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100644 index 0000000..78276c4 Binary files /dev/null and b/TicTacToe_AI/Net/nets/net_6.pt differ diff --git a/TicTacToe_AI/Net/nets/net_7.pt b/TicTacToe_AI/Net/nets/net_7.pt new file mode 100644 index 0000000..f6a987f Binary files /dev/null and b/TicTacToe_AI/Net/nets/net_7.pt differ diff --git a/TicTacToe_AI/Net/nets/net_8.pt b/TicTacToe_AI/Net/nets/net_8.pt new file mode 100644 index 0000000..a369d8b Binary files /dev/null and b/TicTacToe_AI/Net/nets/net_8.pt differ diff --git a/TicTacToe_AI/Net/nets/net_9.pt b/TicTacToe_AI/Net/nets/net_9.pt new file mode 100644 index 0000000..548a56f Binary files /dev/null and b/TicTacToe_AI/Net/nets/net_9.pt differ diff --git a/TicTacToe_AI/Net/pytorch_ai.py b/TicTacToe_AI/Net/pytorch_ai.py index c111016..efea5ae 100644 --- a/TicTacToe_AI/Net/pytorch_ai.py +++ b/TicTacToe_AI/Net/pytorch_ai.py @@ -79,7 +79,7 @@ class Net(torch.nn.Module): return F.log_softmax(x, dim=1) -net = Net() +net = torch.load('./nets/net_3.pt') optimizer = 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482) - defining Unicode char U+01E3 (decimal 483) - defining Unicode char U+01E6 (decimal 486) - defining Unicode char U+01E7 (decimal 487) - defining Unicode char U+01E8 (decimal 488) - defining Unicode char U+01E9 (decimal 489) - defining Unicode char U+01EA (decimal 490) - defining Unicode char U+01EB (decimal 491) - defining Unicode char U+01F0 (decimal 496) - defining Unicode char U+01F4 (decimal 500) - defining Unicode char U+01F5 (decimal 501) - defining Unicode char U+0218 (decimal 536) - defining Unicode char U+0219 (decimal 537) - defining Unicode char U+021A (decimal 538) - defining Unicode char U+021B (decimal 539) - defining Unicode char U+0232 (decimal 562) - defining Unicode char U+0233 (decimal 563) - defining Unicode char U+1E02 (decimal 7682) - defining Unicode char U+1E03 (decimal 7683) - defining Unicode char U+200C (decimal 8204) - defining Unicode char U+2010 (decimal 8208) - defining Unicode char U+2011 (decimal 8209) - defining Unicode char U+2012 (decimal 8210) 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Package csquotes Info: ... configuration file loaded successfully. @@ -644,7 +333,7 @@ File: csquotes.cfg Package csquotes Info: Enabling multilingual quotes. ) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty -Package: hyperref 2018/02/06 v6.86b Hypertext links for LaTeX +Package: hyperref 2018/11/30 v6.88e Hypertext links for LaTeX (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) @@ -659,15 +348,11 @@ Package hobsub Info: Skipping package `ifluatex' (already loaded). Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) Package ifvtex Info: VTeX not detected. 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numeric.bbx standard.bbx + numeric.bbx numeric.cbx biblatex.cfg - ngerman.lbx german.lbx + ngerman.lbx diff --git a/doc/Grundlagen_des_maschinellen_lernens.synctex.gz b/doc/Grundlagen_des_maschinellen_lernens.synctex.gz index 1f2dc2e..fe77a2f 100644 Binary files a/doc/Grundlagen_des_maschinellen_lernens.synctex.gz and b/doc/Grundlagen_des_maschinellen_lernens.synctex.gz differ diff --git a/doc/Grundlagen_des_maschinellen_lernens.tex b/doc/Grundlagen_des_maschinellen_lernens.tex index 70b2bfe..5bb4e43 100644 --- a/doc/Grundlagen_des_maschinellen_lernens.tex +++ b/doc/Grundlagen_des_maschinellen_lernens.tex @@ -15,6 +15,8 @@ \usepackage{listings} % Source code formatting and highlighting \usepackage{fancyhdr} \usepackage{biblatex} +\usepackage{pgfplots} +\usepackage{txfonts} \author{Clemens Dautermann} \title{Grundbegriffe des maschinellen Lernens} @@ -37,7 +39,7 @@ Die wohl bekannteste und am häufigsten zitierte Definiton des maschinellen Lern ---Arthur Samuel, 1959 \end{flushright} \end{quote} -Beim maschinellen lernen werden Computer also nicht mit einem bestimmten Algorythmus programmiert um eine Aufgabe zu lösen, sondern lernen eigenständig diese Aufgabe zu bewältigen. Dies geschieht zumeist, indem das Programm aus einer großen, bereits \glqq gelabelten'', Datenmenge mit Hilfe bestimmter Methoden, die im Folgenden weiter erläutert werden sollen, lernt, gewisse Muster abzuleiten um eine ähnliche Datenmenge selber \glqq labeln'' zu können. Als Label bezeichent man in diesem Fall die gewünschte Ausgabe des Programmes. Dies kann beispielsweise eine Klassifikation sein. Soll das Programm etwa handgeschriebene Ziffern erkennen können, so bezeichnet man das (bearbeitete) Bild der Ziffer als \glqq Input Verctor'' und die Information welche Ziffer der Copmputer hätte erkennen sollen, als \glqq Label ''. Soll jedoch maschinell erlernt werden, ein simuliertes Auto zu fahren, so bestünde der Input Vector aus Sensorinformationen und das Label würde aussagen, in welche Richtung das Lenkrad hätte gedreht werden sollen, wie viel Gas das Programm hätte geben sollen oder andere Steuerungsinformationen. Der Input Vector ist also immer die Eingabe, die der Computer erhält um daraus zu lernen und das Label ist die richtige Antwort, die vom Programm erwartet wurde. Für maschinelles Lernen wird also vor allem eins benötigt: Ein enormer Datensatz, der bereits gelabelt wurde, damit das Programm daraus lernen kann.\newline +Beim maschinellen lernen werden Computer also nicht mit einem bestimmten Algorythmus programmiert um eine Aufgabe zu lösen, sondern lernen eigenständig diese Aufgabe zu bewältigen. Dies geschieht zumeist, indem das Programm aus einer großen, bereits \glqq gelabelten'', Datenmenge mit Hilfe bestimmter Methoden, die im Folgenden weiter erläutert werden sollen, lernt, gewisse Muster abzuleiten um eine ähnliche Datenmenge selber \glqq labeln'' zu können. Als Label bezeichent man in diesem Fall die gewünschte Ausgabe des Programmes. Dies kann beispielsweise eine Klassifikation sein. Soll das Programm etwa handgeschriebene Ziffern erkennen können, so bezeichnet man das (bearbeitete) Bild der Ziffer als \glqq Input Verctor'' und die Information welche Ziffer der Copmputer hätte erkennen sollen, als \glqq Label ''. Soll jedoch maschinell erlernt werden, ein simuliertes Auto zu fahren, so bestünde der Input Vector aus Sensorinformationen und das Label würde aussagen, in welche Richtung das Lenkrad hätte gedreht werden sollen, wie viel Gas das Programm hätte geben sollen oder andere Steuerungsinformationen. Der Input Vector ist also immer die Eingabe, die der Computer erhält um daraus zu lernen und das Label ist die richtige Antwort, die vom Programm erwartet wurde. Für maschinelles Lernen wird also vor allem eins benötigt: Ein enormer Datensatz, der bereits gelabelt wurde, damit das Programm daraus lernen kann.\newline Natürlich werden für maschinelles Lernen trotzdem Algorythmen benötigt. Diese Algorythmen sind jedoch keine problemspezifischen Algorythmen, sondern Algorythmen für maschinelles Lernen. Eine der populärsten Methoden des maschinellen Lernens ist das sogenannte \glqq Neuronale Netz''. \subsection{Einsatzgebiete maschinellen Lernens} \section{Neuronale Netze} @@ -55,7 +57,7 @@ Diese Definition ist stark vereinfacht. Sie enthält ausschließlich die wesentl \end{figure} Ein neuronales Netz ist ähnlich aufgebaut. Es besteht aus \glqq Neuronen'', die eine theoretisch beliebige Anzahl von Eingaben annehmen können und mit einer entsprechenden Ausgabe reagieren, sowie Verbindungen zwischen den Neuronen. Auch das Lernprinzip entspricht dem eines Menschen. Das Netz nimmt immer Zahlen zwischen 0 und 1 als Eingabe an und berechnet eine entsprechende Ausgabe. Es erhält anschließend die Information, wie die richtige Lösung gelautet hätte und lernt dann aus seinen Fehlern, indem es gewisse Werte, die in die Berechnung einfließen, anpasst. Analog lernt ein Mensch, indem er ausprobiert, gegebenenfalls scheitert, anschließend die richtige Antwort durch eine externe Quelle erhält und somit aus seinem Fehler lernt. Im Menschlichen Gehirn verknüpfen sich Dabei oft genutzte neuronale Verbindungen stärker und weniger benutzte Verbindungen bauen sich ab\cite{2}. Die Verstärkung und der Abbau entsprechen dem Ändern der Gewichtung einer Verbindung im neuronalen Netz. Die Gewichtung ist eine Eigenschaft der Verbindung, die eine zentrale Rolle in der Berechnung spielt und soll im folgenden weiter erläutert werden. Diese Ähnlichkeiten sind kein Zufall, sondern viel mehr Intention. Ein neuronales Netz ist nämlich der gezielte Versuch das menschliche Lernen nachzuahmen um maschinelles Lernen zu ermöglichen. \subsection{Der Aufbau eines neuronalen Netzes} -Ein neuronales Netz besteht aus Neuronen und Verbindungen zwischen diesen. Es gibt einen sogenannten \glqq Input Layer'', der die Daten annimmt, eine beliebige Anzahl von sogenannten \glqq Hidden Layers'', in denen das eigentliche Lernen statt findet, und einen sogenannten \glqq Output Layar'', der für die Datenausgabe verantwortlich ist. Die Anzahl der Neuronen ist nach oben nicht begrenzt, wird jedoch zumeist der Aufgabe angepasst. Im Input Layer ist meißt ein Neuron pro Pixel des Eingabebildes vorhanden und im Output Layer ein Neuron pro möglicher Ausgabe. Sollen also 28$\times$28 Pixel große Bilder handgeschriebener Ziffern klassifiziert werden, so gibt es 784 Eingabeneuronen, da jedes Bild 784 Pixel groß ist, und 10 Ausgabeneuronen, da es 10 Ziffern gibt. +Ein neuronales Netz besteht aus Neuronen und Verbindungen zwischen diesen. Es gibt einen sogenannten \glqq Input Layer'', der die Daten, den sogenannten \glqq Input Vector'', annimmt, eine beliebige Anzahl von sogenannten \glqq Hidden Layers'', in denen das eigentliche Lernen statt findet, und einen sogenannten \glqq Output Layar'', der für die Datenausgabe verantwortlich ist. Die Anzahl der Neuronen ist nach oben nicht begrenzt, wird jedoch zumeist der Aufgabe angepasst. Im Input Layer ist meißt ein Neuron pro Pixel des Eingabebildes vorhanden und im Output Layer ein Neuron pro möglicher Ausgabe. Sollen also 28$\times$28 Pixel große Bilder handgeschriebener Ziffern klassifiziert werden, so gibt es 784 Eingabeneuronen, da jedes Bild 784 Pixel groß ist, und 10 Ausgabeneuronen, da es 10 Ziffern gibt. \begin{figure}[h] \centering \includegraphics[width=\linewidth]{../graphics/Neural_Net.png} @@ -68,11 +70,94 @@ Jedes Neuron hat außerdem eine sogenannte Aktivierungsfunktion, die sich von Ne \end{enumerate} Sind alle Neuronen eines Layers jeweils mit allen Neuronen des nächsten Layers verbunden, wird das Layer als \glqq fully connected layer'' bezeichnet. \subsection{Berechnung des Ausgabevektors} +Der Ausgabevektor wird berechnet, indem: +\begin{enumerate} + \item Alle Ausgaben aus der vorherigen Schicht mit der Gewichtung der korrespondierenden Kante multipliziert werden + \item Alle gewichteten Eingabewerte summiert werden + \item Der Bias des Neurons hinzuaddiert wird + \item Die Aktivierungsfunktion auf diesen Wert angewandt wird +\end{enumerate} +Die Aktivierungsfunktion hat dabei die Rolle die Werte zu normieren. Sie sorgt also dafür, dass alle Werte innerhalb des Netzes im Intervall $[0, 1]$ bleiben. Es gibt eine Vielzahl von Aktivierungsfunktionen. Die häufigste ist die sogenannte \glqq Sigmoid'' Funktion: +\begin{figure}[h] +\begin{center} +\begin{tikzpicture} + \begin{axis}[samples=100, width=0.5\linewidth] + \addplot[blue]{tanh(x)}; + \end{axis} +\end{tikzpicture} +\end{center} +\caption{Der Plot der Sigmoid Funktion $\sigma(x)=\frac{e^x}{e^x+1}$} +\end{figure} +\newline +Im Gegensatz dazu haben Gewichtungen typischerweise etwa den doppelten Wert der Eingaben. Alle Were werden jedoch automatisch im Lernprozess angepasst. +\newline +Der Begriff Eingabe- und Ausgabevektor lassen bereits vermuten, dass es sich bei Neuronalen Netzen um Objekte aus dem Bereich der linearen Algebra handelt. Daher wird im Folgenden auch die Notationsweise mit Hilfe von linearer Algebra verwendet. Betrachtet man eine Ausgabe eines Neurons wird diese als $a^ {(layer)}_{neuron}$ bezeichnet. Den Ausgabevektor des Input Layers würde man also folgendermaßen schreiben: +\begin{equation*} +\begin{bmatrix} + a^0_0\\ + a^0_1\\ + a^0_2\\ + \vdots\\ + a^0_n +\end{bmatrix} +\end{equation*} +Die Gewichtungen $w$ der jeweiligen Kanten werden notiert als $w^{(von\ Layer)}_{(zu\ Neuron, von\ Neuron)}$. \glqq von Layer'' bezeichnet dabei das Layer in dem das Neuron liegt, das die Information ausgibt. \glqq zu Neuron'' ist der Index des Neurons im nächsten Layer, das die Information annimmt und \glqq von Neuron'' der Index des Neurons, das die Information abgibt. Die Gewichtung der Kante, die das zweite Neuron im ersten Layer mit dem dritten Neuron im zweiten Layer verebindet würde also als $w^0_{3,2}$ bezeichnet werden. Dabei wird bei null begonnen zu zählen, sodass das erste Layer und das erste Neuron den Index 0 erhält.\newline +Die Gewichtungen aller Verbindungen eines Layers zum nächsten können also als folgende Matrix geschrieben werden: +\begin{equation*} + \begin{bmatrix} + w_{0,0} & w_{0,1} & \cdots & w_{0,n}\\ + w_{1,0} & w_{1,1} & \cdots & w_{1,n}\\ + \vdots & \vdots & \ddots & \vdots\\ + w_{k,0} & w_{k,1} & \cdots & w_{k,n}\\ + \end{bmatrix} +\end{equation*} +Dabei ist $n$ hier die selbe Zahl wie $n$ im Ausgabevektor, da genau so viele Ausgaben vorhanden sein müssen, wie Neuronen in diesem Layer vorhanden sind, da jedes Neuron einen Wert ausgibt.\footnote{Es existieren auch Neuronen, die Daten verwerfen. Diese kommen im hier betrachteten Typ von neuronalem Netz allerdings nicht vor und werden daher der Einfachheit halber außenvor gelassen.}Der Bias Vektor wird genau so wie der Ausgabevektor bezeichnet. +\begin{equation*} +\begin{bmatrix} + b^0_0\\ + b^0_1\\ + b^0_2\\ + \vdots\\ + b^0_n +\end{bmatrix} +\end{equation*} +Beachtet man jetzt noch, dass bei jedem Neuron die Aktivierungsfunktion angewandt werden muss ergibt sich folgende Gleichung für die Berechnung des Ausgabevektors $\vec{o}$ aus einem Einbagevektor $\vec{a}$ durch eine Schicht von Neuronen: +\begin{figure}[h] + \begin{equation*} + \vec{o} = \sigma\left( + \begin{bmatrix} + w_{0,0} & w_{0,1} & \cdots & w_{0,n}\\ + w_{1,0} & w_{1,1} & \cdots & w_{1,n}\\ + \vdots & \vdots & \ddots & \vdots\\ + w_{k,0} & w_{k,1} & \cdots & w_{k,n}\\ + \end{bmatrix} + \begin{bmatrix} + a^0_0\\ + a^0_1\\ + a^0_2\\ + \vdots\\ + a^0_n + \end{bmatrix} + + + \begin{bmatrix} + b^0_0\\ + b^0_1\\ + b^0_2\\ + \vdots\\ + b^0_n + \end{bmatrix} + \right) + \end{equation*} + \caption{Formel zur Berechnung eines Ausgabevektors aus einem Eingabevektor durch ein Layer Neuronen. } +\end{figure} +\newline +Zur Vereinfachung wurde die Funktion hier auf den gesamten Ausgabevektor angewandt. Dies ist korrekt, sofern alle Neuronen eines Layers die selbe Aktivierungsfunktion aufweisen. Dies muss natürlich nicht immer so sein. Sind die Aktivierungsfunktionen der Neuronen eines Layers verschieden, so wird die Aktivierungsfunktion des jeweiligen Neurones separat auf das korrespondierende Element des Vektors $W\cdot\vec{a}+\vec{b}$ angewandt. \subsection{Der Lernprozess} -\subsubsection{Backpropagation} +Der Lernprozess gliedert sich in wenige wesentliche Schritte. Zuerst wird unter Verwendung des oben beschriebenen Prozesses aus einem Eingabevektor ein Ausgabevektor berechnet. Diese Vektoroperation wird im Lernprozess extrem oft durchgeführt, weshalb sich neuronale Netze besonders schnell auf Grafikkarten trainieren lassen. Diese sind für mathematische Operationen im Bereich der linearen Algebra, wie Matritzenmultiplikation oder Addition optimiert und werden daher auch als Vektorprozessoren bezeichnet.\newline +Dieser Ausgabevektor wird nun, mit Hilfe einer Fehlerfunktion, mit dem erwarteten Ausgabevektor verglichen. Dabei ergibt sich ein Skalarfeld, sodass die Fehlerfunktion die Zuordnung $\mathbb{P} \to \varmathbb{R}$ vornimmt, wobei $\mathbb{P}$ alle Variablen des Netzes darstellt. Wenn also das Minimum dieser Fehlerfunktion bestimmt wird, wird der Fehler minimiert und das Netz lernt.\newline +Eine Methode, die hier erläutert werden soll, dieses Minimum zu finden ist das Gradientenverfahren. Nachdem mit Hilfe dieses Verfahrens der Fehler mimnimiert wurde, werden die Variablen des neuronalen Netzes entsprechend angepasst. Diesen Prozess der Fehlerminimierung mittels des Gradientenverfahrens und der anschließenden Anpassung der Werte bezeichnet man auch als \glqq Backpropagation''. Es existieren auch noch andere Verfahren zur Fehlerminimierung, der Einfachheit halber soll hier aber nur Backpropagation erläutert werden. \subsubsection{Fehlerfunktionen} -\subsubsection{SGD} -\subsubsection{Zusammenfassung} +\subsubsection{Gradientenverfahren} \subsection{Verschiedene Layerarten} \subsubsection{Fully connected Layers} \subsubsection{Convolutional Layers} diff --git a/doc/Grundlagen_des_maschinellen_lernens.toc b/doc/Grundlagen_des_maschinellen_lernens.toc index e8d923e..fa96723 100644 --- a/doc/Grundlagen_des_maschinellen_lernens.toc +++ b/doc/Grundlagen_des_maschinellen_lernens.toc @@ -1,62 +1,58 @@ \boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax \babel@toc {ngerman}{} \defcounter {refsection}{0}\relax -\contentsline {section}{\numberline {1}Was ist maschinelles Lernen?}{3}{section.1} +\contentsline {section}{\numberline {1}Was ist maschinelles Lernen?}{3}{section.1}% \defcounter {refsection}{0}\relax 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