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backpropagation memo

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  • 岡谷貴之「深層学習」のBPのメモ
  • scalaのMLP実装用
E_n=E({\bf W}^{(1)},{\bf W}^{(2)},{\bf W}^{(3)},\cdots, {\bf W}^{(n)})
  • n layer MLP
  • $E$はloss function or 目的関数
  • $W$はweight matrix
\begin{eqnarray}
{\bf u}^{(l+1} 
 &=& {\bf W}^{(l+1)} \cdot {\bf z}^{(l)} \\
 &=& {\bf W}^{(l+1)} \cdot f({\bf u}^{(l)})
\end{eqnarray}

ここで $f$はactivation function

\begin{eqnarray}
\frac{ \partial{E} }{ \partial {\bf W}^{(l)} } 
 &=& \frac{ \partial{E} }{ \partial {\bf u}^{(l)} }
  \frac{ \partial {\bf u}^{(l)} }{ \partial {\bf W}^{(l)}} \\       
 &=& \frac{ \partial{E} }{ \partial {\bf u}^{(l)} }
  {\bf z}^{(l-1)} \\       
\end{eqnarray}

ここで,

\begin{eqnarray}
 \frac{ \partial{E} }{ \partial {\bf u}^{(l)} }
  &=& \frac{ \partial{E} }{ \partial {\bf u}^{(l+1)} } 
   \frac{ \partial {\bf u}^{(l+1)} }{ \partial {\bf u}^{(l)} } \\
  &=& \frac{ \partial{E} }{ \partial {\bf u}^{(l+1)} } 
   \left\{ {\bf W}^{(l+1)} \odot f^{'}( {\bf u}^{(l)} ) \right\} 
\end{eqnarray}

ここで

{\bf \delta}^{(l)} =  \frac{ \partial{E} }{ \partial {\bf u}^{(l)} }

とすると

\begin{eqnarray}
\frac{ \partial{E} }{ \partial {\bf W}^{(l)} } 
 &=& {\bf \delta}^{(l)} {\bf z}^{(l-1)} \\       

{\bf \delta}^{(l)} &=& {\bf \delta}^{(l+1)} 
 \left\{ {\bf W}^{(l+1)} \odot f^{'}( {\bf u}^{(l)} ) \right\}  
\end{eqnarray}

となる.

これが逆伝播法

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