\documentclass[a4paper, 12pt, draft=true]{article}
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\usepackage{filecontents}
\usepackage{subcaption}
\usepackage{adjustbox}
\usepackage{xcolor}
\usepackage{tabu}
\usepackage{showframe}
\usepackage{graphicx}
\usepackage{titlecaps}
\usepackage{amssymb}
\usepackage{mathtools}%add-on and patches to amsmath
\usetikzlibrary{calc, 3d}
\usepgfplotslibrary{colorbrewer}

\newcommand\Tstrut{\rule{0pt}{2.6ex}}         % = `top' strut
\newcommand\Bstrut{\rule[-0.9ex]{0pt}{0pt}}   % = `bottom' strut

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\DeclareMathOperator*{\po}{\mathbb{P}\text{-}\mathcal{O}}
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\begin{document}
\newcommand{\plimn}[0]{\plim\limits_{n \to \infty}}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
% \pgfplotsset{
% compat=1.11,
% legend image code/.code={
% \draw[mark repeat=2,mark phase=2]
% plot coordinates {
% (0cm,0cm)
% (0.3cm,0cm)        %% default is (0.3cm,0cm)
% (0.6cm,0cm)         %% default is (0.6cm,0cm)
% };%
% }
% }
% \begin{figure}
%   \begin{subfigure}[h]{\textwidth}
%     \begin{tikzpicture}
%       \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
%                      /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
%         height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
%         ylabel = {Test Accuracy}, cycle
%         list/Dark2, every axis plot/.append style={line width
%           =1.25pt}]
%         % \addplot [dashed] table
%         % [x=epoch, y=accuracy, col sep=comma, mark = none]
%         % {Data/adam_datagen_full.log};
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_1.mean};  
%         % \addplot [dashed] table
%         % [x=epoch, y=accuracy, col sep=comma, mark = none]
%         % {Data/adam_datagen_dropout_02_full.log}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_datagen_1.mean}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_datagen_dropout_02_1.mean};
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_dropout_02_1.mean};

        
%         \addlegendentry{\footnotesize{G.}}
%         \addlegendentry{\footnotesize{G. + D. 0.2}}
%         \addlegendentry{\footnotesize{G. + D. 0.4}}
%         \addlegendentry{\footnotesize{D. 0.2}}
%         \addlegendentry{\footnotesize{D. 0.4}}
%         \addlegendentry{\footnotesize{Default}}
%       \end{axis}
%     \end{tikzpicture}
%     \caption{1 sample per class}
%     \vspace{0.25cm}
%   \end{subfigure}
%   \begin{subfigure}[h]{\textwidth}
%     \begin{tikzpicture}
%       \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
%                      /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
%         height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
%         ylabel = {Test Accuracy}, cycle
%         list/Dark2, every axis plot/.append style={line width
%           =1.25pt}]
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_dropout_00_10.mean}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_dropout_02_10.mean}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_datagen_dropout_00_10.mean}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_datagen_dropout_02_10.mean};

        
%         \addlegendentry{\footnotesize{G.}}
%         \addlegendentry{\footnotesize{G. + D. 0.2}}
%         \addlegendentry{\footnotesize{G. + D. 0.4}}
%         \addlegendentry{\footnotesize{D. 0.2}}
%         \addlegendentry{\footnotesize{D. 0.4}}
%         \addlegendentry{\footnotesize{Default}}
%       \end{axis}
%     \end{tikzpicture}
%     \caption{10 samples per class}
%   \end{subfigure}
%   \begin{subfigure}[h]{\textwidth}
%     \begin{tikzpicture}
%       \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
%                      /pgf/number format/precision=3},tick style = {draw = none}, width = 0.9875\textwidth,
%         height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
%         xlabel = {epoch}, ylabel = {Test Accuracy}, cycle
%         list/Dark2, every axis plot/.append style={line width
%           =1.25pt}, ymin = {0.92}]
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_dropout_00_100.mean};
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_dropout_02_100.mean}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_datagen_dropout_00_100.mean}; 
%         \addplot table
%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
%         {Data/adam_datagen_dropout_02_100.mean}; 
        
%         \addlegendentry{\footnotesize{G.}}
%         \addlegendentry{\footnotesize{G. + D. 0.2}}
%         \addlegendentry{\footnotesize{G. + D. 0.4}}
%         \addlegendentry{\footnotesize{D. 0.2}}
%         \addlegendentry{\footnotesize{D. 0.4}}
%         \addlegendentry{\footnotesize{Default}}
%       \end{axis}
%     \end{tikzpicture}
%     \caption{100 samples per class}
%     \vspace{.25cm}
%   \end{subfigure}
%   \caption{Accuracy for the net given in ... with Dropout (D.),
%     data generation (G.), a combination, or neither (Default) implemented and trained
%     with \textsc{Adam}. For each epoch the 60.000 training samples     
%     were used, or for data generation 10.000 steps with each using
%     batches of 60 generated data points. For each configuration the
%     model was trained 5 times and the average accuracies at each epoch
%     are given in (a). Mean, maximum and minimum values of accuracy on
%     the test and training set are given in (b).}
% \end{figure}
% \begin{table}
%   \centering
%   \begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
%       \Tstrut \Bstrut    & \textsc{Adam} & D. 0.2 & Gen    & Gen.+D. 0.2  \\
%       \hline           
%                          & 
%       \multicolumn{4}{c}{\titlecap{test accuracy for 1 sample}}\Bstrut    \\
%       \cline{2-5}
%        max     \Tstrut   & 0.5633        & 0.5312 & 0.6704 & 0.6604       \\
%        min               & 0.3230        & 0.4224 & 0.4878 & 0.5175       \\
%        mean              & 0.4570        & 0.4714 & 0.5862 & 0.6014       \\
%        var               & 0.0040        & 0.0012 & 0.0036 & 0.0023       \\
%       \hline
%                          & 
%       \multicolumn{4}{c}{\titlecap{test accuracy for 10 samples}}\Bstrut  \\
%       \cline{2-5}
%        max       \Tstrut & 0.8585        & 0.9423 & 0.9310 & 0.9441       \\
%        min               & 0.8148        & 0.9081 & 0.9018 & 0.9061       \\
%        mean              & 0.8377        & 0.9270 & 0.9185 & 0.9232       \\
%        var               & 2.7e-4        & 1.3e-4 & 6e-05  & 1.5e-4       \\
%       \hline
%                          & 
%       \multicolumn{4}{c}{\titlecap{test accuracy for 100 samples}}\Bstrut \\
%       \cline{2-5}
%        max               & 0.9637        & 0.9796 & 0.9810 & 0.9805       \\
%        min               & 0.9506        & 0.9719 & 0.9702 & 0.9727       \\
%        mean              & 0.9582        & 0.9770 & 0.9769 & 0.9783       \\
%        var               & 2e-05         & 1e-05  & 1e-05  & 0            \\
%       \hline
%     \end{tabu}
%     \caption{Values of the test accuracy of the model trained 10 times
%       of random training sets containing 1, 10 and 100 data points per
%       class.}
% \end{table}

% \begin{center}
%   \begin{figure}[h]
%     \centering
%   \begin{subfigure}{\textwidth}
%     \includegraphics[width=\textwidth]{Data/cnn_fashion_fig.pdf}
%     \caption{original\\image}
%   \end{subfigure}
%   \begin{subfigure}{\textwidth}
%     \includegraphics[width=\textwidth]{Data/cnn_fashion_fig1.pdf}
%     \caption{random\\zoom}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist_gen_shear.pdf}
%     \caption{random\\shear}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist_gen_rotation.pdf}
%     \caption{random\\rotation}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist_gen_shift.pdf}
%   \caption{random\\positional shift}
%   \end{subfigure}\\ 
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist5.pdf}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist6.pdf}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist7.pdf}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist8.pdf}
%   \end{subfigure}
%   \begin{subfigure}{0.19\textwidth}
%     \includegraphics[width=\textwidth]{Data/mnist9.pdf}
%   \end{subfigure}
%   \caption{The MNIST data set contains 70.000 images of preprocessed handwritten
%     digits. Of these images 60.000 are used as training images, while
%     the rest are used to validate the models trained.}
% \end{figure}
% \end{center}

% \begin{figure}
%   \begin{adjustbox}{width=\textwidth}
%     \begin{tikzpicture}  
%       \begin{scope}[x = (0:1cm), y=(90:1cm), z=(15:-0.5cm)]
%         \node[canvas is xy plane at z=0, transform shape] at (0,0)
%         {\includegraphics[width=5cm]{Data/klammern_r.jpg}};
%         \node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
%         {\includegraphics[width=5cm]{Data/klammern_g.jpg}};
%         \node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
%         {\includegraphics[width=5cm]{Data/klammern_b.jpg}};
%         \node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
%         {\includegraphics[width=5.3cm]{Data/klammern_rgb.jpg}};
%       \end{scope}
%     \end{tikzpicture}
%   \end{adjustbox}
%   \caption{On the right the red, green and blue chanels of the picture
%     are displayed. In order to better visualize the color channes the
%     black and white picture of each channel has been colored in the
%     respective color. Combining the layers results in the image on the
%     left}
% \end{figure}

% \begin{figure}
%   \centering
%   \begin{subfigure}{\linewidth}
%     \centering
%     \includegraphics[width=\textwidth]{Data/convnet_fig.pdf}
%   \end{subfigure}
%   \begin{subfigure}{.45\linewidth}
%     \centering
%     \begin{tikzpicture}
%       \begin{axis}[enlargelimits=false, width=\textwidth]
%         \addplot[domain=-5:5, samples=100]{tanh(x)};
%       \end{axis}
%     \end{tikzpicture}
%   \end{subfigure}
%   \begin{subfigure}{.45\linewidth}
%     \centering
%     \begin{tikzpicture}
%       \begin{axis}[enlargelimits=false, width=\textwidth,
%         ytick={0,2,4},yticklabels={\hphantom{4.}0,2,4}, ymin=-1]
%         \addplot[domain=-5:5, samples=100]{max(0,x)};
%       \end{axis}
%     \end{tikzpicture}
%   \end{subfigure}
%   \begin{subfigure}{.45\linewidth}
%     \centering
%     \begin{tikzpicture}
%       \begin{axis}[enlargelimits=false, width=\textwidth, ymin=-1,
%         ytick={0,2,4},yticklabels={$\hphantom{-5.}0$,2,4}]
%         \addplot[domain=-5:5, samples=100]{max(0,x)+ 0.1*min(0,x)};
%       \end{axis}
%     \end{tikzpicture}
%   \end{subfigure}
% \end{figure}


% \begin{tikzpicture}
% \begin{axis}[enlargelimits=false]
% \addplot [domain=-5:5, samples=101,unbounded coords=jump]{1/(1+exp(-x)};
% \addplot[domain=-5:5, samples=100]{tanh(x)};
% \addplot[domain=-5:5, samples=100]{max(0,x)};
% \end{axis}
% \end{tikzpicture}

% \begin{tikzpicture}
% \begin{axis}[enlargelimits=false]
% \addplot[domain=-2*pi:2*pi, samples=100]{cos(deg(x))};
% \end{axis}
% \end{tikzpicture}
\newcommand{\abs}[1]{\ensuremath{\left\vert#1\right\vert}}

\[
  \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
  h_{k,n} = \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
      (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
  \left(\sum_{\substack{k \in \kappa \\ \xi_k \in
        [\delta l , \delta(l+1))}} \varphi(\xi_k, v_k) 
    h_{k,n}\right) \approx
\]
\[
  \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
      (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
  \left(\sum_{\substack{k \in \kappa \\ \xi_k \in
        [\delta l , \delta(l+1))}} \left(\varphi(\delta l, v_k)
      \frac{1}{n g_\xi (\delta l)} \pm \frac{\varepsilon}{n}\right) 
    \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
          \delta(l+1))\right\}}}{\abs{\left\{m \in \kappa : \xi_m
          \in [\delta l, \delta(l+1))\right\}}}\right)
\]
\[
  \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
      (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
  \left(\frac{\sum_{\substack{k \in \kappa \\ \xi_k \in
          [\delta l , \delta(l+1))}}\varphi(\delta l,
      v_k)}{\abs{\left\{m \in \kappa : \xi_m
          \in [\delta l, \delta(l+1))\right\}}} 
    \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
          \delta(l+1))\right\}}}{n g_\xi (\delta l)}\right) \pm \varepsilon
\]
The amount of kinks in a given interval of length $\delta$ follows a
binomial distribution,
\[
  \mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
        \delta(l+1))\right\}}\right] = n \int_{\delta
    l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l)
  \pm \delta \tilde{\varepsilon}),
\]
for any $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$, since $g_\xi$ is uniformly continuous on its
support by Assumption..
As the distribution of $v$ is continuous as well we get
$\mathcal{L}(v_k) = \mathcal{L} v| \xi = \delta l) \forall k \in
\kappa : \xi_k \in [\delta l, \delta(l+1))$ for $delta \leq
\delta(\varepsilon, \tilde{\varepsilon})$. Thus we get with the law of
large numbers
\begin{align*}
  &\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
  h_{k,n} \approx\\  
  &\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
  (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
  \}]}}\left(\mathbb{E}[\phi(\xi, v)|\xi=\delta l]
  \stackrel{\mathbb{P}}{\pm}\right) \delta \left(1 \pm
  \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
  \\
  &\approx \left(\sum_{\substack{l \in \mathbb{Z} \\ [\delta
  l, \delta 
  (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
  \}]}}\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \delta
  \stackrel{\mathbb{P}}{\pm}\tilde{\tilde{\varepsilon}}
  \abs{C_{g_\xi}^u - C_{g_\xi}^l}
  \right)\\
  &\phantom{\approx}\cdot \left(1 \pm
  \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
\end{align*}
\newpage



 
\end{document}

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