From 61ea298cd7a428997e350186b5adf1d70f891f02 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Sat, 12 Feb 2022 12:55:12 +0100 Subject: [PATCH] Vorlesung 2 abgeschlossen --- Packages.tex | 3 + .../Linear_Classification.tex | 63 +++++++++++++++++- images/multiclass_linear_classification.png | Bin 0 -> 79923 bytes 3 files changed, 65 insertions(+), 1 deletion(-) create mode 100644 images/multiclass_linear_classification.png diff --git a/Packages.tex b/Packages.tex index e225f32..3aca7e1 100644 --- a/Packages.tex +++ b/Packages.tex @@ -114,6 +114,9 @@ rightsub = \grq% \usepackage{amsmath} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} +\DeclareMathOperator*{\lik}{lik} +\DeclareMathOperator*{\loss}{loss} +\DeclareMathOperator*{\loglik}{loglik} %special symbols \usepackage{fontawesome} \usepackage{amssymb} diff --git a/chapters/Classical_Supervised_Learning/Linear_Classification.tex b/chapters/Classical_Supervised_Learning/Linear_Classification.tex index a671bc7..b7ee6bd 100644 --- a/chapters/Classical_Supervised_Learning/Linear_Classification.tex +++ b/chapters/Classical_Supervised_Learning/Linear_Classification.tex @@ -105,6 +105,12 @@ Auf der Basis dieser Funktion lässt sich eine Verlustfunktion definieren: \begin{equation} \label{eq:sigmoid_loss_function} L(\bm{w}) = \sum_i (\nomeq{sigmoid}(f(\bm{x}_i)) - c_i)^2 = \sum_i (\nomeq{sigmoid}(\bm{w}^T\bm{x}_i + b) - c_i)^2 \end{equation} +Die \noms{sigmoid} hat zudem folgende Eigenschaften: +\begin{itemize} + \item \nomsym{sigmoid} ist begrenzt: $\nomeq{sigmoid}(a)\in(0,1)$ + \item \nomsym{sigmoid} ist symmetrisch: $1- \nomeq{sigmoid}(a) = \nomeq{sigmoid}(-a)$ + \item der Gradient kann einfach bestimmt werden: $\nomeq{sigmoid}'(a) = \frac{\exp(-a)}{(1+\exp(-a))^2} = \nomeq{sigmoid}(a)(1-\nomeq{sigmoid}(a))$ +\end{itemize} \subsubsection{Log-Likelihood}% \label{ssub:Log-Likelihood} @@ -231,4 +237,59 @@ aber auch zielgerichteter als beim \nameref{sub:SDG}. \frac{1}{b}\sum_i l(\bm{x}_i;\bm{\theta})\qquad \bm{\theta}_{t+1} = \bm{\theta}_t - \dfrac{\eta}{b}\sum_i \nabla_{\bm{\theta}} l(\bm{x}_i;\bm{\theta}_t) \end{equation} -WEITER AUF SEITE 166 +\subsection{Gradient for Logisitic Regression}% +\label{sub:Gradient for Logisitic Regression} +Die Loss Function für die Logistic Regression kann dank der Eigenschaften der \noms{sigmoid} (\cref{ssub:Logistic sigmoid function}) einfach aufgestellt werden: +\begin{align} \label{eq:gradient_for_logistic_regression} + \dfrac{\partial\text{loss}_i}{\partial\bm{w}} + &= \dfrac{\partial}{\partial\bm{w}}\left(c_i\log\nomeq{sigmoid}(\bm{w}^T\bm{\phi}(\bm{x}_i)) + (1 - c_i) \log(1-\nomeq{sigmoid}(\bm{w}^T\bm{\phi}(\bm{x}_i)))\right)\\ + &= \dots\text{\color{red}siehe Vorlesung 02 Folie 65}\\ + &= (c_i - \nomeq{sigmoid}(\bm{w}^T\bm{\phi}(\bm{x}_i)))\phi(\bm{x}_i) +\end{align} + +\section{Multiclass Classification}% +\label{sec:Multiclass Classification} +\begin{wrapfigure}{r}{.4\textwidth} + \vspace*{-10mm} + \centering + \includegraphics[width=0.8\linewidth]{multiclass_linear_classification.png} + \caption{Multiclass Linear Classification} + \label{fig:multiclass_linear_classification} + \vspace*{-10mm} +\end{wrapfigure} +Mithilfe der Linearen Klassifikation können auch mehr als zwei Klassen klassifiziert werden. +Hierfür wird die Softmax Likelihood Function verwendet: +\begin{equation} \label{eq:softmax_likelihood_function} + p(c=c_i|\bm{x}) = \frac{\exp(\bm{w}_i^T\bm{\phi}{x})}{\sum_{k=1}^K \exp(\bm{w}_k^T\bm{\phi}(\bm{x}))} +\end{equation} +Hiebei erhält jede Klasse einen eigenen Gewichtsvektor. +Die Klassifizierung erfolgt, +indem geguckt wird, für welche klasse $c_i$ $\bm{w}_i^T\bm{\phi}(\bm{x})$ am höchsten ist. + +Das Problem kann hierbei mithilfe des 1-Hot-Encodings als eine Conditional Multinomial Distribution aufgestellt werden (\cref{sub:Multinomial/Categorical Dristribution}): +\begin{align} \label{eq:multiclass_classification_distribution} + p(c|\bm{x}) &= \prod_{k=1}^K p(c=k|\bm{x})^{\bm{h}_{c,k}}\\ + &= \prod_{k=1}^K \left(\frac{\exp(\bm{w}_k^T\bm{\phi}(\bm{x}_i))}{\sum_{k'=1}^K \exp(\bm{w}_{k'}^T\bm{\phi}(\bm{x}_i))} \right)^{\bm{h}_{c,k}}\\ +\end{align} + +\paragraph{Data log-likelihood}% +\label{par:multiclass_classification:Data log-likelihood} +\begin{align} \label{eq:multiclass_classification:Data log-likelihood} + \loglik(\mathcal{D},\bm{w}_{1:K}) &= \sum_{i=1}^N\log p(c_i|\bm{x}_i)\\ + &= \dots \text{ \color{red}siehe Vorlesung 02 Folie 69}\\ + &= \sum_{i=1}^N\sum_{k=1}^K\bm{h}_{c_i,k}\bm{w}_k^T\bm{\phi}(\bm{x}_i) + - \underbrace{\log\left( \sum_{j=1}^K\exp(\bm{w}_j^T\bm{\phi}(\bm{x}_i))\right)}_\text{independent from $k$} \underbrace{\sum_k\bm{h}_{c_i,k}}_{=1} +\end{align} + +\paragraph{Gradient}% +\label{par:multiclass_classification:Gradient} +\begin{alignat*}{2} \label{eq:multiclass_classification:Gradient} + &\frac{\partial\loss_i}{\partial\bm{w}_k} + &&= \frac{\partial}{\partial \bm{w}_k}\left(\sum_{k=1}^K\bm{h}_{c_i,k}\bm{w}_k^T\bm{\phi}(\bm{x}_i) - \log\left( \sum_{j=1}^K\exp(\bm{w}_j^T\bm{\phi}(\bm{x}_i))\right) \right)\\ + & &&= \dots\text{ \color{red} siehe Übung 2 Aufgabe 1.1 }\\ + & &&= \bm h_{c_i,k} \bm\phi(\bm x_i) - \dfrac{\exp(\bm w_k^T\bm \phi(\bm x_i))}{\sum_j^K \exp(\bm w^T_j \bm \phi(\bm x_i))} \bm \phi(\bm x_i)\\ + \Leftrightarrow&\frac{\partial\loss_i}{\partial\bm{w}} + &&= - \sum_i^N \left( \bm h_{i,k} \bm \phi(\bm x_i) - \dfrac{\exp(\bm w_k^T\bm \phi(\bm x_i))}{\sum_j^K \exp(\bm w^T_j \bm \phi(\bm x_i))} \bm \phi(\bm x_i) \right) \\ + & &&= -\sum_i^N \bm \phi(\bm x_i) \left( h_{i,k} - p_{i,k} \right) +\end{alignat*} + diff --git a/images/multiclass_linear_classification.png b/images/multiclass_linear_classification.png new file mode 100644 index 0000000000000000000000000000000000000000..1713296104a02095ba987c75ed49aa6d21a7bf45 GIT binary patch literal 79923 zcma%ibyQnl&@C+#ij_hsE-hNz-F~=3ad&rj_u_?MK})ex2<}>(pbhTs?k?}r_ulvS 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