}% -\contentsline {subsubsection}{\numberline {2.2.3}Example of information of square loss}{7}% -\contentsline {subsubsection}{\numberline {2.2.4}labels and losses}{9}% -\contentsline {subsubsection}{\numberline {2.2.5}Example TF(idf) documents encoding}{10}% -\contentsline {section}{\numberline {3}Lecture 3 - 07-04-2020}{12}% -\contentsline {subsection}{\numberline {3.1}Overfitting}{14}% -\contentsline {subsubsection}{\numberline {3.1.1}Noise in the data}{14}% -\contentsline {subsection}{\numberline {3.2}Underfitting}{16}% -\contentsline {subsection}{\numberline {3.3}Nearest neighbour}{16}% -\contentsline {section}{\numberline {4}Lecture 4 - 07-04-2020}{18}% -\contentsline {section}{\numberline {5}Lecture 5 - 07-04-2020}{19}% -\contentsline {section}{\numberline {6}Lecture 6 - 07-04-2020}{20}% -\contentsline {section}{\numberline {7}Lecture 7 - 07-04-2020}{21}% -\contentsline {section}{\numberline {8}Lecture 8 - 07-04-2020}{22}% -\contentsline {section}{\numberline {9}Lecture 9 - 07-04-2020}{23}% -\contentsline {section}{\numberline {10}Lecture 10 - 07-04-2020}{24}% -\contentsline {subsection}{\numberline {10.1}TO BE DEFINE}{24}% +\babel@toc {english}{} +\contentsline {section}{\numberline {1}Lecture 1 - 09-03-2020}{4}% +\contentsline {subsection}{\numberline {1.1}Introduction}{4}% +\contentsline {section}{\numberline {2}Lecture 2 - 07-04-2020}{7}% +\contentsline {subsection}{\numberline {2.1}Argomento}{7}% +\contentsline {subsection}{\numberline {2.2}Loss}{7}% +\contentsline {subsubsection}{\numberline {2.2.1}Absolute Loss}{7}% +\contentsline {subsubsection}{\numberline {2.2.2}Square Loss}{8}% +\contentsline {subsubsection}{\numberline {2.2.3}Example of information of square loss}{8}% +\contentsline {subsubsection}{\numberline {2.2.4}labels and losses}{10}% +\contentsline {subsubsection}{\numberline {2.2.5}Example TF(idf) documents encoding}{11}% +\contentsline {section}{\numberline {3}Lecture 3 - 07-04-2020}{13}% +\contentsline {subsection}{\numberline {3.1}Overfitting}{15}% +\contentsline {subsubsection}{\numberline {3.1.1}Noise in the data}{15}% +\contentsline {subsection}{\numberline {3.2}Underfitting}{17}% +\contentsline {subsection}{\numberline {3.3}Nearest neighbour}{17}% +\contentsline {section}{\numberline {4}Lecture 4 - 07-04-2020}{19}% +\contentsline {subsection}{\numberline {4.1}Computing $h_{NN}$}{19}% +\contentsline {subsection}{\numberline {4.2}Tree Predictor}{20}% +\contentsline {section}{\numberline {5}Lecture 5 - 07-04-2020}{23}% +\contentsline {subsection}{\numberline {5.1}Tree Classifier}{23}% +\contentsline {subsection}{\numberline {5.2}Jensen’s inequality}{24}% +\contentsline {subsection}{\numberline {5.3}Tree Predictor}{26}% +\contentsline {subsection}{\numberline {5.4}Statistical model for Machine Learning}{27}% +\contentsline {section}{\numberline {6}Lecture 6 - 07-04-2020}{29}% +\contentsline {section}{\numberline {7}Lecture 7 - 07-04-2020}{30}% +\contentsline {section}{\numberline {8}Lecture 8 - 07-04-2020}{31}% +\contentsline {section}{\numberline {9}Lecture 9 - 07-04-2020}{32}% +\contentsline {section}{\numberline {10}Lecture 10 - 07-04-2020}{33}% +\contentsline {subsection}{\numberline {10.1}TO BE DEFINE}{33}%