|
|
@@ -57,7 +57,7 @@ Let's compute the derivatives of all our models. Throughout the entire paper $n$
|
|
|
\end{tikzpicture}
|
|
|
\end{center}
|
|
|
\begin{align}
|
|
|
- y &= \sigma(xw_1 + yw_2 + b) \\
|
|
|
+ z &= \sigma(xw_1 + yw_2 + b) \\
|
|
|
\sigma(x) &= \frac{1}{1 + e^{-x}} \\
|
|
|
\sigma'(x) &= \sigma(x)(1 - \sigma(x))
|
|
|
\end{align}
|
|
|
@@ -106,8 +106,8 @@ Let's compute the derivatives of all our models. Throughout the entire paper $n$
|
|
|
|
|
|
\begin{align}
|
|
|
a_i^{(1)} &= \sigma(x_iw^{(1)} + b^{(1)}) \\
|
|
|
- \pd[w^{1}]a_1^{(i)} &= a_i^{(1)}(1 - a_i^{(1)})x_i \\
|
|
|
- \pd[b^{1}]a_1^{(i)} &= a_i^{(1)}(1 - a_i^{(1)}) \\
|
|
|
+ \pd[w^{(1)}]a_i^{(1)} &= a_i^{(1)}(1 - a_i^{(1)})x_i \\
|
|
|
+ \pd[b^{1}]a_i^{(1)} &= a_i^{(1)}(1 - a_i^{(1)}) \\
|
|
|
a_i^{(2)} &= \sigma(a_i^{(1)}w^{(2)} + b^{(2)}) \\
|
|
|
\pd[w^{(2)}]a_i^{(2)} &= a_i^{(2)}(1 - a_i^{(2)})a_i^{(1)} \\
|
|
|
\pd[b^{(2)}]a_i^{(2)} &= a_i^{(2)}(1 - a_i^{(2)}) \\
|
|
|
@@ -125,11 +125,11 @@ Let's compute the derivatives of all our models. Throughout the entire paper $n$
|
|
|
\pd[b^{(2)}] C^{(2)} &= \avgsum[i, n] 2(a_i^{(2)} - y_i)a_i^{(2)}(1 - a_i^{(2)}) \\
|
|
|
\pd[a_i^{(1)}]C^{(2)} &= \avgsum[i, n] 2(a_i^{(2)} - y_i)a_i^{(2)}(1 - a_i^{(2)})w^{(2)} \\
|
|
|
e_i &= a_i^{(1)} - \pd[a_i^{(1)}]C^{(2)} \\
|
|
|
- C^{(1)} &= \avgsum[i, n] (a_1^{(i)} - e_i)^2 \\
|
|
|
- \pd[w^{1}]C^{(1)}
|
|
|
- &= \pd[w^{1}]\left(\avgsum[i, n] (a_1^{(i)} - e_i)^2\right) =\\
|
|
|
- &= \avgsum[i, n] \pd[w^{1}]\left((a_1^{(i)} - e_i)^2\right) =\\
|
|
|
- &= \avgsum[i, n] 2(a_1^{(i)} - e_i)\pd[w^{1}]a_1^{(i)} =\\
|
|
|
+ C^{(1)} &= \avgsum[i, n] (a_i^{(1)} - e_i)^2 \\
|
|
|
+ \pd[w^{(1)}]C^{(1)}
|
|
|
+ &= \pd[w^{(1)}]\left(\avgsum[i, n] (a_i^{(1)} - e_i)^2\right) =\\
|
|
|
+ &= \avgsum[i, n] \pd[w^{(1)}]\left((a_i^{(1)} - e_i)^2\right) =\\
|
|
|
+ &= \avgsum[i, n] 2(a_i^{(1)} - e_i)\pd[w^{(1)}]a_i^{(1)} =\\
|
|
|
&= \avgsum[i, n] 2(\pd[a_i^{(1)}]C^{(2)})x_i \\
|
|
|
\pd[b^{1}]C^{(1)} &= \avgsum[i, n] 2(\pd[a_i^{(1)}]C^{(2)})
|
|
|
\end{align}
|
rexim