From 344616294b1a7deb2605c0374355ba14c1fc9a82 Mon Sep 17 00:00:00 2001 From: Clemens Dautermann Date: Mon, 10 Feb 2020 12:00:52 +0100 Subject: [PATCH] more corrections --- doc/Grundlagen_des_maschinellen_lernens.aux | 10 ++++----- doc/Grundlagen_des_maschinellen_lernens.lof | 4 ++-- doc/Grundlagen_des_maschinellen_lernens.log | 6 ++--- doc/Grundlagen_des_maschinellen_lernens.pdf | Bin 1550898 -> 1550895 bytes ...dlagen_des_maschinellen_lernens.synctex.gz | Bin 262274 -> 262083 bytes doc/Grundlagen_des_maschinellen_lernens.tex | 21 +++++++++--------- 6 files changed, 21 insertions(+), 20 deletions(-) diff --git a/doc/Grundlagen_des_maschinellen_lernens.aux b/doc/Grundlagen_des_maschinellen_lernens.aux index aefa978..4894222 100644 --- a/doc/Grundlagen_des_maschinellen_lernens.aux +++ b/doc/Grundlagen_des_maschinellen_lernens.aux @@ -70,12 +70,12 @@ \@writefile{lof}{\defcounter {refsection}{0}\relax }\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Die Gleichung 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zYW5$?1vRIyxokgD{Nk5x{qxtEv0qyHjI2NF=+^n|Ub^A_YX61ycdj>l(ml0kYSZ$a z_1A86ORKC`o$Y#P&hxe%pO%Mi4}Hx0Rde3@7vXyrsoeiqv!&?2jX_QQTFLo~W$x{{ z9c5O#eSULDh49|rr}KY5Gyi$~>ZXU2C0-x@-YRSNR&-?9b{-3S? u|M`C1uhm(NH8r1Sr`r4ee>?sD|3B^f|9s7_|MhS7fA)~Nx2yB*g&6>C;-}RB diff --git a/doc/Grundlagen_des_maschinellen_lernens.tex b/doc/Grundlagen_des_maschinellen_lernens.tex index d34d567..6346394 100644 --- a/doc/Grundlagen_des_maschinellen_lernens.tex +++ b/doc/Grundlagen_des_maschinellen_lernens.tex @@ -288,7 +288,7 @@ Existieren mehr als 2 Klassen handelt es sich also nicht mehr um eine Binärklas Dabei gibt M die Anzahl der Klassen an, c das Label für die Klasse und o die berechnete Klassifizierung für diese Klasse. \subsection{Gradientenverfahren und Backpropagation} \label{Gradient_section} -Das Gradientenverfahren ist ein Verfahren um das Minimum einer Funktion zu finden. Die Funktion, deren Minimum gefunden werden soll ist in diesem Fall die Fehlerfunktion. Diese ist von allen Gewichtungen und Biases des Netzwerkes abhängig, da sie direkt vom Ausgabevektor des Netzes abhängig ist. Der Gradient dieser Funktion ist in Abbildung \ref{Gradient_Function} dargestellt. +Das Gradientenverfahren ist ein Verfahren, um das Minimum einer Funktion zu finden. Die Funktion, deren Minimum gefunden werden soll, ist in diesem Fall die Fehlerfunktion. Diese ist von allen Gewichtungen und Biases des Netzwerkes abhängig, da sie direkt vom Ausgabevektor des Netzes abhängig ist. Der Gradient dieser Funktion ist in Abbildung \ref{Gradient_Function} dargestellt. \begin{figure}[h] \begin{equation*} \nabla C(w_1,b_1,\ldots,w_n,b_n)= @@ -304,8 +304,8 @@ Das Gradientenverfahren ist ein Verfahren um das Minimum einer Funktion zu finde \label{Gradient_Function} \end{figure} \newline -Um also das Ergebnis \glqq richtiger'' zu machen, müssen alle Gewichtungen und Biases negativ zu diesem Gradienten angepasst werden, da der Gradient ja den Hochpunkt angibt. Diese Anpassung erfolgt, indem das Netz vom Ausgabelayer an, deshalb heißt das Verfahren Backpropagation, durchgegangen wird, und die Gewichtungen und Biases angepasst werden.\newline -Oft wird zur Verbildlichung des Gradientenverfahrens die Analogie eines Balles verwendet, der einen Hügel hinunter rollt. Er findet den Tiefpunkt indem er hinab rollt und dabei immer automatisch eine Kraft nach unten wirkt. +Um also das Ergebnis \glqq richtiger'' zu machen, müssen alle Gewichtungen und Biases negativ zu diesem Gradienten angepasst werden, da der Gradient ja in Richtung Hochpunkt zeigt. Diese Anpassung erfolgt, indem das Netz vom Ausgabelayer an deshalb heißt das Verfahren Backpropagation durchgegangen wird, und die Gewichtungen und Biases angepasst werden.\\ +Oft wird zur Verbildlichung des Gradientenverfahrens die Analogie eines Balles verwendet, der einen Hügel hinunter rollt. Er findet den Tiefpunkt, indem er hinab rollt. Dabei wirkt immer automatisch eine Kraft nach unten. \subsubsection{Lernrate} Eine wichtige Rolle dabei spielt die sogenannte \glqq Lernrate'' $\eta$, mit der die Änderung nach der Formel in Abbildung \ref{Learning_Rate_Function} berechnet wird. \begin{figure}[h] @@ -316,24 +316,24 @@ Eine wichtige Rolle dabei spielt die sogenannte \glqq Lernrate'' $\eta$, mit der \label{Learning_Rate_Function} \end{figure} \newline -Diese Lernrate ist notwendig um nicht über das Minimum \glqq hinweg zu springen''. Sollte sie zu groß sein, passiert genau dies, da die Anpassungen der Parameter in zu großen Schritten erfolgt. Sollte sie hingegen zu klein sein, lernt das Netz sehr langsam. Typische Werte sind abhängig von der zu erlernenden Aufgabe, liegen jedoch in der Regel bei etwa 0.01 bis 0.0001 \footnote{Dies ist ein bloßer Erfahrungswert. Maschinelles Lernen erfordert oft sehr viele Versuche, weshalb nicht genau festgelegt werden kann, wann welche Lernrate optimal ist.}. +Diese Lernrate ist notwendig um, nicht über das Minimum \glqq hinweg zu springen''. Sollte sie zu groß sein, passiert genau dies, da die Anpassungen der Parameter in zu großen Schritten erfolgt. Sollte sie hingegen zu klein sein, lernt das Netz sehr langsam. Typische Werte sind abhängig von der zu erlernenden Aufgabe, liegen jedoch in der Regel bei etwa 0.01 bis 0.0001 \footnote{Dies ist ein bloßer Erfahrungswert. Maschinelles Lernen erfordert oft sehr viele Versuche, weshalb nicht genau festgelegt werden kann, wann welche Lernrate optimal ist.}. \begin{figure}[h] \begin{center} \includegraphics[width=0.5\linewidth]{../graphics/gdf_big_lr.png} \end{center} - \caption{$\eta$ ist hier zu groß gewählt} + \caption{$\eta$ ist hier zu groß gewählt.} \label{Learning_Rate_Graphic} \end{figure} \newline -Abbildung \ref{Learning_Rate_Graphic} stellt dar, wieso das Minimum nicht erreicht werden kann, falls die Lernrate zu groß gewählt wurde. Es ist zu sehen, dass der Parameter immer gleich viel geändert wird und dabei das Minimum übersprungen wird, da die Lernrate konstant zu groß ist. Dieses Problem kann behoben werden indem eine adaptive Lernrate verwendet wird. Dabei verringert sich die Lernrate im Laufe des Lernprozesses, sodass zu Beginn die Vorzüge des schnellen Lernens genutzt werden können und am Ende trotzdem ein hoher Grad an Präzision erreicht werden kann. +Abbildung \ref{Learning_Rate_Graphic} stellt dar, wieso das Minimum nicht erreicht werden kann, falls die Lernrate zu groß gewählt wurde. Es ist zu sehen, dass der Parameter immer gleich viel geändert und dabei das Minimum übersprungen wird, da die Lernrate konstant zu groß ist. Dieses Problem kann behoben werden, indem eine adaptive Lernrate verwendet wird. Dabei verringert sich die Lernrate im Laufe des Lernprozesses, sodass zu Beginn die Vorzüge des schnellen Lernens genutzt werden können und am Ende trotzdem ein hoher Grad an Präzision erreicht werden kann. \subsection{Verschiedene Layerarten} -Mit Hilfe von maschinellem Lernen lassen sich eine Vielzahl von Aufgaben bewältigen. Entsprechend komplex müssen Neuronale Netze aber auch sein. Demzufolge ist es notwendig, Neuronen zu entwickeln, die andere Fähigkeiten aufweisenl, als das einfache oben im sogenannten \glqq Linear Layer'' verwendete Neuron. Da man in der Regel nur eine Art von Neuron in einem Layer verwendet, wird das gesamte Layer nach der verwendeten Neuronenart benannt. Die unten beschriebenen Layerarten werden vor allem in einer Klasse von neuronalen Netzen verwendet, die als \glqq Convolutional neural networks'' bezeichnet werden. Sie werden meißt im Bereich der komplexen fragmentbasierten Bilderkennung eingesetzt, da sie besonders gut geeignet sind um Kanten oder gewisse Teile eines Bildes, wie zum Beispiel Merkmale eines Gesichtes, zu erkennen. +Mit Hilfe von maschinellem Lernen lassen sich eine Vielzahl von Aufgaben bewältigen. Entsprechend komplex müssen Neuronale Netze aber auch sein. Daher ist es notwendig, Neuronen zu entwickeln, die andere Fähigkeiten aufweisen, als das einfache im sogenannten \glqq Linear Layer'' verwendete Neuron. Da man in der Regel nur eine Art von Neuronen in einem Layer verwendet, wird das gesamte Layer nach der verwendeten Neuronenart benannt. Die unten beschriebenen Layerarten werden vor allem in einer Klasse von neuronalen Netzen verwendet, die als \glqq Convolutional neural networks'' bezeichnet werden. Sie werden meist im Bereich der komplexen fragmentbasierten Bilderkennung eingesetzt, da sie besonders gut geeignet sind, um Kanten oder gewisse Teile eines Bildes wie zum Beispiel Merkmale eines Gesichtes zu erkennen. \subsubsection{Convolutional Layers} -Convolutional Layers weisen eine fundamental andere Funktionsweise als lineare Layers auf. Sie nehmen zwar ebenfalls rationale Zahlen an und geben rationale Zahlen aus \footnote{Im Folgenden werden 2 Dimensionale convolutional Layers betrachtet, da diese einfacher vorstellbar sind. Sie nehmen dann eine Matrix rationaler Zahlen an und geben auch eine Matrix rationaler Zahlen aus. Dies korrespondiert mit dem Anwendungsbereich der Erkennung von schwarz weiß Bildern.}, berechnen die Ausgabe jedoch nicht nur mit Hilfe einer Aktivierungsfunktion sondern unter der Verwendung sogenannter \glqq Filter''. Diese Filter sind eine $m\times n$ große Matrix, die auch als \glqq Kernel'' bezeichnet wird. Der Kernel wird dabei über die Eingabematrix bewegt (daher der Name convolution) und erzeugt eine Ausgabematrix. Dafür wird der betrachtete Abschnitt der Eingabematrix $A$ und des Kernels $B$ skalar multipliziert wobei das Skalarprodukt als Frobenius-Skalarprodukt, also als +Convolutional Layers weisen eine fundamental andere Funktionsweise als lineare Layers auf. Sie nehmen zwar ebenfalls rationale Zahlen an und geben rationale Zahlen aus \footnote{Im Folgenden werden zweidimensionale convolutional Layers betrachtet, da diese einfacher vorstellbar sind. Sie nehmen dann eine Matrix rationaler Zahlen an und geben auch eine Matrix rationaler Zahlen aus. Dies korrespondiert mit dem Anwendungsbereich der Erkennung von schwarz-weiß-Bildern.}, berechnen die Ausgabe jedoch nicht nur mit Hilfe einer Aktivierungsfunktion, sondern unter der Verwendung sogenannter \glqq Filter''. Diese Filter sind eine $m\times n$ große Matrix, die auch als \glqq Kernel'' bezeichnet wird. Der Kernel wird dabei über die Eingabematrix bewegt (daher der Name convolution) und erzeugt eine Ausgabematrix. Dafür wird der betrachtete Abschnitt der Eingabematrix $A$ und des Kernels $B$ skalar multipliziert, wobei das Skalarprodukt als Frobenius-Skalarprodukt, also als \begin{equation*} \langle A, B\rangle=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{ij}b_{ij} \end{equation*} -definiert ist. Die Matritzen werden also Komponentenweise multipliziert und diese Produkte dann summiert.\newline +definiert ist. Die Matritzen werden also komponentenweise multipliziert und diese Produkte dann summiert.\newline Dies ist in Abbildung \ref{Convolution_illustration} verbildlicht. \begin{figure}[h] \begin{center} @@ -431,7 +431,8 @@ beispielsweise zur einfachen Kantenerkennung genutzt werden. Zur Veranschaulichu \end{figure} \\ Der jeweils dunkel dargestellte Bereich kann als das identifiziert werden, was vom convolutional Layer als Kante erkannt wurde. Hier werden eindeutige Limitationen deutlich: Es kann nur als Kante erkannt werden, was auch eindeutig senkrecht oder waagerecht ist. Außerdem kann es zu Fehlentscheidungen kommen.\\ -Die Kernels werden natürlich nicht per Hand initialisiert und angepasst, sondern setzen sich aus Parameternm zusammen, die im Laufe des Lernprozesses durch das Netz anpassbar sind. Das Netz kann also die Filtermatrix selber verändern. Die Filter werden meißt mit Zufallswerten initialisiert und dann während des Lernens angepasst. Ferner muss ein Kernel auch nicht immer drei Einheiten breit sein, sondern kann jede Größe $\ge2$ annehmen. Je nachdem, wie sich der Kernel über die Eingabematrix bewegt, ist außerdem ein sogenanntes \glqq Padding'' nötig, da gegebenenfalls Werte betrachtet werden müssten, die nicht in der Eingabematrix liegen. In der Regel werden daher alle Werte, die nicht in der Eingabematrix vorhanden sind durch 0 ersetzt. Das Padding ist in Abbildung \ref{Convolution_illustration} als weiß in der Eingabematrix dargestellt. Es ist eine Art \glqq Rand aus Nullen'', der um dasd Bild gelegt wird.\\ +Die Kernels werden natürlich nicht per Hand initialisiert und angepasst, sondern setzen sich aus Parametern zusammen, die im Laufe des Lernprozesses durch das Netz anpassbar sind. Das Netz kann also die Filtermatrix selber verändern. Die Filter werden meist mit Zufallswerten initialisiert und dann während des Lernens angepasst. Ferner muss ein Kernel auch nicht immer drei Einheiten breit sein, sondern kann jede Größe $\ge2$ annehmen. Je nachdem, wie sich der Kernel über die Eingabematrix bewegt, ist außerdem ein sogenanntes \glqq Padding'' nötig, da gegebenenfalls Werte betrachtet werden müssten, die nicht in der Eingabematrix liegen. In der Regel werden daher alle Werte, die nicht in der Eingabematrix vorhanden sind, durch 0 ersetzt. Das Padding ist in Abbildung \ref{Convolution_illustration} in der Eingabematrix weiß dargestellt. Es ist eine Art \glqq Rand aus Nullen'', der um das Bild gelegt wird. +\\ \\ Hintereinander können convolutional Layers auch ganze Elemente eines Bildes erkennen. Erkennt das erste Layer wie oben gezeigt beispielsweise Kanten, so kann das Layer darauf Kombinationen aus diesen, wie beispielsweise Ecken oder Winkel, erkennen. Wie gefilterte Bilder für sogenannte \glqq High-Level-Features'' aussehen können ist in Abbilldung \ref{HL_features_conv} dargestellt. Die Ausgabebilder von Convolutional Layers werden als \glqq Feature map'' bezeichnet. \begin{figure}[h]