From fc03541d2c3cfc395236c0bd980ec4506f53f961 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Fri, 5 Feb 2021 23:11:41 +0100 Subject: [PATCH] =?UTF-8?q?fehlerbasiertes=20lernen=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Acronyms.tex | 2 + .../Fehlerbasiertes Lernen.tex | 155 +++++++++++++++++- .../Ähnlichkeitsbasiertes Lernen.tex | 2 +- images/backpropagation_inner_layer.png | Bin 0 -> 32775 bytes images/knn_ausgabefunktionen.png | Bin 0 -> 98374 bytes images/knn_dense_layer.png | Bin 0 -> 63803 bytes images/knn_netzwerkarchitektur.png | Bin 0 -> 89497 bytes images/knn_neuron.png | Bin 0 -> 15824 bytes parts/Maschinelles Lernen.tex | 1 + 9 files changed, 158 insertions(+), 2 deletions(-) create mode 100644 images/backpropagation_inner_layer.png create mode 100644 images/knn_ausgabefunktionen.png create mode 100644 images/knn_dense_layer.png create mode 100644 images/knn_netzwerkarchitektur.png create mode 100644 images/knn_neuron.png diff --git a/Acronyms.tex b/Acronyms.tex index d7f124c..467da36 100644 --- a/Acronyms.tex +++ b/Acronyms.tex @@ -27,4 +27,6 @@ \acro{MFPC}{Modified-Fuzzy-Pattern-Classifier} \acro{FPGA}{Field Programmable Gate Array} \acro{ID3}{Iterative Dichotomizer 3} + \acro{KNN}{Künstliches Neuronales Netz} + \acrodefplural{KNN}{Künstliche Neuronale Netze} \end{acronym} \ No newline at end of file diff --git a/chapters/Maschinelles Lernen/Fehlerbasiertes Lernen.tex b/chapters/Maschinelles Lernen/Fehlerbasiertes Lernen.tex index 3489452..da357e7 100644 --- a/chapters/Maschinelles Lernen/Fehlerbasiertes Lernen.tex +++ b/chapters/Maschinelles Lernen/Fehlerbasiertes Lernen.tex @@ -1,2 +1,155 @@ \chapter{Fehlerbasiertes Lernen} -\label{error-based learning} \ No newline at end of file +\label{error-based learning} + Beim \say{fehlerbasiertem Lernen} wird ein bereits parametrisiertes Prädiktionsmodell zufällig initialisiert. + Anschließend wird das Modell iterativ auf Basis einer Kostenfunktion (Fehlerfunktion) so angepasst, + dass die Kosten/ der Fehler minimiert wird. + Diese Art des Lernens setzt 2 Dinge voraus: + \begin{enumerate} + \item Fehlerfunktion für Modelle + \item Optimierungsalgorithmus zur Anpassung der Parameter + \end{enumerate} + Dieser Ansatz des Lernens wird vorallem bei \acsp{KNN}(\ref{knn}) benutzt. + + \section{\acfp{KNN}} + \label{knn} + Wie man schon am Namen erkennen kann sind \aclp{KNN} Modelle, + deren Informationsverarbeitung, der eines menschlichen Gehirns nachempfungen sind. + Sie setzen sich auch vernetzten einfachen, parallel arbeitenden Einheiten zusammen. + Den \textbf{Neuronen}. + Durch die Anpassung und Gewichtung der Verbindungen zwischen den Neuronen kann ein \ac{KNN} angelernt werden. + + \subsection{formales Neuron} + \label{knn: neuron} + Formal ist ein Neuron eines \ac{KNN} beschrieben als: + \begin{center} + \includegraphics[width = .5\textwidth]{knn_neuron.png} + \end{center} + Es setzt sich zusammen aus: + \begin{itemize} + \item Eingangsvektor $e$: + \large + $$e=(e_1,\dots,e_N,1)^T$$ + \normalsize + \item Gewichtsvektor $g$: + \large + $$g=(g_1,\dots,g_N,-B)^T$$ + \normalsize + \item Aktivitätszustand $z$: + \large + $$z=\sum^{N+1}_{i=1}g_i\cdot e_i = g^T e$$ + \normalsize + \item Ausgabefunktion $f$: + \large + $$a = f(z)$$ + \normalsize + Für die Ausgabefunktion werden unterschiedliche Funktionen verwendet.\\ + \includegraphics[width = \textwidth]{knn_ausgabefunktionen.png} + \end{itemize} + + \subsection{Netzwerkarchitektur} + \label{knn: architecture} + Theoretisch lässt sich jedes Problem, dass von einem \ac{KNN} gelöst werden kann mit nur einem + dense Layer (\ref{knn: dense layer}) und einer Ausgabeschicht lösen. + Allerdings steigt bei diesem Ansatz die Anzahl der benötigten Neuronen häufig mit exponentiell mit dem Anstieg der Dimensionen. + Daher werden meist \acp{KNN} mit mehreren verborgenen Schichten (hidden layers) verwendet. + \begin{center} + \includegraphics[width = .8\textwidth]{knn_netzwerkarchitektur.png} + \end{center} + + \subsubsection{Dense Layer} + \label{knn: dense layer} + Unter einem \say{Dense Layer} versteht man bei eim \ac{KNN} eine vollvernetzte Schicht. + Das bedeutet, dass jedes Neuron in dieser Schicht mit jedem Neuron der vorhergegangenen Schicht verknüpft ist. + \begin{center} + \includegraphics[width = .6\textwidth]{knn_dense_layer.png} + \end{center} + + \subsection{Fehlerfunktion} + \label{knn: error function} + Zwei der am häufigst verwendeten Fehlerfunktionen sind:\\ + ($t_i\in\{0,1\}:$vorgegebenes Ausgangssignal; $a_i\in]0,1[:$ Ausgangssignal des Netzes) + \paragraph{Quadratische Abweichung} + \large + $$f(a,t)=\sum_i \frac{1}{2}(a_i-t_i)^2$$ + \normalsize + + \paragraph{Kategorische Kreuzentropie} + \large + $$f(a,t)=-\sum_i t_i\ln(a_i)$$ + \normalsize + + \subsection{Parameteroptimierung} + \label{knn: parameteroptimierung} + Die Parameter werden iterativ so verändert/ optimiert, dass die Fehlerfunktion (\ref{knn: error function}) minimal wird. + + \subsubsection{Gradientenabstieg} + \label{knn: gradientenabstieg} + Das Gradientenverfahren ist ein Verfahren zur Parameteroptimierung. + Die Idee für das Verfahren leitet sich vom Hill-Climbing Algorithmus (\ref{hill climbing}) ableitet. + Der größte Unterschied liegt darin, dass der Parameterraum nicht diskret sondern kontinuierlich ist. + Das bedeutet, das jeder Punkt unendlich viele Nachbarn hat. + Durch die Bestimmung des \textbf{Gradienten} kann die Richtung des steilsten Abstiegs angegeben werden: + \large + $$\Delta g= \begin{bmatrix} + \frac{\delta g}{\delta w_1}\\ + \frac{\delta g}{\delta w_2}\\ + \vdots\\ + \frac{\delta g}{\delta w_n} + \end{bmatrix} + $$ + \normalsize + Zu beginn werden die Gewichte $w$ des Neuronen definiert (zufällig oder anders). + Anschließend werden sie bei jeder Iteration über + \large + $$w\gets w+\alpha\cdot \Delta g(w)$$ + \normalsize + umdefiniert. + Hierbei ist $\alpha$ ein Hyperparamter, der die \textbf{Lernrate} definiert.\\ + Zu erwähnen ist noch, dass auch hier wie beim Hill-Climbing (\ref{hill climbing}) die Gefahr besteht, + dass lediglich ein lokales anstatt dem globalen Minimum gefunden wird. + + \subsubsection{Backpropagation} + \label{knn: backpropagation} + Da der Fehler, den die Fehlerfunktion angibt auf alle Schichten des \ac{KNN} zurückzuführen ist, + muss auch die Optimierung Rückwärtsgerichtet in allen Schichten stattfinden. + Man spricht hierbei von einer \say{Backpropagation}.\\ + Um ein Ausgangsneuron zu updaten muss die Fehlerfunktion + $$f(a,t) = \sum_i \frac{1}{2}(a_i-t_i)^2$$ + erstmal auf den Ausgabewert des jeweiligen Neurons abgeleitet werden: + $$\frac{\delta f}{\delta a_i}=a_i-t$$ + Wenn man dies nun auf die Zustandsfunktion des Neurons ableitet erhält man: + $$\frac{\delta f}{\delta z_i} = \frac{\delta f}{\delta a_i}\cdot\frac{\delta a_i}{\delta z_i}$$ + Um die Änderung für eine Gewichtung $g_{ij}$ zu bestimmen muss die Fehlerfunktion nun zunächst auf dieses $g_{ij}$ abgeleitet werden: + $$\frac{\delta f}{\delta g_{ij}} = \frac{\delta f}{\delta z_i} \cdot\frac{\delta z_i}{\delta g_{ij}} = \frac{\delta f}{\delta a_i}\cdot\frac{\delta a_i}{\delta z_i}\cdot\frac{\delta z_i}{\delta g_{ij}}$$ + Nun kann man durch Einsetzen der verwendeten $f$, $a_i$ und $z_i$ die Formel für die Änderung von $g_{ij}$ aufstellen.\\ + \textbf{Beispiel:} + \Large + \begin{tabbing} + $f(a,t)=\sum_i\frac{1}{2}(a_i-t_i)^2$\hspace{5mm}\=und\hspace{5mm}\=$\frac{\delta f}{\delta a_i}=a_i-t_i$\\\\ + $a_i =\frac{1}{a+e^{z_i}}$ \>und \>$\frac{\delta a_i}{\delta z_i}=a_i(1-a_i)$\\\\ + $z_i=\sum^{N+1}_{j=1}g_{ij}\cdot a_i$ \>und \>$\frac{\delta z_i}{\delta g_{ij}}=a_j$ + \end{tabbing} + \normalsize + Hieraus ergibt sich: + $$\frac{\delta f}{\delta g_{ij}} = \frac{\delta f}{\delta z_i} \cdot\frac{\delta z_i}{\delta g_{ij}} = \frac{\delta f}{\delta a_i}\cdot\frac{\delta a_i}{\delta z_i}\cdot\frac{\delta z_i}{\delta g_{ij}}=(a_i-t_i)\cdot a_i(1-a_i)\cdot a_j$$ + Da uns alle diese Werte bekannt sind können wir $g_{ij}$ updaten: + \large + $$g_{ij}\gets g_{ij} + \alpha\cdot\left((a_i-t_i)\cdot a_i(1-a_i)\cdot a_j\right)$$ + \normalsize + Um die Backpropagation nun bei den anderen Neuronen auf tieferen Schichten anzuwenden muss auf diese abgeleitet werden: + \begin{align*} + \frac{\delta f}{\delta a_i}&=(a_i-t_i)\\ + \frac{\delta f}{\delta z_i}&=\frac{\delta f}{\delta a_i}\cdot\frac{\delta a_i}{\delta z_i}=a_i(1-a_i)\frac{\delta f}{\delta a_i}\\ + \frac{\delta f}{\delta g_{ij}}&=\frac{\delta f}{\delta z_i}\cdot\frac{\delta z_i}{\delta g_{ij}} = a_j\frac{\delta f}{\delta z_i}\\ + \frac{\delta f}{\delta a_j}&=\sum_i\frac{\delta f}{\delta z_i}\cdot\frac{\delta z_i}{\delta a_j}=\sum_i g_{ij}\frac{\delta f}{\delta z_i} + \end{align*} + Die Summe wird durch die Mehrzahl der Nachfolgenden Neuronen von $B_j$ berursacht. + \begin{center} + \includegraphics[width = .4\textwidth]{backpropagation_inner_layer.png} + \end{center} + + \subsection{Normalisierung} + \label{knn: normalisierung} + Auch bei \acp{KNN} ist es sinnvoll die Eingangs-Features zu normalisieren (\ref{k-nearest-neighbour: normalisierung}). + Hierdurch wird verhindert, + dass die Features alleine durch ihren Wertebereich einen ungleichmäßigen Einfluss auf die Ausgabe der Neuronen haben. \ No newline at end of file diff --git a/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex b/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex index be0ec09..ff8c880 100644 --- a/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex +++ b/chapters/Maschinelles Lernen/Ähnlichkeitsbasiertes Lernen.tex @@ -35,7 +35,7 @@ \end{cases}$$ \subsection{Normalisierung} - \label{k-nearest-neighbour} + \label{k-nearest-neighbour: normalisierung} Da unterschiedliche Wertebereiche der verschiedenen Features einen großen Einfluss auf den K-Nächsten Nachbarn haben müssen sie normalisiert werden. 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