Presentation.tex

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\usepackage{hyperref}
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\usepackage{mathtools}
\newtheorem{theo}{Theorem}
\usepackage{pgfplots}
\pgfplotsset{compat=1.9}

\usepackage{multicol}

\logo{\includegraphics[scale=0.3]{tumlogo.pdf}} 

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\title{Proseminar: Data Mining}
\subtitle{Neuronale Netze: Grundlagen}


\author{Lukas Krenz}
\date{12. Juni 2015}
\institute{Technische Universität München}
\begin{document}

\begin{frame}
\titlepage
\end{frame}
 \begin{frame}{Die Magie}
      
  \begin{columns}[c]
    \begin{column}[c]{.33\textwidth}
    \includegraphics[height=3.66cm]{figures/guitar.png}

    man in black shirt is playing guitar.
    \end{column}
    \begin{column}[c]{.33\textwidth}
    \includegraphics[height=3.66cm]{figures/dogbar.png}

    black and white dog jumps over bar.
    \end{column}
    \begin{column}[c]{.33\textwidth}

    \includegraphics[height=3.66cm]{figures/teddy.jpeg}

    a woman holding a teddy bear in front of a mirror.
    \end{column}  
  \end{columns}

  Quelle: Deep Visual-Semantic Alignments for Generating Image Descriptions, CVPR 2015
 \end{frame}

\begin{frame}{Was sind Neuronale Netze?}
Neuronale Netze sind ein:
\begin{itemize}
\item graphisches
\item nicht-lineares
\item flexibles
\end{itemize}
Modell für maschinelles Lernen.
\end{frame}

\begin{frame}{Topologie}
\begin{figure}[ht!]
\label{fig:MLP}
  \centering
    \input{graph_neural_network}
  \caption{Ein 2-schichtiges MLP.}
\end{figure}

\end{frame}


\begin{frame}{Aktivierungsfunktionen}
\begin{columns}[c]
  \begin{column}[c]{.6\textwidth}
  \begin{figure}[ht!]
  \centering
  \caption{Die logistische Funktion $\sigma_1$}
    \begin{tikzpicture}[scale=0.80]

        \begin{axis}%
        [
            grid=major,     
            xmin=-6,
            xmax=6,
            axis x line=bottom,
            ytick={0,.5,1},
            ymax=1,
            axis y line=middle,
        ]
            \addplot%
            [
                blue,%
                mark=none,
                samples=100,
                domain=-6:6,
            ]
            (x,{1/(1+exp(-x))});    
            \end{axis}
    \end{tikzpicture}
\end{figure}
  \end{column}  
  \begin{column}[c]{.5\textwidth}
    \begin{align}
      \sigma_1(x) & =  \frac{1}{1+e^{-x}} \\
      \sigma_2(x) & =  \tanh(x) = \frac{1-e^{-2x}}{1+e^{-2x}} \\
      \sigma_3(x) & =  \max(0,x)
    \end{align}
    \end{column}
\end{columns} 

\end{frame}

\begin{frame}{Feedforward}
\begin{figure}[ht!]
%\label{fig:MLP}
  \centering
    \input{graph_neural_network} %TODO: Create smaller version of graphic without annotations.
\end{figure}

\begin{align}
	\text{net}_j & = \sum_{i} w_{ji} y_i = w_j^t x \\
	y_j & = \sigma (\text{net}_j)
\end{align} 
\end{frame}

\begin{frame}{Backprop}
\begin{align}
	E &= \sum_{k=1}^K \sum_{i=1}^N \left( y_i^k - t_i^k \right)^2 \\
	  &=  \sum_n E_n(y_1, \ldots, y_c)
\end{align}

\begin{equation}
\frac{\partial E^n}{\partial w_{ij}} = \frac{\partial E^n}{\partial net_j}  \frac{\partial net_j }{\partial w_{ji}}
\end{equation}

\end{frame}

\begin{frame}{Backprop (ctd.)}


\begin{align}
  \frac{\partial net_j }{\partial w_{ji}} & =  y_i
  \\
  \frac{\partial E^n}{\partial net_j} & =  \delta_j
  \\
  \frac{\partial E^n}{\partial w_{i,j}} &= y_i  \delta_j
\end{align}



\begin{equation}
\label{eq:backpropagation}
\delta_j =  \begin{cases}
               \sigma ' (net_j) (y_j - t_j)           & \text{wenn j Ausgabeneuron ist}\\
               \sigma ' (net_j) \sum_k w_{kj} \delta_k     & \text{wenn j verdecktes Neuron ist}
           \end{cases} 
\end{equation} 
\end{frame}
\begin{frame}{Optimierung}


\begin{columns}[c]
  \begin{column}[c]{.3\textwidth}
  \begin{figure}[ht!]
  \centering
  \caption{Die Rosenbrock Funktion $(1-x)^2 + 100(y - x^2)^2$}
\input{3dplot.tex}
\end{figure}
  \end{column}  
  \begin{column}[c]{.7\textwidth}
    \begin{align}
    x_{n+1} &=x_n- \eta  \nabla F(x_n) \\
    \Delta_w^t &= - \eta  \nabla_w E(w;t)
    \end{align}
    \end{column}
\end{columns} 
\end{frame}

\begin{frame}{Diskussion und Ausblick}
\begin{columns}[T]
  \begin{column}[T]{.5\textwidth}
        Vorteile:
        \begin{itemize}
          \item mächtig
          \item flexibel
        \end{itemize}
  \end{column}  
  \begin{column}[T]{.5\textwidth}
        Nachteile
        \begin{itemize}
          \item viele Parameter, kompliziert anzupassen
          \item langsam
        \end{itemize}
    \end{column}
\end{columns} 
In der \alert{aktuellen} Forschung relevant: Netze anderer Topologien 
\end{frame}
\end{document}