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Modification of backpropagation_explaination.md

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julienChemillier 2023-02-17 14:52:29 +01:00
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doc/cnn/backpropagation_explaination.md
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@ -12,11 +12,9 @@ Relations entre les différents variables:
$l_2 = \dfrac{e^{a_2}}{e^{a_1}+e^{a_2}+e^{a_3}}$
$l_3 = \dfrac{e^{a_3}}{e^{a_1}+e^{a_2}+e^{a_3}}$
$E = \dfrac{1}{2}((l_1-o_1)^2+(l_2-o_2)^2+(l_3-o_3)^2)$
- $\dfrac{\partial E}{\partial l1} = o_1 - l_1$
$\dfrac{\partial E}{\partial l2} = o_2 - l_2$
$\dfrac{\partial E}{\partial l3} = o_3 - l_3$
- $\dfrac{\partial l_1}{\partial a_1} = l_1(1-l_1)$
$\dfrac{\partial E}{\partial a_1} = \dfrac{\partial E}{\partial l_1} \dfrac{\partial l_1}{\partial a_1} = (o_1-l_1)l_1(1-l_1)$
- $\forall i$, $\dfrac{\partial E}{\partial li} = o_i - l_i$
- $\forall i$, $\dfrac{\partial l_i}{\partial a_i} = l_i(1-l_i)$
$\dfrac{\partial E}{\partial a_i} = \dfrac{\partial E}{\partial l_i} \dfrac{\partial l_i}{\partial a_i}$
> Derivatives:
$\dfrac{\partial E}{\partial a_i} = \dfrac{\partial E}{\partial l_i} \dfrac{\partial l_i}{\partial a_i} = (o_i-l_i)l_1(1-l_i)$
@ -31,21 +29,21 @@ $\dfrac{\partial E}{\partial b_i} = \dfrac{\partial E}{\partial a_i}$
Soit f la fonction d'activation de la première couche (qui transforme les $a_i$ en $l_i$) et g la fonction d'activation de la deuxième couche (qui transforme les $c_i$ en $d_i$).
- $d_1 =g(c_1)$
$d_2 = g(c2)$
$d_2 = g(c_2)$
$c_1 = w_{11}l_1 + w_{21}l_2 + w_{31}l_3 + b'_1$
$c_2 = w_{12}l_1 + w_{22}l_2 + w_{32}l_3 + b'_2$
$c_2 = w_{12}l_1 + w_{22}l_2 + w_{32}l_3 + b'_2$
$l_1 = f(a_1)$
$l_2 = f(a_2)$
$l_3 = f(a_3)$
- $\dfrac{\partial E}{\partial a_1} = \dfrac{\partial E_{c_1}}{\partial c_1} \dfrac{\partial c_1}{\partial l_1} \dfrac{\partial l_1}{\partial a_1} + \dfrac{\partial E_{c_2}}{\partial c_2} \dfrac{\partial c_2}{\partial l_1} \dfrac{\partial l_1}{\partial a_1}$
$\dfrac{\partial c_2}{\partial l_1} = w_{12}$
$\dfrac{\partial c_1}{\partial l_1} = w_{11}$
$\dfrac{\partial l_1}{\partial a_1} = f'(a_1)$
> Dérivées:
$\dfrac{\partial E}{\partial b_j} = \dfrac{\partial E}{\partial l_i} $
$\dfrac{\partial E}{\partial w_{ij}} = \dfrac{\partial E}{\partial c_j}l_i$
$\dfrac{\partial E}{\partial a_i} = \dfrac{\partial E_{c_1}}{\partial c_1} w_{i1} + \dfrac{\partial E_{c_2}}{\partial c_2} w_{i2}$
$\dfrac{\partial E}{\partial a_i} = (\dfrac{\partial E_{c_1}}{\partial c_1} w_{i1} + \dfrac{\partial E_{c_2}}{\partial c_2} w_{i2}$ )$f'(a_i)$
---
---
@ -59,6 +57,7 @@ $\forall i,j: \space b_{i j} = \dfrac{a_{2i \space 2j} + a_{2i+1 \space 2j} + a_
> Dérivées:
$\forall i,j: \space \dfrac{\partial E}{\partial a_{i \space j}} = \dfrac{1}{4} \dfrac{\partial E}{\partial b_{k \space l}} $
où $k = \Big\lfloor \dfrac{i}{2} \Big\rfloor$ et $l = \Big\lfloor \dfrac{j}{2} \Big\rfloor$
---
---
@ -75,4 +74,4 @@ $ $
> Dérivées:
$\dfrac{\partial E}{\partial b_{i,j}} = \dfrac{\partial E}{\partial c_{i, j}}$
$\dfrac{\partial E}{\partial k_{i,j}} = \sum\limits_{p=0}^{2} \sum\limits_{l=0}^{2} \Big( \dfrac{\partial E}{\partial c_{k \space l}} a_{i+p, j+l}\Big)$
$\dfrac{\partial E}{\partial a_{i,j}} = \sum\limits_{k=max(0, k\_size-1)}^{min(k\_size, dim\_input-j)} \sum\limits_{l=max(0, k\ _size-1)}^{min(k\_size, dim\_input-k)} \dfrac{}{}$
$\dfrac{\partial E}{\partial a_{i,j}} = \sum\limits_{k=max(0, k\_size-1)}^{min(k\_size, dim\_input-j)} \sum\limits_{l=max(0, k\_size-1)}^{min(k\_size, dim\_input-k)} \dfrac{}{}$