From 36d34ea23914a80432f457d371406edf553c74ef Mon Sep 17 00:00:00 2001 From: Andreaierardi Date: Mon, 8 Jun 2020 10:24:00 +0200 Subject: [PATCH] lecture 23 --- .../Machine Learning/Notes/img/lez23-img1.JPG | Bin 0 -> 12054 bytes .../Machine Learning/Notes/img/lez23-img2.JPG | Bin 0 -> 22906 bytes .../Machine Learning/Notes/img/lez23-img3.JPG | Bin 0 -> 15122 bytes .../Machine Learning/Notes/img/lez23-img4.JPG | Bin 0 -> 12732 bytes .../Machine Learning/Notes/img/lez23-img5.JPG | Bin 0 -> 33934 bytes .../Notes/lectures/lecture23.aux | 13 + .../Notes/lectures/lecture23.log | 437 ++++++++++++ .../Notes/lectures/lecture23.pdf | Bin 0 -> 250882 bytes .../Notes/lectures/lecture23.synctex.gz | Bin 0 -> 22053 bytes .../Notes/lectures/lecture23.tex | 161 ++++- .../Notes/lectures/lecture24.tex | 6 + .../Machine Learning/Notes/main.aux | 375 ++++++----- .../Machine Learning/Notes/main.lof | 171 ++--- .../Machine Learning/Notes/main.log | 625 ++++++++++-------- .../Machine Learning/Notes/main.pdf 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Intelligence/Machine Learning/Notes/lectures/lecture23.tex index 9489df939..341ae2147 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/lectures/lecture23.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/lectures/lecture23.tex @@ -2,6 +2,165 @@ \begin{document} \chapter{Lecture 23 - 08-06-2020} +\bred{Bagging} +$$ +h_1,...,h_t \qquad \hat{\ell}(f) \leq e^{-2 \, T \gamma^2} \qquad \gamma_t > \gamma>0 +$$ +Under the assumption that $ \{ h_t(x_z) \neq y_z \} $ \qquad $ \gamma_t = \frac{1}{2} - \hat{\ell}_s(h_t)$ are independent +$$ +f = sgn ( \sum_{i=1}^T h_i) \qquad Bagging +$$ +\section{Boosting} +$$ +f = sgn ( \sum_{i=1}^T w_i h_i) \qquad Boosting +$$ +The hard thing here is how to compute the weights. +\\ +$ +h_1, ..., h_t \qquad X \rightarrow \{ -1,+1 \} +$ +\begin{figure}[h] + \centering + \includegraphics[width=0.3\linewidth]{../img/lez23-img1.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +\\ +$$ +\hat{\ell}(f)\sum^m_{t=1} I \{ y_t g(x_t) \leq 0 \} +\leq +\frac{1}{m} \sum_{t=1}^m e^{-g(x_t) y_t +} = $$ +$$ +g = \sum_{i = 1}^T w_i h_i \textit{and we substitute g} \qquad and \qquad f = sgn(g) +$$ +$$ += \ \frac{1}{m} \sum^m_{t=1} e^{-y_t \, \sum_{t=1}^T w_i h_i(x_i)} \qquad L_i(t) = h_i (x_t) y_t \in \{-1,+1 \} i = 1, ..., T +$$ +$ +L_i(z) +$ where $Z$ uniform over $\{1,...,m\}$ +$$ +\hat(\ell)(f) \leq \frac{1}{m} \sum_{t=1}^m e^{- \sum_{t=1}^T w_i L_i (t)} = \barra{E} \left[ e^{-\sum_{t=1}^T w_i L_i (t)}\right] +$$ +$$ +\barra{E} \left[ \prod_{t=1}^T e^{- w_i L_i}\right] =^? \prod^T_{t=1} \barra{E} \left[ e^{-w_i Li} \right] +$$ +Ok if $Li$ are independent +\\$E\left[XY \right] = \barra{E}[X] \, \barra{E} [Y]$ +\\ +$X,Y$ are independent +\\ +$\barra{E}_i $ is a probability $P_i$ and $P_i$ is sum $\{1,...m\}$ +\\ +$$ +\hat{\ell}(f)\ \leq \ \prod_{i=1}^T \barra{E}_i \left[ e^{- w_i L_i} \right] +\ = \ \prod_{i=1}^T \left( e^{w_i} P_i (L_i = 1) + e^{w_i} P_i\left(L_i=1 \right) \right) \ = \ \prod_{i=1}^T \left( e^{-w_i}(1 - \epsilon_i) + e^{-w_i} \varepsilon_i \right)$$ +$ L_i(z) \qquad z \sim P_i$ +$$ +\varepsilon_i = P_i(L_i = -1) = \sum_{t=1}^m I \{y_t h_i(x_t) \leq 0 \} P_i(t) \qquad \textbf{weighted training error of $h_i$} +$$ +$$ +F(w) = e^{-w} (1- \varepsilon) + e^w \varepsilon \qquad F'(w) = 0 \Leftrightarrow w = \frac{1}{2} \ln \frac{1-\varepsilon}{\varepsilon} \qquad 0< \varepsilon< 1 +$$ +$$ +P_i(t) > 0 \quad \forall i,t \qquad \varepsilon_i = \frac{1}{2} \Rightarrow w_i = 0 +$$ +$$ +\varepsilon_i >\frac{1}{2} \Rightarrow w_i < 0 \qquad \qquad +\varepsilon_i < \frac{1}{2} \Rightarrow w_i > 0 +$$ - +$$ +\hat{\ell}(f) \ \leq \ \prod_{i=1}^T \sqrt[]{4 \, \varepsilon_i (1-\varepsilon_i)} +$$ +$$ +\gamma_i = \frac{1}{2} - \varepsilon_i \qquad \textit{edge over random guessing $ 0 < \varepsilon_i <1$} +$$ +$$ +1 + x \leq e^x \quad \forall x \in \barra{R} \quad \hat{\ell}(f) \leq \prod_{i=1}^T \sqrt[]{4 \, \varepsilon_i(1 - \varepsilon_i)} \ = \ +\prod_{i=1}^T \sqrt[]{1- 4 \gamma^2} = +$$ +$$\ = \ \prod_{i = 1}^T 4 (\frac{1}{2} - \gamma_i)(\frac{1}{2}+ \gamma_i) \ = \prod_{i=1}^T e^{- 2 \gamma_i^2} = e^{- 2 \sum_{i=1}^T \gamma_i^2} \leq e^{-2 T \gamma^2} +$$ +$ +If \quad |\gamma_i| > \gamma > 0 \quad i = 1, ..., T$ +$$ +\hat{\ell}_s(f) = 0 \ \Leftrightarrow \ e^{-\varepsilon T \gamma^2}< \frac{1}{m} \ \Leftrightarrow \ T > \frac{ \ln m}{2 \gamma^2} +$$ +$$ +E \left[ \prod_i e^{-w_i L_i }\right] \ = \ \prod_i E \left[ e^{-w_i L_i } \right] +$$ +$P_i,...,P_T \quad P_1(t) = \frac{1}{m}$ \quad $ t = 1,...m$ +$$ +P_{i+1}(t) = \frac{P_i(t) e^{-w_i L_i(t)}}{E_i \left[ e^{-w_i L_i} \right] } \qquad \sum_t P_{i+1}(t) = \frac{1}{E_i [e^{-w_i L_i}]} +\sum_t P_i(t) e^{...} +$$ +MANCAaaa +$$ +e^{-w_i L_i(t)} = E_i [e^{-w_i L_i}] \frac{P_{i+1}}{P_i(t)} +$$ +$$ +E[\prod_{i=1}^T e^{-w_i L_i}] = \frac{1}{m} \sum_{t=1}^m \prod_{i=1}^T e^{-w_i L_i(t)} = \frac{1}{m} \sum_t l\left( \prod_i E [e^{- w_i L_i}] \frac{P_{t+1}(t)}{P_i(t)}\right) \ = +$$ +$$ +\frac{1}{m} \sum_t \left( \prod_i E_i [e^{-w_i L_i} \right) \frac{P_{t+1}(t)}{P_1(t)} = \left( \prod_i E_i[e^{-w_i L_i}] \right) \red{\frac{1}{m} \sum_t \frac{P_{t+1}(t)}{y m}} +$$ +where \bred{red} cancel out since $= 1$ +\newpage +\section{Adaboost} +It is a meta learning algorithm. +\begin{figure}[h] + \centering + \includegraphics[width=0.5\linewidth]{../img/lez23-img2.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +\\ +Initialize $P_i(t) = \frac{1}{m} \quad t=1,...,m$\\ +For $i = 1,...,T$\\ +1) Feed $A$ with $S$ wrighted by $P_i$ and get $h_1$\\ +2) $w_i = \frac{1}{2} \ln \frac{\varepsilon_i}{1-\varepsilon_i}$ +\\ +3) Compute $P_{i+1}$ +\\ +Output $\sum_i w_i h_i$ +\\\\ +What should $A$ do? \\ +1) $A$ should pay attention to $P_i$ \\ +2) More precisely $A$ should output $h_i$ s.t. $|\gamma_i|$ is as big as possible \\ where $ | \gamma | \rightarrow \frac{1}{2} \varepsilon_i$ +\begin{figure}[h] + \centering + \includegraphics[width=0.5\linewidth]{../img/lez23-img3.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +$$ +P_{i+1}= \frac{P_i(t) e^{-w_i L_i(t)} }{E_i[ \quad ]} +$$ +$$ +L_i(t) = 1 \ \Leftrightarrow h_t(x_t) = y_t \qquad w_i > 0 +$$ +\\ +\begin{figure}[h] + \centering + \includegraphics[width=0.3\linewidth]{../img/lez23-img4.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +Typically $h_i$ (classifiers) are simple +\\ +\bred{Decision stamps}: +$$ +h(x) = \pm \ sgn (x_i- \tau) +$$ +$i$ if is feature index, $\tau \in \barra{R}$\\ +At the end boosting is gonna look like this +\\ +\begin{figure}[h] + \centering + \includegraphics[width=0.5\linewidth]{../img/lez23-img5.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ \end{document} \ No newline at end of file diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/lectures/lecture24.tex 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Underfull \hbox (badness 10000) in paragraph at lines 5--17 @@ -1151,7 +1151,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 48--52 [] -[40 +[41 ] Underfull \hbox (badness 10000) in paragraph at lines 53--57 @@ -1171,7 +1171,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 62--66 [] - + File: lectures/../img/lez6-img0.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez6-img0.JPG used on input line 85. @@ -1186,7 +1186,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 94--100 [] -[41 <./lectures/../img/lez6-img0.JPG>] +[42 <./lectures/../img/lez6-img0.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 105--107 [] @@ -1202,25 +1202,25 @@ Overfull \hbox (13.10564pt too wide) detected at line 133 [][] [] -[42] - +[43] + File: lectures/../img/lez6-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez6-img1.JPG used on input line 144. 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Underfull \hbox (badness 10000) in paragraph at lines 4--6 @@ -1294,10 +1294,10 @@ Overfull \hbox (0.51405pt too wide) detected at line 54 ] \OMS/cmsy/m/n/12  \OML/cmm/m/it/12 e[] [] -[47 +[48 ] - + File: lectures/../img/lez7-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez7-img1.JPG used on input line 73. @@ -1317,7 +1317,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 81--87 [] - + File: lectures/../img/lez7-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez7-img2.JPG used on input line 98. @@ -1331,7 +1331,7 @@ Overfull \hbox (27.96684pt too wide) detected at line 111 U/msb/m/n/12 P [] \OMS/cmsy/m/n/12  [] -[48 <./lectures/../img/lez7-img1.JPG>] +[49 <./lectures/../img/lez7-img1.JPG>] Overfull \hbox (4.4292pt too wide) detected at line 118 \OMS/cmsy/m/n/12  \U/msb/m/n/12 P [] \OT1/cmr/m/n/12 + \U/msb/m/n/12 P [] \OM S/cmsy/m/n/12  \OT1/cmr/m/n/12 2 \OMS/cmsy/m/n/12  \OML/cmm/m/it/12 e[] \OM @@ -1343,7 +1343,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 122--125 [] - + File: lectures/../img/lez7-img3.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez7-img3.JPG used on input line 132. @@ -1354,7 +1354,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 127--145 [] -[49 <./lectures/../img/lez7-img2.JPG> <./lectures/../img/lez7-img3.JPG>] +[50 <./lectures/../img/lez7-img2.JPG> <./lectures/../img/lez7-img3.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 147--159 [] @@ -1370,13 +1370,13 @@ Overfull \hbox (76.34352pt too wide) detected at line 161 `[] [] [][]\OMS/cmsy/m/n/12 ! 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Underfull \hbox (badness 10000) in paragraph at lines 10--13 @@ -1479,7 +1479,7 @@ Overfull \hbox (74.73193pt too wide) detected at line 27 /n/12 + [] [] [] - + File: lectures/../img/lez8-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez8-img1.JPG used on input line 32. @@ -1499,7 +1499,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 42--47 [] -[54 +[55 <./lectures/../img/lez8-img1.JPG>] Overfull \hbox (22.02261pt too wide) detected at line 51 @@ -1508,7 +1508,7 @@ T1/cmr/m/n/12 (\OML/cmm/m/it/12 h\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 \OML/cmm/ m/it/12 `[] [] \OMS/cmsy/m/n/12  [] [] [] - + File: lectures/../img/lez8-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez8-img2.JPG used on input line 55. @@ -1548,24 +1548,24 @@ Underfull \hbox (badness 10000) in paragraph at lines 74--104 [] -[55 <./lectures/../img/lez8-img2.JPG>] [56] +[56 <./lectures/../img/lez8-img2.JPG>] [57] Underfull \hbox (badness 10000) in paragraph at lines 125--133 [] - + File: lectures/../img/lez8-img3.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez8-img3.JPG used on input line 144. (pdftex.def) Requested size: 195.0pt x 59.7786pt. - + File: lectures/../img/lez8-img4.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez8-img4.JPG used on input line 165. (pdftex.def) Requested size: 234.00238pt x 85.03378pt. -[57 <./lectures/../img/lez8-img3.JPG>] +[58 <./lectures/../img/lez8-img3.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 183--186 [] @@ -1575,13 +1575,13 @@ Underfull \hbox (badness 10000) in paragraph at lines 188--195 [] - + File: lectures/../img/lez8-img5.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez8-img5.JPG used on input line 206. (pdftex.def) Requested size: 312.00119pt x 84.1262pt. -[58 <./lectures/../img/lez8-img4.JPG>]) [59 <./lectures/../img/lez8-img5.JPG>] +[59 <./lectures/../img/lez8-img4.JPG>]) [60 <./lectures/../img/lez8-img5.JPG>] (lectures/lecture9.tex Chapter 9. @@ -1594,7 +1594,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 9--12 [] - + File: lectures/../img/lez9-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez9-img1.JPG used on input line 27. @@ -1609,7 +1609,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 18--40 [] -[60 +[61 <./lectures/../img/lez9-img1.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 42--55 @@ -1632,7 +1632,7 @@ Overfull \hbox (72.7222pt too wide) detected at line 63 OMS/cmsy/m/n/12  []  [] [] - + File: lectures/../img/lez9-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez9-img2.JPG used on input line 70. @@ -1642,7 +1642,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 66--74 [] -[61 <./lectures/../img/lez9-img2.JPG>] +[62 <./lectures/../img/lez9-img2.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 84--90 [] @@ -1657,7 +1657,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 103--112 [] -[62] +[63] Underfull \hbox (badness 10000) in paragraph at lines 145--148 [] @@ -1697,12 +1697,12 @@ Underfull \hbox (badness 10000) in paragraph at lines 190--197 [] -[63] +[64] Underfull \hbox (badness 10000) in paragraph at lines 204--207 [] - + File: lectures/../img/lez9-img3.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez9-img3.JPG used on input line 215. @@ -1718,7 +1718,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 209--119 [] -[64 <./lectures/../img/lez9-img3.JPG>] [65] (lectures/lecture10.tex +[65 <./lectures/../img/lez9-img3.JPG>] [66] (lectures/lecture10.tex Chapter 10. LaTeX Font Warning: Command \small invalid in math mode on input line 11. @@ -1774,7 +1774,7 @@ Overfull \hbox (4.56644pt too wide) detected at line 53 cmr/m/n/12 = [] -[66 +[67 ] Overfull \hbox (17.46814pt too wide) detected at line 61 @@ -1797,7 +1797,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 77--80 [] - + File: lectures/../img/lez10-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez10-img1.JPG used on input line 84. @@ -1816,8 +1816,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 109--112 [] -[67] - +[68] + File: lectures/../img/lez10-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez10-img2.JPG used on input line 131 @@ -1827,10 +1827,10 @@ Package pdftex.def Info: lectures/../img/lez10-img2.JPG used on input line 131 LaTeX Warning: `h' float specifier changed to `ht'. -) [68 <./lectures/../img/lez10-img1.JPG>] (lectures/lecture11.tex [69 <./lectur +) [69 <./lectures/../img/lez10-img1.JPG>] (lectures/lecture11.tex [70 <./lectur es/../img/lez10-img2.JPG>] Chapter 11. - + File: lectures/../img/lez11-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez11-img1.JPG used on input line 21. @@ -1851,10 +1851,10 @@ Underfull \hbox (badness 10000) in paragraph at lines 9--39 [] -[70 +[71 <./lectures/../img/lez11-img1.JPG>] - + File: lectures/../img/lez11-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez11-img2.JPG used on input line 53. @@ -1891,8 +1891,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 123--125 [] -[71 <./lectures/../img/lez11-img2.JPG>] - +[72 <./lectures/../img/lez11-img2.JPG>] + File: lectures/../img/lez11-img3.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez11-img3.JPG used on input line 135 @@ -1913,7 +1913,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 166--170 [] -[72 <./lectures/../img/lez11-img3.JPG>] +[73 <./lectures/../img/lez11-img3.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 193--195 [] @@ -1971,7 +1971,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 233--251 [] -[73] +[74] Underfull \hbox (badness 10000) in paragraph at lines 253--256 [] @@ -2006,8 +2006,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 270--287 [] -[74] - +[75] + File: lectures/../img/lez11-img4.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez11-img4.JPG used on input line 311 @@ -2048,7 +2048,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 296--333 [] -[75]) [76 <./lectures/../img/lez11-img4.JPG>] (lectures/lecture12.tex +[76]) [77 <./lectures/../img/lez11-img4.JPG>] (lectures/lecture12.tex Chapter 12. Underfull \hbox (badness 10000) in paragraph at lines 8--17 @@ -2060,7 +2060,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 8--17 [] - + File: lectures/../img/lez12-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img1.JPG used on input line 35. @@ -2079,19 +2079,19 @@ Underfull \hbox (badness 10000) in paragraph at lines 32--44 LaTeX Warning: `h' float specifier changed to `ht'. -[77 +[78 ] Underfull \hbox (badness 10000) in paragraph at lines 49--60 [] -[78 <./lectures/../img/lez12-img1.JPG>] +[79 <./lectures/../img/lez12-img1.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 65--67 [] - + File: lectures/../img/lez12-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img2.JPG used on input line 83. @@ -2102,7 +2102,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 80--88 [] - + File: lectures/../img/lez12-img3.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img3.JPG used on input line 108 @@ -2111,14 +2111,14 @@ Package pdftex.def Info: lectures/../img/lez12-img3.JPG used on input line 108 LaTeX Warning: `h' float specifier changed to `ht'. - + File: lectures/../img/lez12-img4.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img4.JPG used on input line 124 . (pdftex.def) Requested size: 155.99762pt x 130.43692pt. -[79 <./lectures/../img/lez12-img2.JPG>] - +[80 <./lectures/../img/lez12-img2.JPG>] + File: lectures/../img/lez12-img5.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img5.JPG used on input line 147 @@ -2128,8 +2128,8 @@ Package pdftex.def Info: lectures/../img/lez12-img5.JPG used on input line 147 LaTeX Warning: `h' float specifier changed to `ht'. -[80 <./lectures/../img/lez12-img3.JPG> <./lectures/../img/lez12-img4.JPG>] - +[81 <./lectures/../img/lez12-img3.JPG> <./lectures/../img/lez12-img4.JPG>] + File: lectures/../img/lez12-img6.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img6.JPG used on input line 156 @@ -2140,8 +2140,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 173--176 [] -[81 <./lectures/../img/lez12-img5.JPG> <./lectures/../img/lez12-img6.JPG>] - +[82 <./lectures/../img/lez12-img5.JPG> <./lectures/../img/lez12-img6.JPG>] + File: lectures/../img/lez12-img7.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez12-img7.JPG used on input line 180 @@ -2167,7 +2167,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 199--210 [] -[82 <./lectures/../img/lez12-img7.JPG>]) [83] (lectures/lecture13.tex +[83 <./lectures/../img/lez12-img7.JPG>]) [84] (lectures/lecture13.tex Chapter 13. Underfull \hbox (badness 10000) in paragraph at lines 8--10 @@ -2213,7 +2213,7 @@ Overfull \hbox (17.38333pt too wide) detected at line 38 /12 c \OMS/cmsy/m/n/12  [] [] [] [] - + File: lectures/../img/lez13-img1.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez13-img1.JPG used on input line 51. @@ -2224,16 +2224,16 @@ Underfull \hbox (badness 10000) in paragraph at lines 42--57 [] -[84 +[85 ] - + File: lectures/../img/lez13-img2.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez13-img2.JPG used on input line 65. (pdftex.def) Requested size: 77.99881pt x 83.57153pt. - + File: lectures/../img/lez13-img3.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez13-img3.JPG used on input line 74. @@ -2252,8 +2252,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 59--81 LaTeX Warning: `h' float specifier changed to `ht'. -[85 <./lectures/../img/lez13-img1.JPG> <./lectures/../img/lez13-img2.JPG>] - +[86 <./lectures/../img/lez13-img1.JPG> <./lectures/../img/lez13-img2.JPG>] + File: lectures/../img/lez13-img4.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez13-img4.JPG used on input line 93. @@ -2269,8 +2269,8 @@ Underfull \hbox (badness 10000) in paragraph at lines 86--100 [] -[86 <./lectures/../img/lez13-img3.JPG> <./lectures/../img/lez13-img4.JPG>] - +[87 <./lectures/../img/lez13-img3.JPG> <./lectures/../img/lez13-img4.JPG>] + File: lectures/../img/lez13-img5.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez13-img5.JPG used on input line 120 @@ -2291,7 +2291,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 102--127 [] -[87 <./lectures/../img/lez13-img5.JPG>] +[88 <./lectures/../img/lez13-img5.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 134--138 [] @@ -2311,7 +2311,7 @@ Underfull \hbox (badness 10000) in paragraph at lines 143--148 [] - + File: lectures/../img/lez13-img6.JPG Graphic file (type jpg) Package pdftex.def Info: lectures/../img/lez13-img6.JPG used on input line 152 @@ -2327,12 +2327,12 @@ Underfull \hbox (badness 10000) in paragraph at lines 149--167 [] -[88 <./lectures/../img/lez13-img6.JPG>] +[89 <./lectures/../img/lez13-img6.JPG>] Underfull \hbox (badness 10000) in paragraph at lines 194--197 [] -) [89] [90] (lectures/lecture14.tex +) [90] [91] (lectures/lecture14.tex Chapter 14. 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Learning/Notes/main.tex index b99811058..4cb9da109 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/main.tex +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/main.tex @@ -155,5 +155,9 @@ \newpage \subfile{lectures/lecture22} \newpage +\subfile{lectures/lecture23} +\newpage +\subfile{lectures/lecture24} +\newpage \end{document} %This is never printed \ No newline at end of file diff --git a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/main.toc b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/main.toc index 60d1b24ec..b78aebbd9 100644 --- a/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/main.toc +++ b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/Notes/main.toc @@ -1,100 +1,104 @@ \babel@toc {english}{} -\contentsline {chapter}{\numberline {1}Lecture 1 - 09-03-2020}{7}% -\contentsline {section}{\numberline {1.1}Introduction of the course}{7}% -\contentsline {section}{\numberline {1.2}Examples}{7}% -\contentsline {subsection}{\numberline {1.2.1}Spam filtering}{10}% -\contentsline {chapter}{\numberline {2}Lecture 2 - 10-03-2020}{11}% -\contentsline {section}{\numberline {2.1}Argomento}{11}% -\contentsline {section}{\numberline {2.2}Loss}{11}% -\contentsline {subsection}{\numberline {2.2.1}Absolute Loss}{11}% -\contentsline {subsection}{\numberline {2.2.2}Square Loss}{12}% -\contentsline {subsection}{\numberline {2.2.3}Example of information of square loss}{13}% -\contentsline {subsection}{\numberline {2.2.4}labels and losses}{14}% -\contentsline {subsection}{\numberline {2.2.5}Example TF(idf) documents encoding}{16}% -\contentsline {chapter}{\numberline {3}Lecture 3 - 16-03-2020}{18}% -\contentsline {section}{\numberline {3.1}Overfitting}{20}% -\contentsline {subsection}{\numberline {3.1.1}Noise in the data}{20}% -\contentsline {section}{\numberline {3.2}Underfitting}{22}% -\contentsline {section}{\numberline {3.3}Nearest neighbour}{22}% -\contentsline {chapter}{\numberline {4}Lecture 4 - 17-03-2020}{25}% -\contentsline {section}{\numberline {4.1}Computing $h_{NN}$}{25}% -\contentsline {section}{\numberline {4.2}Tree Predictor}{27}% -\contentsline {chapter}{\numberline {5}Lecture 5 - 23-03-2020}{31}% -\contentsline {section}{\numberline {5.1}Tree Classifier}{31}% -\contentsline {section}{\numberline {5.2}Jensen’s inequality}{33}% -\contentsline {section}{\numberline {5.3}Tree Predictor}{37}% -\contentsline {section}{\numberline {5.4}Statistical model for Machine Learning}{38}% -\contentsline {chapter}{\numberline {6}Lecture 6 - 24-03-2020}{40}% -\contentsline {section}{\numberline {6.1}Bayes Optimal Predictor}{40}% -\contentsline {subsection}{\numberline {6.1.1}Square Loss}{41}% -\contentsline {subsection}{\numberline {6.1.2}Zero-one loss for binary classification}{42}% -\contentsline {section}{\numberline {6.2}Bayes Risk}{45}% -\contentsline {chapter}{\numberline {7}Lecture 7 - 30-03-2020}{47}% -\contentsline {section}{\numberline {7.1}Chernoff-Hoffding bound}{47}% -\contentsline {section}{\numberline {7.2}Union Bound}{48}% -\contentsline {section}{\numberline {7.3}Studying overfitting of a ERM}{52}% -\contentsline {chapter}{\numberline {8}Lecture 8 - 31-03-2020}{54}% -\contentsline {section}{\numberline {8.1}The problem of estimating risk in practise}{55}% -\contentsline {section}{\numberline {8.2}Cross-validation}{57}% -\contentsline {section}{\numberline {8.3}Nested cross validation}{59}% -\contentsline {chapter}{\numberline {9}Lecture 9 - 06-04-2020}{60}% -\contentsline {section}{\numberline {9.1}Tree predictors}{60}% -\contentsline {subsection}{\numberline {9.1.1}Catalan Number}{62}% -\contentsline {chapter}{\numberline {10}Lecture 10 - 07-04-2020}{66}% -\contentsline {section}{\numberline {10.1}TO BE DEFINE}{66}% -\contentsline {section}{\numberline {10.2}MANCANO 20 MINUTI DI LEZIONE}{66}% -\contentsline {section}{\numberline {10.3}Compare risk for zero-one loss}{68}% -\contentsline {chapter}{\numberline {11}Lecture 11 - 20-04-2020}{70}% -\contentsline {section}{\numberline {11.1}Analysis of $K_{NN}$}{70}% -\contentsline {subsection}{\numberline {11.1.1}Study of $K_{NN}$}{73}% -\contentsline {subsection}{\numberline {11.1.2}study of trees}{74}% -\contentsline {section}{\numberline {11.2}Non-parametric Algorithms}{75}% -\contentsline {subsection}{\numberline {11.2.1}Example of parametric algorithms}{76}% -\contentsline {chapter}{\numberline {12}Lecture 12 - 21-04-2020}{77}% -\contentsline {section}{\numberline {12.1}Non parametrics algorithms}{77}% -\contentsline {subsection}{\numberline {12.1.1}Theorem: No free lunch}{77}% -\contentsline {section}{\numberline {12.2}Highly Parametric Learning Algorithm}{79}% -\contentsline {subsection}{\numberline {12.2.1}Linear Predictors}{79}% -\contentsline {subsection}{\numberline {12.2.2}MinDisagreement}{83}% -\contentsline {chapter}{\numberline {13}Lecture 13 - 27-04-2020}{84}% -\contentsline {section}{\numberline {13.1}Linear prediction}{84}% -\contentsline {subsection}{\numberline {13.1.1}MinDisOpt}{84}% -\contentsline {section}{\numberline {13.2}The Perception Algorithm}{87}% -\contentsline {subsection}{\numberline {13.2.1}Perception convergence Theorem}{88}% -\contentsline {chapter}{\numberline {14}Lecture 14 - 28-04-2020}{91}% -\contentsline {section}{\numberline {14.1}Linear Regression}{91}% -\contentsline {subsection}{\numberline {14.1.1}The problem of linear regression}{91}% -\contentsline {subsection}{\numberline {14.1.2}Ridge regression}{92}% -\contentsline {section}{\numberline {14.2}Percetron}{93}% -\contentsline {subsection}{\numberline {14.2.1}Online Learning }{94}% -\contentsline {subsection}{\numberline {14.2.2}Online Gradiant Descent (OGD)}{96}% -\contentsline {chapter}{\numberline {15}Lecture 15 - 04-05-2020}{97}% -\contentsline {section}{\numberline {15.1}Regret analysis of OGD}{97}% -\contentsline {subsection}{\numberline {15.1.1}Projected OGD}{98}% -\contentsline {chapter}{\numberline {16}Lecture 16 - 05-05-2020}{102}% -\contentsline {section}{\numberline {16.1}Analysis of Perceptron in the non-separable case using OGD framework.}{102}% -\contentsline {subsection}{\numberline {16.1.1}Strongly convex loss functions}{106}% -\contentsline {chapter}{\numberline {17}Lecture 17 - 11-05-2020}{108}% -\contentsline {section}{\numberline {17.1}Strongly convex loss functions}{108}% -\contentsline {subsection}{\numberline {17.1.1}OGD for Strongly Convex losses}{108}% -\contentsline {subsection}{\numberline {17.1.2}Relate sequential risk and statistical risk}{109}% -\contentsline {chapter}{\numberline {18}Lecture 18 - 12-05-2020}{112}% -\contentsline {section}{\numberline {18.1}Kernel functions}{112}% -\contentsline {subsection}{\numberline {18.1.1}Feature expansion}{112}% -\contentsline {subsection}{\numberline {18.1.2}Kernels implements feature expansion (Efficiently}{113}% -\contentsline {section}{\numberline {18.2}Gaussian Kernel}{114}% -\contentsline {chapter}{\numberline {19}Lecture 19 - 18-05-2020}{117}% -\contentsline {section}{\numberline {19.1}Support Vector Machine (SVM)}{120}% -\contentsline {chapter}{\numberline {20}Lecture 20 - 19-05-2020}{121}% -\contentsline {section}{\numberline {20.1}Support Vector Machine Analysis}{121}% -\contentsline {subsection}{\numberline {20.1.1}Fritz John Optimality Conditions}{121}% -\contentsline {subsection}{\numberline {20.1.2}Non-separable case}{122}% -\contentsline {section}{\numberline {20.2}Pegasos: OGD to solve SVM}{124}% -\contentsline {chapter}{\numberline {21}Lecture 21 - 25-05-2020}{126}% -\contentsline {section}{\numberline {21.1}Pegasos in Kernel space}{126}% -\contentsline {section}{\numberline {21.2}Stability}{126}% -\contentsline {chapter}{\numberline {22}Lecture 22 - 26-05-2020}{131}% -\contentsline {section}{\numberline {22.1}Continous of Pegasos}{131}% -\contentsline {section}{\numberline {22.2}Boosting and ensemble predictors }{132}% -\contentsline {subsection}{\numberline {22.2.1}Bagging}{134}% -\contentsline {subsection}{\numberline {22.2.2}Random Forest}{134}% +\contentsline {chapter}{\numberline {1}Lecture 1 - 09-03-2020}{8}% +\contentsline {section}{\numberline {1.1}Introduction of the course}{8}% +\contentsline {section}{\numberline {1.2}Examples}{8}% +\contentsline {subsection}{\numberline {1.2.1}Spam filtering}{11}% +\contentsline {chapter}{\numberline {2}Lecture 2 - 10-03-2020}{12}% +\contentsline {section}{\numberline {2.1}Argomento}{12}% +\contentsline {section}{\numberline {2.2}Loss}{12}% +\contentsline {subsection}{\numberline {2.2.1}Absolute Loss}{12}% +\contentsline {subsection}{\numberline {2.2.2}Square Loss}{13}% +\contentsline {subsection}{\numberline {2.2.3}Example of information of square loss}{14}% +\contentsline {subsection}{\numberline {2.2.4}labels and losses}{15}% +\contentsline {subsection}{\numberline {2.2.5}Example TF(idf) documents encoding}{17}% +\contentsline {chapter}{\numberline {3}Lecture 3 - 16-03-2020}{19}% +\contentsline {section}{\numberline {3.1}Overfitting}{21}% +\contentsline {subsection}{\numberline {3.1.1}Noise in the data}{21}% +\contentsline {section}{\numberline {3.2}Underfitting}{23}% +\contentsline {section}{\numberline {3.3}Nearest neighbour}{23}% +\contentsline {chapter}{\numberline {4}Lecture 4 - 17-03-2020}{26}% +\contentsline {section}{\numberline {4.1}Computing $h_{NN}$}{26}% +\contentsline {section}{\numberline {4.2}Tree Predictor}{28}% +\contentsline {chapter}{\numberline {5}Lecture 5 - 23-03-2020}{32}% +\contentsline {section}{\numberline {5.1}Tree Classifier}{32}% +\contentsline {section}{\numberline {5.2}Jensen’s inequality}{34}% +\contentsline {section}{\numberline {5.3}Tree Predictor}{38}% +\contentsline {section}{\numberline {5.4}Statistical model for Machine Learning}{39}% +\contentsline {chapter}{\numberline {6}Lecture 6 - 24-03-2020}{41}% +\contentsline {section}{\numberline {6.1}Bayes Optimal Predictor}{41}% +\contentsline {subsection}{\numberline {6.1.1}Square Loss}{42}% +\contentsline {subsection}{\numberline {6.1.2}Zero-one loss for binary classification}{43}% +\contentsline {section}{\numberline {6.2}Bayes Risk}{46}% +\contentsline {chapter}{\numberline {7}Lecture 7 - 30-03-2020}{48}% +\contentsline {section}{\numberline {7.1}Chernoff-Hoffding bound}{48}% +\contentsline {section}{\numberline {7.2}Union Bound}{49}% +\contentsline {section}{\numberline {7.3}Studying overfitting of a ERM}{53}% +\contentsline {chapter}{\numberline {8}Lecture 8 - 31-03-2020}{55}% +\contentsline {section}{\numberline {8.1}The problem of estimating risk in practise}{56}% +\contentsline {section}{\numberline {8.2}Cross-validation}{58}% +\contentsline {section}{\numberline {8.3}Nested cross validation}{60}% +\contentsline {chapter}{\numberline {9}Lecture 9 - 06-04-2020}{61}% +\contentsline {section}{\numberline {9.1}Tree predictors}{61}% +\contentsline {subsection}{\numberline {9.1.1}Catalan Number}{63}% +\contentsline {chapter}{\numberline {10}Lecture 10 - 07-04-2020}{67}% +\contentsline {section}{\numberline {10.1}TO BE DEFINE}{67}% +\contentsline {section}{\numberline {10.2}MANCANO 20 MINUTI DI LEZIONE}{67}% +\contentsline {section}{\numberline {10.3}Compare risk for zero-one loss}{69}% +\contentsline {chapter}{\numberline {11}Lecture 11 - 20-04-2020}{71}% +\contentsline {section}{\numberline {11.1}Analysis of $K_{NN}$}{71}% +\contentsline {subsection}{\numberline {11.1.1}Study of $K_{NN}$}{74}% +\contentsline {subsection}{\numberline {11.1.2}study of trees}{75}% +\contentsline {section}{\numberline {11.2}Non-parametric Algorithms}{76}% +\contentsline {subsection}{\numberline {11.2.1}Example of parametric algorithms}{77}% +\contentsline {chapter}{\numberline {12}Lecture 12 - 21-04-2020}{78}% +\contentsline {section}{\numberline {12.1}Non parametrics algorithms}{78}% +\contentsline {subsection}{\numberline {12.1.1}Theorem: No free lunch}{78}% +\contentsline {section}{\numberline {12.2}Highly Parametric Learning Algorithm}{80}% +\contentsline {subsection}{\numberline {12.2.1}Linear Predictors}{80}% +\contentsline {subsection}{\numberline {12.2.2}MinDisagreement}{84}% +\contentsline {chapter}{\numberline {13}Lecture 13 - 27-04-2020}{85}% +\contentsline {section}{\numberline {13.1}Linear prediction}{85}% +\contentsline {subsection}{\numberline {13.1.1}MinDisOpt}{85}% +\contentsline {section}{\numberline {13.2}The Perception Algorithm}{88}% +\contentsline {subsection}{\numberline {13.2.1}Perception convergence Theorem}{89}% +\contentsline {chapter}{\numberline {14}Lecture 14 - 28-04-2020}{92}% +\contentsline {section}{\numberline {14.1}Linear Regression}{92}% +\contentsline {subsection}{\numberline {14.1.1}The problem of linear regression}{92}% +\contentsline {subsection}{\numberline {14.1.2}Ridge regression}{93}% +\contentsline {section}{\numberline {14.2}Percetron}{94}% +\contentsline {subsection}{\numberline {14.2.1}Online Learning }{95}% +\contentsline {subsection}{\numberline {14.2.2}Online Gradiant Descent (OGD)}{97}% +\contentsline {chapter}{\numberline {15}Lecture 15 - 04-05-2020}{98}% +\contentsline {section}{\numberline {15.1}Regret analysis of OGD}{98}% +\contentsline {subsection}{\numberline {15.1.1}Projected OGD}{99}% +\contentsline {chapter}{\numberline {16}Lecture 16 - 05-05-2020}{103}% +\contentsline {section}{\numberline {16.1}Analysis of Perceptron in the non-separable case using OGD framework.}{103}% +\contentsline {subsection}{\numberline {16.1.1}Strongly convex loss functions}{107}% +\contentsline {chapter}{\numberline {17}Lecture 17 - 11-05-2020}{109}% +\contentsline {section}{\numberline {17.1}Strongly convex loss functions}{109}% +\contentsline {subsection}{\numberline {17.1.1}OGD for Strongly Convex losses}{109}% +\contentsline {subsection}{\numberline {17.1.2}Relate sequential risk and statistical risk}{110}% +\contentsline {chapter}{\numberline {18}Lecture 18 - 12-05-2020}{113}% +\contentsline {section}{\numberline {18.1}Kernel functions}{113}% +\contentsline {subsection}{\numberline {18.1.1}Feature expansion}{113}% +\contentsline {subsection}{\numberline {18.1.2}Kernels implements feature expansion (Efficiently}{114}% +\contentsline {section}{\numberline {18.2}Gaussian Kernel}{115}% +\contentsline {chapter}{\numberline {19}Lecture 19 - 18-05-2020}{118}% +\contentsline {section}{\numberline {19.1}Support Vector Machine (SVM)}{121}% +\contentsline {chapter}{\numberline {20}Lecture 20 - 19-05-2020}{122}% +\contentsline {section}{\numberline {20.1}Support Vector Machine Analysis}{122}% +\contentsline {subsection}{\numberline {20.1.1}Fritz John Optimality Conditions}{122}% +\contentsline {subsection}{\numberline {20.1.2}Non-separable case}{123}% +\contentsline {section}{\numberline {20.2}Pegasos: OGD to solve SVM}{125}% +\contentsline {chapter}{\numberline {21}Lecture 21 - 25-05-2020}{127}% +\contentsline {section}{\numberline {21.1}Pegasos in Kernel space}{127}% +\contentsline {section}{\numberline {21.2}Stability}{127}% +\contentsline {chapter}{\numberline {22}Lecture 22 - 26-05-2020}{132}% +\contentsline {section}{\numberline {22.1}Continous of Pegasos}{132}% +\contentsline {section}{\numberline {22.2}Boosting and ensemble predictors }{133}% +\contentsline {subsection}{\numberline {22.2.1}Bagging}{135}% +\contentsline {subsection}{\numberline {22.2.2}Random Forest}{135}% +\contentsline {chapter}{\numberline {23}Lecture 23 - 08-06-2020}{137}% +\contentsline {section}{\numberline {23.1}Boosting}{137}% +\contentsline {section}{\numberline {23.2}Adaboost}{140}% +\contentsline {chapter}{\numberline {24}Lecture 24 - 09-06-2020}{142}%

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