From 6cd2e8ff48462bef1d80d0f9e657967579b05b91 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 15 Jan 2024 21:27:12 +0100 Subject: [PATCH] updating slides --- doc/pub/week1/html/week1-bs.html | 187 +++---- doc/pub/week1/html/week1-reveal.html | 148 ++---- doc/pub/week1/html/week1-solarized.html | 168 ++---- doc/pub/week1/html/week1.html | 168 ++---- doc/pub/week1/ipynb/ipynb-week1-src.tar.gz | Bin 2624762 -> 2624762 bytes doc/pub/week1/ipynb/week1.ipynb | 574 ++++++++++----------- doc/pub/week1/pdf/week1.pdf | Bin 3154397 -> 3152919 bytes doc/src/week1/week1.do.txt | 122 ++--- 8 files changed, 516 insertions(+), 851 deletions(-) diff --git a/doc/pub/week1/html/week1-bs.html b/doc/pub/week1/html/week1-bs.html index fb5e53be..b3796b40 100644 --- a/doc/pub/week1/html/week1-bs.html +++ b/doc/pub/week1/html/week1-bs.html @@ -259,6 +259,10 @@ None, 'setting-up-the-equations-for-a-neural-network'), ('Definitions', 2, None, 'definitions'), + ('Inputs to tje activation function', + 2, + None, + 'inputs-to-tje-activation-function'), ('Derivatives and the chain rule', 2, None, @@ -271,6 +275,10 @@ 2, None, 'bringing-it-together-first-back-propagation-equation'), + ('Analyzing the last results', + 2, + None, + 'analyzing-the-last-results'), ('More considerations', 2, None, 'more-considerations'), ('Derivatives in terms of $z_j^L$', 2, @@ -281,11 +289,15 @@ 2, None, 'final-back-propagating-equation'), - ('Setting up the Back propagation algorithm', + ('Using the chain rule and summing over all $k$ entries', + 2, + None, + 'using-the-chain-rule-and-summing-over-all-k-entries'), + ('Setting up the back propagation algorithm', 2, None, 'setting-up-the-back-propagation-algorithm'), - ('Setting up the Back propagation algorithm, part 2', + ('Setting up the back propagation algorithm, part 2', 2, None, 'setting-up-the-back-propagation-algorithm-part-2'), @@ -293,11 +305,6 @@ 2, None, 'setting-up-the-back-propagation-algorithm-part-3'), - ('Setting up the Back propagation algorithm, final ' - 'considerations', - 2, - None, - 'setting-up-the-back-propagation-algorithm-final-considerations'), ('Updating the gradients', 2, None, 'updating-the-gradients')]} end of tocinfo --> @@ -402,17 +409,19 @@
  • Other parameters
  • Setting up the equations for a neural network
  • Definitions
    • +
  • Inputs to tje activation function
  • Derivatives and the chain rule
  • Derivative of the cost function
  • Bringing it together, first back propagation equation
    • +
  • Analyzing the last results
  • More considerations
  • Derivatives in terms of \( z_j^L \)
  • Bringing it together
  • Final back propagating equation
    • -
  • Setting up the Back propagation algorithm
    • -
  • Setting up the Back propagation algorithm, part 2
    • +
  • Using the chain rule and summing over all \( k \) entries
    • +
  • Setting up the back propagation algorithm
    • +
  • Setting up the back propagation algorithm, part 2
  • Setting up the Back propagation algorithm, part 3
    • -
  • Setting up the Back propagation algorithm, final considerations
  • Updating the gradients
    • @@ -1580,22 +1589,19 @@

    Setting up

    $$ -{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2, $$ -

    where the $t_i$s are our \( n \) targets (the values we want to +

    where the $y_i$s are our \( n \) targets (the values we want to reproduce), while the outputs of the network after having propagated -all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with \( K \) -classes. +all inputs \( \boldsymbol{x} \) are given by \( \boldsymbol{\tilde{y}}_i \).

    Definitions

    -

    With our definition of the targets \( \hat{t} \), the outputs of the -network \( \hat{y} \) and the inputs \( \hat{x} \) we +

    With our definition of the targets \( \boldsymbol{y} \), the outputs of the +network \( \boldsymbol{\tilde{y}} \) and the inputs \( \boldsymbol{x} \) we define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the \( l \)-th layer as a function of the bias, the weights which add up from the previous layer \( l-1 \) and the forward passes/outputs @@ -1616,8 +1622,12 @@

    Definitions

    \hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. $$ -

    With the activation values \( \hat{z}^l \) we can in turn define the -output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our + + +

    Inputs to tje activation function

    + +

    With the activation values \( \boldsymbol{z}^l \) we can in turn define the +output of layer \( l \) as \( \boldsymbol{a}^l = f(\boldsymbol{z}^l) \) where \( f \) is our activation function. In the examples here we will use the sigmoid function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers and their nodes. It means we have @@ -1654,18 +1664,18 @@

    Derivative of the cost f

    Let us specialize to the output layer \( l=L \). Our cost function is

    $$ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2, $$

    The derivative of this function with respect to the weights is

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +\frac{\partial{\cal C}(\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, $$

    The last partial derivative can easily be computed and reads (by applying the chain rule)

    $$ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}. $$ @@ -1674,19 +1684,23 @@

    Bri

    We have thus

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial{\cal C}((\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)a_j^L(1-a_j^L)a_k^{L-1}, $$

    Defining

    $$ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - y_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, $$

    and using the Hadamard product of two vectors we can write this as

    $$ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. +\boldsymbol{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\boldsymbol{a}^L)}. $$ + + +

    Analyzing the last results

    +

    This is an important expression. The second term on the right handside measures how fast the cost function is changing as a function of the $j$th output activation. If, for example, the cost function doesn't depend @@ -1714,7 +1728,7 @@

    More considerations

    With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. +\frac{\partial{\cal C}}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. $$ @@ -1739,10 +1753,6 @@

    Bringing it together

    We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are

    -
    -
    - - $$ \begin{equation} \frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, @@ -1766,8 +1776,6 @@

    Bringing it together

    \label{_auto3} \end{equation} $$ -
    -
    @@ -1778,8 +1786,12 @@

    Final back propagating e \delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. $$ -

    We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have

    +

    We want to express this in terms of the equations for layer \( l+1 \).

    + + +

    Using the chain rule and summing over all \( k \) entries

    +

    We obtain

    $$ \delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, $$ @@ -1799,65 +1811,43 @@

    Final back propagating e

    We are now ready to set up the algorithm for back propagation and learning the weights and biases.

    -

    Setting up the Back propagation algorithm

    +

    Setting up the back propagation algorithm

    The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    -
    -
    - -

    First, we set up the input data \( \hat{x} \) and the activations +

    First, we set up the input data \( \hat{x} \) and the activations \( \hat{z}_1 \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^1 \).

    -
    -
    - -
    -
    - -

    Secondly, we perform then the feed forward till we reach the output +

    Secondly, we perform then the feed forward till we reach the output layer and compute all \( \hat{z}_l \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^l \) for \( l=2,3,\dots,L \).

    -
    -
    - -

    Setting up the Back propagation algorithm, part 2

    +

    Setting up the back propagation algorithm, part 2

    -
    -
    -

    Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all

    $$ \delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. $$ -
    -
    - -
    -
    -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    $$ \delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). $$ -
    -

    Setting up the Back propagation algorithm, part 3

    -
    -
    - -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    Finally, we update the weights and the biases using gradient descent +for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases +according to the rules +

    + $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ @@ -1866,71 +1856,18 @@

    Setting $$ b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ -

    -
    - - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    - - -

    Setting up the Back propagation algorithm, final considerations

    - -

    The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    - -
    -
    - -

    First, we set up the input data \( \boldsymbol{x} \) and the activations -\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and -the pertinent outputs \( \boldsymbol{a}^1 \). -

    -
    -
    - - -
    -
    - -

    Secondly, we perform then the feed forward till we reach the output -layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the -activation function and the pertinent outputs \( \boldsymbol{a}^l \) for -\( l=2,3,\dots,L \). -

    -
    -
    - - -
    -
    - -

    Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all

    -$$ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -$$ -
    -
    +

    with \( \eta \) being the learning rate.

    Updating the gradients

    -
    -
    - -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    +

    With the back propagate error for each \( l=L-1,L-2,\dots,1 \) as

    $$ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), $$ -
    -
    - -
    -
    - -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases according to the rules

    $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ @@ -1939,13 +1876,7 @@

    Updating the gradients

    $$ b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ -
    -
    - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    diff --git a/doc/pub/week1/html/week1-reveal.html b/doc/pub/week1/html/week1-reveal.html index c9df4e9d..bceb433b 100644 --- a/doc/pub/week1/html/week1-reveal.html +++ b/doc/pub/week1/html/week1-reveal.html @@ -1481,24 +1481,21 @@

    Setting up the equations

     
    $$ -{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2, $$

     
    -

    where the $t_i$s are our \( n \) targets (the values we want to +

    where the $y_i$s are our \( n \) targets (the values we want to reproduce), while the outputs of the network after having propagated -all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with \( K \) -classes. +all inputs \( \boldsymbol{x} \) are given by \( \boldsymbol{\tilde{y}}_i \).

    Definitions

    -

    With our definition of the targets \( \hat{t} \), the outputs of the -network \( \hat{y} \) and the inputs \( \hat{x} \) we +

    With our definition of the targets \( \boldsymbol{y} \), the outputs of the +network \( \boldsymbol{\tilde{y}} \) and the inputs \( \boldsymbol{x} \) we define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the \( l \)-th layer as a function of the bias, the weights which add up from the previous layer \( l-1 \) and the forward passes/outputs @@ -1522,9 +1519,13 @@

    Definitions

    \hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. $$

     
    +

    + +
    +

    Inputs to tje activation function

    -

    With the activation values \( \hat{z}^l \) we can in turn define the -output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our +

    With the activation values \( \boldsymbol{z}^l \) we can in turn define the +output of layer \( l \) as \( \boldsymbol{a}^l = f(\boldsymbol{z}^l) \) where \( f \) is our activation function. In the examples here we will use the sigmoid function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers and their nodes. It means we have @@ -1570,7 +1571,7 @@

    Derivative of the cost function

    Let us specialize to the output layer \( l=L \). Our cost function is

     
    $$ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2, $$

     
    @@ -1578,14 +1579,14 @@

    Derivative of the cost function

     
    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +\frac{\partial{\cal C}(\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, $$

     

    The last partial derivative can easily be computed and reads (by applying the chain rule)

     
    $$ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}. $$

     

    @@ -1596,23 +1597,27 @@

    Bringing it togeth

    We have thus

     
    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial{\cal C}((\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)a_j^L(1-a_j^L)a_k^{L-1}, $$

     

    Defining

     
    $$ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - y_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, $$

     

    and using the Hadamard product of two vectors we can write this as

     
    $$ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. +\boldsymbol{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\boldsymbol{a}^L)}. $$

     
    + + +

    +

    Analyzing the last results

    This is an important expression. The second term on the right handside measures how fast the cost function is changing as a function of the $j$th @@ -1645,7 +1650,7 @@

    More considerations

    With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely

     
    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. +\frac{\partial{\cal C}}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. $$

     

    @@ -1676,10 +1681,6 @@

    Bringing it together

    We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are

    -
    -The starting equations -

    -

     
    $$ \begin{equation} @@ -1709,7 +1710,6 @@

    Bringing it together

    \end{equation} $$

     
    -

    @@ -1722,8 +1722,13 @@

    Final back propagating equation

    $$

     
    -

    We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have

    +

    We want to express this in terms of the equations for layer \( l+1 \).

    +
    + +
    +

    Using the chain rule and summing over all \( k \) entries

    +

    We obtain

     
    $$ \delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, @@ -1750,65 +1755,48 @@

    Final back propagating equation

    -

    Setting up the Back propagation algorithm

    +

    Setting up the back propagation algorithm

    The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    -
    - -

    -

    First, we set up the input data \( \hat{x} \) and the activations +

    First, we set up the input data \( \hat{x} \) and the activations \( \hat{z}_1 \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^1 \).

    -
    - -
    - -

    -

    Secondly, we perform then the feed forward till we reach the output +

    Secondly, we perform then the feed forward till we reach the output layer and compute all \( \hat{z}_l \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^l \) for \( l=2,3,\dots,L \).

    -
    -

    Setting up the Back propagation algorithm, part 2

    +

    Setting up the back propagation algorithm, part 2

    -
    - -

    Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all

     
    $$ \delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. $$

     
    -

    - -
    - -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

     
    $$ \delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). $$

     
    -

    Setting up the Back propagation algorithm, part 3

    -
    - -

    -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    Finally, we update the weights and the biases using gradient descent +for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases +according to the rules +

    +

     
    $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, @@ -1820,70 +1808,21 @@

    Setting up the Back pr b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$

     
    -

    - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    -
    -
    -

    Setting up the Back propagation algorithm, final considerations

    - -

    The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    - -
    - -

    -

    First, we set up the input data \( \boldsymbol{x} \) and the activations -\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and -the pertinent outputs \( \boldsymbol{a}^1 \). -

    -
    - - -
    - -

    -

    Secondly, we perform then the feed forward till we reach the output -layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the -activation function and the pertinent outputs \( \boldsymbol{a}^l \) for -\( l=2,3,\dots,L \). -

    -
    - - -
    - -

    -

    Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all

    -

     
    -$$ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -$$ -

     
    -

    +

    with \( \eta \) being the learning rate.

    Updating the gradients

    -
    - -

    -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    +

    With the back propagate error for each \( l=L-1,L-2,\dots,1 \) as

     
    $$ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), $$

     
    -

    - -
    - -

    -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases according to the rules

     
    $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, @@ -1895,11 +1834,6 @@

    Updating the gradients

    b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$

     
    -

    - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    diff --git a/doc/pub/week1/html/week1-solarized.html b/doc/pub/week1/html/week1-solarized.html index 83978612..a4f37c4f 100644 --- a/doc/pub/week1/html/week1-solarized.html +++ b/doc/pub/week1/html/week1-solarized.html @@ -286,6 +286,10 @@ None, 'setting-up-the-equations-for-a-neural-network'), ('Definitions', 2, None, 'definitions'), + ('Inputs to tje activation function', + 2, + None, + 'inputs-to-tje-activation-function'), ('Derivatives and the chain rule', 2, None, @@ -298,6 +302,10 @@ 2, None, 'bringing-it-together-first-back-propagation-equation'), + ('Analyzing the last results', + 2, + None, + 'analyzing-the-last-results'), ('More considerations', 2, None, 'more-considerations'), ('Derivatives in terms of $z_j^L$', 2, @@ -308,11 +316,15 @@ 2, None, 'final-back-propagating-equation'), - ('Setting up the Back propagation algorithm', + ('Using the chain rule and summing over all $k$ entries', + 2, + None, + 'using-the-chain-rule-and-summing-over-all-k-entries'), + ('Setting up the back propagation algorithm', 2, None, 'setting-up-the-back-propagation-algorithm'), - ('Setting up the Back propagation algorithm, part 2', + ('Setting up the back propagation algorithm, part 2', 2, None, 'setting-up-the-back-propagation-algorithm-part-2'), @@ -320,11 +332,6 @@ 2, None, 'setting-up-the-back-propagation-algorithm-part-3'), - ('Setting up the Back propagation algorithm, final ' - 'considerations', - 2, - None, - 'setting-up-the-back-propagation-algorithm-final-considerations'), ('Updating the gradients', 2, None, 'updating-the-gradients')]} end of tocinfo --> @@ -1475,22 +1482,19 @@

    Setting up the equations

    $$ -{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2, $$ -

    where the $t_i$s are our \( n \) targets (the values we want to +

    where the $y_i$s are our \( n \) targets (the values we want to reproduce), while the outputs of the network after having propagated -all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with \( K \) -classes. +all inputs \( \boldsymbol{x} \) are given by \( \boldsymbol{\tilde{y}}_i \).











    Definitions

    -

    With our definition of the targets \( \hat{t} \), the outputs of the -network \( \hat{y} \) and the inputs \( \hat{x} \) we +

    With our definition of the targets \( \boldsymbol{y} \), the outputs of the +network \( \boldsymbol{\tilde{y}} \) and the inputs \( \boldsymbol{x} \) we define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the \( l \)-th layer as a function of the bias, the weights which add up from the previous layer \( l-1 \) and the forward passes/outputs @@ -1511,8 +1515,12 @@

    Definitions

    \hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. $$ -

    With the activation values \( \hat{z}^l \) we can in turn define the -output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our + +









    +

    Inputs to tje activation function

    + +

    With the activation values \( \boldsymbol{z}^l \) we can in turn define the +output of layer \( l \) as \( \boldsymbol{a}^l = f(\boldsymbol{z}^l) \) where \( f \) is our activation function. In the examples here we will use the sigmoid function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers and their nodes. It means we have @@ -1549,18 +1557,18 @@

    Derivative of the cost function

    Let us specialize to the output layer \( l=L \). Our cost function is

    $$ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2, $$

    The derivative of this function with respect to the weights is

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +\frac{\partial{\cal C}(\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, $$

    The last partial derivative can easily be computed and reads (by applying the chain rule)

    $$ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}. $$ @@ -1569,19 +1577,23 @@

    Bringing it togeth

    We have thus

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial{\cal C}((\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)a_j^L(1-a_j^L)a_k^{L-1}, $$

    Defining

    $$ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - y_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, $$

    and using the Hadamard product of two vectors we can write this as

    $$ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. +\boldsymbol{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\boldsymbol{a}^L)}. $$ + +









    +

    Analyzing the last results

    +

    This is an important expression. The second term on the right handside measures how fast the cost function is changing as a function of the $j$th output activation. If, for example, the cost function doesn't depend @@ -1609,7 +1621,7 @@

    More considerations

    With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. +\frac{\partial{\cal C}}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. $$ @@ -1634,10 +1646,6 @@

    Bringing it together

    We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are

    -
    -The starting equations -

    - $$ \begin{equation} \frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, @@ -1661,7 +1669,6 @@

    Bringing it together

    \label{_auto3} \end{equation} $$ -










    @@ -1672,8 +1679,12 @@

    Final back propagating equation

    \delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. $$ -

    We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have

    +

    We want to express this in terms of the equations for layer \( l+1 \).

    + +









    +

    Using the chain rule and summing over all \( k \) entries

    +

    We obtain

    $$ \delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, $$ @@ -1693,61 +1704,43 @@

    Final back propagating equation

    We are now ready to set up the algorithm for back propagation and learning the weights and biases.











    -

    Setting up the Back propagation algorithm

    +

    Setting up the back propagation algorithm

    The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    -
    - -

    -

    First, we set up the input data \( \hat{x} \) and the activations +

    First, we set up the input data \( \hat{x} \) and the activations \( \hat{z}_1 \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^1 \).

    -
    - -
    - -

    -

    Secondly, we perform then the feed forward till we reach the output +

    Secondly, we perform then the feed forward till we reach the output layer and compute all \( \hat{z}_l \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^l \) for \( l=2,3,\dots,L \).

    -
    -









    -

    Setting up the Back propagation algorithm, part 2

    +

    Setting up the back propagation algorithm, part 2

    -
    - -

    Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all

    $$ \delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. $$ -
    - -
    - -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    $$ \delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). $$ -










    Setting up the Back propagation algorithm, part 3

    -
    - -

    -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    Finally, we update the weights and the biases using gradient descent +for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases +according to the rules +

    + $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ @@ -1756,66 +1749,18 @@

    Setting up the Back pr $$ b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ -

    - - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    - -









    -

    Setting up the Back propagation algorithm, final considerations

    - -

    The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    - -
    - -

    -

    First, we set up the input data \( \boldsymbol{x} \) and the activations -\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and -the pertinent outputs \( \boldsymbol{a}^1 \). -

    -
    - - -
    - -

    -

    Secondly, we perform then the feed forward till we reach the output -layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the -activation function and the pertinent outputs \( \boldsymbol{a}^l \) for -\( l=2,3,\dots,L \). -

    -
    - - -
    - -

    -

    Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all

    -$$ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -$$ -
    +

    with \( \eta \) being the learning rate.











    Updating the gradients

    -
    - -

    -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    +

    With the back propagate error for each \( l=L-1,L-2,\dots,1 \) as

    $$ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), $$ -
    - -
    - -

    -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases according to the rules

    $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ @@ -1824,12 +1769,7 @@

    Updating the gradients

    $$ b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ -
    - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week1/html/week1.html b/doc/pub/week1/html/week1.html index 7c06db08..41b60bae 100644 --- a/doc/pub/week1/html/week1.html +++ b/doc/pub/week1/html/week1.html @@ -363,6 +363,10 @@ None, 'setting-up-the-equations-for-a-neural-network'), ('Definitions', 2, None, 'definitions'), + ('Inputs to tje activation function', + 2, + None, + 'inputs-to-tje-activation-function'), ('Derivatives and the chain rule', 2, None, @@ -375,6 +379,10 @@ 2, None, 'bringing-it-together-first-back-propagation-equation'), + ('Analyzing the last results', + 2, + None, + 'analyzing-the-last-results'), ('More considerations', 2, None, 'more-considerations'), ('Derivatives in terms of $z_j^L$', 2, @@ -385,11 +393,15 @@ 2, None, 'final-back-propagating-equation'), - ('Setting up the Back propagation algorithm', + ('Using the chain rule and summing over all $k$ entries', + 2, + None, + 'using-the-chain-rule-and-summing-over-all-k-entries'), + ('Setting up the back propagation algorithm', 2, None, 'setting-up-the-back-propagation-algorithm'), - ('Setting up the Back propagation algorithm, part 2', + ('Setting up the back propagation algorithm, part 2', 2, None, 'setting-up-the-back-propagation-algorithm-part-2'), @@ -397,11 +409,6 @@ 2, None, 'setting-up-the-back-propagation-algorithm-part-3'), - ('Setting up the Back propagation algorithm, final ' - 'considerations', - 2, - None, - 'setting-up-the-back-propagation-algorithm-final-considerations'), ('Updating the gradients', 2, None, 'updating-the-gradients')]} end of tocinfo --> @@ -1552,22 +1559,19 @@

    Setting up the equations

    $$ -{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2, $$ -

    where the $t_i$s are our \( n \) targets (the values we want to +

    where the $y_i$s are our \( n \) targets (the values we want to reproduce), while the outputs of the network after having propagated -all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with \( K \) -classes. +all inputs \( \boldsymbol{x} \) are given by \( \boldsymbol{\tilde{y}}_i \).











    Definitions

    -

    With our definition of the targets \( \hat{t} \), the outputs of the -network \( \hat{y} \) and the inputs \( \hat{x} \) we +

    With our definition of the targets \( \boldsymbol{y} \), the outputs of the +network \( \boldsymbol{\tilde{y}} \) and the inputs \( \boldsymbol{x} \) we define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the \( l \)-th layer as a function of the bias, the weights which add up from the previous layer \( l-1 \) and the forward passes/outputs @@ -1588,8 +1592,12 @@

    Definitions

    \hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. $$ -

    With the activation values \( \hat{z}^l \) we can in turn define the -output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our + +









    +

    Inputs to tje activation function

    + +

    With the activation values \( \boldsymbol{z}^l \) we can in turn define the +output of layer \( l \) as \( \boldsymbol{a}^l = f(\boldsymbol{z}^l) \) where \( f \) is our activation function. In the examples here we will use the sigmoid function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers and their nodes. It means we have @@ -1626,18 +1634,18 @@

    Derivative of the cost function

    Let us specialize to the output layer \( l=L \). Our cost function is

    $$ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, +{\cal C}(\boldsymbol{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2, $$

    The derivative of this function with respect to the weights is

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +\frac{\partial{\cal C}(\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, $$

    The last partial derivative can easily be computed and reads (by applying the chain rule)

    $$ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}. $$ @@ -1646,19 +1654,23 @@

    Bringing it togeth

    We have thus

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial{\cal C}((\boldsymbol{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)a_j^L(1-a_j^L)a_k^{L-1}, $$

    Defining

    $$ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - y_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, $$

    and using the Hadamard product of two vectors we can write this as

    $$ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. +\boldsymbol{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\boldsymbol{a}^L)}. $$ + +









    +

    Analyzing the last results

    +

    This is an important expression. The second term on the right handside measures how fast the cost function is changing as a function of the $j$th output activation. If, for example, the cost function doesn't depend @@ -1686,7 +1698,7 @@

    More considerations

    With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely

    $$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. +\frac{\partial{\cal C}}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. $$ @@ -1711,10 +1723,6 @@

    Bringing it together

    We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are

    -
    -The starting equations -

    - $$ \begin{equation} \frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, @@ -1738,7 +1746,6 @@

    Bringing it together

    \label{_auto3} \end{equation} $$ -










    @@ -1749,8 +1756,12 @@

    Final back propagating equation

    \delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. $$ -

    We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have

    +

    We want to express this in terms of the equations for layer \( l+1 \).

    + +









    +

    Using the chain rule and summing over all \( k \) entries

    +

    We obtain

    $$ \delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, $$ @@ -1770,61 +1781,43 @@

    Final back propagating equation

    We are now ready to set up the algorithm for back propagation and learning the weights and biases.











    -

    Setting up the Back propagation algorithm

    +

    Setting up the back propagation algorithm

    The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    -
    - -

    -

    First, we set up the input data \( \hat{x} \) and the activations +

    First, we set up the input data \( \hat{x} \) and the activations \( \hat{z}_1 \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^1 \).

    -
    - -
    - -

    -

    Secondly, we perform then the feed forward till we reach the output +

    Secondly, we perform then the feed forward till we reach the output layer and compute all \( \hat{z}_l \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^l \) for \( l=2,3,\dots,L \).

    -
    -









    -

    Setting up the Back propagation algorithm, part 2

    +

    Setting up the back propagation algorithm, part 2

    -
    - -

    Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all

    $$ \delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. $$ -
    - -
    - -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    $$ \delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). $$ -










    Setting up the Back propagation algorithm, part 3

    -
    - -

    -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    Finally, we update the weights and the biases using gradient descent +for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases +according to the rules +

    + $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ @@ -1833,66 +1826,18 @@

    Setting up the Back pr $$ b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ -

    - - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    - -









    -

    Setting up the Back propagation algorithm, final considerations

    - -

    The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

    - -
    - -

    -

    First, we set up the input data \( \boldsymbol{x} \) and the activations -\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and -the pertinent outputs \( \boldsymbol{a}^1 \). -

    -
    - - -
    - -

    -

    Secondly, we perform then the feed forward till we reach the output -layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the -activation function and the pertinent outputs \( \boldsymbol{a}^l \) for -\( l=2,3,\dots,L \). -

    -
    - - -
    - -

    -

    Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all

    -$$ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -$$ -
    +

    with \( \eta \) being the learning rate.











    Updating the gradients

    -
    - -

    -

    Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as

    +

    With the back propagate error for each \( l=L-1,L-2,\dots,1 \) as

    $$ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), $$ -
    - -
    - -

    -

    Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules

    +

    we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,1 \) and update the weights and biases according to the rules

    $$ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, $$ @@ -1901,12 +1846,7 @@

    Updating the gradients

    $$ b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, $$ -
    - -

    The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. -

    © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week1/ipynb/ipynb-week1-src.tar.gz b/doc/pub/week1/ipynb/ipynb-week1-src.tar.gz index 88082c5e85172ab520b000227a432a3e2ef07bfd..6cd7c61433ce12c6a7102d305280df3cf050293d 100644 GIT binary patch delta 139 zcmWN_y8?j#06(Q73slkU+W;N>BRI z&Ojm=%1Fi%%S5J<$V^h1%R(|)%1YL`Q_K*HrweoA{8|06u^?TltK-$ug zuJojtP$KEeK!y^_NX9aeM5Z#6R5F>%LYA_UwQOW7JK4)Yaw+5}Cpk+gm0aZd;r5OH Dq_-hY diff --git a/doc/pub/week1/ipynb/week1.ipynb b/doc/pub/week1/ipynb/week1.ipynb index 81f9445e..74996037 100644 --- a/doc/pub/week1/ipynb/week1.ipynb +++ b/doc/pub/week1/ipynb/week1.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "4320b4a1", + "id": "fcc6f7f7", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "4e28dae4", + "id": "1e7100ab", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "3e300193", + "id": "eef90275", "metadata": { "editable": true }, @@ -43,7 +43,7 @@ }, { "cell_type": "markdown", - "id": "66a491f7", + "id": "4b59a81b", "metadata": { "editable": true }, @@ -61,7 +61,7 @@ }, { "cell_type": "markdown", - "id": "d8907311", + "id": "e33689b5", "metadata": { "editable": true }, @@ -97,7 +97,7 @@ }, { "cell_type": "markdown", - "id": "1aa2e78e", + "id": "5da650f6", "metadata": { "editable": true }, @@ -113,7 +113,7 @@ }, { "cell_type": "markdown", - "id": "0e013cd9", + "id": "d30dc02c", "metadata": { "editable": true }, @@ -136,7 +136,7 @@ }, { "cell_type": "markdown", - "id": "6829fcc9", + "id": "5b072b26", "metadata": { "editable": true }, @@ -152,7 +152,7 @@ }, { "cell_type": "markdown", - "id": "722521f9", + "id": "08decd18", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "eae20b92", + "id": "41440cf8", "metadata": { "editable": true }, @@ -186,7 +186,7 @@ }, { "cell_type": "markdown", - "id": "06cebda5", + "id": "57fa58bc", "metadata": { "editable": true }, @@ -206,7 +206,7 @@ }, { "cell_type": "markdown", - "id": "09675d7a", + "id": "faa8a182", "metadata": { "editable": true }, @@ -225,7 +225,7 @@ }, { "cell_type": "markdown", - "id": "e4b40616", + "id": "458087cf", "metadata": { "editable": true }, @@ -243,7 +243,7 @@ }, { "cell_type": "markdown", - "id": "12913b33", + "id": "c16e338d", "metadata": { "editable": true }, @@ -265,7 +265,7 @@ }, { "cell_type": "markdown", - "id": "df48c2e7", + "id": "599bb230", "metadata": { "editable": true }, @@ -288,7 +288,7 @@ }, { "cell_type": "markdown", - "id": "c5a59f0c", + "id": "0fddab7e", "metadata": { "editable": true }, @@ -304,7 +304,7 @@ }, { "cell_type": "markdown", - "id": "0a8f1137", + "id": "0e6656d4", "metadata": { "editable": true }, @@ -330,7 +330,7 @@ }, { "cell_type": "markdown", - "id": "fd888494", + "id": "4781b12f", "metadata": { "editable": true }, @@ -346,7 +346,7 @@ }, { "cell_type": "markdown", - "id": "25e974bc", + "id": "90177fea", "metadata": { "editable": true }, @@ -362,7 +362,7 @@ }, { "cell_type": "markdown", - "id": "62edc1e1", + "id": "de16686e", "metadata": { "editable": true }, @@ -382,7 +382,7 @@ }, { "cell_type": "markdown", - "id": "6a7ede58", + "id": "4ffb766a", "metadata": { "editable": true }, @@ -398,7 +398,7 @@ }, { "cell_type": "markdown", - "id": "ecaa3c0d", + "id": "f3183d87", "metadata": { "editable": true }, @@ -416,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "66a2360b", + "id": "51f13830", "metadata": { "editable": true }, @@ -432,7 +432,7 @@ }, { "cell_type": "markdown", - "id": "d8f2772d", + "id": "fbfd3a76", "metadata": { "editable": true }, @@ -444,7 +444,7 @@ }, { "cell_type": "markdown", - "id": "c4564c51", + "id": "16865eff", "metadata": { "editable": true }, @@ -462,7 +462,7 @@ }, { "cell_type": "markdown", - "id": "e10b17a7", + "id": "20995064", "metadata": { "editable": true }, @@ -474,7 +474,7 @@ }, { "cell_type": "markdown", - "id": "9dbfd14c", + "id": "2dbc3418", "metadata": { "editable": true }, @@ -486,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "c4c97e30", + "id": "8db378c8", "metadata": { "editable": true }, @@ -498,7 +498,7 @@ }, { "cell_type": "markdown", - "id": "713dc432", + "id": "d629aae3", "metadata": { "editable": true }, @@ -508,7 +508,7 @@ }, { "cell_type": "markdown", - "id": "3581a429", + "id": "8569ed0f", "metadata": { "editable": true }, @@ -520,7 +520,7 @@ }, { "cell_type": "markdown", - "id": "39f963a3", + "id": "d7afc473", "metadata": { "editable": true }, @@ -530,7 +530,7 @@ }, { "cell_type": "markdown", - "id": "6040d9f8", + "id": "25c09815", "metadata": { "editable": true }, @@ -542,7 +542,7 @@ }, { "cell_type": "markdown", - "id": "057958de", + "id": "44a7fd46", "metadata": { "editable": true }, @@ -554,7 +554,7 @@ }, { "cell_type": "markdown", - "id": "839d28b6", + "id": "f86d02ac", "metadata": { "editable": true }, @@ -564,7 +564,7 @@ }, { "cell_type": "markdown", - "id": "f2a17acd", + "id": "37912f62", "metadata": { "editable": true }, @@ -576,7 +576,7 @@ }, { "cell_type": "markdown", - "id": "3d4a645f", + "id": "f761828c", "metadata": { "editable": true }, @@ -586,7 +586,7 @@ }, { "cell_type": "markdown", - "id": "f80b8072", + "id": "d8b42487", "metadata": { "editable": true }, @@ -598,7 +598,7 @@ }, { "cell_type": "markdown", - "id": "e3b48953", + "id": "d21a6415", "metadata": { "editable": true }, @@ -610,7 +610,7 @@ }, { "cell_type": "markdown", - "id": "78067d8b", + "id": "86fbebd4", "metadata": { "editable": true }, @@ -620,7 +620,7 @@ }, { "cell_type": "markdown", - "id": "af1edd1f", + "id": "9dcb1dcb", "metadata": { "editable": true }, @@ -633,7 +633,7 @@ }, { "cell_type": "markdown", - "id": "99d1c97d", + "id": "81a077d7", "metadata": { "editable": true }, @@ -643,7 +643,7 @@ }, { "cell_type": "markdown", - "id": "17c04539", + "id": "9585b276", "metadata": { "editable": true }, @@ -655,7 +655,7 @@ }, { "cell_type": "markdown", - "id": "c831d1a3", + "id": "f7e3f8f7", "metadata": { "editable": true }, @@ -670,7 +670,7 @@ }, { "cell_type": "markdown", - "id": "c2871973", + "id": "4a165419", "metadata": { "editable": true }, @@ -683,7 +683,7 @@ }, { "cell_type": "markdown", - "id": "7f51cbbb", + "id": "b9bd529d", "metadata": { "editable": true }, @@ -695,7 +695,7 @@ }, { "cell_type": "markdown", - "id": "0429287d", + "id": "a85ad75f", "metadata": { "editable": true }, @@ -707,7 +707,7 @@ }, { "cell_type": "markdown", - "id": "f5fd3ebb", + "id": "4ca9ef86", "metadata": { "editable": true }, @@ -719,7 +719,7 @@ }, { "cell_type": "markdown", - "id": "ab377e04", + "id": "9719144a", "metadata": { "editable": true }, @@ -729,7 +729,7 @@ }, { "cell_type": "markdown", - "id": "20effe57", + "id": "c3e13391", "metadata": { "editable": true }, @@ -742,7 +742,7 @@ }, { "cell_type": "markdown", - "id": "09ec86f6", + "id": "62ea75aa", "metadata": { "editable": true }, @@ -753,7 +753,7 @@ }, { "cell_type": "markdown", - "id": "6b302097", + "id": "5a7f8000", "metadata": { "editable": true }, @@ -765,7 +765,7 @@ }, { "cell_type": "markdown", - "id": "3b41c760", + "id": "0317aa3e", "metadata": { "editable": true }, @@ -776,7 +776,7 @@ }, { "cell_type": "markdown", - "id": "b9b21938", + "id": "25ce2c4c", "metadata": { "editable": true }, @@ -790,7 +790,7 @@ }, { "cell_type": "markdown", - "id": "02b7197c", + "id": "0540e4f8", "metadata": { "editable": true }, @@ -800,7 +800,7 @@ }, { "cell_type": "markdown", - "id": "1d3fee6d", + "id": "612e87aa", "metadata": { "editable": true }, @@ -812,7 +812,7 @@ }, { "cell_type": "markdown", - "id": "e6ff02fd", + "id": "867e4f08", "metadata": { "editable": true }, @@ -822,7 +822,7 @@ }, { "cell_type": "markdown", - "id": "7cd823d8", + "id": "f26bcd09", "metadata": { "editable": true }, @@ -836,7 +836,7 @@ }, { "cell_type": "markdown", - "id": "2b40d9fc", + "id": "5b739e65", "metadata": { "editable": true }, @@ -848,7 +848,7 @@ }, { "cell_type": "markdown", - "id": "16c99897", + "id": "d1277c88", "metadata": { "editable": true }, @@ -859,7 +859,7 @@ }, { "cell_type": "markdown", - "id": "7dbe5801", + "id": "2edba55a", "metadata": { "editable": true }, @@ -873,7 +873,7 @@ }, { "cell_type": "markdown", - "id": "2021e74c", + "id": "48ad1100", "metadata": { "editable": true }, @@ -884,7 +884,7 @@ }, { "cell_type": "markdown", - "id": "ba22386d", + "id": "5b2a972a", "metadata": { "editable": true }, @@ -896,7 +896,7 @@ }, { "cell_type": "markdown", - "id": "45b474b6", + "id": "f01c985e", "metadata": { "editable": true }, @@ -906,7 +906,7 @@ }, { "cell_type": "markdown", - "id": "443dea3b", + "id": "9e189ab9", "metadata": { "editable": true }, @@ -918,7 +918,7 @@ }, { "cell_type": "markdown", - "id": "afd29081", + "id": "c0bad804", "metadata": { "editable": true }, @@ -928,7 +928,7 @@ }, { "cell_type": "markdown", - "id": "87a24bf9", + "id": "ba855e4b", "metadata": { "editable": true }, @@ -943,7 +943,7 @@ }, { "cell_type": "markdown", - "id": "91b91975", + "id": "45026dd0", "metadata": { "editable": true }, @@ -955,7 +955,7 @@ }, { "cell_type": "markdown", - "id": "0b975bac", + "id": "041d9a49", "metadata": { "editable": true }, @@ -966,7 +966,7 @@ }, { "cell_type": "markdown", - "id": "d43a8352", + "id": "f26ae58b", "metadata": { "editable": true }, @@ -978,7 +978,7 @@ }, { "cell_type": "markdown", - "id": "5513e434", + "id": "4b045765", "metadata": { "editable": true }, @@ -989,7 +989,7 @@ }, { "cell_type": "markdown", - "id": "54676b42", + "id": "2272c38f", "metadata": { "editable": true }, @@ -1001,7 +1001,7 @@ }, { "cell_type": "markdown", - "id": "ab329304", + "id": "43b8f49b", "metadata": { "editable": true }, @@ -1011,7 +1011,7 @@ }, { "cell_type": "markdown", - "id": "6dade8f1", + "id": "ebbc9e2d", "metadata": { "editable": true }, @@ -1023,7 +1023,7 @@ }, { "cell_type": "markdown", - "id": "dc5cb70d", + "id": "26f242cb", "metadata": { "editable": true }, @@ -1033,7 +1033,7 @@ }, { "cell_type": "markdown", - "id": "2e6c935f", + "id": "8b788b85", "metadata": { "editable": true }, @@ -1045,7 +1045,7 @@ }, { "cell_type": "markdown", - "id": "f4645d72", + "id": "b4729e36", "metadata": { "editable": true }, @@ -1055,7 +1055,7 @@ }, { "cell_type": "markdown", - "id": "b936614e", + "id": "bebfd109", "metadata": { "editable": true }, @@ -1072,7 +1072,7 @@ }, { "cell_type": "markdown", - "id": "927d9ce8", + "id": "ed851734", "metadata": { "editable": true }, @@ -1088,7 +1088,7 @@ }, { "cell_type": "markdown", - "id": "0b83882c", + "id": "4676d2f8", "metadata": { "editable": true }, @@ -1104,7 +1104,7 @@ }, { "cell_type": "markdown", - "id": "4b167325", + "id": "0aa50d69", "metadata": { "editable": true }, @@ -1128,7 +1128,7 @@ }, { "cell_type": "markdown", - "id": "1e5c9c72", + "id": "14fac089", "metadata": { "editable": true }, @@ -1144,7 +1144,7 @@ }, { "cell_type": "markdown", - "id": "174ffd64", + "id": "85066d35", "metadata": { "editable": true }, @@ -1159,7 +1159,7 @@ }, { "cell_type": "markdown", - "id": "a2bffedb", + "id": "1ff54312", "metadata": { "editable": true }, @@ -1183,7 +1183,7 @@ }, { "cell_type": "markdown", - "id": "416e2acf", + "id": "8fcc30c7", "metadata": { "editable": true }, @@ -1194,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "706e047f", + "id": "f8208016", "metadata": { "editable": true }, @@ -1206,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "95ebfc88", + "id": "a7333648", "metadata": { "editable": true }, @@ -1216,7 +1216,7 @@ }, { "cell_type": "markdown", - "id": "788f0513", + "id": "04b809f2", "metadata": { "editable": true }, @@ -1228,7 +1228,7 @@ }, { "cell_type": "markdown", - "id": "983ec1b3", + "id": "35c6d449", "metadata": { "editable": true }, @@ -1238,7 +1238,7 @@ }, { "cell_type": "markdown", - "id": "0478cfae", + "id": "02ed1e44", "metadata": { "editable": true }, @@ -1255,7 +1255,7 @@ }, { "cell_type": "markdown", - "id": "2f7a6f35", + "id": "2726c365", "metadata": { "editable": true }, @@ -1271,7 +1271,7 @@ }, { "cell_type": "markdown", - "id": "5e4cd821", + "id": "1a158d2b", "metadata": { "editable": true }, @@ -1283,7 +1283,7 @@ }, { "cell_type": "markdown", - "id": "01e1e34b", + "id": "92e435b2", "metadata": { "editable": true }, @@ -1293,7 +1293,7 @@ }, { "cell_type": "markdown", - "id": "130dc6ae", + "id": "5ca8cb95", "metadata": { "editable": true }, @@ -1307,7 +1307,7 @@ }, { "cell_type": "markdown", - "id": "073f6823", + "id": "f7ccba75", "metadata": { "editable": true }, @@ -1319,7 +1319,7 @@ }, { "cell_type": "markdown", - "id": "f4293d54", + "id": "9e6a39dd", "metadata": { "editable": true }, @@ -1333,7 +1333,7 @@ }, { "cell_type": "markdown", - "id": "73625652", + "id": "8bc5b83b", "metadata": { "editable": true }, @@ -1345,7 +1345,7 @@ }, { "cell_type": "markdown", - "id": "507af02f", + "id": "6cacf1ae", "metadata": { "editable": true }, @@ -1355,7 +1355,7 @@ }, { "cell_type": "markdown", - "id": "ce5a5a85", + "id": "72fa175a", "metadata": { "editable": true }, @@ -1367,7 +1367,7 @@ }, { "cell_type": "markdown", - "id": "ae408292", + "id": "e69baaf2", "metadata": { "editable": true }, @@ -1377,7 +1377,7 @@ }, { "cell_type": "markdown", - "id": "d04eb159", + "id": "2bf65fb9", "metadata": { "editable": true }, @@ -1389,7 +1389,7 @@ }, { "cell_type": "markdown", - "id": "c5cfcaf7", + "id": "167833fc", "metadata": { "editable": true }, @@ -1408,7 +1408,7 @@ }, { "cell_type": "markdown", - "id": "fe04f2ad", + "id": "54a4c7c6", "metadata": { "editable": true }, @@ -1421,7 +1421,7 @@ }, { "cell_type": "markdown", - "id": "1e32111c", + "id": "8ae2f0fc", "metadata": { "editable": true }, @@ -1431,7 +1431,7 @@ }, { "cell_type": "markdown", - "id": "6649f0b8", + "id": "9572a7da", "metadata": { "editable": true }, @@ -1443,7 +1443,7 @@ }, { "cell_type": "markdown", - "id": "ba618c2c", + "id": "7cdbb0a0", "metadata": { "editable": true }, @@ -1462,7 +1462,7 @@ }, { "cell_type": "markdown", - "id": "6dd790c2", + "id": "5ff0320b", "metadata": { "editable": true }, @@ -1484,7 +1484,7 @@ }, { "cell_type": "markdown", - "id": "f706146b", + "id": "2b30a102", "metadata": { "editable": true }, @@ -1496,7 +1496,7 @@ }, { "cell_type": "markdown", - "id": "8ebeabbc", + "id": "9259245f", "metadata": { "editable": true }, @@ -1508,7 +1508,7 @@ }, { "cell_type": "markdown", - "id": "a97aa34b", + "id": "0a01b003", "metadata": { "editable": true }, @@ -1523,7 +1523,7 @@ }, { "cell_type": "markdown", - "id": "048eb9d4", + "id": "59ed49ed", "metadata": { "editable": true }, @@ -1541,7 +1541,7 @@ }, { "cell_type": "markdown", - "id": "121bf07e", + "id": "ad9a1b18", "metadata": { "editable": true }, @@ -1561,7 +1561,7 @@ }, { "cell_type": "markdown", - "id": "5d8e9037", + "id": "0b3cd96e", "metadata": { "editable": true }, @@ -1573,7 +1573,7 @@ }, { "cell_type": "markdown", - "id": "51bc5f5b", + "id": "40fa1452", "metadata": { "editable": true }, @@ -1585,7 +1585,7 @@ }, { "cell_type": "markdown", - "id": "2d5e294d", + "id": "58692225", "metadata": { "editable": true }, @@ -1595,7 +1595,7 @@ }, { "cell_type": "markdown", - "id": "939d0a55", + "id": "fe9f809e", "metadata": { "editable": true }, @@ -1607,7 +1607,7 @@ }, { "cell_type": "markdown", - "id": "f570ac90", + "id": "439867db", "metadata": { "editable": true }, @@ -1617,7 +1617,7 @@ }, { "cell_type": "markdown", - "id": "009f4e96", + "id": "f2cb0e82", "metadata": { "editable": true }, @@ -1629,7 +1629,7 @@ }, { "cell_type": "markdown", - "id": "a8139423", + "id": "2aba0c7e", "metadata": { "editable": true }, @@ -1639,7 +1639,7 @@ }, { "cell_type": "markdown", - "id": "20390079", + "id": "1dfab70b", "metadata": { "editable": true }, @@ -1651,7 +1651,7 @@ }, { "cell_type": "markdown", - "id": "9e8fc2e5", + "id": "0ff4e530", "metadata": { "editable": true }, @@ -1665,7 +1665,7 @@ }, { "cell_type": "markdown", - "id": "7ff12a50", + "id": "4ef9710b", "metadata": { "editable": true }, @@ -1684,7 +1684,7 @@ }, { "cell_type": "markdown", - "id": "6da7044a", + "id": "b37c800f", "metadata": { "editable": true }, @@ -1694,7 +1694,7 @@ }, { "cell_type": "markdown", - "id": "9b65a19d", + "id": "a8bb360d", "metadata": { "editable": true }, @@ -1706,7 +1706,7 @@ }, { "cell_type": "markdown", - "id": "7d5c7ed6", + "id": "8364ba08", "metadata": { "editable": true }, @@ -1725,7 +1725,7 @@ }, { "cell_type": "markdown", - "id": "38810b88", + "id": "3e5040e3", "metadata": { "editable": true }, @@ -1735,7 +1735,7 @@ }, { "cell_type": "markdown", - "id": "a371bcfc", + "id": "b5096aa2", "metadata": { "editable": true }, @@ -1748,7 +1748,7 @@ }, { "cell_type": "markdown", - "id": "9f305f27", + "id": "16fdafee", "metadata": { "editable": true }, @@ -1758,7 +1758,7 @@ }, { "cell_type": "markdown", - "id": "e385a971", + "id": "10e4e6fe", "metadata": { "editable": true }, @@ -1769,7 +1769,7 @@ }, { "cell_type": "markdown", - "id": "9da2f063", + "id": "604468f7", "metadata": { "editable": true }, @@ -1781,7 +1781,7 @@ }, { "cell_type": "markdown", - "id": "073a76c0", + "id": "96f6b399", "metadata": { "editable": true }, @@ -1791,7 +1791,7 @@ }, { "cell_type": "markdown", - "id": "1fbe78bc", + "id": "43608437", "metadata": { "editable": true }, @@ -1816,7 +1816,7 @@ }, { "cell_type": "markdown", - "id": "664e154e", + "id": "40aaa76b", "metadata": { "editable": true }, @@ -1844,7 +1844,7 @@ }, { "cell_type": "markdown", - "id": "517f499c", + "id": "7658ab1c", "metadata": { "editable": true }, @@ -1870,7 +1870,7 @@ }, { "cell_type": "markdown", - "id": "b063094b", + "id": "23bb1a54", "metadata": { "editable": true }, @@ -1882,7 +1882,7 @@ }, { "cell_type": "markdown", - "id": "b009e439", + "id": "597c8f3c", "metadata": { "editable": true }, @@ -1894,7 +1894,7 @@ }, { "cell_type": "markdown", - "id": "a03f04b3", + "id": "99a7a59e", "metadata": { "editable": true }, @@ -1904,7 +1904,7 @@ }, { "cell_type": "markdown", - "id": "983fa2a5", + "id": "251b9027", "metadata": { "editable": true }, @@ -1916,7 +1916,7 @@ }, { "cell_type": "markdown", - "id": "d89f4b01", + "id": "c9b81034", "metadata": { "editable": true }, @@ -1928,7 +1928,7 @@ }, { "cell_type": "markdown", - "id": "f6493197", + "id": "8bfec184", "metadata": { "editable": true }, @@ -1940,7 +1940,7 @@ }, { "cell_type": "markdown", - "id": "7417e746", + "id": "b8d4568b", "metadata": { "editable": true }, @@ -1950,7 +1950,7 @@ }, { "cell_type": "markdown", - "id": "ae684067", + "id": "73970bf7", "metadata": { "editable": true }, @@ -1962,7 +1962,7 @@ }, { "cell_type": "markdown", - "id": "5094a5a8", + "id": "f28dea6e", "metadata": { "editable": true }, @@ -1972,7 +1972,7 @@ }, { "cell_type": "markdown", - "id": "19810edd", + "id": "ca070216", "metadata": { "editable": true }, @@ -1984,7 +1984,7 @@ }, { "cell_type": "markdown", - "id": "649d5c0c", + "id": "4f91c2de", "metadata": { "editable": true }, @@ -1996,7 +1996,7 @@ }, { "cell_type": "markdown", - "id": "1b684c8d", + "id": "8f8f3863", "metadata": { "editable": true }, @@ -2008,7 +2008,7 @@ }, { "cell_type": "markdown", - "id": "628cdc31", + "id": "9fd62926", "metadata": { "editable": true }, @@ -2018,7 +2018,7 @@ }, { "cell_type": "markdown", - "id": "9e9adf2c", + "id": "c4ad376f", "metadata": { "editable": true }, @@ -2030,7 +2030,7 @@ }, { "cell_type": "markdown", - "id": "1474846d", + "id": "4db27b21", "metadata": { "editable": true }, @@ -2041,7 +2041,7 @@ }, { "cell_type": "markdown", - "id": "1eca1e5c", + "id": "55a2b442", "metadata": { "editable": true }, @@ -2053,7 +2053,7 @@ }, { "cell_type": "markdown", - "id": "19147098", + "id": "4abbd2ad", "metadata": { "editable": true }, @@ -2063,7 +2063,7 @@ }, { "cell_type": "markdown", - "id": "67bf9119", + "id": "1ac87f54", "metadata": { "editable": true }, @@ -2075,7 +2075,7 @@ }, { "cell_type": "markdown", - "id": "604ff323", + "id": "141392b6", "metadata": { "editable": true }, @@ -2085,7 +2085,7 @@ }, { "cell_type": "markdown", - "id": "3b6c39c6", + "id": "c9cb1f95", "metadata": { "editable": true }, @@ -2097,7 +2097,7 @@ }, { "cell_type": "markdown", - "id": "077837d9", + "id": "a59c8aaf", "metadata": { "editable": true }, @@ -2118,7 +2118,7 @@ }, { "cell_type": "markdown", - "id": "02639901", + "id": "bf5b964d", "metadata": { "editable": true }, @@ -2140,7 +2140,7 @@ }, { "cell_type": "markdown", - "id": "3898692a", + "id": "d10460e6", "metadata": { "editable": true }, @@ -2167,7 +2167,7 @@ }, { "cell_type": "markdown", - "id": "91b801ce", + "id": "764b98ce", "metadata": { "editable": true }, @@ -2184,7 +2184,7 @@ }, { "cell_type": "markdown", - "id": "fee3f920", + "id": "650b2586", "metadata": { "editable": true }, @@ -2196,7 +2196,7 @@ }, { "cell_type": "markdown", - "id": "2c75a68c", + "id": "1349d83b", "metadata": { "editable": true }, @@ -2216,7 +2216,7 @@ }, { "cell_type": "markdown", - "id": "61fcad7e", + "id": "96406bf0", "metadata": { "editable": true }, @@ -2245,7 +2245,7 @@ }, { "cell_type": "markdown", - "id": "ab4441a1", + "id": "13cc0146", "metadata": { "editable": true }, @@ -2259,7 +2259,7 @@ }, { "cell_type": "markdown", - "id": "b062353c", + "id": "53b94748", "metadata": { "editable": true }, @@ -2276,42 +2276,39 @@ }, { "cell_type": "markdown", - "id": "aaef8a3d", + "id": "41d0322a", "metadata": { "editable": true }, "source": [ "$$\n", - "{\\cal C}(\\hat{W}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2,\n", + "{\\cal C}(\\boldsymbol{\\Theta}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - \\tilde{y}_i\\right)^2,\n", "$$" ] }, { "cell_type": "markdown", - "id": "0522096e", + "id": "f5c84939", "metadata": { "editable": true }, "source": [ - "where the $t_i$s are our $n$ targets (the values we want to\n", + "where the $y_i$s are our $n$ targets (the values we want to\n", "reproduce), while the outputs of the network after having propagated\n", - "all inputs $\\hat{x}$ are given by $y_i$. Below we will demonstrate\n", - "how the basic equations arising from the back propagation algorithm\n", - "can be modified in order to study classification problems with $K$\n", - "classes." + "all inputs $\\boldsymbol{x}$ are given by $\\boldsymbol{\\tilde{y}}_i$." ] }, { "cell_type": "markdown", - "id": "5c26c834", + "id": "6b755763", "metadata": { "editable": true }, "source": [ "## Definitions\n", "\n", - "With our definition of the targets $\\hat{t}$, the outputs of the\n", - "network $\\hat{y}$ and the inputs $\\hat{x}$ we\n", + "With our definition of the targets $\\boldsymbol{y}$, the outputs of the\n", + "network $\\boldsymbol{\\tilde{y}}$ and the inputs $\\boldsymbol{x}$ we\n", "define now the activation $z_j^l$ of node/neuron/unit $j$ of the\n", "$l$-th layer as a function of the bias, the weights which add up from\n", "the previous layer $l-1$ and the forward passes/outputs\n", @@ -2320,7 +2317,7 @@ }, { "cell_type": "markdown", - "id": "5acfb987", + "id": "b7ae4c93", "metadata": { "editable": true }, @@ -2332,7 +2329,7 @@ }, { "cell_type": "markdown", - "id": "a09a55ea", + "id": "95b179ce", "metadata": { "editable": true }, @@ -2345,7 +2342,7 @@ }, { "cell_type": "markdown", - "id": "b2e5b321", + "id": "7020d8b0", "metadata": { "editable": true }, @@ -2357,13 +2354,15 @@ }, { "cell_type": "markdown", - "id": "d72fef8d", + "id": "8366fb38", "metadata": { "editable": true }, "source": [ - "With the activation values $\\hat{z}^l$ we can in turn define the\n", - "output of layer $l$ as $\\hat{a}^l = f(\\hat{z}^l)$ where $f$ is our\n", + "## Inputs to tje activation function\n", + "\n", + "With the activation values $\\boldsymbol{z}^l$ we can in turn define the\n", + "output of layer $l$ as $\\boldsymbol{a}^l = f(\\boldsymbol{z}^l)$ where $f$ is our\n", "activation function. In the examples here we will use the sigmoid\n", "function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers\n", "and their nodes. It means we have" @@ -2371,7 +2370,7 @@ }, { "cell_type": "markdown", - "id": "985fa159", + "id": "d740a7bf", "metadata": { "editable": true }, @@ -2383,7 +2382,7 @@ }, { "cell_type": "markdown", - "id": "f604e895", + "id": "1d2ebbd5", "metadata": { "editable": true }, @@ -2395,7 +2394,7 @@ }, { "cell_type": "markdown", - "id": "30552830", + "id": "ec6341d5", "metadata": { "editable": true }, @@ -2407,7 +2406,7 @@ }, { "cell_type": "markdown", - "id": "dedb85bb", + "id": "b6bdbe01", "metadata": { "editable": true }, @@ -2417,7 +2416,7 @@ }, { "cell_type": "markdown", - "id": "e2b0f10a", + "id": "cd92ba61", "metadata": { "editable": true }, @@ -2429,7 +2428,7 @@ }, { "cell_type": "markdown", - "id": "90b643f0", + "id": "0f4860e8", "metadata": { "editable": true }, @@ -2439,7 +2438,7 @@ }, { "cell_type": "markdown", - "id": "4f4b11a3", + "id": "4c3a18b3", "metadata": { "editable": true }, @@ -2451,7 +2450,7 @@ }, { "cell_type": "markdown", - "id": "02ec7cbb", + "id": "4ece12a5", "metadata": { "editable": true }, @@ -2465,19 +2464,19 @@ }, { "cell_type": "markdown", - "id": "2078e934", + "id": "4104e482", "metadata": { "editable": true }, "source": [ "$$\n", - "{\\cal C}(\\hat{W^L}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - t_i\\right)^2,\n", + "{\\cal C}(\\boldsymbol{\\Theta}^L) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - \\tilde{y}_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - y_i\\right)^2,\n", "$$" ] }, { "cell_type": "markdown", - "id": "5d6fe1a6", + "id": "4fa9192d", "metadata": { "editable": true }, @@ -2487,19 +2486,19 @@ }, { "cell_type": "markdown", - "id": "b1bfd605", + "id": "c3d0038b", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n", + "\\frac{\\partial{\\cal C}(\\boldsymbol{\\Theta}^L)}{\\partial w_{jk}^L} = \\left(a_j^L - y_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n", "$$" ] }, { "cell_type": "markdown", - "id": "b03519e9", + "id": "c2f5dcfa", "metadata": { "editable": true }, @@ -2509,19 +2508,19 @@ }, { "cell_type": "markdown", - "id": "5348b212", + "id": "e49090ce", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},\n", + "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "7c3fd36c", + "id": "344365f7", "metadata": { "editable": true }, @@ -2533,19 +2532,19 @@ }, { "cell_type": "markdown", - "id": "a86efd84", + "id": "9145df31", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", + "\\frac{\\partial{\\cal C}((\\boldsymbol{\\Theta}^L)}{\\partial w_{jk}^L} = \\left(a_j^L - y_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", "$$" ] }, { "cell_type": "markdown", - "id": "6039061e", + "id": "7af5f61c", "metadata": { "editable": true }, @@ -2555,19 +2554,19 @@ }, { "cell_type": "markdown", - "id": "3cd929b7", + "id": "908bec69", "metadata": { "editable": true }, "source": [ "$$\n", - "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - t_j\\right) = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", + "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - y_j\\right) = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", "$$" ] }, { "cell_type": "markdown", - "id": "19bc6c08", + "id": "0277e537", "metadata": { "editable": true }, @@ -2577,23 +2576,25 @@ }, { "cell_type": "markdown", - "id": "567909ed", + "id": "5f5eb1a8", "metadata": { "editable": true }, "source": [ "$$\n", - "\\hat{\\delta}^L = f'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\hat{a}^L)}.\n", + "\\boldsymbol{\\delta}^L = f'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\boldsymbol{a}^L)}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "4af79549", + "id": "b3cfe16b", "metadata": { "editable": true }, "source": [ + "## Analyzing the last results\n", + "\n", "This is an important expression. The second term on the right handside\n", "measures how fast the cost function is changing as a function of the $j$th\n", "output activation. If, for example, the cost function doesn't depend\n", @@ -2605,7 +2606,7 @@ }, { "cell_type": "markdown", - "id": "c6cfb413", + "id": "1b750f1d", "metadata": { "editable": true }, @@ -2623,7 +2624,7 @@ }, { "cell_type": "markdown", - "id": "0edb4ed3", + "id": "52df8c1e", "metadata": { "editable": true }, @@ -2635,7 +2636,7 @@ }, { "cell_type": "markdown", - "id": "ff4a9c6e", + "id": "dc70df7c", "metadata": { "editable": true }, @@ -2645,19 +2646,19 @@ }, { "cell_type": "markdown", - "id": "bbf3a975", + "id": "94070518", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", + "\\frac{\\partial{\\cal C}}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "bba3716b", + "id": "a8006c9c", "metadata": { "editable": true }, @@ -2669,7 +2670,7 @@ }, { "cell_type": "markdown", - "id": "b973d14f", + "id": "fb90872d", "metadata": { "editable": true }, @@ -2681,7 +2682,7 @@ }, { "cell_type": "markdown", - "id": "c08afe8e", + "id": "048e84fd", "metadata": { "editable": true }, @@ -2691,7 +2692,7 @@ }, { "cell_type": "markdown", - "id": "198a2d36", + "id": "cdd867fd", "metadata": { "editable": true }, @@ -2703,7 +2704,7 @@ }, { "cell_type": "markdown", - "id": "8d58a7ea", + "id": "88415bff", "metadata": { "editable": true }, @@ -2713,21 +2714,19 @@ }, { "cell_type": "markdown", - "id": "ce9c4ffb", + "id": "e748189f", "metadata": { "editable": true }, "source": [ "## Bringing it together\n", "\n", - "We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are\n", - "\n", - "**The starting equations.**" + "We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are" ] }, { "cell_type": "markdown", - "id": "6633f82a", + "id": "7a40af52", "metadata": { "editable": true }, @@ -2745,7 +2744,7 @@ }, { "cell_type": "markdown", - "id": "ee2956f9", + "id": "8cf04e16", "metadata": { "editable": true }, @@ -2755,7 +2754,7 @@ }, { "cell_type": "markdown", - "id": "888a5c5e", + "id": "e1c592fd", "metadata": { "editable": true }, @@ -2773,7 +2772,7 @@ }, { "cell_type": "markdown", - "id": "770ca792", + "id": "e5f442c2", "metadata": { "editable": true }, @@ -2783,7 +2782,7 @@ }, { "cell_type": "markdown", - "id": "6450f307", + "id": "5c5b0b75", "metadata": { "editable": true }, @@ -2801,7 +2800,7 @@ }, { "cell_type": "markdown", - "id": "0b0a6b2f", + "id": "2a46a89c", "metadata": { "editable": true }, @@ -2813,7 +2812,7 @@ }, { "cell_type": "markdown", - "id": "3c82c708", + "id": "1e8f5355", "metadata": { "editable": true }, @@ -2825,17 +2824,29 @@ }, { "cell_type": "markdown", - "id": "7859a3cf", + "id": "092c449a", + "metadata": { + "editable": true + }, + "source": [ + "We want to express this in terms of the equations for layer $l+1$." + ] + }, + { + "cell_type": "markdown", + "id": "cfc2e20b", "metadata": { "editable": true }, "source": [ - "We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have" + "## Using the chain rule and summing over all $k$ entries\n", + "\n", + "We obtain" ] }, { "cell_type": "markdown", - "id": "ce1ecc95", + "id": "6538d6f5", "metadata": { "editable": true }, @@ -2847,7 +2858,7 @@ }, { "cell_type": "markdown", - "id": "08817904", + "id": "e8e0f5dd", "metadata": { "editable": true }, @@ -2857,7 +2868,7 @@ }, { "cell_type": "markdown", - "id": "5a18714a", + "id": "11160d18", "metadata": { "editable": true }, @@ -2869,7 +2880,7 @@ }, { "cell_type": "markdown", - "id": "5bcb264f", + "id": "b30fdbce", "metadata": { "editable": true }, @@ -2879,7 +2890,7 @@ }, { "cell_type": "markdown", - "id": "a1ab6d71", + "id": "1d6a73e3", "metadata": { "editable": true }, @@ -2891,7 +2902,7 @@ }, { "cell_type": "markdown", - "id": "74ac928f", + "id": "ddbfba54", "metadata": { "editable": true }, @@ -2903,20 +2914,20 @@ }, { "cell_type": "markdown", - "id": "c1d9a800", + "id": "40e6bbd9", "metadata": { "editable": true }, "source": [ - "## Setting up the Back propagation algorithm\n", + "## Setting up the back propagation algorithm\n", "\n", "The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n", "\n", - "First, we set up the input data $\\hat{x}$ and the activations\n", + "**First**, we set up the input data $\\hat{x}$ and the activations\n", "$\\hat{z}_1$ of the input layer and compute the activation function and\n", "the pertinent outputs $\\hat{a}^1$.\n", "\n", - "Secondly, we perform then the feed forward till we reach the output\n", + "**Secondly**, we perform then the feed forward till we reach the output\n", "layer and compute all $\\hat{z}_l$ of the input layer and compute the\n", "activation function and the pertinent outputs $\\hat{a}^l$ for\n", "$l=2,3,\\dots,L$." @@ -2924,19 +2935,19 @@ }, { "cell_type": "markdown", - "id": "dd14afca", + "id": "c169304e", "metadata": { "editable": true }, "source": [ - "## Setting up the Back propagation algorithm, part 2\n", + "## Setting up the back propagation algorithm, part 2\n", "\n", "Thereafter we compute the ouput error $\\hat{\\delta}^L$ by computing all" ] }, { "cell_type": "markdown", - "id": "f5753db9", + "id": "12ee2fcb", "metadata": { "editable": true }, @@ -2948,7 +2959,7 @@ }, { "cell_type": "markdown", - "id": "ad5f3e2c", + "id": "ca5a7e18", "metadata": { "editable": true }, @@ -2958,7 +2969,7 @@ }, { "cell_type": "markdown", - "id": "dcfa414d", + "id": "47aee3e4", "metadata": { "editable": true }, @@ -2970,19 +2981,21 @@ }, { "cell_type": "markdown", - "id": "a16b5fcf", + "id": "d56cb6ea", "metadata": { "editable": true }, "source": [ "## Setting up the Back propagation algorithm, part 3\n", "\n", - "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules" + "Finally, we update the weights and the biases using gradient descent\n", + "for each $l=L-1,L-2,\\dots,1$ and update the weights and biases\n", + "according to the rules" ] }, { "cell_type": "markdown", - "id": "2ca1a8ee", + "id": "8849613b", "metadata": { "editable": true }, @@ -2994,7 +3007,7 @@ }, { "cell_type": "markdown", - "id": "fe92be6a", + "id": "a1b74037", "metadata": { "editable": true }, @@ -3006,87 +3019,51 @@ }, { "cell_type": "markdown", - "id": "2a6e532a", + "id": "900569e4", "metadata": { "editable": true }, "source": [ - "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n", - "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training." + "with $\\eta$ being the learning rate." ] }, { "cell_type": "markdown", - "id": "4fa8cee9", - "metadata": { - "editable": true - }, - "source": [ - "## Setting up the Back propagation algorithm, final considerations\n", - "\n", - "The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n", - "\n", - "First, we set up the input data $\\boldsymbol{x}$ and the activations\n", - "$\\boldsymbol{z}_1$ of the input layer and compute the activation function and\n", - "the pertinent outputs $\\boldsymbol{a}^1$.\n", - "\n", - "Secondly, we perform then the feed forward till we reach the output\n", - "layer and compute all $\\boldsymbol{z}_l$ of the input layer and compute the\n", - "activation function and the pertinent outputs $\\boldsymbol{a}^l$ for\n", - "$l=2,3,\\dots,L$.\n", - "\n", - "Thereafter we compute the ouput error $\\boldsymbol{\\delta}^L$ by computing all" - ] - }, - { - "cell_type": "markdown", - "id": "203c265b", - "metadata": { - "editable": true - }, - "source": [ - "$$\n", - "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "2fb48f44", + "id": "26dac645", "metadata": { "editable": true }, "source": [ "## Updating the gradients\n", "\n", - "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,2$ as" + "With the back propagate error for each $l=L-1,L-2,\\dots,1$ as" ] }, { "cell_type": "markdown", - "id": "bc3bc72b", + "id": "53c4e37d", "metadata": { "editable": true }, "source": [ "$$\n", - "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).\n", + "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),\n", "$$" ] }, { "cell_type": "markdown", - "id": "ea6c8276", + "id": "dce1efb3", "metadata": { "editable": true }, "source": [ - "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules" + "we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,1$ and update the weights and biases according to the rules" ] }, { "cell_type": "markdown", - "id": "65221b07", + "id": "f7fd0c67", "metadata": { "editable": true }, @@ -3098,7 +3075,7 @@ }, { "cell_type": "markdown", - "id": "01434784", + "id": "1ee56f72", "metadata": { "editable": true }, @@ -3107,17 +3084,6 @@ "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", "$$" ] - }, - { - "cell_type": "markdown", - "id": "61c6bb9f", - "metadata": { - "editable": true - }, - "source": [ - "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n", - "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training." - ] } ], "metadata": {}, diff --git a/doc/pub/week1/pdf/week1.pdf b/doc/pub/week1/pdf/week1.pdf index a1cfaa57ccdb887a723713927acb43cb1b43a839..79d8103e4358964f1a47f8d3c2d24da8cee2116f 100644 GIT binary patch delta 78473 zcmXuKV|-n0+dLerv2EM7ZKJVmX9qjB&8D$!JB^(kqe)|{!Q1P(|L>vw)#Gshfr z&e|Rx0MG0g01pO&10jHrKqw$I5C#Yfgag6@5rE%-h(IJDG7trb3Pb~<12KS@KrA3O 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the $y_i$s are our $n$ targets (the values we want to reproduce), while the outputs of the network after having propagated -all inputs $\hat{x}$ are given by $y_i$. Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with $K$ -classes. +all inputs $\bm{x}$ are given by $\bm{\tilde{y}}_i$. !split ===== Definitions ===== -With our definition of the targets $\hat{t}$, the outputs of the -network $\hat{y}$ and the inputs $\hat{x}$ we +With our definition of the targets $\bm{y}$, the outputs of the +network $\bm{\tilde{y}}$ and the inputs $\bm{x}$ we define now the activation $z_j^l$ of node/neuron/unit $j$ of the $l$-th layer as a function of the bias, the weights which add up from the previous layer $l-1$ and the forward passes/outputs @@ -1131,8 +1128,11 @@ compact form as the matrix-vector products we discussed earlier, \] !et -With the activation values $\hat{z}^l$ we can in turn define the -output of layer $l$ as $\hat{a}^l = f(\hat{z}^l)$ where $f$ is our +!split +===== Inputs to tje activation function ===== + +With the activation values $\bm{z}^l$ we can in turn define the +output of layer $l$ as $\bm{a}^l = f(\bm{z}^l)$ where $f$ is our activation function. In the examples here we will use the sigmoid function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers and their nodes. It means we have @@ -1176,20 +1176,20 @@ With these definitions we can now compute the derivative of the cost function in Let us specialize to the output layer $l=L$. Our cost function is !bt \[ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, +{\cal C}(\bm{\Theta}^L) = \frac{1}{2}\sum_{i=1}^n\left(y_i - \tilde{y}_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - y_i\right)^2, \] !et The derivative of this function with respect to the weights is !bt \[ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, +\frac{\partial{\cal C}(\bm{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, \] !et The last partial derivative can easily be computed and reads (by applying the chain rule) !bt \[ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}. \] !et @@ -1201,23 +1201,26 @@ The last partial derivative can easily be computed and reads (by applying the ch We have thus !bt \[ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, +\frac{\partial{\cal C}((\bm{\Theta}^L)}{\partial w_{jk}^L} = \left(a_j^L - y_j\right)a_j^L(1-a_j^L)a_k^{L-1}, \] !et Defining !bt \[ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, +\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - y_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, \] !et and using the Hadamard product of two vectors we can write this as !bt \[ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. +\bm{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\bm{a}^L)}. \] !et +!split +===== Analyzing the last results ===== + This is an important expression. The second term on the right handside measures how fast the cost function is changing as a function of the $j$th output activation. If, for example, the cost function doesn't depend @@ -1247,7 +1250,7 @@ trouble in calculating With the definition of $\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely !bt \[ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. +\frac{\partial{\cal C}}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. \] !et @@ -1274,8 +1277,6 @@ That is, the error $\delta_j^L$ is exactly equal to the rate of change of the co We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are -!bblock The starting equations - !bt \begin{equation} \frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, @@ -1294,8 +1295,6 @@ and \delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}, \end{equation} !et -!eblock - !split @@ -1307,8 +1306,12 @@ We have that (replacing $L$ with a general layer $l$) \delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. \] !et -We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have +We want to express this in terms of the equations for layer $l+1$. +!split +===== Using the chain rule and summing over all $k$ entries ===== + +We obtain !bt \[ \delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, @@ -1331,53 +1334,46 @@ This is our final equation. We are now ready to set up the algorithm for back propagation and learning the weights and biases. !split -===== Setting up the Back propagation algorithm ===== - - +===== Setting up the back propagation algorithm ===== The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm. -!bblock -First, we set up the input data $\hat{x}$ and the activations +_First_, we set up the input data $\hat{x}$ and the activations $\hat{z}_1$ of the input layer and compute the activation function and the pertinent outputs $\hat{a}^1$. -!eblock -!bblock -Secondly, we perform then the feed forward till we reach the output +_Secondly_, we perform then the feed forward till we reach the output layer and compute all $\hat{z}_l$ of the input layer and compute the activation function and the pertinent outputs $\hat{a}^l$ for $l=2,3,\dots,L$. -!eblock + !split -===== Setting up the Back propagation algorithm, part 2 ===== +===== Setting up the back propagation algorithm, part 2 ===== -!bblock Thereafter we compute the ouput error $\hat{\delta}^L$ by computing all !bt \[ \delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. \] !et -!eblock -!bblock Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as !bt \[ \delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). \] !et -!eblock !split ===== Setting up the Back propagation algorithm, part 3 ===== -!bblock -Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules +Finally, we update the weights and the biases using gradient descent +for each $l=L-1,L-2,\dots,1$ and update the weights and biases +according to the rules + !bt \[ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, @@ -1389,56 +1385,18 @@ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, \] !et -!eblock - -The parameter $\eta$ is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. - - -!split -===== Setting up the Back propagation algorithm, final considerations ===== - - - -The four equations above provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm. - -!bblock -First, we set up the input data $\bm{x}$ and the activations -$\bm{z}_1$ of the input layer and compute the activation function and -the pertinent outputs $\bm{a}^1$. -!eblock - -!bblock -Secondly, we perform then the feed forward till we reach the output -layer and compute all $\bm{z}_l$ of the input layer and compute the -activation function and the pertinent outputs $\bm{a}^l$ for -$l=2,3,\dots,L$. -!eblock - -!bblock -Thereafter we compute the ouput error $\bm{\delta}^L$ by computing all -!bt -\[ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -\] -!et -!eblock +with $\eta$ being the learning rate. !split ===== Updating the gradients ===== - -!bblock -Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as +With the back propagate error for each $l=L-1,L-2,\dots,1$ as !bt \[ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), \] !et -!eblock - -!bblock -Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules +we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,1$ and update the weights and biases according to the rules !bt \[ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, @@ -1450,8 +1408,4 @@ w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, \] !et -!eblock - -The parameter $\eta$ is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.