From 5a2303eb23af6373f54fe05b7dcfd8567cb22809 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Wed, 16 Feb 2022 21:48:44 +0100 Subject: [PATCH] =?UTF-8?q?Neural=20Networks=20Grundlagen=20hinzugef=C3=BC?= =?UTF-8?q?gt.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Glossary.tex | 4 + ML_Zusammenfassung.tex | 1 + Packages.tex | 3 + chapters/Neural_Networks/Basics.tex | 169 ++++++++++++++++++ images/Leaky_ReLU_activation_function.png | Bin 0 -> 673239 bytes images/ReLU_activation_function.png | Bin 0 -> 587783 bytes images/activation_functions.png | Bin 0 -> 196436 bytes ...ntial_linear_units_activation_function.png | Bin 0 -> 572901 bytes images/feature_learning.png | Bin 0 -> 28894 bytes images/feedforward_neural_network.png | Bin 0 -> 164350 bytes ...feedforward_neural_network_composition.png | Bin 0 -> 49646 bytes images/perceptron.png | Bin 0 -> 69058 bytes images/sigmoid_activation_function.png | Bin 0 -> 680107 bytes images/tanh_activation_function.png | Bin 0 -> 387940 bytes 14 files changed, 177 insertions(+) create mode 100644 chapters/Neural_Networks/Basics.tex create mode 100644 images/Leaky_ReLU_activation_function.png create mode 100644 images/ReLU_activation_function.png create mode 100644 images/activation_functions.png create mode 100644 images/exponential_linear_units_activation_function.png create mode 100644 images/feature_learning.png create mode 100644 images/feedforward_neural_network.png create mode 100644 images/feedforward_neural_network_composition.png create mode 100644 images/perceptron.png create mode 100644 images/sigmoid_activation_function.png create mode 100644 images/tanh_activation_function.png diff --git a/Glossary.tex b/Glossary.tex index b39acd2..0f494ee 100644 --- a/Glossary.tex +++ b/Glossary.tex @@ -80,6 +80,9 @@ \newacronym{RBF}{RBF}{Radial Basis Function Kernel} \newacronym{SVM}{SVM}{Support Vector Machine} \newacronym{ARD}{ARD}{Automatic Relevance Determination} +\newacronym{MLP}{MLP}{Multi-Layer Perceptron} +\newacronym{ReLU}{ReLU}{Rectified Linear Unit} +\newacronym{ELU}{ELU}{Exponential Linear Units} %-------------------- %nomenclature @@ -134,6 +137,7 @@ \newnom{gaussian_distribution}{Gaußsche Normalverteilung}{\mathcal{N}}{} \newnom{gaussian_process}{Gaußscher Prozess}{\mathcal{GP}}{} \newnom{hyper_parameters}{Hyper-Parameter}{\bm{\beta}}{} +\newnom{activation_function}{Aktivierungsfunktion}{\phi}{} \shorthandoff{"} \makeglossaries diff --git a/ML_Zusammenfassung.tex b/ML_Zusammenfassung.tex index 5c26169..87e70af 100644 --- a/ML_Zusammenfassung.tex +++ b/ML_Zusammenfassung.tex @@ -51,6 +51,7 @@ \part{Neural Networks} \label{part:Neural Networks} + \input{chapters/Neural_Networks/Basics.tex} \input{chapters/Neural_Networks/Neural_Networks_and_Backpropagation.tex} \input{chapters/Neural_Networks/CNNs_and_LSTMs.tex} diff --git a/Packages.tex b/Packages.tex index a3d3b6c..095db22 100644 --- a/Packages.tex +++ b/Packages.tex @@ -22,6 +22,8 @@ \usepackage{svg} %subfigures \usepackage{subcaption} +%borders around images +\usepackage[export]{adjustbox} %-------------------- %german quotation @@ -117,6 +119,7 @@ rightsub = \grq% \DeclareMathOperator*{\lik}{lik} \DeclareMathOperator*{\loss}{loss} \DeclareMathOperator*{\loglik}{loglik} +\DeclareMathOperator*{\softmax}{softmax} %special symbols \usepackage{fontawesome} \usepackage{amssymb} diff --git a/chapters/Neural_Networks/Basics.tex b/chapters/Neural_Networks/Basics.tex new file mode 100644 index 0000000..0b54e5f --- /dev/null +++ b/chapters/Neural_Networks/Basics.tex @@ -0,0 +1,169 @@ +\chapter{Basics}% +\label{cha:Neural Networks:Basics} +Neuronale Netze sind vom menschlichen Gehirn inspiriert. +Dieses besteht aus einer Vielzahl ($\approx 10^11$) Neuronen, +die untereinander stark verknüpft sind ($\approx 10^4$ Verknüpfungen pro Neuron). +Das Signal der eingehenden Neuronen muss stark genug sein, +damit ein angesteuertes Neuron ein Signal ausgibt. + +Diesem Prinzip folgend bestehen auch Neuronale Netze aus einfachen Bestandteilen (den Neuronen\slash\,Perzeptronen). +Ein Neuron hat dabei mehrere Eingänge und einen Ausgang. +Die Berechnung des Ausgabewertes ergibt sich durch + +\begin{equation} \label{eq:perceptron_output_function} + y = \nomeq{activation_function}(\bm{w}^T\bm x + b) +\end{equation} +\begin{wrapfigure}{r}{.5\textwidth} + \vspace*{-5mm} + \centering + \includegraphics[width=0.8\linewidth]{perceptron.png} + \caption{Aufbau eines Perzeptron} + \label{fig:perceptron} + \vspace*{-20mm} +\end{wrapfigure} +\begin{itemize} + \item $y$:\tabto{15mm}Output + \item \nomsym{activation_function}:\tabto{15mm}\noms{activation_function} + \item $\bm w$:\tabto{15mm}Gewichte + \item $\bm x$:\tabto{15mm} Inputs + \item $b$:\tabto{15mm}Bias +\end{itemize} +Ein Beispiel für eine solche Output-Funktion ist schon aus \cref{cha:Linear Regression} bekannt. + +\section{Feedforward Neural Networks}% +\label{sec:Feedforward Neural Networks} +\begin{wrapfigure}{r}{.4\textwidth} + \vspace*{-10mm} + \centering + \includegraphics[width=\linewidth]{feedforward_neural_network.png} + \caption{Feedforward Neural Network} + \label{fig:feedforward_neural_network} +\end{wrapfigure} +Ein Feedforward Neural Network ist ein Neurales Netzwerk, +bei dem die einzelnen Neuronen in Schichten aufgebaut sind, +wobei jedes Neutronen nur Inputs aus der darunterliegenden Schicht erhält und an die darüberliegende Schicht weitergibt. +Feedforward Neural Networks werden daher auch als \glspl{MLP} bezeichnet. +Das Netzwerk hat daher die Form eines gerichteten azyklischen Graphens (Directed Acyclic Graph). + +Hierbei werden die Eingabewerte an die Eingabeschicht (Input Layer) übergeben und die Ausgabe des Neuralen Netzwerkes erfolgt an der Output Layer. +Alle Schichten zwischen Input und Output Layer werden als verdeckte Schichten (Hidden Layer) bezeichnet. + +Die Gewichte zwischen den einzelnen Layern bilden eine Matrix $\bm W$. +Der Ausgabevektor $\bm y$, +bestehend aus den Ausgabewerten der in der Schicht enthaltenen Neuronen, +ergibt sich durch: + +\begin{equation} \label{eq:output_function_feedforward_layer} + \bm y = \nomeq{activation_function}(\bm W\bm x + \bm b) +\end{equation} +\begin{wrapfigure}{r}{.3\textwidth} + \vspace*{-8mm} + \centering + \includegraphics[width=0.8\linewidth]{feedforward_neural_network_composition.png} + \caption{Feedforward Neural Network mit Funktionen} + \label{fig:feedforward_neural_network_composition} + \vspace*{-20mm} +\end{wrapfigure} +Da jede Layer einen Vektor als Eingabe und Ausgabe hat, +kann man sie auch als vektorwertige Funktion schreiben +\begin{align} \label{eq:feedforward_neural_network_vector_valued_function} + \bm h^{(1)} &= f^{(1)}(\bm x)\\ + \bm h^{(2)} &= f^{(2)}(\bm h^{(1)})\\ + \vdots\\ + \bm y &= f^{(L)}(\bm h^{(L-1)}) +\end{align} +Daher kann man die Ausgabe eines Feedforward Neural Networks auch als Komposition der Funktionen der einzelnen Schichten betrachten. +\begin{equation} \label{eq:feedforward_neural_network_composition} + \bm y = f^{(L)}\circ f^{(L-1)}\circ\dots f^{(1)}(\bm x) +\end{equation} +Hieraus ergibt sich allerdings auch, +dass jedes mehrschichtige Feedforward Neural Network auch durch ein Netzwerk mit nur einer Schicht ersetzt werden kann (Universal Representation). +Hierzu wären allerdings eine exponentielle Anzahl an Neuronen nötig, +weshalb in der Praxis mehrschichtige Netzwerke verwendet werden (Compact Representation). + +\subsection{Example: XOR}% +\label{sub:Example: XOR} +({\color{red}siehe Vorlesung 08 Folie 25 und 26}) + +\section{\nomf{activation_function}}% +\label{sec:Activation Function} +Die \nomf{activation_function} gibt vor, +ab welchem Schwellwert das Produkt aus Eingangswerten und Gewichten zu relevanten Ausgabewert führt. +In den meisten Fällen wird die \glsxtrshort{ReLU} \noms{activation_function} verwendet, +wobei es sich auch lohnt, die Leaky \glsxtrshort{ReLU} oder \glsxtrshort{ELU} auszubrobieren. +Die Sigmoid Funktion (\cref{ssub:Logistic sigmoid function}) sollte ausschließlich als \noms{activation_function} in Klassifikationsproblemen verwendet werden.\\ +\includegraphics[scale=.7]{sigmoid_activation_function.png}\\ +\hrule{\textwidth,1mm} +\includegraphics[scale=.7]{tanh_activation_function.png}\\ +\hrule{\textwidth,1mm} +\includegraphics[scale=.7]{ReLU_activation_function.png}\\ +\hrule{\textwidth,1mm} +\includegraphics[scale=.7]{Leaky_ReLU_activation_function.png}\\ +\hrule{\textwidth,1mm} +\includegraphics[scale=.7]{exponential_linear_units_activation_function.png}\\ + + +\section{Optimization}% +\label{sec:Optimization} +Die Optimierung der Parameter $\bm \theta$ erfolgen nach allgemeinen ML Rezept: +\begin{equation} \label{eq:neural_network_parameter_optimization} + \bm\theta^* = \argmin_{\bm\theta} \sum_{i=1}^N l(\bm x_i,\bm\theta) + \nomeq{regularization_factor}\text{ penalty}(\bm\theta) +\end{equation} +Welche Loss Function $l$ verwendet werden sollte hängt dabei vom Anwendungsgebiet ab: +\begin{table}[H] + \centering + \caption{Loss Functions für verschiedene Anwendungsaufgaben} + \label{tab:loss_functions_for_different_tasks} + \begin{tabularx}{\textwidth}{|c|Y|Y|} + \hline + & \bfseries Deterministic & \bfseries Probabilistic \\ + \hline + \hline + \multicolumn{3}{|c|}{\bfseries Regression} \\ + \hline + \bfseries Output Layer & + \begin{tabular}{@{}c@{}} linear \\ $\bm f = \bm W^{(L)}\bm h^{(L-1)}+\bm b^{(L)}$ \end{tabular} & + \begin{tabular}{@{}c@{}} linear Gaussian \\ $p(\bm y|\bm x) = \nomeq{gaussian_distribution}(\bm y|\bm W^{(L)}\bm h^{(L-1)} + \bm b^{(L)}, \nomeq{covariance})$\end{tabular}\\ + \hline + \bfseries Loss & + \begin{tabular}{@{}c@{}} squared error \\ $l_i(\bm x_i,\bm\theta) = \frac{1}{2}(\bm f(\bm x_i)-\bm y_i)^2$\end{tabular} & + \begin{tabular}{@{}c@{}} negative log-likelihood \\ $l_i(\bm x_i,\bm\theta) = -\log\nomeq{gaussian_distribution}(\bm y_i|\bm\mu(\bm x_i),\nomeq{covariance})$ \end{tabular}\\ + \hline + \hline + \multicolumn{3}{|c|}{\bfseries Binary Classification} \\ + \hline + \bfseries Output Layer & + \begin{tabular}{@{}c@{}} linear \\ $\bm f = \bm W^{(L)}\bm h^{(L-1)}+ b^{(L)}$ \end{tabular} & + \begin{tabular}{@{}c@{}} sigmoid \\ $f = \sigma(\bm W^{(L)}\bm h^{(L-1)} + b^{(L)})$\end{tabular}\\ + \hline + \bfseries Loss & + \begin{tabular}{@{}c@{}} \nameref{sub:Hinge Loss} \\ $l(\bm x_i, \bm\theta) = \max(0,1-y_i f(\bm x_i))$\end{tabular} & + \begin{tabular}{@{}c@{}} negative log-likelihood \\ $l_i(\bm x_i,\bm\theta) = -c_i \log f(\bm x_i)$\end{tabular}\\ + \hline + \hline + \multicolumn{3}{|c|}{\bfseries Multi-class Classification} \\ + \hline + \bfseries Output Layer & + \begin{tabular}{@{}c@{}} linear \\ $\bm f = \bm W^{(L)}\bm h^{(L-1)} + \bm b^{(L)}$\end{tabular} & + \begin{tabular}{@{}c@{}} sigmoid \\ $\bm f = \softmax(\bm W^{(L)}\bm h^{(L-1)} + \bm b^{(L)})$\end{tabular}\\ + \hline + \bfseries Loss & + \begin{tabular}{@{}c@{}} Multi-class SVM loss \\ {\color{red} in diesem Fach nicht behandelt} \end{tabular} & + \begin{tabular}{@{}c@{}} negative log-likelihood \\ $l_i(\bm x_i,\bm\theta) = -\sum_{k=1}^K \bm h_{c_i,k} \log y_k(\bm x_i)$\end{tabular}\\ + \hline + \end{tabularx} +\end{table} + +\section{Feature Learning}% +\label{sec:Feature Learning} +Die letzte Schicht eines Neural Networks macht im Prinzip nur eine \nameref{cha:Linear Regression}, +welche durch die vorhergegangenen Schichten vorbereitet wird. +\begin{figure}[H] + \centering + \includegraphics[width=0.7\textwidth]{feature_learning.png} + \caption{Schichten vor dem letzten Layer bereiten die Daten für eine \nameref{cha:Linear Regression} vor} + \label{fig:feature_learning} +\end{figure} +\subsection{Beispiel}% +\label{sub:Feature Learning:Beispiel} +{\color{red} siehe Vorlesung 08 Folie 35} diff --git a/images/Leaky_ReLU_activation_function.png b/images/Leaky_ReLU_activation_function.png new file mode 100644 index 0000000000000000000000000000000000000000..a4abb5135201ef8345fcce8a3323990627dc8a15 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