From 23f64e1719c1a90e7a94ff80b1c7a875941cb7b7 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Sun, 13 Feb 2022 23:38:31 +0100 Subject: [PATCH] =?UTF-8?q?Lagrangian=20Optimization=20hinzugef=C3=BCgt.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Glossary.tex | 2 +- ML_Zusammenfassung.tex | 1 + .../Support_Vector_Machines.tex | 5 + .../Constraint Optimization.tex | 66 + images/constraint_optimization.png | Bin 0 -> 256457 bytes images/lagrangian_optimization_example.pdf | Bin 0 -> 82305 bytes images/lagrangian_optimization_example.png | Bin 0 -> 463410 bytes images/lagrangian_optimization_example.svg | 2545 +++++++++++++++++ 8 files changed, 2618 insertions(+), 1 deletion(-) create mode 100644 chapters/Mathematische_Grundlagen/Constraint Optimization.tex create mode 100644 images/constraint_optimization.png create mode 100644 images/lagrangian_optimization_example.pdf create mode 100644 images/lagrangian_optimization_example.png create mode 100644 images/lagrangian_optimization_example.svg diff --git a/Glossary.tex b/Glossary.tex index 1b4270e..94e1f33 100644 --- a/Glossary.tex +++ b/Glossary.tex @@ -76,7 +76,7 @@ \newacronym{knn}{k"=NN}{k"=Nearest Neighbors} \newacronym{RSS}{RSS}{Residual Sum of Squares} \newacronym{CART}{CART}{Classification an Regression Trees} -\newacronym{DNN}{DNN}{Dynamic Neural Network} +\newacronym{DNN}{DNN}{Deep Neural Network} \newacronym{RBF}{RBF}{Radial Basis Function Kernel} \newacronym{SVM}{SVM}{Support Vector Machine} diff --git a/ML_Zusammenfassung.tex b/ML_Zusammenfassung.tex index 3cba9c4..475df11 100644 --- a/ML_Zusammenfassung.tex +++ b/ML_Zusammenfassung.tex @@ -37,6 +37,7 @@ \input{chapters/Mathematische_Grundlagen/Probability_Theory.tex} \input{chapters/Mathematische_Grundlagen/Kernel_Basics.tex} \input{chapters/Mathematische_Grundlagen/Sub-Gradients.tex} + \input{chapters/Mathematische_Grundlagen/Constraint Optimization.tex} \part{Classical Supervised Learning} \label{part:Classical Supervised Learning} diff --git a/chapters/Kernel_Methods/Support_Vector_Machines.tex b/chapters/Kernel_Methods/Support_Vector_Machines.tex index 008c1c9..769b634 100644 --- a/chapters/Kernel_Methods/Support_Vector_Machines.tex +++ b/chapters/Kernel_Methods/Support_Vector_Machines.tex @@ -156,3 +156,8 @@ Im Falle des Hinge Loss bedeutet das: \section{Anwendungsbeispiele}% \label{sec:Anwendungsbeispiele} {\color{red} siehe Vorlesung 06 Folien 34 ff.} + +\section{\glsxtrshortpl{SVM} with Kernels}% +\label{sec:SVMs with Kernels} + + diff --git a/chapters/Mathematische_Grundlagen/Constraint Optimization.tex b/chapters/Mathematische_Grundlagen/Constraint Optimization.tex new file mode 100644 index 0000000..74ce145 --- /dev/null +++ b/chapters/Mathematische_Grundlagen/Constraint Optimization.tex @@ -0,0 +1,66 @@ +\chapter{Constraint Optimization}% +\label{cha:Constraint Optimization} +Um zu erklären, +was es mit der Constraint Optimization auf sich hat, +lohnt es sich ein einfaches Beispiel zu betrachten + +Es soll $\argmin_x x^2$ für $x\ge b$ ermittelt werden. +Abhängig davon, +wie $b$ gewählt wird ist das Minimum gleich dem globalen Minimum oder gleich dem Grenzwert: +\begin{center} + \includegraphics[width = .6\textwidth]{constraint_optimization.png} +\end{center} + +\section{Lagrangian Multipliers}% +\label{sec:Lagrangian Multipliers} +Eine Möglichkeit ein Constraint Optimization Problem (für konvexe Funktionen) zu lösen sind die Lagrangian Multiplier. +Hierzu wird die Langrangesche Funktion gebildet: +\begin{equation} \label{eq:Lagrangian} + L = \text{objective} - \text{multiplier}\cdot \text{constraint} +\end{equation} +Hierbei ist die \say{objective} die Funtion die Optimiert werden soll. +Allgemein schreibt man für eine zu minimierende Funktion $f(\bm x)$ und eine Menge von Constraints $h_i(\bm x)\le b_i$ ($i=1\dots K$): +\begin{equation} \label{eq:lagrangian_function} + L(\bm x,\bm\lambda)=f(\bm x)-\sum_{i=1}^K\lambda_i(h_i(\bm x) - b_i)\qquad \lambda_i\ge 0\forall i=1\dots K +\end{equation} +Die Lagrangian Optimization erfolgt anschließend durch: +\begin{equation} \label{eq:lagrangian_optimization} + \min_{\bm x} \max_{\bm\lambda} L(\bm x,\bm\lambda) +\end{equation} +Wenn sowohl die Funktion $f$ als auch die Constraints $h$ konvex sind dürfen $\min_{\bm x}$ und $\max_{\bm\lambda}$ \say{getauscht} werden. +Man spricht hierbei dann von einem Dual Optimization Problem +\begin{equation} \label{eq:dual_optimization_problem} + \bm\lambda^*=\argmax_{\bm\lambda} g(\bm\lambda), g(\bm\lambda)= \min_{\bm x}L(\bm x,\bm\lambda) +\end{equation} +Hieraus ergibt sich der folgende Ablauf für die Lagrangian Optimization +\begin{mybox} + \textbf{\large Lagrangian Optimization}\\ + \begin{enumerate} + \item Die Langrangesche Funktion ermitteln + $$L(\bm x,\bm\lambda) = f(\bm x)-\sum_{i=1}^K \lambda_i(h_i(\bm x) - b_i)$$ + \item Die optimale Lösung für die Grundparameter (primal parameters) ermitteln + $$\frac{\partial L(\bm x,\bm\lambda)}{\partial x}=0\rightarrow\bm x^* = f(\bm\lambda)$$ + \item $\bm x^*$ wieder in die Langrangesche Funktion einsetzen um die Dual Function zu erhalten + $$ g(\bm\lambda) = L(f(\bm\lambda),\bm\lambda)$$ + \item Die optimale Lösung für die Dual Function ermitteln + $$\bm\lambda^* = \argmax_{\bm\lambda} g(\bm\lambda), \lambda_i\ge 0 \forall i$$ + \item Die Optimale Gesamtlösung errechnen + $$\bm x^* = f(\bm\lambda^*)$$ + \end{enumerate} +\end{mybox} + +\subsection{Erklärung am Beispiel}% +\label{sub:Erklärung am Beispiel} +Für unser Beispiel ($\argmin_x x^2, x\ge b$) ergibt sich die Langrangesche Funktion: +\begin{equation} \label{eq:lagrangian_example} + L(x,\lambda) = \underbrace{x^2}_{\text{objective}}-\underbrace{\lambda}_{\text{multiplier}} \cdot \underbrace{(x-b)}_{\text{constraint}} +\end{equation} +Das hat den Effekt, dass $\min_x$ und $\max_\lambda$ gegeneinander ankämpfen. +Die Anwendung der Lagrangian Optimization führt im Beispiel zu: +\begin{itemize} + \item $xb$: 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