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\begin{document} 
\section{askish nevronikou diktiou}
esto ena nevrotiko diktio me mia isodo $x$, ena nevrona kai mia exodo $y$. h sinartisi evergopoihshs
einai h sigmoeidis:
\begin{equation}
         \sigma(z) = \frac{1}{1+e^{-z}}
\end{equation}
h exodos tou nevrona einai:
\begin{equation}
         y=\sigma(wx+b)
\end{equation}
opou $w$ einai to varos kai $b$ h polosi. dinontai ta dedomena ekpedeushs tou diktiou:
\begin{center}
\begin{tabular}{|c|c|}
\hline
        $x$ & $y$ \\
\hline
        1 & 1 \\
\hline
        0 & 0 \\
\end{tabular}
\end{center}
va vrethoun oi times $w$ $b$ pou na elaxistopoioun
\begin{equation}
          E=\frac{1}{2} \sum_{i=1}^2 (y_i-\hat{y}_i)^2
\end{equation}
\subsection*{lisi} 
ypologizoume thn exodo tou diktiou:
\begin{itemize}          
\item gia $x=1$ : $\hat{y}_1=\sigma(w+b)$
\item gia $x=0$ : $\hat{y}_2=\sigma(b)$
\end{itemize}
tetragoniko sphalma:
\begin{equation}
E=\frac{1}{2} \left[1 -\sigma(w+b)^2 +
(0 - \sigma(b))^2 \right]
\end{equation}
gia elaxistopoihsh paragogou pernoume merikes paragwgous tou $E$ pros $w$ kai $b$ kai ises me miden.

\begin{equation}
\frac{\partial E}{\partial w}=0, \quad
\frac{\partial E}{\partial b}=0
\end{equation}
h epilisi dini $w$, $b$.
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