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@ -163,22 +163,6 @@ Coding conventions and basic OOP was part of the course. All of my German presen
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and a big, but algorithmically not challenging project. To be honest,
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I only fixed some Java bugs.}\\
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%----------------------------------------------------------------------------------------
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% WORK EXPERIENCE -2-
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{\raggedleft\textsc{2011}\par}
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{\raggedright\large Student research assistant at \textsc{ Institute of Toxicology and Genetics}, KIT\\
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\textit{participating in a university research project}\\[5pt]}
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\normalsize{In summer 2011 I worked for over a month for a
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research project at KIT. I have written bash scripts for file
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conversions, fixed some bugs and re-written a slow Mathematica script
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in a much faster Python version. But it quickly turned out that
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this project had a lot of C++ source which was rarely commented or
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documented. I realized, that I wouldn't have time for this project
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after beginning my studies at university.}\\
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%----------------------------------------------------------------------------------------
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% WORK EXPERIENCE -4-
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@ -208,7 +192,7 @@ after beginning my studies at university.}\\
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\colorbox{shade}{\textcolor{text1}{
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\begin{tabular}{c|p{7cm}}
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\raisebox{-4pt}{\textifsymbol{18}} & Parkstraße 17, 76131 Karlsruhe \\ % Address
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\raisebox{-4pt}{\textifsymbol{18}} & Alte Allee 107, 81245 Munich \\ % Address
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\raisebox{-3pt}{\Mobilefone} & +49 $($1636$)$ 28 04 91 \\ % Phone number
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\raisebox{-1pt}{\Letter} & \href{mailto:info@martin-thoma.de}{info@martin-thoma.de} \\ % Email address
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\Keyboard & \href{http://martin-thoma.com}{martin-thoma.com} \\ % Website
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@ -331,6 +315,22 @@ Good Knowledge & \textsc{Python}\\ \\
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%----------------------------------------------------------------------------------------
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\section{Work Experience}
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%----------------------------------------------------------------------------------------
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% WORK EXPERIENCE -2-
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{\raggedleft\textsc{2011}\par}
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{\raggedright\large Student research assistant at \textsc{ Institute of Toxicology and Genetics}, KIT\\
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\textit{participating in a university research project}\\[5pt]}
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\normalsize{In summer 2011 I worked for over a month for a
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research project at KIT. I have written bash scripts for file
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conversions, fixed some bugs and re-written a slow Mathematica script
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in a much faster Python version. But it quickly turned out that
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this project had a lot of C++ source which was rarely commented or
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documented. I realized, that I wouldn't have time for this project
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after beginning my studies at university.}\\
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%----------------------------------------------------------------------------------------
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% WORK EXPERIENCE -3-
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Binary file not shown.
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@ -1,7 +1,8 @@
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\begin{abstract}
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This paper reviews the most common activation functions for convolution neural
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networks. They are evaluated on TODO dataset and possible reasons for the
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differences in their performance are given.
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networks. They are evaluated on the Asirra, GTSRB, HASYv2, STL-10, CIFAR-10,
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CIFAR-100 and MNIST dataset. Possible reasons for the differences in their
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performance are given.
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New state of the art results are achieved for TODO.
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New state of the art results are achieved for Asirra, GTSRB, HASYv2 and STL-10.
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\end{abstract}
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@ -7,17 +7,17 @@
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\centering
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\hspace*{-1cm}\begin{tabular}{lllll}
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\toprule
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Name & Function $\varphi(x)$ & Range of Values & $\varphi'(x)$ \\\midrule % & Used by
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Sign function$^\dagger$ & $\begin{cases}+1 &\text{if } x \geq 0\\-1 &\text{if } x < 0\end{cases}$ & $\Set{-1,1}$ & $0$ \\%& \cite{971754} \\
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\parbox[t]{2.6cm}{Heaviside\\step function$^\dagger$} & $\begin{cases}+1 &\text{if } x > 0\\0 &\text{if } x < 0\end{cases}$ & $\Set{0, 1}$ & $0$ \\%& \cite{mcculloch1943logical}\\
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Logistic function & $\frac{1}{1+e^{-x}}$ & $[0, 1]$ & $\frac{e^x}{(e^x +1)^2}$ \\%& \cite{duch1999survey} \\
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Tanh & $\frac{e^x - e^{-x}}{e^x + e^{-x}} = \tanh(x)$ & $[-1, 1]$ & $\sech^2(x)$ \\%& \cite{LeNet-5,Thoma:2014}\\
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\gls{ReLU}$^\dagger$ & $\max(0, x)$ & $[0, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\0 &\text{if } x < 0\end{cases}$ \\%& \cite{AlexNet-2012}\\
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\parbox[t]{2.6cm}{\gls{LReLU}$^\dagger$\footnotemark\\(\gls{PReLU})} & $\varphi(x) = \max(\alpha x, x)$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\\alpha &\text{if } x < 0\end{cases}$ \\%& \cite{maas2013rectifier,he2015delving} \\
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Softplus & $\log(e^x + 1)$ & $(0, +\infty)$ & $\frac{e^x}{e^x + 1}$ \\%& \cite{dugas2001incorporating,glorot2011deep} \\
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\gls{ELU} & $\begin{cases}x &\text{if } x > 0\\\alpha (e^x - 1) &\text{if } x \leq 0\end{cases}$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\\alpha e^x &\text{otherwise}\end{cases}$ \\%& \cite{clevert2015fast} \\
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Softmax$^\ddagger$ & $o(\mathbf{x})_j = \frac{e^{x_j}}{\sum_{k=1}^K e^{x_k}}$ & $[0, 1]^K$ & $o(\mathbf{x})_j \cdot \frac{\sum_{k=1}^K e^{x_k} - e^{x_j}}{\sum_{k=1}^K e^{x_k}}$ \\%& \cite{AlexNet-2012,Thoma:2014}\\
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Maxout$^\ddagger$ & $o(\mathbf{x}) = \max_{x \in \mathbf{x}} x$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x_i = \max \mathbf{x}\\0 &\text{otherwise}\end{cases}$ \\%& \cite{goodfellow2013maxout} \\
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Name & Function $\varphi(x)$ & Range of Values & $\varphi'(x)$ & Used by \\\midrule %
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Sign function$^\dagger$ & $\begin{cases}+1 &\text{if } x \geq 0\\-1 &\text{if } x < 0\end{cases}$ & $\Set{-1,1}$ & $0$ & \cite{971754} \\
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\parbox[t]{2.6cm}{Heaviside\\step function$^\dagger$} & $\begin{cases}+1 &\text{if } x > 0\\0 &\text{if } x < 0\end{cases}$ & $\Set{0, 1}$ & $0$ & \cite{mcculloch1943logical}\\
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Logistic function & $\frac{1}{1+e^{-x}}$ & $[0, 1]$ & $\frac{e^x}{(e^x +1)^2}$ & \cite{duch1999survey} \\
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Tanh & $\frac{e^x - e^{-x}}{e^x + e^{-x}} = \tanh(x)$ & $[-1, 1]$ & $\sech^2(x)$ & \cite{LeNet-5,Thoma:2014}\\
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\gls{ReLU}$^\dagger$ & $\max(0, x)$ & $[0, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\0 &\text{if } x < 0\end{cases}$ & \cite{AlexNet-2012}\\
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\parbox[t]{2.6cm}{\gls{LReLU}$^\dagger$\footnotemark\\(\gls{PReLU})} & $\varphi(x) = \max(\alpha x, x)$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\\alpha &\text{if } x < 0\end{cases}$ & \cite{maas2013rectifier,he2015delving} \\
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Softplus & $\log(e^x + 1)$ & $(0, +\infty)$ & $\frac{e^x}{e^x + 1}$ & \cite{dugas2001incorporating,glorot2011deep} \\
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\gls{ELU} & $\begin{cases}x &\text{if } x > 0\\\alpha (e^x - 1) &\text{if } x \leq 0\end{cases}$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\\alpha e^x &\text{otherwise}\end{cases}$ & \cite{clevert2015fast} \\
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Softmax$^\ddagger$ & $o(\mathbf{x})_j = \frac{e^{x_j}}{\sum_{k=1}^K e^{x_k}}$ & $[0, 1]^K$ & $o(\mathbf{x})_j \cdot \frac{\sum_{k=1}^K e^{x_k} - e^{x_j}}{\sum_{k=1}^K e^{x_k}}$ & \cite{AlexNet-2012,Thoma:2014}\\
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Maxout$^\ddagger$ & $o(\mathbf{x}) = \max_{x \in \mathbf{x}} x$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x_i = \max \mathbf{x}\\0 &\text{otherwise}\end{cases}$ & \cite{goodfellow2013maxout} \\
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\bottomrule
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\end{tabular}
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\caption[Activation functions]{Overview of activation functions. Functions
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@ -63,13 +63,11 @@
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\end{tabular}
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\caption[Activation function evaluation results on CIFAR-100]{Training and
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test accuracy of adjusted baseline models trained with different
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activation functions on CIFAR-100. For LReLU, $\alpha = 0.3$ was
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activation functions on CIFAR-100. For \gls{LReLU}, $\alpha = 0.3$ was
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chosen.}
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\label{table:CIFAR-100-accuracies-activation-functions}
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\end{table}
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\glsreset{LReLU}
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\begin{table}[H]
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\centering
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\setlength\tabcolsep{1.5pt}
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@ -91,7 +89,7 @@
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\end{tabular}
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\caption[Activation function evaluation results on HASYv2]{Test accuracy of
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adjusted baseline models trained with different activation
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functions on HASYv2. For LReLU, $\alpha = 0.3$ was chosen.}
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functions on HASYv2. For \gls{LReLU}, $\alpha = 0.3$ was chosen.}
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\label{table:HASYv2-accuracies-activation-functions}
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\end{table}
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@ -116,8 +114,93 @@
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\end{tabular}
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\caption[Activation function evaluation results on STL-10]{Test accuracy of
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adjusted baseline models trained with different activation
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functions on STL-10. For LReLU, $\alpha = 0.3$ was chosen.}
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functions on STL-10. For \gls{LReLU}, $\alpha = 0.3$ was chosen.}
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\label{table:STL-10-accuracies-activation-functions}
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\end{table}
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\begin{table}[H]
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\centering
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\hspace*{-1cm}\begin{tabular}{lllll}
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\toprule
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Name & Function $\varphi(x)$ & Range of Values & $\varphi'(x)$ \\\midrule % & Used by
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Sign function$^\dagger$ & $\begin{cases}+1 &\text{if } x \geq 0\\-1 &\text{if } x < 0\end{cases}$ & $\Set{-1,1}$ & $0$ \\%& \cite{971754} \\
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\parbox[t]{2.6cm}{Heaviside\\step function$^\dagger$} & $\begin{cases}+1 &\text{if } x > 0\\0 &\text{if } x < 0\end{cases}$ & $\Set{0, 1}$ & $0$ \\%& \cite{mcculloch1943logical}\\
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Logistic function & $\frac{1}{1+e^{-x}}$ & $[0, 1]$ & $\frac{e^x}{(e^x +1)^2}$ \\%& \cite{duch1999survey} \\
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Tanh & $\frac{e^x - e^{-x}}{e^x + e^{-x}} = \tanh(x)$ & $[-1, 1]$ & $\sech^2(x)$ \\%& \cite{LeNet-5,Thoma:2014}\\
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\gls{ReLU}$^\dagger$ & $\max(0, x)$ & $[0, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\0 &\text{if } x < 0\end{cases}$ \\%& \cite{AlexNet-2012}\\
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\parbox[t]{2.6cm}{\gls{LReLU}$^\dagger$\footnotemark\\(\gls{PReLU})} & $\varphi(x) = \max(\alpha x, x)$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\\alpha &\text{if } x < 0\end{cases}$ \\%& \cite{maas2013rectifier,he2015delving} \\
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Softplus & $\log(e^x + 1)$ & $(0, +\infty)$ & $\frac{e^x}{e^x + 1}$ \\%& \cite{dugas2001incorporating,glorot2011deep} \\
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\gls{ELU} & $\begin{cases}x &\text{if } x > 0\\\alpha (e^x - 1) &\text{if } x \leq 0\end{cases}$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x > 0\\\alpha e^x &\text{otherwise}\end{cases}$ \\%& \cite{clevert2015fast} \\
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Softmax$^\ddagger$ & $o(\mathbf{x})_j = \frac{e^{x_j}}{\sum_{k=1}^K e^{x_k}}$ & $[0, 1]^K$ & $o(\mathbf{x})_j \cdot \frac{\sum_{k=1}^K e^{x_k} - e^{x_j}}{\sum_{k=1}^K e^{x_k}}$ \\%& \cite{AlexNet-2012,Thoma:2014}\\
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Maxout$^\ddagger$ & $o(\mathbf{x}) = \max_{x \in \mathbf{x}} x$ & $(-\infty, +\infty)$ & $\begin{cases}1 &\text{if } x_i = \max \mathbf{x}\\0 &\text{otherwise}\end{cases}$ \\%& \cite{goodfellow2013maxout} \\
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\bottomrule
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\end{tabular}
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\caption[Activation functions]{Overview of activation functions. Functions
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marked with $\dagger$ are not differentiable at 0 and functions
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marked with $\ddagger$ operate on all elements of a layer
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simultaneously. The hyperparameters $\alpha \in (0, 1)$ of Leaky
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ReLU and ELU are typically $\alpha = 0.01$. Other activation
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function like randomized leaky ReLUs exist~\cite{xu2015empirical},
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but are far less commonly used.\\
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Some functions are smoothed versions of others, like the logistic
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function for the Heaviside step function, tanh for the sign
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function, softplus for ReLU.\\
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Softmax is the standard activation function for the last layer of
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a classification network as it produces a probability
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distribution. See \Cref{fig:activation-functions-plot} for a plot
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of some of them.}
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\label{table:activation-functions-overview}
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\end{table}
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\footnotetext{$\alpha$ is a hyperparameter in leaky ReLU, but a learnable parameter in the parametric ReLU function.}
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\begin{figure}[ht]
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\centering
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\begin{tikzpicture}
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\definecolor{color1}{HTML}{E66101}
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\definecolor{color2}{HTML}{FDB863}
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\definecolor{color3}{HTML}{B2ABD2}
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\definecolor{color4}{HTML}{5E3C99}
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\begin{axis}[
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legend pos=north west,
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legend cell align={left},
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axis x line=middle,
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axis y line=middle,
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x tick label style={/pgf/number format/fixed,
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/pgf/number format/fixed zerofill,
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/pgf/number format/precision=1},
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y tick label style={/pgf/number format/fixed,
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/pgf/number format/fixed zerofill,
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/pgf/number format/precision=1},
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grid = major,
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width=16cm,
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height=8cm,
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grid style={dashed, gray!30},
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xmin=-2, % start the diagram at this x-coordinate
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xmax= 2, % end the diagram at this x-coordinate
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ymin=-1, % start the diagram at this y-coordinate
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ymax= 2, % end the diagram at this y-coordinate
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xlabel=x,
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ylabel=y,
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tick align=outside,
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enlargelimits=false]
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\addplot[domain=-2:2, color1, ultra thick,samples=500] {1/(1+exp(-x))};
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\addplot[domain=-2:2, color2, ultra thick,samples=500] {tanh(x)};
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\addplot[domain=-2:2, color4, ultra thick,samples=500] {max(0, x)};
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\addplot[domain=-2:2, color4, ultra thick,samples=500, dashed] {ln(exp(x) + 1)};
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\addplot[domain=-2:2, color3, ultra thick,samples=500, dotted] {max(x, exp(x) - 1)};
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\addlegendentry{$\varphi_1(x)=\frac{1}{1+e^{-x}}$}
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\addlegendentry{$\varphi_2(x)=\tanh(x)$}
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\addlegendentry{$\varphi_3(x)=\max(0, x)$}
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\addlegendentry{$\varphi_4(x)=\log(e^x + 1)$}
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\addlegendentry{$\varphi_5(x)=\max(x, e^x - 1)$}
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\end{axis}
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\end{tikzpicture}
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\caption[Activation functions]{Activation functions plotted in $[-2, +2]$.
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$\tanh$ and ELU are able to produce negative numbers. The image of
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ELU, ReLU and Softplus is not bound on the positive side, whereas
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$\tanh$ and the logistic function are always below~1.}
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\label{fig:activation-functions-plot}
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\end{figure}
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\glsreset{LReLU}
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\twocolumn
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@ -1,24 +1,42 @@
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%!TEX root = main.tex
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\section{Introduction}
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TODO\cite{Thoma:2014}
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Artificial neural networks have dozends of hyperparameters which influence
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their behaviour during training and evaluation time. One parameter is the
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choice of activation functions. While in principle every neuron could have a
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different activation function, in practice networks only use two activation
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functions: The softmax function for the output layer in order to obtain a
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probability distribution over the possible classes and one activation function
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for all other neurons.
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\section{Terminology}
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TODO
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Activation functions should have the following properties:
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\begin{itemize}
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\item \textbf{Non-linearity}: A linear activation function in a simple feed
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forward network leads to a linear function. This means no matter how
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many layers the network uses, there is an equivalent network with
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only the input and the output layer. Please note that \glspl{CNN} are
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different. Padding and pooling are also non-linear operations.
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\item \textbf{Differentiability}: Activation functions need to be
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differentiable in order to be able to apply gradient descent. It is
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not necessary that they are differentiable at any point. In practice,
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the gradient at non-differentiable points can simply be set to zero
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in order to prevent weight updates at this point.
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\item \textbf{Non-zero gradient}: The sign function is not suitable for
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gradient descent based optimizers as its gradient is zero at all
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differentiable points. An activation function should have infinitely
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many points with non-zero gradient.
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\end{itemize}
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\section{Activation Functions}
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Nonlinear, differentiable activation functions are important for neural
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networks to allow them to learn nonlinear decision boundaries. One of the
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simplest and most widely used activation functions for \glspl{CNN} is
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\gls{ReLU}~\cite{AlexNet-2012}, but others such as
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One of the simplest and most widely used activation functions for \glspl{CNN}
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is \gls{ReLU}~\cite{AlexNet-2012}, but others such as
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\gls{ELU}~\cite{clevert2015fast}, \gls{PReLU}~\cite{he2015delving}, softplus~\cite{7280459}
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and softsign~\cite{bergstra2009quadratic} have been proposed. The baseline uses
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\gls{ELU}.
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and softsign~\cite{bergstra2009quadratic} have been proposed.
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Activation functions differ in the range of values and the derivative. The
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definitions and other comparisons of eleven activation functions are given
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in~\cref{table:activation-functions-overview}.
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\section{Important Differences of Proposed Activation Functions}
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Theoretical explanations why one activation function is preferable to another
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in some scenarios are the following:
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\begin{itemize}
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@ -96,6 +114,7 @@ in~\cref{table:HASYv2-accuracies-activation-functions}. For both datasets, the
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logistic function has a much shorter training time and a noticeably lower test
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accuracy.
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\glsunset{LReLU}
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\begin{table}[H]
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\centering
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\begin{tabular}{lccc}
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@ -111,7 +130,7 @@ accuracy.
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ReLU & \cellcolor{yellow!25}Yes\footnotemark & \cellcolor{red!25} No & \cellcolor{yellow!25}Half-sided \\
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Softplus & \cellcolor{green!25}No & \cellcolor{red!25} No & \cellcolor{yellow!25}Half-sided \\
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S2ReLU & \cellcolor{green!25}No & \cellcolor{green!25}Yes & \cellcolor{green!25} No \\
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LReLU/PReLU & \cellcolor{green!25}No & \cellcolor{green!25}Yes & \cellcolor{green!25} No \\
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\gls{LReLU}/PReLU & \cellcolor{green!25}No & \cellcolor{green!25}Yes & \cellcolor{green!25} No \\
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ELU & \cellcolor{green!25}No & \cellcolor{green!25}Yes & \cellcolor{green!25} No \\
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\bottomrule
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\end{tabular}
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@ -120,8 +139,6 @@ accuracy.
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\end{table}
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\footnotetext{The dying ReLU problem is similar to the vanishing gradient problem.}
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\glsunset{LReLU}
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\begin{table}[H]
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\centering
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\begin{tabular}{lccclllll}
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|
|
@ -173,4 +190,5 @@ accuracy.
|
|||
functions on MNIST.}
|
||||
\label{table:MNIST-accuracies-activation-functions}
|
||||
\end{table}
|
||||
\glsreset{LReLU}
|
||||
\glsreset{LReLU}
|
||||
|
||||
|
|
|
|||
|
|
@ -7,7 +7,15 @@
|
|||
\usepackage{amsmath,amssymb}
|
||||
\usepackage[table]{xcolor}
|
||||
\usepackage[absolute,overlay]{textpos}
|
||||
\usepackage{pgfplots}
|
||||
\pgfplotsset{compat=1.13}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows.meta}
|
||||
\usetikzlibrary{decorations.pathreplacing}
|
||||
\usetikzlibrary{positioning}
|
||||
\usetikzlibrary{decorations.text}
|
||||
\usetikzlibrary{decorations.pathmorphing}
|
||||
\usetikzlibrary{shapes.multipart, calc}
|
||||
\usepackage{csquotes}
|
||||
\usepackage[binary-units,group-separator={,}]{siunitx}
|
||||
\sisetup{per-mode=fraction,
|
||||
|
|
@ -59,7 +67,7 @@
|
|||
\usepackage{braket} % needed for \Set
|
||||
\usepackage{algorithm,algpseudocode}
|
||||
|
||||
\usepackage[xindy,toc,section=chapter,numberedsection=autolabel]{glossaries}
|
||||
\usepackage[xindy,toc,section=section]{glossaries}
|
||||
|
||||
% Make document nicer
|
||||
\DeclareMathOperator*{\argmin}{arg\,min}
|
||||
|
|
@ -93,6 +101,7 @@
|
|||
\input{content}
|
||||
\bibliographystyle{IEEEtranSA}
|
||||
\bibliography{bibliography}
|
||||
\printglossaries%
|
||||
\input{appendix}
|
||||
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue