From f78b9da0a87a1b73bb4691cb2a5c36f2b2088393 Mon Sep 17 00:00:00 2001 From: Andreaierardi Date: Tue, 19 May 2020 10:26:30 +0200 Subject: [PATCH] up lect 20 --- .../Machine Learning/img/lez20-img1.JPG | Bin 0 -> 9772 bytes .../Machine Learning/img/lez20-img2.JPG | Bin 0 -> 9772 bytes .../Machine Learning/img/lez20-img3.JPG | Bin 0 -> 13383 bytes .../Machine Learning/lectures/lecture20.aux | 13 + .../Machine Learning/lectures/lecture20.log | 423 ++++++++++++++++++ .../Machine Learning/lectures/lecture20.pdf | Bin 0 -> 206546 bytes .../lectures/lecture20.synctex.gz | Bin 0 -> 32615 bytes .../Machine Learning/lectures/lecture20.tex | 205 +++++++++ .../Machine Learning/main.aux | 7 + .../Machine Learning/main.lof | 3 + .../Machine Learning/main.log | 92 +++- .../Machine Learning/main.pdf | Bin 2630747 -> 2680957 bytes .../Machine Learning/main.synctex.gz | Bin 617840 -> 652074 bytes .../Machine Learning/main.toc | 4 + 14 files changed, 737 insertions(+), 10 deletions(-) create mode 100644 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\frac{u^*}{\gamma^*} + $$ + $$ + \gamma^2 \| w\|^2 = 1 \textit{ is redundant!} \qquad y_t w^T x_t \geq 1 \ t=1,...,m + $$ +What we do with $w^*$? +\\ +\subsection{Fritz John Optimality Conditions} +$$ +\min_{w \in \barra{R}^d} f(w) \quad s.t \quad g_t(w) \leq 0 \ t=1,...,m \qquad \textit{ $f,g_1,..,g_m$ all differentiable} +$$ +If $w_0$ is optimal solution, then $\exists \alpha = (\alpha_1,...,\alpha_m) \in \barra{R}^m$ +\\ +$$ +\nabla f(w_0) + \sum_{t \in I} \alpha_t \nabla g_t (w_0) = 0 \qquad I = \{ t : g_t(w_0) = 0 \} +$$ +$$ +f(n) = \frac{1}{2} \| w \|^2 \qquad g_t(w)= 1 - y_t \, w^T \, x_t, \ \nabla g_t(w^*) = - y_t \, x_t +$$ +$w^*$ SVM solution +$$ +w^* - \sum_{t \in I} \alpha_t y_t x_t = 0 \ \ \Leftrightarrow \ \ w^* = \sum_{t \in I} \alpha_t y_t x_t +$$ +where $f(n) = \nabla f(w^*)$ +$$ +I = \{ \, t : y_t(w^*)^T \, x_t = 1 \} \textbf{ support vectors} +$$ +We want a generalisation of this two non separable training set. +\\ +\subsection{Non-separable case} +$$ +\min_{w \in \barra{R}^d} \frac{1}{2} + \| w\|^2 \qquad s.t \quad y_t\, w^T \, x_t \ \geq 1$$ +We cannot satisfy all the constraints since are inconsistent. Maybe we can try to satisfy the most possible constrain so: +$$ +\min_{w \in \barra{R}^d} \frac{1}{2} + \| w\|^2 + \frac{1}{2} \sum_{t=1}^m \xi_t \qquad + y_t\, w^T \, x_t \ \geq 1- \xi_t +$$ +where $\xi_t$ slack variables and $ \xi > 0$ +We want $\xi_t$ given $w$: +$$ +\xi_t = \begin{cases} +1- y_t \, w^T \, x_t \qquad if \ y_t \, w^T x_t < 1 +\\ +0 \qquad otherwise +\end{cases} \qquad \xi_t = \left[ 1- y_t \, w^T \, x_t \right] = h_t(w) \ \textit{ hinge loss} +$$ +We replate this in the first equation and we get a convex function plus $\lambda$-SC function: +$$ +\min_{w \in \barra{R}^d} F(w) \qquad F(w) = \frac{1}{m} \sum_{t = 1}^m h_t(w) + \frac{1}{2} +\| w\|^2 $$ +And this is unconstrained and $F(W)$ is $\lambda$-S.C. +\\\\ +I also want to check my shape of the function is not changing. +\\ +Assume I can write the solution as: +$$ +w^* = \sum_{t=1}^m \alpha_t \, y_t \, x_t + u +$$ +where $u$ is orthogonal to each of $x_1,...,x_m$ +$ + \sum_{t=1}^m \alpha_t \, y_t \, x_t \ = \ v + $ +$$ + w^* = v + u \qquad v = w^* - u \quad \|v \| \leq \| w^*\| +$$ +\begin{figure}[h] + \centering + \includegraphics[width=0.3\linewidth]{../img/lez20-img1.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +\\ +Now I can check the hinge loss: +$$ +h_t(v) = \left[ 1- y_t(w^*)^* x_t+ \red{y_t \, u^T \, x_t} \right]_+ = h_t(w^*) +$$ +Since $y_t \, u^T \, x_t = 0$ this cancel out and we get the hinge loss. +$$ +F(v) = \frac{1}{m} \sum_t h_t(w^*) + \frac{1}{2} \| w\|^2 \leq F(w^*) +$$ +$$ +w^* = \sum_{t=1}^m \alpha_t y_t x_t \qquad \alpha_t \neq 0 \ \Leftrightarrow \ h_t(w^*) > 0 +$$ +Including $t : y_t(w^*)^T x_t = 1$\\ +\begin{figure}[h] + \centering + \includegraphics[width=0.2\linewidth]{../img/lez20-img2.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +\\ +Support vector are those in which I need slack variables in order to be satisfied. +\begin{figure}[h] + \centering + \includegraphics[width=0.5\linewidth]{../img/lez20-img3.JPG} + \caption{} + %\label{fig:} +\end{figure}\\ +\\ +$$ +F(w) = \frac{1}{m} \sum_{t=1}^m h_t(w) + \frac{1}{2} \| w\|^2 = \frac{1}{m} \sum_{t=1}^m \ell_t(w) +$$ +MANCA FORMULAA\\ +We need to minimise the hinge loss and we use Pegasos. +\section{Pegasos: OGD to solve SVM} +Stochastic gradiant descent. +\\\\ +Parameters: $ \lambda > 0$, $T$ number of rounds +\\ +Set $w_1 =(0,...,0)$ +\\ +For $t = 1,...,T$ +\\ +1) Draw $(x_{zt}, y_{zt})$ at random from training set +\\ +2) $w_{t+1} = w_t - \eta_t \nabla \ell_{zt}(w_t)$ +\\ +Output $\bar{w} = \frac{1}{T} \sum_t w_t$ +$$ +\ell_{zt}(w) = h_{zt} (w) + \frac{1}{2} \| w\|^2 \qquad w^* = arg \min_{w \in \barra{R}^*} \left( \frac{1}{m} \sum_{t=1}^m h_t(w) + \frac{\lambda}{2} \| w\|^2\right) +$$ +$\forall s_1,...,s_T$ realisation of $z_1,...,z_T$ +$$ +\frac{1}{T} \sum_{t=1}^m \ell_{st}(w_t) \leq \frac{1}{T}\sum_{t=1}^m \ell_{st} (w^*) + \frac{G^2}{2 \, \lambda \, T} \ln (T+1) \quad OGD\ Analysis +$$ +$$ +G = \max_t \| \nabla \ell_{st}(w_t) \| +$$ +In general $G$ is random. +$$ +F(\bar{w}) \leq F(w^*) + \varepsilon \qquad \| \bar{w}- w^*\| \ \leq^? \ | F(\bar{w}) - F(w^*) | \ \leq \ L \| \bar{w} - w^* \| +$$ +where $F$ is the average of the losses: $F(w_t) = \frac{1}{m} \sum_{s=1}^m \ell_s(w_t)$ \\ +So we use Liptstik solution. +$$ +\barra{E} \, [\ell_{zt}(w_t) | z_1,...,z_{t-1}] = \frac{1}{m} \sum_{s=1}^m \ell_s(w_t) \qquad \barra{E} \left[ \, X \, \right] = \barra{E} \left[ \, \barra{E} \left[ \, X | Y \, \right] \, \right] +$$ +Now we use Jensen inequality: +$$ +\barra{E} \, [ F(\bar{w} ) ] \ \leq^{J} \barra{E} \, \left[ \frac{1}{T} \sum_{t=1}^T F(w_t) \right] \ = \ \barra{E} \, \left[ \frac{1}{T} \sum_{t=1}^T \barra{E} \, \left[ \, \ell_{zt}(w_t) \, | \, z_1,...,z_{t-1} \, \right] \, \right] +\ = $$ +$$ +\ = \ \barra{E} \left[ \frac{1}{T} \sum^T_{t=1} \ell_{zt} (w_t) \right] \ \leq \ \barra{E} \left[ \frac{1}{T} \sum_{t=1}^T \ell_{zt}(w^*) \right] + \frac{\barra{E}[G^2]}{2 \, \lambda \, T} \ln (T+1) = +$$ +$$ += \barra{E} \left[ \frac{1}{T} \sum_{t=1}^T \barra{E} [\ell_{zt}(w^*) | z_1,...,z_{t-1} ]\right] + \frac{\barra{E} [G^2]}{2 \, \lambda \, T} \ \ln \, (T+1) = +$$ +$$ += \barra{E} \left[ \frac{1}{T} \sum_{t=1}^T F(w^*) \right] + \frac{E[G^2]}{2 \, \lambda \, T} \ln (T+1) = \ = F(w^*) + \frac{\barra{E} [G^2]}{2 \, \lambda \, T} \ln (T+1) +$$ +$\barra{E} [G^2] \leq ? +$ +We are bounding $G^2 \ \forall s_1,..,s_T$ +$$ +\nabla \ell_{st}(w_t) = - y_{st} \, x_{st} \, I\{ h_{st}(w_t) >0\} + \lambda \, w\qquad \ell_s(w) = h_t(w) + \frac{1}{2} \|w\|^2 +$$ +$$ +v_t = y_{st} \, x_{st} \, I \{ h_{st} (w_t) >0 \}, \quad \nabla \ell_{st}(w_t) = - v_t + \lambda w_t \quad \eta_t = \frac{1}{\lambda \, t} +$$ +$$ +w_{t+1} = w_t - \eta_t \nabla \ell_t(w_t) = w_t+ \eta_t \, v_t -\eta_t \, \lambda \, w_t = \left(1- \frac{1}{t} \right) w_t + \frac{1}{\lambda \, t} v_t= +$$ +$$ +\|\nabla \ell_{st}(w_t) \| \leq \| v_t \| + \lambda \| w_t \| \leq X + \lambda \|w_t\| \qquad X = \max_t \| x_t\| +$$ +$$ +w_{t+1} = \left( 1- \frac{1}{t} \right) w_t + \frac{1}{\lambda \, t} v_t \qquad w_1 = (0,...0) \quad w_t = \sum_t \beta_t v_t +$$ +Fix $s ] (lectures/lecture20.tex Chapter 20. -) [121 -] (main.bbl +Underfull \hbox (badness 10000) in paragraph at lines 23--26 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 29--32 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 45--48 + + [] + +[121 + +] +Overfull \hbox (33.66243pt too wide) detected at line 66 +\OML/cmm/m/it/12 [] \OT1/cmr/m/n/12 = [] \OML/cmm/m/it/12 [] \OT1/cmr/m/n/12 + = [] = \OML/cmm/m/it/12 h[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 w\OT1/cmr/m/n/12 +) [] + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 70--76 + + [] + + +File: lectures/../img/lez20-img1.JPG Graphic file (type jpg) + +Package pdftex.def Info: lectures/../img/lez20-img1.JPG used on input line 88. + +(pdftex.def) Requested size: 117.00119pt x 70.84105pt. + +Underfull \hbox (badness 10000) in paragraph at lines 85--94 + + [] + +[122 <./lectures/../img/lez20-img1.JPG>] + +File: lectures/../img/lez20-img2.JPG Graphic file (type jpg) + +Package pdftex.def Info: lectures/../img/lez20-img2.JPG used on input line 107 +. +(pdftex.def) Requested size: 77.99881pt x 62.51866pt. + +File: lectures/../img/lez20-img3.JPG Graphic file (type jpg) + +Package pdftex.def Info: lectures/../img/lez20-img3.JPG used on input line 115 +. +(pdftex.def) Requested size: 195.0pt x 94.35489pt. + +Underfull \hbox (badness 10000) in paragraph at lines 103--120 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 103--120 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 103--120 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 126--139 + + [] + +[123 <./lectures/../img/lez20-img2.JPG> <./lectures/../img/lez20-img3.JPG>] +[124]) [125] (main.bbl LaTeX Warning: Empty `thebibliography' environment on input line 3. -) [122 +) [126 ] (main.aux) ) Here is how much of TeX's memory you used: - 5696 strings out of 480934 - 87332 string characters out of 2909670 + 5714 strings out of 480934 + 87854 string characters out of 2909670 336249 words of memory out of 3000000 - 21286 multiletter control sequences out of 15000+200000 + 21301 multiletter control sequences out of 15000+200000 561784 words of font info for 96 fonts, out of 3000000 for 9000 1141 hyphenation exceptions out of 8191 34i,13n,42p,348b,361s stack positions out of 5000i,500n,10000p,200000b,50000s @@ -3063,9 +3135,9 @@ s/cm/cmr8.pfb> -Output written on main.pdf (123 pages, 2630747 bytes). +Output written on main.pdf (127 pages, 2680957 bytes). 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