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# 『ゼロから作るDeep Learning』5章 メモ

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# 5章　誤差逆伝播法

## 導入される数学の概念

• 計算グラフ (computational graph)
• 順伝播 (forward propagation)
• 逆伝播 (backward propagation)
• アフィン (Affine) 変換
幾何学の分野で，ニューラルネットワークの順伝播で行う行列の内積のこと

まず，
$\begin{eqnarray} y &=& \dfrac {1} { 1+\exp \left( -x\right)} \tag{5.9} \end{eqnarray}$

なので，式 $(5.9)$ より
$$y^{2}= \dfrac {1^{2}} {\left( 1+\exp \left( -x\right) \right) ^{2}}$$

を準備しておく．

\begin{eqnarray}
\dfrac {\partial L} {dy}y^{2}\exp \left( -x\right)
&=&
\dfrac {\partial L} {dy}\dfrac {1} {\left( 1+\exp \left( -x\right) \right) ^{2}}\exp \left( -x\right)\\
&=&
\dfrac {\partial L} {dy}\dfrac {1^{2}} {\left( 1+\exp \left( -x\right) \right)}\dfrac{\exp \left( -x\right)}{\left( 1+\exp \left( -x\right) \right)}\\
&=&
\dfrac {\partial L} {dy}y\dfrac{\exp \left( -x\right)}{ 1+\exp \left( -x\right)}\\
&=&
\dfrac {\partial L} {dy}y\left(\dfrac {\exp \left( -x\right)} { 1+\exp \left( -x\right)}+\dfrac {1} { 1+\exp \left( -x\right)}-\dfrac {1} { 1+\exp \left( -x\right)}\right)\\
&=&
\dfrac {\partial L} {dy}y\left(\dfrac {1+\exp \left( -x\right)} { 1+\exp \left( -x\right)}-\dfrac {1} { 1+\exp \left( -x\right)}\right)\\
&=&
\dfrac {\partial L} {dy}y\left(1-\dfrac {1} { 1+\exp \left( -x\right)}\right)\\
&=&
\dfrac {\partial L} {dy}y(1-y)\\
\end{eqnarray}


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