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lecture 23
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\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}
|
||||
|
|
@ -0,0 +1,6 @@
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\begin{document}
|
||||
|
||||
\chapter{Lecture 24 - 09-06-2020}
|
||||
|
||||
\end{document}
|
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\babel@toc {english}{}
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||||
\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}%
|
||||
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|
||||
\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}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\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}%
|
||||
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|
||||
\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}%
|
||||
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|
||||
\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}%
|
||||
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|
||||
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|
||||
\contentsline {subsection}{\numberline {12.1.1}Theorem: No free lunch}{77}%
|
||||
\contentsline {section}{\numberline {12.2}Highly Parametric Learning Algorithm}{79}%
|
||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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||||
\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}%
|
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\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}%
|
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\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}%
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\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}%
|
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\contentsline {subsection}{\numberline {20.1.2}Non-separable case}{122}%
|
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\contentsline {section}{\numberline {20.2}Pegasos: OGD to solve SVM}{124}%
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\contentsline {chapter}{\numberline {21}Lecture 21 - 25-05-2020}{126}%
|
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\contentsline {section}{\numberline {21.1}Pegasos in Kernel space}{126}%
|
||||
\contentsline {section}{\numberline {21.2}Stability}{126}%
|
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\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}%
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\contentsline {subsection}{\numberline {22.2.1}Bagging}{134}%
|
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\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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Loading…
Reference in New Issue
Block a user