From 5c0a72801782c3302988fb95923f8cd4436f6833 Mon Sep 17 00:00:00 2001 From: julienChemillier Date: Fri, 17 Feb 2023 14:52:29 +0100 Subject: [PATCH] Modification of backpropagation_explaination.md --- doc/cnn/backpropagation_explaination.md | 19 +++++++++---------- 1 file changed, 9 insertions(+), 10 deletions(-) diff --git a/doc/cnn/backpropagation_explaination.md b/doc/cnn/backpropagation_explaination.md index cbeecc6..102ac58 100644 --- a/doc/cnn/backpropagation_explaination.md +++ b/doc/cnn/backpropagation_explaination.md @@ -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: $\dfrac{\partial E}{\partial b_{i,j}} = \dfrac{\partial E}{\partial c_{i, j}}$ $\dfrac{\partial E}{\partial k_{i,j}} = \sum\limits_{p=0}^{2} \sum\limits_{l=0}^{2} \Big( \dfrac{\partial E}{\partial c_{k \space l}} a_{i+p, j+l}\Big)$ -$\dfrac{\partial E}{\partial a_{i,j}} = \sum\limits_{k=max(0, k\_size-1)}^{min(k\_size, dim\_input-j)} \sum\limits_{l=max(0, k\ _size-1)}^{min(k\_size, dim\_input-k)} \dfrac{}{}$ +$\dfrac{\partial E}{\partial a_{i,j}} = \sum\limits_{k=max(0, k\_size-1)}^{min(k\_size, dim\_input-j)} \sum\limits_{l=max(0, k\_size-1)}^{min(k\_size, dim\_input-k)} \dfrac{}{}$