[Tex/LaTex] Plotting softmax activation function

Question

In the MSE below, I define a function logsumexp as [declare function={logsumexp(\x)=\log(\sum{\exp^{\x_i}});}] to help in plotting the softmax activation function.

When I used the function, to add plot \addplot[blue,smooth] {exp(x) /logsumexp(x))}; everything messed-up.

MSE: (\addplot line commented out for the softmax function)

\documentclass[11pt]{article}

\usepackage{subfigure}
\usepackage{pgfplots}
\usepackage[top=3cm,left=3cm,right=3cm,bottom=3cm]{geometry}
% Scriptsize axis style.
\pgfplotsset{every axis/.append style={tick label style={/pgf/number format/fixed},font=\scriptsize,ylabel near ticks,xlabel near ticks,grid=major}}

\pgfplotsset{compat=1.16}

\begin{document}
\begin{figure}[t!]
    \centering
    \subfigure[sigmoid activation function.]{
            \begin{tikzpicture}[declare function={sigma(\x)=1/(1+exp(-\x));}]
            \begin{axis}[width=5.5cm,height=6cm,ylabel=$\sigma(z)$,xlabel=$z$,ymin=0,ymax=1.25,xmin=-5,xmax=5]
                \addplot[blue,smooth] {1/(1+exp(-x))};
            \end{axis}
        \end{tikzpicture}
    }
    \subfigure[Softmax activation function. ]{
        \begin{tikzpicture}[declare function={logsumexp(\x)=\log(\sum{\exp^{\x_i}});}]
            \begin{axis}[width=5.5cm,height=6cm,ylabel=$ \sigma(z)_j$,xlabel=$z$,ymin=-1.25,ymax=1.25,xmin=-5,xmax=5]
               %\addplot[blue,smooth] {exp(x) /logsumexp(x))};

            \end{axis}
        \end{tikzpicture}
    }
        \caption[Activation functions.]{Sigmoid and Softmax activation functions}
        \label{fig:sigmoid-tanh}
\end{figure}
logsumexp function: $logsumexp(x)=\log(\sum{\exp^{x_i}})$
\end{document}

when \addplot in uncommented, everything messed-up. What am I missing?

Check that your $'s match around math expressions. If they do, then you've probably used a symbol in normal text that needs to be in math mode. Symbols such as subscripts ( _ ), integrals ( \int ), Greek letters ( \alpha, \beta, \delta ), and modifiers (\vec{x}, \tilde{x} ) must be written in math mode. See the full list here.If you intended to use mathematics mode, then use $ … $ for 'inline math mode', $$ … $$ for 'display math mode' or alternatively \begin{math} … \end{math}.

EDIT

Giving an example with some values for x.

import numpy as np
x = [1.2, 2.5, 3.1, 4.4, 1.6, 2.4, 3.6]
np.exp(x) / np.sum(np.exp(x))
array([0.01933382, 0.07094152, 0.12926387, 0.47430749, 0.02884267,
       0.06419054, 0.21312009])

Answer 1

As for your question, as mentioned in the comments, \exp and \log are commands that typeset these functions, you probably want to use the built in functions exp and ln instead. Likewise, \sum is a command that typesets a sum symbol, but unlike in the previous cases there is no built-in function. So one has to build it. The following contains an example. (If that's not the precise function you want, you want to provide a precise explanation that defines the function in common terms, i.e. using a mathematical language and not some python code.)

\documentclass[11pt]{article}
\usepackage[top=3cm,left=3cm,right=3cm,bottom=3cm]{geometry}
\usepackage{amsmath}
\DeclareMathOperator{\logsumexp}{logsumexp}
\usepackage{subcaption}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\pgfplotsset{every axis/.append style={tick label style={/pgf/number format/fixed},font=\scriptsize,ylabel near ticks,xlabel near ticks,grid=major}}

\pgfmathdeclarefunction{sumexp}{3}{%
\begingroup%
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}%
\pgfmathsetmacro{\myx}{#1}%
\pgfmathtruncatemacro{\myxmin}{#2}%
\pgfmathtruncatemacro{\myxmax}{#3}%
\pgfmathsetmacro{\mysum}{0}%
\pgfplotsforeachungrouped\XX in {\myxmin,...,\myxmax}%
{\pgfmathsetmacro{\mysum}{\mysum+exp(\XX)}}%
\pgfmathparse{\mysum+exp(#1)}%
\pgfmathsmuggle\pgfmathresult\endgroup%
}%

\begin{document}
\begin{figure}[t!]
    \centering
    \begin{subfigure}[t]{0.45\textwidth}
            \begin{tikzpicture}[declare function={sigma(\x)=1/(1+exp(-\x));}]
            \begin{axis}[width=5.5cm,height=6cm,ylabel=$\sigma(z)$,xlabel=$z$,ymin=0,ymax=1.25,xmin=-5,xmax=5]
                \addplot[blue,smooth] {1/(1+exp(-x))};
            \end{axis}
        \end{tikzpicture}
     \caption{Sigmoid activation function.}
    \end{subfigure} 
    \begin{subfigure}[t]{0.45\textwidth}    
        \begin{tikzpicture}
            \begin{axis}[width=5.5cm,height=6cm,ylabel=$\sigma(z)_j$,
            xlabel=$z$,     xmin=-5,xmax=5]
               \addplot[blue,domain=-5:5,samples=51] 
               {exp(x)/sumexp(x,-4,0)};
            \end{axis}
        \end{tikzpicture}
      \caption{Softmax activation function.}
     \end{subfigure}  
   \caption[Activation functions.]{Sigmoid and Softmax activation functions}
   \label{fig:sigmoid-tanh}
\end{figure}
\end{document}