From a0aa01c2b0d540181f92bef893f4d44b750a39e9 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 2 Feb 2024 05:48:49 +0100 Subject: [PATCH] update --- doc/pub/week3/html/week3-bs.html | 134 +- doc/pub/week3/html/week3-reveal.html | 126 +- doc/pub/week3/html/week3-solarized.html | 126 +- doc/pub/week3/html/week3.html | 126 +- doc/pub/week3/ipynb/ipynb-week3-src.tar.gz | Bin 51145 -> 51145 bytes doc/pub/week3/ipynb/week3.ipynb | 820 +-- doc/pub/week3/pdf/week3.pdf | Bin 529475 -> 535846 bytes .../.ipynb_checkpoints/week3-checkpoint.ipynb | 4405 +++++++++++++++++ doc/src/week3/figures/simple.pdf | Bin 0 -> 1814058 bytes doc/src/week3/week3.do.txt | 47 +- 10 files changed, 5461 insertions(+), 323 deletions(-) create mode 100644 doc/src/week3/.ipynb_checkpoints/week3-checkpoint.ipynb create mode 100644 doc/src/week3/figures/simple.pdf diff --git a/doc/pub/week3/html/week3-bs.html b/doc/pub/week3/html/week3-bs.html index b84230e2..6fc30599 100644 --- a/doc/pub/week3/html/week3-bs.html +++ b/doc/pub/week3/html/week3-bs.html @@ -153,6 +153,10 @@ None, 'new-expression-for-the-derivative'), ('Final derivatives', 2, None, 'final-derivatives'), + ('In general not this simple', + 2, + None, + 'in-general-not-this-simple'), ('Automatic differentiation', 2, None, @@ -170,6 +174,22 @@ ('The derivatives', 2, None, 'the-derivatives'), ('Important observations', 2, None, 'important-observations'), ('The training', 2, None, 'the-training'), + ('Code examples for the simple models', + 2, + None, + 'code-examples-for-the-simple-models'), + ('Simple neural network and the back propagation equations', + 2, + None, + 'simple-neural-network-and-the-back-propagation-equations'), + ('The ouput layer', 2, None, 'the-ouput-layer'), + ('Compact expressions', 2, None, 'compact-expressions'), + ('For the output layer', 2, None, 'for-the-output-layer'), + ('Explicit derivatives', 2, None, 'explicit-derivatives'), + ('Setting up the equations for the optimization', + 2, + None, + 'setting-up-the-equations-for-the-optimization'), ('Getting serious, the back propagation equations for a neural ' 'network', 2, @@ -358,6 +378,7 @@
  • Defining intermediate operations
  • New expression for the derivative
  • Final derivatives
    • +
  • In general not this simple
  • Automatic differentiation
  • Chain rule
  • First network example, simple percepetron with one input
    • @@ -366,6 +387,13 @@
  • The derivatives
  • Important observations
  • The training
    • +
  • Code examples for the simple models
    • +
  • Simple neural network and the back propagation equations
    • +
  • The ouput layer
    • +
  • Compact expressions
    • +
  • For the output layer
    • +
  • Explicit derivatives
    • +
  • Setting up the equations for the optimization
  • Getting serious, the back propagation equations for a neural network
  • Analyzing the last results
  • More considerations
    • @@ -1141,12 +1169,21 @@

    Final derivatives

    and requires only three operations if we can reuse all intermediate variables.

    + +

    In general not this simple

    + +

    In general, see the generalization below, unless we can obtain simple +analytical expressions which we can simplify further, the final +implementation of automatic differentiation involves repeated +calculations (and thereby operations) of derivatives of elementary +functions. +

    +

    Automatic differentiation

    We can make this example more formal. Automatic differentiation is a -formalization of the previous example (see graph from whiteboard -notes). +formalization of the previous example (see graph).

    We define \( \boldsymbol{x}\in x_1,\dots, x_l \) input variables to a given function \( f(\boldsymbol{x}) \) and \( x_{l+1},\dots, x_L \) intermediate variables.

    @@ -1188,7 +1225,12 @@

    Chain rule

    First network example, simple percepetron with one input

    -

    As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable \( x \) and one target value \( y \). We define an activation function \( \sigma_1 \) which takes as input

    +

    As yet another example we define now a simple perceptron model with +all quantities given by scalars. We consider only one input variable +\( x \) and one target value \( y \). We define an activation function +\( \sigma_1 \) which takes as input +

    + $$ z_1 = w_1x+b_1, $$ @@ -1237,7 +1279,7 @@

    Optimizing the parameters

    Adding a hidden layer

    -

    We change our simple model to a (see graph on whiteboard notes) +

    We change our simple model to (see graph) network with just one hidden layer but with scalar variables only.

    @@ -1304,7 +1346,7 @@

    The training

    The training of the parameters is done through various gradient descent approximations with

    $$ -w_{i}\leftarrow = w_{i}- \eta \delta_i a_{i-1}, +w_{i}\leftarrow w_{i}- \eta \delta_i a_{i-1}, $$

    and

    @@ -1318,6 +1360,88 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.

    + +

    Code examples for the simple models

    + + +

    Simple neural network and the back propagation equations

    + +

    Let us now try to increase our level of ambitions and attempt to set +up the equations for a neural network with two input nodes, one hidden +layer with two hidden nodes and one utput layers. +

    + +

    We need to define the following parameters and variables with the input layer (layer \( (0) \)) +where we label the nodes \( x_0 \) and \( x_1 \) +

    +$$ +x_0 = a_0^{(0)} \wedge x_1 = a_1^{(0)}. +$$ + +

    The hidden layer (layer \( (1) \)) has nodes which yield the outputs \( a_0^{(1)} \) and \( a_1^{(1)} \)) with weight \( \boldsymbol{w} \) and bias \( \boldsymbol{b} \) parameters

    +$$ +w_{ij}^{(1)}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\right\} \wedge b^{(1)}=\left\{b_0^{(1)},b_1^{(1)}\right\}. +$$ + + + +

    The ouput layer

    + +

    Finally, we have the ouput layer given by layer label \( (2) \) with output \( a^{(2)} \) and weights and biases to be determined

    +$$ +w_{i}^{(2)}=\left\{w_{0}^{(2)},w_{1}^{(2)}\right\} \wedge b^{(2)}. +$$ + +

    Our output is \( \tilde{y}=a^{(2)} \) and we define a generic cost function \( C(a^{(2)},y;\boldsymbol{\Theta}) \) where \( y \) is the target value (a scalar here). +The parameters we need to optimize are given by +

    +$$ +\boldsymbol{\Theta}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\right\}. +$$ + + + +

    Compact expressions

    + +

    We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions. +The inputs to the first hidden layer are +

    +$$ +\begin{bmatrix}z_0^{(1)} \\ z_1^{(1)} \end{bmatrix}=\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\ w_{10}^{(1)} &w_{11}^{(1)} \end{bmatrix}\begin{bmatrix}a_0^{(0)} \\ a_1^{(0)} \end{bmatrix}+\begin{bmatrix}b_0^{(1)} \\ b_1^{(1)} \end{bmatrix}, +$$ + +

    with outputs

    +$$ +\begin{bmatrix}a_0^{(1)} \\ a_1^{(1)} \end{bmatrix}=\begin{bmatrix}\sigma^{(1)}(z_0^{(1)}) \\ \sigma^{(1)}(z_1^{(1)}) \end{bmatrix}. +$$ + + + +

    For the output layer

    + +

    For the final output layer we have the inputs to the final activation function

    +$$ +z^{(2)} = w_{0}^{(2)}a_0^{(1)} +w_{1}^{(2)}a_1^{(1)}+b^{(2)}, +$$ + +

    resulting in the output

    +$$ +a^{(2)}=\sigma^{(2)}(z^{(2)}). +$$ + + + +

    Explicit derivatives

    + +

    In total we have nine parameters which we need to train. +Using the chain rule (or just the back propagation algorithm) we can find all derivatives. For the output layer we have then +

    + + +

    Setting up the equations for the optimization

    + +

    For th

    +

    Getting serious, the back propagation equations for a neural network

    diff --git a/doc/pub/week3/html/week3-reveal.html b/doc/pub/week3/html/week3-reveal.html index f8cd5fc0..5c9865c7 100644 --- a/doc/pub/week3/html/week3-reveal.html +++ b/doc/pub/week3/html/week3-reveal.html @@ -1009,12 +1009,22 @@

    Final derivatives

    and requires only three operations if we can reuse all intermediate variables.

    +
    +

    In general not this simple

    + +

    In general, see the generalization below, unless we can obtain simple +analytical expressions which we can simplify further, the final +implementation of automatic differentiation involves repeated +calculations (and thereby operations) of derivatives of elementary +functions. +

    +
    +

    Automatic differentiation

    We can make this example more formal. Automatic differentiation is a -formalization of the previous example (see graph from whiteboard -notes). +formalization of the previous example (see graph).

    We define \( \boldsymbol{x}\in x_1,\dots, x_l \) input variables to a given function \( f(\boldsymbol{x}) \) and \( x_{l+1},\dots, x_L \) intermediate variables.

    @@ -1064,7 +1074,12 @@

    Chain rule

    First network example, simple percepetron with one input

    -

    As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable \( x \) and one target value \( y \). We define an activation function \( \sigma_1 \) which takes as input

    +

    As yet another example we define now a simple perceptron model with +all quantities given by scalars. We consider only one input variable +\( x \) and one target value \( y \). We define an activation function +\( \sigma_1 \) which takes as input +

    +

     
    $$ z_1 = w_1x+b_1, @@ -1125,7 +1140,7 @@

    Optimizing the parameters

    Adding a hidden layer

    -

    We change our simple model to a (see graph on whiteboard notes) +

    We change our simple model to (see graph) network with just one hidden layer but with scalar variables only.

    @@ -1208,7 +1223,7 @@

    The training

     
    $$ -w_{i}\leftarrow = w_{i}- \eta \delta_i a_{i-1}, +w_{i}\leftarrow w_{i}- \eta \delta_i a_{i-1}, $$

     
    @@ -1226,6 +1241,107 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.

    +
    +

    Code examples for the simple models

    +
    + +
    +

    Simple neural network and the back propagation equations

    + +

    Let us now try to increase our level of ambitions and attempt to set +up the equations for a neural network with two input nodes, one hidden +layer with two hidden nodes and one utput layers. +

    + +

    We need to define the following parameters and variables with the input layer (layer \( (0) \)) +where we label the nodes \( x_0 \) and \( x_1 \) +

    +

     
    +$$ +x_0 = a_0^{(0)} \wedge x_1 = a_1^{(0)}. +$$ +

     
    + +

    The hidden layer (layer \( (1) \)) has nodes which yield the outputs \( a_0^{(1)} \) and \( a_1^{(1)} \)) with weight \( \boldsymbol{w} \) and bias \( \boldsymbol{b} \) parameters

    +

     
    +$$ +w_{ij}^{(1)}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\right\} \wedge b^{(1)}=\left\{b_0^{(1)},b_1^{(1)}\right\}. +$$ +

     
    +

    + +
    +

    The ouput layer

    + +

    Finally, we have the ouput layer given by layer label \( (2) \) with output \( a^{(2)} \) and weights and biases to be determined

    +

     
    +$$ +w_{i}^{(2)}=\left\{w_{0}^{(2)},w_{1}^{(2)}\right\} \wedge b^{(2)}. +$$ +

     
    + +

    Our output is \( \tilde{y}=a^{(2)} \) and we define a generic cost function \( C(a^{(2)},y;\boldsymbol{\Theta}) \) where \( y \) is the target value (a scalar here). +The parameters we need to optimize are given by +

    +

     
    +$$ +\boldsymbol{\Theta}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\right\}. +$$ +

     
    +

    + +
    +

    Compact expressions

    + +

    We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions. +The inputs to the first hidden layer are +

    +

     
    +$$ +\begin{bmatrix}z_0^{(1)} \\ z_1^{(1)} \end{bmatrix}=\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\ w_{10}^{(1)} &w_{11}^{(1)} \end{bmatrix}\begin{bmatrix}a_0^{(0)} \\ a_1^{(0)} \end{bmatrix}+\begin{bmatrix}b_0^{(1)} \\ b_1^{(1)} \end{bmatrix}, +$$ +

     
    + +

    with outputs

    +

     
    +$$ +\begin{bmatrix}a_0^{(1)} \\ a_1^{(1)} \end{bmatrix}=\begin{bmatrix}\sigma^{(1)}(z_0^{(1)}) \\ \sigma^{(1)}(z_1^{(1)}) \end{bmatrix}. +$$ +

     
    +

    + +
    +

    For the output layer

    + +

    For the final output layer we have the inputs to the final activation function

    +

     
    +$$ +z^{(2)} = w_{0}^{(2)}a_0^{(1)} +w_{1}^{(2)}a_1^{(1)}+b^{(2)}, +$$ +

     
    + +

    resulting in the output

    +

     
    +$$ +a^{(2)}=\sigma^{(2)}(z^{(2)}). +$$ +

     
    +

    + +
    +

    Explicit derivatives

    + +

    In total we have nine parameters which we need to train. +Using the chain rule (or just the back propagation algorithm) we can find all derivatives. For the output layer we have then +

    +
    + +
    +

    Setting up the equations for the optimization

    + +

    For th

    +
    +

    Getting serious, the back propagation equations for a neural network

    diff --git a/doc/pub/week3/html/week3-solarized.html b/doc/pub/week3/html/week3-solarized.html index b0ba7605..9b99b120 100644 --- a/doc/pub/week3/html/week3-solarized.html +++ b/doc/pub/week3/html/week3-solarized.html @@ -180,6 +180,10 @@ None, 'new-expression-for-the-derivative'), ('Final derivatives', 2, None, 'final-derivatives'), + ('In general not this simple', + 2, + None, + 'in-general-not-this-simple'), ('Automatic differentiation', 2, None, @@ -197,6 +201,22 @@ ('The derivatives', 2, None, 'the-derivatives'), ('Important observations', 2, None, 'important-observations'), ('The training', 2, None, 'the-training'), + ('Code examples for the simple models', + 2, + None, + 'code-examples-for-the-simple-models'), + ('Simple neural network and the back propagation equations', + 2, + None, + 'simple-neural-network-and-the-back-propagation-equations'), + ('The ouput layer', 2, None, 'the-ouput-layer'), + ('Compact expressions', 2, None, 'compact-expressions'), + ('For the output layer', 2, None, 'for-the-output-layer'), + ('Explicit derivatives', 2, None, 'explicit-derivatives'), + ('Setting up the equations for the optimization', + 2, + None, + 'setting-up-the-equations-for-the-optimization'), ('Getting serious, the back propagation equations for a neural ' 'network', 2, @@ -1048,12 +1068,21 @@

    Final derivatives

    and requires only three operations if we can reuse all intermediate variables.

    +









    +

    In general not this simple

    + +

    In general, see the generalization below, unless we can obtain simple +analytical expressions which we can simplify further, the final +implementation of automatic differentiation involves repeated +calculations (and thereby operations) of derivatives of elementary +functions. +

    +









    Automatic differentiation

    We can make this example more formal. Automatic differentiation is a -formalization of the previous example (see graph from whiteboard -notes). +formalization of the previous example (see graph).

    We define \( \boldsymbol{x}\in x_1,\dots, x_l \) input variables to a given function \( f(\boldsymbol{x}) \) and \( x_{l+1},\dots, x_L \) intermediate variables.

    @@ -1095,7 +1124,12 @@

    Chain rule











    First network example, simple percepetron with one input

    -

    As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable \( x \) and one target value \( y \). We define an activation function \( \sigma_1 \) which takes as input

    +

    As yet another example we define now a simple perceptron model with +all quantities given by scalars. We consider only one input variable +\( x \) and one target value \( y \). We define an activation function +\( \sigma_1 \) which takes as input +

    + $$ z_1 = w_1x+b_1, $$ @@ -1144,7 +1178,7 @@

    Optimizing the parameters











    Adding a hidden layer

    -

    We change our simple model to a (see graph on whiteboard notes) +

    We change our simple model to (see graph) network with just one hidden layer but with scalar variables only.

    @@ -1210,7 +1244,7 @@

    The training

    The training of the parameters is done through various gradient descent approximations with

    $$ -w_{i}\leftarrow = w_{i}- \eta \delta_i a_{i-1}, +w_{i}\leftarrow w_{i}- \eta \delta_i a_{i-1}, $$

    and

    @@ -1224,6 +1258,88 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.

    +









    +

    Code examples for the simple models

    + +









    +

    Simple neural network and the back propagation equations

    + +

    Let us now try to increase our level of ambitions and attempt to set +up the equations for a neural network with two input nodes, one hidden +layer with two hidden nodes and one utput layers. +

    + +

    We need to define the following parameters and variables with the input layer (layer \( (0) \)) +where we label the nodes \( x_0 \) and \( x_1 \) +

    +$$ +x_0 = a_0^{(0)} \wedge x_1 = a_1^{(0)}. +$$ + +

    The hidden layer (layer \( (1) \)) has nodes which yield the outputs \( a_0^{(1)} \) and \( a_1^{(1)} \)) with weight \( \boldsymbol{w} \) and bias \( \boldsymbol{b} \) parameters

    +$$ +w_{ij}^{(1)}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\right\} \wedge b^{(1)}=\left\{b_0^{(1)},b_1^{(1)}\right\}. +$$ + + +









    +

    The ouput layer

    + +

    Finally, we have the ouput layer given by layer label \( (2) \) with output \( a^{(2)} \) and weights and biases to be determined

    +$$ +w_{i}^{(2)}=\left\{w_{0}^{(2)},w_{1}^{(2)}\right\} \wedge b^{(2)}. +$$ + +

    Our output is \( \tilde{y}=a^{(2)} \) and we define a generic cost function \( C(a^{(2)},y;\boldsymbol{\Theta}) \) where \( y \) is the target value (a scalar here). +The parameters we need to optimize are given by +

    +$$ +\boldsymbol{\Theta}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\right\}. +$$ + + +









    +

    Compact expressions

    + +

    We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions. +The inputs to the first hidden layer are +

    +$$ +\begin{bmatrix}z_0^{(1)} \\ z_1^{(1)} \end{bmatrix}=\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\ w_{10}^{(1)} &w_{11}^{(1)} \end{bmatrix}\begin{bmatrix}a_0^{(0)} \\ a_1^{(0)} \end{bmatrix}+\begin{bmatrix}b_0^{(1)} \\ b_1^{(1)} \end{bmatrix}, +$$ + +

    with outputs

    +$$ +\begin{bmatrix}a_0^{(1)} \\ a_1^{(1)} \end{bmatrix}=\begin{bmatrix}\sigma^{(1)}(z_0^{(1)}) \\ \sigma^{(1)}(z_1^{(1)}) \end{bmatrix}. +$$ + + +









    +

    For the output layer

    + +

    For the final output layer we have the inputs to the final activation function

    +$$ +z^{(2)} = w_{0}^{(2)}a_0^{(1)} +w_{1}^{(2)}a_1^{(1)}+b^{(2)}, +$$ + +

    resulting in the output

    +$$ +a^{(2)}=\sigma^{(2)}(z^{(2)}). +$$ + + +









    +

    Explicit derivatives

    + +

    In total we have nine parameters which we need to train. +Using the chain rule (or just the back propagation algorithm) we can find all derivatives. For the output layer we have then +

    + +









    +

    Setting up the equations for the optimization

    + +

    For th

    +









    Getting serious, the back propagation equations for a neural network

    diff --git a/doc/pub/week3/html/week3.html b/doc/pub/week3/html/week3.html index 9196b6c8..1c251661 100644 --- a/doc/pub/week3/html/week3.html +++ b/doc/pub/week3/html/week3.html @@ -257,6 +257,10 @@ None, 'new-expression-for-the-derivative'), ('Final derivatives', 2, None, 'final-derivatives'), + ('In general not this simple', + 2, + None, + 'in-general-not-this-simple'), ('Automatic differentiation', 2, None, @@ -274,6 +278,22 @@ ('The derivatives', 2, None, 'the-derivatives'), ('Important observations', 2, None, 'important-observations'), ('The training', 2, None, 'the-training'), + ('Code examples for the simple models', + 2, + None, + 'code-examples-for-the-simple-models'), + ('Simple neural network and the back propagation equations', + 2, + None, + 'simple-neural-network-and-the-back-propagation-equations'), + ('The ouput layer', 2, None, 'the-ouput-layer'), + ('Compact expressions', 2, None, 'compact-expressions'), + ('For the output layer', 2, None, 'for-the-output-layer'), + ('Explicit derivatives', 2, None, 'explicit-derivatives'), + ('Setting up the equations for the optimization', + 2, + None, + 'setting-up-the-equations-for-the-optimization'), ('Getting serious, the back propagation equations for a neural ' 'network', 2, @@ -1125,12 +1145,21 @@

    Final derivatives

    and requires only three operations if we can reuse all intermediate variables.

    +









    +

    In general not this simple

    + +

    In general, see the generalization below, unless we can obtain simple +analytical expressions which we can simplify further, the final +implementation of automatic differentiation involves repeated +calculations (and thereby operations) of derivatives of elementary +functions. +

    +









    Automatic differentiation

    We can make this example more formal. Automatic differentiation is a -formalization of the previous example (see graph from whiteboard -notes). +formalization of the previous example (see graph).

    We define \( \boldsymbol{x}\in x_1,\dots, x_l \) input variables to a given function \( f(\boldsymbol{x}) \) and \( x_{l+1},\dots, x_L \) intermediate variables.

    @@ -1172,7 +1201,12 @@

    Chain rule











    First network example, simple percepetron with one input

    -

    As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable \( x \) and one target value \( y \). We define an activation function \( \sigma_1 \) which takes as input

    +

    As yet another example we define now a simple perceptron model with +all quantities given by scalars. We consider only one input variable +\( x \) and one target value \( y \). We define an activation function +\( \sigma_1 \) which takes as input +

    + $$ z_1 = w_1x+b_1, $$ @@ -1221,7 +1255,7 @@

    Optimizing the parameters











    Adding a hidden layer

    -

    We change our simple model to a (see graph on whiteboard notes) +

    We change our simple model to (see graph) network with just one hidden layer but with scalar variables only.

    @@ -1287,7 +1321,7 @@

    The training

    The training of the parameters is done through various gradient descent approximations with

    $$ -w_{i}\leftarrow = w_{i}- \eta \delta_i a_{i-1}, +w_{i}\leftarrow w_{i}- \eta \delta_i a_{i-1}, $$

    and

    @@ -1301,6 +1335,88 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.

    +









    +

    Code examples for the simple models

    + +









    +

    Simple neural network and the back propagation equations

    + +

    Let us now try to increase our level of ambitions and attempt to set +up the equations for a neural network with two input nodes, one hidden +layer with two hidden nodes and one utput layers. +

    + +

    We need to define the following parameters and variables with the input layer (layer \( (0) \)) +where we label the nodes \( x_0 \) and \( x_1 \) +

    +$$ +x_0 = a_0^{(0)} \wedge x_1 = a_1^{(0)}. +$$ + +

    The hidden layer (layer \( (1) \)) has nodes which yield the outputs \( a_0^{(1)} \) and \( a_1^{(1)} \)) with weight \( \boldsymbol{w} \) and bias \( \boldsymbol{b} \) parameters

    +$$ +w_{ij}^{(1)}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\right\} \wedge b^{(1)}=\left\{b_0^{(1)},b_1^{(1)}\right\}. +$$ + + +









    +

    The ouput layer

    + +

    Finally, we have the ouput layer given by layer label \( (2) \) with output \( a^{(2)} \) and weights and biases to be determined

    +$$ +w_{i}^{(2)}=\left\{w_{0}^{(2)},w_{1}^{(2)}\right\} \wedge b^{(2)}. +$$ + +

    Our output is \( \tilde{y}=a^{(2)} \) and we define a generic cost function \( C(a^{(2)},y;\boldsymbol{\Theta}) \) where \( y \) is the target value (a scalar here). +The parameters we need to optimize are given by +

    +$$ +\boldsymbol{\Theta}=\left\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\right\}. +$$ + + +









    +

    Compact expressions

    + +

    We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions. +The inputs to the first hidden layer are +

    +$$ +\begin{bmatrix}z_0^{(1)} \\ z_1^{(1)} \end{bmatrix}=\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\ w_{10}^{(1)} &w_{11}^{(1)} \end{bmatrix}\begin{bmatrix}a_0^{(0)} \\ a_1^{(0)} \end{bmatrix}+\begin{bmatrix}b_0^{(1)} \\ b_1^{(1)} \end{bmatrix}, +$$ + +

    with outputs

    +$$ +\begin{bmatrix}a_0^{(1)} \\ a_1^{(1)} \end{bmatrix}=\begin{bmatrix}\sigma^{(1)}(z_0^{(1)}) \\ \sigma^{(1)}(z_1^{(1)}) \end{bmatrix}. +$$ + + +









    +

    For the output layer

    + +

    For the final output layer we have the inputs to the final activation function

    +$$ +z^{(2)} = w_{0}^{(2)}a_0^{(1)} +w_{1}^{(2)}a_1^{(1)}+b^{(2)}, +$$ + +

    resulting in the output

    +$$ +a^{(2)}=\sigma^{(2)}(z^{(2)}). +$$ + + +









    +

    Explicit derivatives

    + +

    In total we have nine parameters which we need to train. +Using the chain rule (or just the back propagation algorithm) we can find all derivatives. For the output layer we have then +

    + +









    +

    Setting up the equations for the optimization

    + +

    For th

    +









    Getting serious, the back propagation equations for a neural network

    diff --git a/doc/pub/week3/ipynb/ipynb-week3-src.tar.gz b/doc/pub/week3/ipynb/ipynb-week3-src.tar.gz index 838b24ec87db52c54155a22fdea67c0a832ff351..079a34a9914d1993e1e742cd4be8650ce5561994 100644 GIT binary patch delta 18 ZcmX@v&wR3S=SKEJ#{oU=2D|_O diff --git a/doc/pub/week3/ipynb/week3.ipynb b/doc/pub/week3/ipynb/week3.ipynb index a9e53d42..45fe2569 100644 --- a/doc/pub/week3/ipynb/week3.ipynb +++ b/doc/pub/week3/ipynb/week3.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "8915be3d", + "id": "0ddf6ac0", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "fec342db", + "id": "5e1860e3", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "fe2dcb89", + "id": "ffba166a", "metadata": { "editable": true }, @@ -47,7 +47,7 @@ }, { "cell_type": "markdown", - "id": "111f7b7e", + "id": "e86e3d42", "metadata": { "editable": true }, @@ -63,7 +63,7 @@ }, { "cell_type": "markdown", - "id": "dbdbcb30", + "id": "99ff04b4", "metadata": { "editable": true }, @@ -81,7 +81,7 @@ }, { "cell_type": "markdown", - "id": "319519a5", + "id": "ef9140f7", "metadata": { "editable": true }, @@ -95,7 +95,7 @@ }, { "cell_type": "markdown", - "id": "111dffd6", + "id": "fec74532", "metadata": { "editable": true }, @@ -114,7 +114,7 @@ }, { "cell_type": "markdown", - "id": "625a14dc", + "id": "8afea69d", "metadata": { "editable": true }, @@ -138,7 +138,7 @@ }, { "cell_type": "markdown", - "id": "60a99ac9", + "id": "c5ebf063", "metadata": { "editable": true }, @@ -165,7 +165,7 @@ }, { "cell_type": "markdown", - "id": "056ff7d4", + "id": "bf486cce", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "963e0389", + "id": "fa50fa29", "metadata": { "editable": true }, @@ -194,7 +194,7 @@ }, { "cell_type": "markdown", - "id": "8961e599", + "id": "f0b49977", "metadata": { "editable": true }, @@ -205,7 +205,7 @@ }, { "cell_type": "markdown", - "id": "8af0c1b7", + "id": "73c3edd2", "metadata": { "editable": true }, @@ -223,7 +223,7 @@ }, { "cell_type": "markdown", - "id": "07484429", + "id": "1bcfbd2f", "metadata": { "editable": true }, @@ -252,7 +252,7 @@ }, { "cell_type": "markdown", - "id": "67fbadbb", + "id": "6d5213b7", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "9f8733eb", + "id": "9569d8ef", "metadata": { "editable": true }, @@ -282,7 +282,7 @@ }, { "cell_type": "markdown", - "id": "b5ff69af", + "id": "38eef3f2", "metadata": { "editable": true }, @@ -294,7 +294,7 @@ }, { "cell_type": "markdown", - "id": "cd25ca5b", + "id": "fcf0708b", "metadata": { "editable": true }, @@ -308,7 +308,7 @@ }, { "cell_type": "markdown", - "id": "95feb967", + "id": "b07615b3", "metadata": { "editable": true }, @@ -320,7 +320,7 @@ }, { "cell_type": "markdown", - "id": "ad8e3d52", + "id": "82bb7457", "metadata": { "editable": true }, @@ -336,7 +336,7 @@ }, { "cell_type": "markdown", - "id": "9ee4c83f", + "id": "2a2cdf76", "metadata": { "editable": true }, @@ -352,7 +352,7 @@ }, { "cell_type": "markdown", - "id": "245edc97", + "id": "568c1e95", "metadata": { "editable": true }, @@ -364,7 +364,7 @@ }, { "cell_type": "markdown", - "id": "cace207a", + "id": "5339302d", "metadata": { "editable": true }, @@ -374,7 +374,7 @@ }, { "cell_type": "markdown", - "id": "22b74b99", + "id": "bb3341b7", "metadata": { "editable": true }, @@ -386,7 +386,7 @@ }, { "cell_type": "markdown", - "id": "92f6806d", + "id": "21b5f540", "metadata": { "editable": true }, @@ -404,7 +404,7 @@ }, { "cell_type": "markdown", - "id": "cb988299", + "id": "cce1685d", "metadata": { "editable": true }, @@ -417,7 +417,7 @@ }, { "cell_type": "markdown", - "id": "e30501e3", + "id": "65a520f2", "metadata": { "editable": true }, @@ -434,7 +434,7 @@ }, { "cell_type": "markdown", - "id": "20149b42", + "id": "b85ade18", "metadata": { "editable": true }, @@ -446,7 +446,7 @@ }, { "cell_type": "markdown", - "id": "50c4304b", + "id": "df448c2b", "metadata": { "editable": true }, @@ -458,7 +458,7 @@ }, { "cell_type": "markdown", - "id": "25a0e0f1", + "id": "a4af64d8", "metadata": { "editable": true }, @@ -474,7 +474,7 @@ }, { "cell_type": "markdown", - "id": "926fc4a1", + "id": "4529702f", "metadata": { "editable": true }, @@ -491,7 +491,7 @@ }, { "cell_type": "markdown", - "id": "fcb52c3d", + "id": "03bef104", "metadata": { "editable": true }, @@ -503,7 +503,7 @@ }, { "cell_type": "markdown", - "id": "50b2ca44", + "id": "24a26da6", "metadata": { "editable": true }, @@ -516,7 +516,7 @@ }, { "cell_type": "markdown", - "id": "e6639ea0", + "id": "52cfa34a", "metadata": { "editable": true }, @@ -528,7 +528,7 @@ }, { "cell_type": "markdown", - "id": "2190cd62", + "id": "fbced099", "metadata": { "editable": true }, @@ -544,7 +544,7 @@ }, { "cell_type": "markdown", - "id": "e5213195", + "id": "ec346551", "metadata": { "editable": true }, @@ -556,7 +556,7 @@ }, { "cell_type": "markdown", - "id": "09db4516", + "id": "11a7b0a7", "metadata": { "editable": true }, @@ -568,7 +568,7 @@ }, { "cell_type": "markdown", - "id": "8ffa7496", + "id": "45579322", "metadata": { "editable": true }, @@ -580,7 +580,7 @@ }, { "cell_type": "markdown", - "id": "80b73384", + "id": "0edc5d7d", "metadata": { "editable": true }, @@ -590,7 +590,7 @@ }, { "cell_type": "markdown", - "id": "cc9a931b", + "id": "283b302b", "metadata": { "editable": true }, @@ -602,7 +602,7 @@ }, { "cell_type": "markdown", - "id": "8424ea2d", + "id": "b71be914", "metadata": { "editable": true }, @@ -612,7 +612,7 @@ }, { "cell_type": "markdown", - "id": "b0bcbeab", + "id": "98217797", "metadata": { "editable": true }, @@ -624,7 +624,7 @@ }, { "cell_type": "markdown", - "id": "bcd96873", + "id": "b5098f9e", "metadata": { "editable": true }, @@ -638,7 +638,7 @@ }, { "cell_type": "markdown", - "id": "9c9061e5", + "id": "e5c9d090", "metadata": { "editable": true }, @@ -650,7 +650,7 @@ }, { "cell_type": "markdown", - "id": "907c6e9d", + "id": "94c240d5", "metadata": { "editable": true }, @@ -660,7 +660,7 @@ }, { "cell_type": "markdown", - "id": "51f6c41f", + "id": "2d2c2360", "metadata": { "editable": true }, @@ -672,7 +672,7 @@ }, { "cell_type": "markdown", - "id": "998ce6e2", + "id": "d3af7d6b", "metadata": { "editable": true }, @@ -682,7 +682,7 @@ }, { "cell_type": "markdown", - "id": "6ca796fa", + "id": "decd5dfe", "metadata": { "editable": true }, @@ -694,7 +694,7 @@ }, { "cell_type": "markdown", - "id": "7e9722cc", + "id": "0d420b46", "metadata": { "editable": true }, @@ -709,7 +709,7 @@ }, { "cell_type": "markdown", - "id": "a31b27a0", + "id": "904b606b", "metadata": { "editable": true }, @@ -721,7 +721,7 @@ }, { "cell_type": "markdown", - "id": "baa5a5cc", + "id": "3415dc6e", "metadata": { "editable": true }, @@ -733,7 +733,7 @@ }, { "cell_type": "markdown", - "id": "39f05c9a", + "id": "f991c51b", "metadata": { "editable": true }, @@ -745,7 +745,7 @@ }, { "cell_type": "markdown", - "id": "418b57d8", + "id": "c13d5c48", "metadata": { "editable": true }, @@ -757,7 +757,7 @@ }, { "cell_type": "markdown", - "id": "f65b51b1", + "id": "7b96ec04", "metadata": { "editable": true }, @@ -767,7 +767,7 @@ }, { "cell_type": "markdown", - "id": "edb10e24", + "id": "308036c2", "metadata": { "editable": true }, @@ -779,7 +779,7 @@ }, { "cell_type": "markdown", - "id": "fcceac51", + "id": "a3d2d41f", "metadata": { "editable": true }, @@ -789,7 +789,7 @@ }, { "cell_type": "markdown", - "id": "c78ae883", + "id": "c41a3968", "metadata": { "editable": true }, @@ -801,7 +801,7 @@ }, { "cell_type": "markdown", - "id": "53111b97", + "id": "36b1a16c", "metadata": { "editable": true }, @@ -821,7 +821,7 @@ }, { "cell_type": "markdown", - "id": "879b5184", + "id": "305406af", "metadata": { "editable": true }, @@ -833,7 +833,7 @@ }, { "cell_type": "markdown", - "id": "7fc020b2", + "id": "bf161871", "metadata": { "editable": true }, @@ -845,7 +845,7 @@ }, { "cell_type": "markdown", - "id": "06ffaa24", + "id": "89cfb04d", "metadata": { "editable": true }, @@ -855,7 +855,7 @@ }, { "cell_type": "markdown", - "id": "2ce16392", + "id": "ecafb334", "metadata": { "editable": true }, @@ -867,7 +867,7 @@ }, { "cell_type": "markdown", - "id": "94359507", + "id": "4ad5adda", "metadata": { "editable": true }, @@ -878,7 +878,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "8dfcb4c2", + "id": "4e12434c", "metadata": { "collapsed": false, "editable": true @@ -897,7 +897,7 @@ }, { "cell_type": "markdown", - "id": "ff23144d", + "id": "8fd52ae2", "metadata": { "editable": true }, @@ -908,7 +908,7 @@ }, { "cell_type": "markdown", - "id": "2b8120e8", + "id": "1e53cb11", "metadata": { "editable": true }, @@ -920,7 +920,7 @@ }, { "cell_type": "markdown", - "id": "75e8df5c", + "id": "5eab8eaa", "metadata": { "editable": true }, @@ -930,7 +930,7 @@ }, { "cell_type": "markdown", - "id": "b010cede", + "id": "746ca186", "metadata": { "editable": true }, @@ -942,7 +942,7 @@ }, { "cell_type": "markdown", - "id": "bbf51a38", + "id": "65a27079", "metadata": { "editable": true }, @@ -955,7 +955,7 @@ }, { "cell_type": "markdown", - "id": "2bc52086", + "id": "9d1a4553", "metadata": { "editable": true }, @@ -967,7 +967,7 @@ }, { "cell_type": "markdown", - "id": "d153d464", + "id": "8c4bc805", "metadata": { "editable": true }, @@ -979,7 +979,7 @@ }, { "cell_type": "markdown", - "id": "7ef5fde3", + "id": "2dca93e5", "metadata": { "editable": true }, @@ -996,7 +996,7 @@ }, { "cell_type": "markdown", - "id": "76c15f7f", + "id": "5b5f2ad3", "metadata": { "editable": true }, @@ -1008,7 +1008,7 @@ }, { "cell_type": "markdown", - "id": "003060b4", + "id": "4db46996", "metadata": { "editable": true }, @@ -1020,7 +1020,7 @@ }, { "cell_type": "markdown", - "id": "bafaab22", + "id": "8a1e5ed6", "metadata": { "editable": true }, @@ -1030,7 +1030,7 @@ }, { "cell_type": "markdown", - "id": "e1242659", + "id": "be32c63e", "metadata": { "editable": true }, @@ -1042,7 +1042,7 @@ }, { "cell_type": "markdown", - "id": "bc4340c2", + "id": "ab2f3ff4", "metadata": { "editable": true }, @@ -1052,7 +1052,7 @@ }, { "cell_type": "markdown", - "id": "8e0e4969", + "id": "fd6eb999", "metadata": { "editable": true }, @@ -1064,7 +1064,7 @@ }, { "cell_type": "markdown", - "id": "4f94e032", + "id": "f938782d", "metadata": { "editable": true }, @@ -1074,7 +1074,7 @@ }, { "cell_type": "markdown", - "id": "46bf127c", + "id": "db50b6c1", "metadata": { "editable": true }, @@ -1086,7 +1086,7 @@ }, { "cell_type": "markdown", - "id": "a04879ac", + "id": "211687d9", "metadata": { "editable": true }, @@ -1096,7 +1096,7 @@ }, { "cell_type": "markdown", - "id": "7d00bd6e", + "id": "0a3e0dce", "metadata": { "editable": true }, @@ -1108,7 +1108,7 @@ }, { "cell_type": "markdown", - "id": "47264885", + "id": "111fa90c", "metadata": { "editable": true }, @@ -1120,7 +1120,7 @@ }, { "cell_type": "markdown", - "id": "36eda3e9", + "id": "0b27beea", "metadata": { "editable": true }, @@ -1130,7 +1130,7 @@ }, { "cell_type": "markdown", - "id": "9588adaa", + "id": "65aeb9f4", "metadata": { "editable": true }, @@ -1142,7 +1142,7 @@ }, { "cell_type": "markdown", - "id": "577fbe29", + "id": "30efe746", "metadata": { "editable": true }, @@ -1152,7 +1152,7 @@ }, { "cell_type": "markdown", - "id": "4b0989d4", + "id": "c1a84179", "metadata": { "editable": true }, @@ -1164,7 +1164,7 @@ }, { "cell_type": "markdown", - "id": "a8608d86", + "id": "d55f83de", "metadata": { "editable": true }, @@ -1182,7 +1182,7 @@ }, { "cell_type": "markdown", - "id": "3e5481e6", + "id": "5bbf99a9", "metadata": { "editable": true }, @@ -1194,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "aa76fa19", + "id": "6a4502c1", "metadata": { "editable": true }, @@ -1206,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "af444f42", + "id": "6c3de3f4", "metadata": { "editable": true }, @@ -1216,7 +1216,7 @@ }, { "cell_type": "markdown", - "id": "9bf7630b", + "id": "7fd5444d", "metadata": { "editable": true }, @@ -1228,7 +1228,7 @@ }, { "cell_type": "markdown", - "id": "81e13615", + "id": "bc122a94", "metadata": { "editable": true }, @@ -1239,7 +1239,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "9f8ac8c9", + "id": "e2ee4896", "metadata": { "collapsed": false, "editable": true @@ -1258,7 +1258,7 @@ }, { "cell_type": "markdown", - "id": "907f83d7", + "id": "e999732a", "metadata": { "editable": true }, @@ -1274,7 +1274,7 @@ }, { "cell_type": "markdown", - "id": "35cb5206", + "id": "7c010c91", "metadata": { "editable": true }, @@ -1287,7 +1287,7 @@ }, { "cell_type": "markdown", - "id": "1f69074a", + "id": "244b1d3f", "metadata": { "editable": true }, @@ -1299,7 +1299,7 @@ }, { "cell_type": "markdown", - "id": "9d249e6f", + "id": "5eb35189", "metadata": { "editable": true }, @@ -1309,7 +1309,7 @@ }, { "cell_type": "markdown", - "id": "4b28a138", + "id": "6886a427", "metadata": { "editable": true }, @@ -1321,7 +1321,7 @@ }, { "cell_type": "markdown", - "id": "8dcd732c", + "id": "54633caa", "metadata": { "editable": true }, @@ -1331,7 +1331,7 @@ }, { "cell_type": "markdown", - "id": "207713bb", + "id": "b0400d4a", "metadata": { "editable": true }, @@ -1343,7 +1343,7 @@ }, { "cell_type": "markdown", - "id": "962b7823", + "id": "cc0dbb76", "metadata": { "editable": true }, @@ -1353,7 +1353,7 @@ }, { "cell_type": "markdown", - "id": "73d61dd0", + "id": "141fe2a0", "metadata": { "editable": true }, @@ -1365,7 +1365,7 @@ }, { "cell_type": "markdown", - "id": "0640c804", + "id": "ea9f1cf9", "metadata": { "editable": true }, @@ -1377,7 +1377,7 @@ }, { "cell_type": "markdown", - "id": "740411b1", + "id": "8901fbc3", "metadata": { "editable": true }, @@ -1389,7 +1389,7 @@ }, { "cell_type": "markdown", - "id": "d82213b4", + "id": "c24313d2", "metadata": { "editable": true }, @@ -1399,7 +1399,7 @@ }, { "cell_type": "markdown", - "id": "41b917d4", + "id": "05ecb8cb", "metadata": { "editable": true }, @@ -1411,7 +1411,7 @@ }, { "cell_type": "markdown", - "id": "ff3d3eaa", + "id": "64f3968f", "metadata": { "editable": true }, @@ -1421,7 +1421,7 @@ }, { "cell_type": "markdown", - "id": "1309968a", + "id": "da00fc99", "metadata": { "editable": true }, @@ -1433,7 +1433,7 @@ }, { "cell_type": "markdown", - "id": "17e7d953", + "id": "8f423494", "metadata": { "editable": true }, @@ -1443,7 +1443,7 @@ }, { "cell_type": "markdown", - "id": "a308a4a7", + "id": "95a5a2f5", "metadata": { "editable": true }, @@ -1455,7 +1455,7 @@ }, { "cell_type": "markdown", - "id": "bb59edfe", + "id": "423ac388", "metadata": { "editable": true }, @@ -1465,7 +1465,7 @@ }, { "cell_type": "markdown", - "id": "f60aad8a", + "id": "445b7109", "metadata": { "editable": true }, @@ -1477,7 +1477,7 @@ }, { "cell_type": "markdown", - "id": "c19a8b42", + "id": "95ab69fd", "metadata": { "editable": true }, @@ -1487,7 +1487,7 @@ }, { "cell_type": "markdown", - "id": "c877e9a6", + "id": "8521a931", "metadata": { "editable": true }, @@ -1499,7 +1499,7 @@ }, { "cell_type": "markdown", - "id": "275d4348", + "id": "e22ec49c", "metadata": { "editable": true }, @@ -1510,7 +1510,7 @@ }, { "cell_type": "markdown", - "id": "872f6952", + "id": "585d5a9b", "metadata": { "editable": true }, @@ -1522,7 +1522,7 @@ }, { "cell_type": "markdown", - "id": "358adf18", + "id": "bb25c720", "metadata": { "editable": true }, @@ -1534,7 +1534,7 @@ }, { "cell_type": "markdown", - "id": "5c965049", + "id": "e734364d", "metadata": { "editable": true }, @@ -1547,7 +1547,7 @@ }, { "cell_type": "markdown", - "id": "a7add5b9", + "id": "961b0305", "metadata": { "editable": true }, @@ -1557,7 +1557,7 @@ }, { "cell_type": "markdown", - "id": "ee38d5eb", + "id": "129c5cae", "metadata": { "editable": true }, @@ -1569,7 +1569,7 @@ }, { "cell_type": "markdown", - "id": "d6a3a38e", + "id": "ba63a379", "metadata": { "editable": true }, @@ -1579,7 +1579,7 @@ }, { "cell_type": "markdown", - "id": "fb991ff8", + "id": "0338d001", "metadata": { "editable": true }, @@ -1591,7 +1591,7 @@ }, { "cell_type": "markdown", - "id": "dfc9aec8", + "id": "cdd78c25", "metadata": { "editable": true }, @@ -1601,7 +1601,23 @@ }, { "cell_type": "markdown", - "id": "c8578162", + "id": "40ce00e1", + "metadata": { + "editable": true + }, + "source": [ + "## In general not this simple\n", + "\n", + "In general, see the generalization below, unless we can obtain simple\n", + "analytical expressions which we can simplify further, the final\n", + "implementation of automatic differentiation involves repeated\n", + "calculations (and thereby operations) of derivatives of elementary\n", + "functions." + ] + }, + { + "cell_type": "markdown", + "id": "b36d8850", "metadata": { "editable": true }, @@ -1609,8 +1625,7 @@ "## Automatic differentiation\n", "\n", "We can make this example more formal. Automatic differentiation is a\n", - "formalization of the previous example (see graph from whiteboard\n", - "notes).\n", + "formalization of the previous example (see graph).\n", "\n", "We define $\\boldsymbol{x}\\in x_1,\\dots, x_l$ input variables to a given function $f(\\boldsymbol{x})$ and $x_{l+1},\\dots, x_L$ intermediate variables.\n", "\n", @@ -1619,7 +1634,7 @@ }, { "cell_type": "markdown", - "id": "95cd9460", + "id": "c481d2fe", "metadata": { "editable": true }, @@ -1631,7 +1646,7 @@ }, { "cell_type": "markdown", - "id": "e701a20c", + "id": "01dd943d", "metadata": { "editable": true }, @@ -1644,7 +1659,7 @@ }, { "cell_type": "markdown", - "id": "04e5ccf2", + "id": "eb2d65d6", "metadata": { "editable": true }, @@ -1657,7 +1672,7 @@ }, { "cell_type": "markdown", - "id": "53a5ded2", + "id": "f94ae276", "metadata": { "editable": true }, @@ -1669,7 +1684,7 @@ }, { "cell_type": "markdown", - "id": "ab36b239", + "id": "da76dfc1", "metadata": { "editable": true }, @@ -1679,7 +1694,7 @@ }, { "cell_type": "markdown", - "id": "e4241af0", + "id": "9d6d85ed", "metadata": { "editable": true }, @@ -1691,7 +1706,7 @@ }, { "cell_type": "markdown", - "id": "4d24fa4e", + "id": "b88aec02", "metadata": { "editable": true }, @@ -1706,19 +1721,22 @@ }, { "cell_type": "markdown", - "id": "a13220a5", + "id": "bc2d6567", "metadata": { "editable": true }, "source": [ "## First network example, simple percepetron with one input\n", "\n", - "As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable $x$ and one target value $y$. We define an activation function $\\sigma_1$ which takes as input" + "As yet another example we define now a simple perceptron model with\n", + "all quantities given by scalars. We consider only one input variable\n", + "$x$ and one target value $y$. We define an activation function\n", + "$\\sigma_1$ which takes as input" ] }, { "cell_type": "markdown", - "id": "dcba4620", + "id": "cb9322f2", "metadata": { "editable": true }, @@ -1730,7 +1748,7 @@ }, { "cell_type": "markdown", - "id": "ca100656", + "id": "80b529af", "metadata": { "editable": true }, @@ -1744,7 +1762,7 @@ }, { "cell_type": "markdown", - "id": "1c1cb6ee", + "id": "96868f06", "metadata": { "editable": true }, @@ -1756,7 +1774,7 @@ }, { "cell_type": "markdown", - "id": "73697ad3", + "id": "a59b3404", "metadata": { "editable": true }, @@ -1772,7 +1790,7 @@ }, { "cell_type": "markdown", - "id": "ba627d2a", + "id": "e9d763fe", "metadata": { "editable": true }, @@ -1784,7 +1802,7 @@ }, { "cell_type": "markdown", - "id": "ef3dd990", + "id": "f5136d8a", "metadata": { "editable": true }, @@ -1794,7 +1812,7 @@ }, { "cell_type": "markdown", - "id": "ceb4f0da", + "id": "28c8e221", "metadata": { "editable": true }, @@ -1806,7 +1824,7 @@ }, { "cell_type": "markdown", - "id": "9950a7e7", + "id": "e3982595", "metadata": { "editable": true }, @@ -1816,7 +1834,7 @@ }, { "cell_type": "markdown", - "id": "39948179", + "id": "0058685c", "metadata": { "editable": true }, @@ -1828,7 +1846,7 @@ }, { "cell_type": "markdown", - "id": "a3f86d2c", + "id": "4e5f7c8d", "metadata": { "editable": true }, @@ -1838,7 +1856,7 @@ }, { "cell_type": "markdown", - "id": "9221e84e", + "id": "e6f7258c", "metadata": { "editable": true }, @@ -1850,14 +1868,14 @@ }, { "cell_type": "markdown", - "id": "b011a759", + "id": "4e967bdf", "metadata": { "editable": true }, "source": [ "## Adding a hidden layer\n", "\n", - "We change our simple model to a (see graph on whiteboard notes)\n", + "We change our simple model to (see graph)\n", "network with just one hidden layer but with scalar variables only.\n", "\n", "Our output variable changes to $a_2$ and $a_1$ is now the output from the hidden node and $a_0=x$.\n", @@ -1866,7 +1884,7 @@ }, { "cell_type": "markdown", - "id": "3fd2991f", + "id": "69ecfcb0", "metadata": { "editable": true }, @@ -1878,7 +1896,7 @@ }, { "cell_type": "markdown", - "id": "2f33d9e7", + "id": "f669e5e3", "metadata": { "editable": true }, @@ -1890,7 +1908,7 @@ }, { "cell_type": "markdown", - "id": "e3fdcc77", + "id": "87b80d8e", "metadata": { "editable": true }, @@ -1900,7 +1918,7 @@ }, { "cell_type": "markdown", - "id": "3ce33278", + "id": "9adf15a8", "metadata": { "editable": true }, @@ -1912,7 +1930,7 @@ }, { "cell_type": "markdown", - "id": "832411d8", + "id": "91bc47a6", "metadata": { "editable": true }, @@ -1922,7 +1940,7 @@ }, { "cell_type": "markdown", - "id": "946f4d4e", + "id": "678a1fac", "metadata": { "editable": true }, @@ -1934,7 +1952,7 @@ }, { "cell_type": "markdown", - "id": "f66f1cdc", + "id": "05b1b5f4", "metadata": { "editable": true }, @@ -1946,7 +1964,7 @@ }, { "cell_type": "markdown", - "id": "3b9d434f", + "id": "7c85574f", "metadata": { "editable": true }, @@ -1958,7 +1976,7 @@ }, { "cell_type": "markdown", - "id": "79e18bd7", + "id": "c30e25b3", "metadata": { "editable": true }, @@ -1970,7 +1988,7 @@ }, { "cell_type": "markdown", - "id": "40c6cf7c", + "id": "ee140437", "metadata": { "editable": true }, @@ -1982,7 +2000,7 @@ }, { "cell_type": "markdown", - "id": "b8069b08", + "id": "3ccd1703", "metadata": { "editable": true }, @@ -1992,7 +2010,7 @@ }, { "cell_type": "markdown", - "id": "8ccd1ed1", + "id": "10db5c89", "metadata": { "editable": true }, @@ -2008,7 +2026,7 @@ }, { "cell_type": "markdown", - "id": "0c41997e", + "id": "673f0a42", "metadata": { "editable": true }, @@ -2020,19 +2038,19 @@ }, { "cell_type": "markdown", - "id": "6bafd3f9", + "id": "6cc366ca", "metadata": { "editable": true }, "source": [ "$$\n", - "w_{i}\\leftarrow = w_{i}- \\eta \\delta_i a_{i-1},\n", + "w_{i}\\leftarrow w_{i}- \\eta \\delta_i a_{i-1},\n", "$$" ] }, { "cell_type": "markdown", - "id": "6df81343", + "id": "69e30bd7", "metadata": { "editable": true }, @@ -2042,7 +2060,7 @@ }, { "cell_type": "markdown", - "id": "6cc519ae", + "id": "2715262b", "metadata": { "editable": true }, @@ -2054,7 +2072,7 @@ }, { "cell_type": "markdown", - "id": "702ee2c8", + "id": "c02b8d51", "metadata": { "editable": true }, @@ -2068,7 +2086,233 @@ }, { "cell_type": "markdown", - "id": "4ef68b5b", + "id": "2e7e59eb", + "metadata": { + "editable": true + }, + "source": [ + "## Code examples for the simple models" + ] + }, + { + "cell_type": "markdown", + "id": "d5c58a0e", + "metadata": { + "editable": true + }, + "source": [ + "## Simple neural network and the back propagation equations\n", + "\n", + "Let us now try to increase our level of ambitions and attempt to set\n", + "up the equations for a neural network with two input nodes, one hidden\n", + "layer with two hidden nodes and one utput layers.\n", + "\n", + "We need to define the following parameters and variables with the input layer (layer $(0)$) \n", + "where we label the nodes $x_0$ and $x_1$" + ] + }, + { + "cell_type": "markdown", + "id": "1cf327e4", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "x_0 = a_0^{(0)} \\wedge x_1 = a_1^{(0)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e0e224f9", + "metadata": { + "editable": true + }, + "source": [ + "The hidden layer (layer $(1)$) has nodes which yield the outputs $a_0^{(1)}$ and $a_1^{(1)}$) with weight $\\boldsymbol{w}$ and bias $\\boldsymbol{b}$ parameters" + ] + }, + { + "cell_type": "markdown", + "id": "9ceeb0da", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "w_{ij}^{(1)}=\\left\\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\\right\\} \\wedge b^{(1)}=\\left\\{b_0^{(1)},b_1^{(1)}\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2c258e69", + "metadata": { + "editable": true + }, + "source": [ + "## The ouput layer\n", + "\n", + "Finally, we have the ouput layer given by layer label $(2)$ with output $a^{(2)}$ and weights and biases to be determined" + ] + }, + { + "cell_type": "markdown", + "id": "5f634655", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "w_{i}^{(2)}=\\left\\{w_{0}^{(2)},w_{1}^{(2)}\\right\\} \\wedge b^{(2)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "69e8fdf0", + "metadata": { + "editable": true + }, + "source": [ + "Our output is $\\tilde{y}=a^{(2)}$ and we define a generic cost function $C(a^{(2)},y;\\boldsymbol{\\Theta})$ where $y$ is the target value (a scalar here).\n", + "The parameters we need to optimize are given by" + ] + }, + { + "cell_type": "markdown", + "id": "9edff35a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\Theta}=\\left\\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "076735bb", + "metadata": { + "editable": true + }, + "source": [ + "## Compact expressions\n", + "\n", + "We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions.\n", + "The inputs to the first hidden layer are" + ] + }, + { + "cell_type": "markdown", + "id": "2628a490", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{bmatrix}z_0^{(1)} \\\\ z_1^{(1)} \\end{bmatrix}=\\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\\\ w_{10}^{(1)} &w_{11}^{(1)} \\end{bmatrix}\\begin{bmatrix}a_0^{(0)} \\\\ a_1^{(0)} \\end{bmatrix}+\\begin{bmatrix}b_0^{(1)} \\\\ b_1^{(1)} \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e9aacc95", + "metadata": { + "editable": true + }, + "source": [ + "with outputs" + ] + }, + { + "cell_type": "markdown", + "id": "e9434bbb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{bmatrix}a_0^{(1)} \\\\ a_1^{(1)} \\end{bmatrix}=\\begin{bmatrix}\\sigma^{(1)}(z_0^{(1)}) \\\\ \\sigma^{(1)}(z_1^{(1)}) \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5d20d323", + "metadata": { + "editable": true + }, + "source": [ + "## For the output layer\n", + "\n", + "For the final output layer we have the inputs to the final activation function" + ] + }, + { + "cell_type": "markdown", + "id": "8b323c72", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "z^{(2)} = w_{0}^{(2)}a_0^{(1)} +w_{1}^{(2)}a_1^{(1)}+b^{(2)},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c52b2643", + "metadata": { + "editable": true + }, + "source": [ + "resulting in the output" + ] + }, + { + "cell_type": "markdown", + "id": "24591cba", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "a^{(2)}=\\sigma^{(2)}(z^{(2)}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f7e3831c", + "metadata": { + "editable": true + }, + "source": [ + "## Explicit derivatives\n", + "\n", + "In total we have nine parameters which we need to train.\n", + "Using the chain rule (or just the back propagation algorithm) we can find all derivatives. For the output layer we have then" + ] + }, + { + "cell_type": "markdown", + "id": "c45b1840", + "metadata": { + "editable": true + }, + "source": [ + "## Setting up the equations for the optimization\n", + "\n", + "For th" + ] + }, + { + "cell_type": "markdown", + "id": "2911a5cb", "metadata": { "editable": true }, @@ -2082,7 +2326,7 @@ }, { "cell_type": "markdown", - "id": "18d15df8", + "id": "c9b03fc9", "metadata": { "editable": true }, @@ -2094,7 +2338,7 @@ }, { "cell_type": "markdown", - "id": "6c88a6c7", + "id": "14554427", "metadata": { "editable": true }, @@ -2104,7 +2348,7 @@ }, { "cell_type": "markdown", - "id": "cfc5fe3f", + "id": "8bec084e", "metadata": { "editable": true }, @@ -2116,7 +2360,7 @@ }, { "cell_type": "markdown", - "id": "685b6c92", + "id": "c9c39755", "metadata": { "editable": true }, @@ -2126,7 +2370,7 @@ }, { "cell_type": "markdown", - "id": "b4d0957b", + "id": "17c76ed4", "metadata": { "editable": true }, @@ -2138,7 +2382,7 @@ }, { "cell_type": "markdown", - "id": "b961fd6a", + "id": "74ddc88b", "metadata": { "editable": true }, @@ -2156,7 +2400,7 @@ }, { "cell_type": "markdown", - "id": "2b71af4f", + "id": "b031c31c", "metadata": { "editable": true }, @@ -2174,7 +2418,7 @@ }, { "cell_type": "markdown", - "id": "6ff8b7a0", + "id": "fd1f83cd", "metadata": { "editable": true }, @@ -2186,7 +2430,7 @@ }, { "cell_type": "markdown", - "id": "d25227bb", + "id": "d31e795c", "metadata": { "editable": true }, @@ -2196,7 +2440,7 @@ }, { "cell_type": "markdown", - "id": "80b44cfa", + "id": "9f190962", "metadata": { "editable": true }, @@ -2208,7 +2452,7 @@ }, { "cell_type": "markdown", - "id": "84e2cc8c", + "id": "36d5d29e", "metadata": { "editable": true }, @@ -2220,7 +2464,7 @@ }, { "cell_type": "markdown", - "id": "4fad9c89", + "id": "919cf9cf", "metadata": { "editable": true }, @@ -2232,7 +2476,7 @@ }, { "cell_type": "markdown", - "id": "dc2c4185", + "id": "f78a863a", "metadata": { "editable": true }, @@ -2242,7 +2486,7 @@ }, { "cell_type": "markdown", - "id": "adb54afb", + "id": "a4ed2ab8", "metadata": { "editable": true }, @@ -2254,7 +2498,7 @@ }, { "cell_type": "markdown", - "id": "e969a773", + "id": "84987650", "metadata": { "editable": true }, @@ -2264,7 +2508,7 @@ }, { "cell_type": "markdown", - "id": "450cd91f", + "id": "cf03afa3", "metadata": { "editable": true }, @@ -2276,7 +2520,7 @@ }, { "cell_type": "markdown", - "id": "bf38b491", + "id": "ec3990d1", "metadata": { "editable": true }, @@ -2294,7 +2538,7 @@ }, { "cell_type": "markdown", - "id": "b068ea9c", + "id": "bf7d7789", "metadata": { "editable": true }, @@ -2304,7 +2548,7 @@ }, { "cell_type": "markdown", - "id": "2e64e05f", + "id": "b665fa01", "metadata": { "editable": true }, @@ -2322,7 +2566,7 @@ }, { "cell_type": "markdown", - "id": "6c0422e8", + "id": "89f2f4a2", "metadata": { "editable": true }, @@ -2332,7 +2576,7 @@ }, { "cell_type": "markdown", - "id": "689123cb", + "id": "1e988025", "metadata": { "editable": true }, @@ -2350,7 +2594,7 @@ }, { "cell_type": "markdown", - "id": "57ee3448", + "id": "3a094edb", "metadata": { "editable": true }, @@ -2362,7 +2606,7 @@ }, { "cell_type": "markdown", - "id": "723ec30a", + "id": "7871f49e", "metadata": { "editable": true }, @@ -2374,7 +2618,7 @@ }, { "cell_type": "markdown", - "id": "04cbecd3", + "id": "c0b3e3b3", "metadata": { "editable": true }, @@ -2384,7 +2628,7 @@ }, { "cell_type": "markdown", - "id": "b135ec11", + "id": "f4f23a2c", "metadata": { "editable": true }, @@ -2396,7 +2640,7 @@ }, { "cell_type": "markdown", - "id": "2cf6dde6", + "id": "fc7f8cd3", "metadata": { "editable": true }, @@ -2408,7 +2652,7 @@ }, { "cell_type": "markdown", - "id": "01776615", + "id": "5c55bec2", "metadata": { "editable": true }, @@ -2418,7 +2662,7 @@ }, { "cell_type": "markdown", - "id": "087c4ece", + "id": "fe082d20", "metadata": { "editable": true }, @@ -2430,7 +2674,7 @@ }, { "cell_type": "markdown", - "id": "bd979998", + "id": "33829c24", "metadata": { "editable": true }, @@ -2440,7 +2684,7 @@ }, { "cell_type": "markdown", - "id": "667761ea", + "id": "f0f86ecf", "metadata": { "editable": true }, @@ -2452,7 +2696,7 @@ }, { "cell_type": "markdown", - "id": "51597515", + "id": "91aae55f", "metadata": { "editable": true }, @@ -2464,7 +2708,7 @@ }, { "cell_type": "markdown", - "id": "95bc8546", + "id": "ff8e481e", "metadata": { "editable": true }, @@ -2487,7 +2731,7 @@ }, { "cell_type": "markdown", - "id": "92aef9b4", + "id": "5c5da9c3", "metadata": { "editable": true }, @@ -2499,7 +2743,7 @@ }, { "cell_type": "markdown", - "id": "d7ec5a5a", + "id": "40911753", "metadata": { "editable": true }, @@ -2511,7 +2755,7 @@ }, { "cell_type": "markdown", - "id": "075dc16d", + "id": "856128ab", "metadata": { "editable": true }, @@ -2521,7 +2765,7 @@ }, { "cell_type": "markdown", - "id": "afc0c6ba", + "id": "34fe5da7", "metadata": { "editable": true }, @@ -2533,7 +2777,7 @@ }, { "cell_type": "markdown", - "id": "90699b64", + "id": "4a9ff0cc", "metadata": { "editable": true }, @@ -2547,7 +2791,7 @@ }, { "cell_type": "markdown", - "id": "7778eda5", + "id": "23497353", "metadata": { "editable": true }, @@ -2559,7 +2803,7 @@ }, { "cell_type": "markdown", - "id": "d4e442e8", + "id": "6ec3c291", "metadata": { "editable": true }, @@ -2571,7 +2815,7 @@ }, { "cell_type": "markdown", - "id": "716705aa", + "id": "06a12980", "metadata": { "editable": true }, @@ -2581,7 +2825,7 @@ }, { "cell_type": "markdown", - "id": "9d45c586", + "id": "2066cd70", "metadata": { "editable": true }, @@ -2593,7 +2837,7 @@ }, { "cell_type": "markdown", - "id": "d97b6df8", + "id": "69e4f843", "metadata": { "editable": true }, @@ -2605,7 +2849,7 @@ }, { "cell_type": "markdown", - "id": "9d5b0c48", + "id": "30681f91", "metadata": { "editable": true }, @@ -2615,7 +2859,7 @@ }, { "cell_type": "markdown", - "id": "f9464957", + "id": "21a50c42", "metadata": { "editable": true }, @@ -2627,7 +2871,7 @@ }, { "cell_type": "markdown", - "id": "78b37719", + "id": "321d5425", "metadata": { "editable": true }, @@ -2639,7 +2883,7 @@ }, { "cell_type": "markdown", - "id": "c992c573", + "id": "05fc3334", "metadata": { "editable": true }, @@ -2668,7 +2912,7 @@ }, { "cell_type": "markdown", - "id": "646a658a", + "id": "cba959a3", "metadata": { "editable": true }, @@ -2694,7 +2938,7 @@ }, { "cell_type": "markdown", - "id": "6c96ca53", + "id": "e71a0657", "metadata": { "editable": true }, @@ -2724,7 +2968,7 @@ }, { "cell_type": "markdown", - "id": "9212cbb9", + "id": "66af926f", "metadata": { "editable": true }, @@ -2756,7 +3000,7 @@ }, { "cell_type": "markdown", - "id": "1c37d150", + "id": "dd0be212", "metadata": { "editable": true }, @@ -2784,7 +3028,7 @@ }, { "cell_type": "markdown", - "id": "2256d8e2", + "id": "728db2cf", "metadata": { "editable": true }, @@ -2804,7 +3048,7 @@ }, { "cell_type": "markdown", - "id": "7d4f20fb", + "id": "b6168ece", "metadata": { "editable": true }, @@ -2829,7 +3073,7 @@ }, { "cell_type": "markdown", - "id": "589ede45", + "id": "c840023c", "metadata": { "editable": true }, @@ -2841,7 +3085,7 @@ }, { "cell_type": "markdown", - "id": "5e373a3c", + "id": "d3db0dad", "metadata": { "editable": true }, @@ -2863,7 +3107,7 @@ }, { "cell_type": "markdown", - "id": "37f7b205", + "id": "17ea53df", "metadata": { "editable": true }, @@ -2885,7 +3129,7 @@ }, { "cell_type": "markdown", - "id": "a8895092", + "id": "316d1b83", "metadata": { "editable": true }, @@ -2911,7 +3155,7 @@ }, { "cell_type": "markdown", - "id": "0cc8ccf8", + "id": "f070429b", "metadata": { "editable": true }, @@ -2931,7 +3175,7 @@ }, { "cell_type": "markdown", - "id": "d3c3d77e", + "id": "220dd249", "metadata": { "editable": true }, @@ -2951,7 +3195,7 @@ }, { "cell_type": "markdown", - "id": "e4ef3c44", + "id": "5ae08182", "metadata": { "editable": true }, @@ -2977,7 +3221,7 @@ }, { "cell_type": "markdown", - "id": "bdcd9c2a", + "id": "697731e3", "metadata": { "editable": true }, @@ -3006,7 +3250,7 @@ }, { "cell_type": "markdown", - "id": "12fc172e", + "id": "80ca5f58", "metadata": { "editable": true }, @@ -3024,7 +3268,7 @@ }, { "cell_type": "markdown", - "id": "e1893183", + "id": "5dc02d47", "metadata": { "editable": true }, @@ -3040,7 +3284,7 @@ }, { "cell_type": "markdown", - "id": "cc9e29cb", + "id": "32bea05f", "metadata": { "editable": true }, @@ -3052,7 +3296,7 @@ }, { "cell_type": "markdown", - "id": "09a008a1", + "id": "832db7ee", "metadata": { "editable": true }, @@ -3066,7 +3310,7 @@ }, { "cell_type": "markdown", - "id": "0763c08d", + "id": "b653322c", "metadata": { "editable": true }, @@ -3086,7 +3330,7 @@ }, { "cell_type": "markdown", - "id": "f88cfdec", + "id": "5da91eaa", "metadata": { "editable": true }, @@ -3107,7 +3351,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "8cfde848", + "id": "e6d07675", "metadata": { "collapsed": false, "editable": true @@ -3248,7 +3492,7 @@ }, { "cell_type": "markdown", - "id": "db428e3c", + "id": "9a2dbfe9", "metadata": { "editable": true }, @@ -3264,7 +3508,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "2fb12f9f", + "id": "9f161a66", "metadata": { "collapsed": false, "editable": true @@ -3277,7 +3521,7 @@ }, { "cell_type": "markdown", - "id": "c546a093", + "id": "e68c9c7d", "metadata": { "editable": true }, @@ -3289,7 +3533,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "176f5b06", + "id": "604c9998", "metadata": { "collapsed": false, "editable": true @@ -3311,7 +3555,7 @@ }, { "cell_type": "markdown", - "id": "f8742c6f", + "id": "c9bfc10d", "metadata": { "editable": true }, @@ -3327,7 +3571,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "c8575866", + "id": "d8f49e20", "metadata": { "collapsed": false, "editable": true @@ -3365,7 +3609,7 @@ }, { "cell_type": "markdown", - "id": "683b630d", + "id": "df48f82a", "metadata": { "editable": true }, @@ -3378,7 +3622,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "31150cfc", + "id": "f403c12a", "metadata": { "collapsed": false, "editable": true @@ -3399,7 +3643,7 @@ }, { "cell_type": "markdown", - "id": "d77dda8d", + "id": "5623d0ea", "metadata": { "editable": true }, @@ -3415,7 +3659,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "050ba6e4", + "id": "3d62db42", "metadata": { "collapsed": false, "editable": true @@ -3473,7 +3717,7 @@ }, { "cell_type": "markdown", - "id": "b119382e", + "id": "88821a06", "metadata": { "editable": true }, @@ -3488,7 +3732,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "5657fb72", + "id": "12cbc833", "metadata": { "collapsed": false, "editable": true @@ -3509,7 +3753,7 @@ }, { "cell_type": "markdown", - "id": "fea8599d", + "id": "319a1a88", "metadata": { "editable": true }, @@ -3533,7 +3777,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "40b99950", + "id": "54868d01", "metadata": { "collapsed": false, "editable": true @@ -4005,7 +4249,7 @@ }, { "cell_type": "markdown", - "id": "0f05b3ba", + "id": "d748d83c", "metadata": { "editable": true }, @@ -4017,7 +4261,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "a506dacc", + "id": "910f47c9", "metadata": { "collapsed": false, "editable": true @@ -4061,7 +4305,7 @@ }, { "cell_type": "markdown", - "id": "3549dc80", + "id": "0c94d0f3", "metadata": { "editable": true }, @@ -4077,7 +4321,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "fefae858", + "id": "56b585f0", "metadata": { "collapsed": false, "editable": true @@ -4092,7 +4336,7 @@ }, { "cell_type": "markdown", - "id": "831120ea", + "id": "9ddb5c57", "metadata": { "editable": true }, @@ -4103,7 +4347,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "e8594b5a", + "id": "ad3e44f8", "metadata": { "collapsed": false, "editable": true @@ -4118,7 +4362,7 @@ }, { "cell_type": "markdown", - "id": "539d39f4", + "id": "dd207f7e", "metadata": { "editable": true }, @@ -4134,7 +4378,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "2004b61b", + "id": "116d71d6", "metadata": { "collapsed": false, "editable": true @@ -4148,7 +4392,7 @@ }, { "cell_type": "markdown", - "id": "a7fe6ddc", + "id": "2d9ecbf1", "metadata": { "editable": true }, @@ -4163,7 +4407,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "0a94909f", + "id": "88bf89ec", "metadata": { "collapsed": false, "editable": true @@ -4189,7 +4433,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "b3e5a3fd", + "id": "e3aadb3f", "metadata": { "collapsed": false, "editable": true @@ -4204,7 +4448,7 @@ }, { "cell_type": "markdown", - "id": "2304d744", + "id": "bf00c4e9", "metadata": { "editable": true }, @@ -4215,7 +4459,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "9df2287c", + "id": "63e8039f", "metadata": { "collapsed": false, "editable": true @@ -4230,7 +4474,7 @@ }, { "cell_type": "markdown", - "id": "66b9b602", + "id": "bfe13e9d", "metadata": { "editable": true }, @@ -4241,7 +4485,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "01cce111", + "id": "4b1bfe90", "metadata": { "collapsed": false, "editable": true @@ -4261,7 +4505,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "3c5abba7", + "id": "5f19bb11", "metadata": { "collapsed": false, "editable": true @@ -4276,7 +4520,7 @@ }, { "cell_type": "markdown", - "id": "240ac236", + "id": "6ea49d0f", "metadata": { "editable": true }, @@ -4291,7 +4535,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "bbd3572a", + "id": "07da3155", "metadata": { "collapsed": false, "editable": true @@ -4328,7 +4572,7 @@ }, { "cell_type": "markdown", - "id": "b142b5d2", + "id": "0961ab9b", "metadata": { "editable": true }, @@ -4341,7 +4585,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "f87f30d1", + "id": "0668907c", "metadata": { "collapsed": false, "editable": true @@ -4364,7 +4608,7 @@ }, { "cell_type": "markdown", - "id": "819b8944", + "id": "71069c49", "metadata": { "editable": true }, @@ -4374,7 +4618,7 @@ }, { "cell_type": "markdown", - "id": "feb82ce5", + "id": "bac25152", "metadata": { "editable": true }, @@ -4392,7 +4636,7 @@ }, { "cell_type": "markdown", - "id": "c01dc408", + "id": "c1d400f8", "metadata": { "editable": true }, @@ -4427,7 +4671,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "69c6af80", + "id": "12fe850e", "metadata": { "collapsed": false, "editable": true @@ -4439,7 +4683,7 @@ }, { "cell_type": "markdown", - "id": "347ae423", + "id": "0a6ab0cc", "metadata": { "editable": true }, @@ -4451,7 +4695,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "440b5c63", + "id": "a0e49457", "metadata": { "collapsed": false, "editable": true @@ -4464,7 +4708,7 @@ }, { "cell_type": "markdown", - "id": "d483bcd1", + "id": "c4ddfcbe", "metadata": { "editable": true }, @@ -4475,7 +4719,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "8e94bd59", + "id": "35b9c060", "metadata": { "collapsed": false, "editable": true @@ -4488,7 +4732,7 @@ }, { "cell_type": "markdown", - "id": "59f15944", + "id": "895ba32c", "metadata": { "editable": true }, @@ -4503,7 +4747,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "22c373db", + "id": "2e995a7a", "metadata": { "collapsed": false, "editable": true @@ -4515,7 +4759,7 @@ }, { "cell_type": "markdown", - "id": "06650c68", + "id": "43bbc8a4", "metadata": { "editable": true }, @@ -4527,7 +4771,7 @@ }, { "cell_type": "markdown", - "id": "078b1574", + "id": "e5685595", "metadata": { "editable": true }, @@ -4540,7 +4784,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "bf578e87", + "id": "c6023e61", "metadata": { "collapsed": false, "editable": true @@ -4597,7 +4841,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "f273ebcf", + "id": "1bae0551", "metadata": { "collapsed": false, "editable": true @@ -4626,7 +4870,7 @@ { "cell_type": "code", "execution_count": 28, - 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"cell_type": "markdown", + "id": "aab2170a", + "metadata": {}, + "source": [ + "\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "1bd81cd9", + "metadata": {}, + "source": [ + "# January 29-February 2 : Advanced machine learning and data analysis for the physical sciences\n", + "**Morten Hjorth-Jensen**, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan, USA\n", + "\n", + "Date: **January 30**" + ] + }, + { + "cell_type": "markdown", + "id": "29ea92e5", + "metadata": {}, + "source": [ + "## Overview of third week\n", + "\n", + "1. Discussion of possible projects\n", + "\n", + "2. Review of neural networks and automatic differentiation\n", + "\n", + "3. Discussion of codes\n", + "\n", + "4. [Video of lecture](https://youtu.be/OUTFo0oJadU)\n", + "\n", + "5. [Link to material for project suggestions](https://github.com/CompPhysics/AdvancedMachineLearning/tree/main/doc/Projects/2024/ProjectProposals)" + ] + }, + { + "cell_type": "markdown", + "id": "a2cc2fa2", + "metadata": {}, + "source": [ + "## Mathematics of deep learning\n", + "\n", + "**Two recent books online.**\n", + "\n", + "1. [The Modern Mathematics of Deep Learning, by Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen](https://arxiv.org/abs/2105.04026), published as [Mathematical Aspects of Deep Learning, pp. 1-111. Cambridge University Press, 2022](https://doi.org/10.1017/9781009025096.002)\n", + "\n", + "2. [Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory, Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger](https://doi.org/10.48550/arXiv.2310.20360)" + ] + }, + { + "cell_type": "markdown", + "id": "cecd85dc", + "metadata": {}, + "source": [ + "## Reminder on books with hands-on material and codes\n", + "* [Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch](https://sebastianraschka.com/blog/2022/ml-pytorch-book.html)\n", + "\n", + "* [David Foster, Generative Deep Learning with TensorFlow](https://www.oreilly.com/library/view/generative-deep-learning/9781098134174/ch01.html)\n", + "\n", + "* [Bali and Gavras, Generative AI with Python and TensorFlow 2](https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2)\n", + "\n", + "All three books have GitHub addresses from where one can download all codes. We will borrow most of the material from these three texts as well as \n", + "from Goodfellow, Bengio and Courville's text [Deep Learning](https://www.deeplearningbook.org/)" + ] + }, + { + "cell_type": "markdown", + "id": "cedb5428", + "metadata": {}, + "source": [ + "## Reading recommendations\n", + "\n", + "1. Rashkca et al., chapter 11, jupyter-notebook sent separately, from [GitHub](https://github.com/rasbt/machine-learning-book)\n", + "\n", + "2. Goodfellow et al, chapter 6 and 7 contain most of the neural network background." + ] + }, + { + "cell_type": "markdown", + "id": "ecd18965", + "metadata": {}, + "source": [ + "## Mathematics of deep learning and neural networks\n", + "\n", + "Neural networks, in its so-called feed-forward form, where each\n", + "iterations contains a feed-forward stage and a back-propgagation\n", + "stage, consist of series of affine matrix-matrix and matrix-vector\n", + "multiplications. The unknown parameters (the so-called biases and\n", + "weights which deternine the architecture of a neural network), are\n", + "uptaded iteratively using the so-called back-propagation algorithm.\n", + "This algorithm corresponds to the so-called reverse mode of \n", + "automatic differentation." + ] + }, + { + "cell_type": "markdown", + "id": "e822b4b7", + "metadata": {}, + "source": [ + "## Basics of an NN\n", + "\n", + "A neural network consists of a series of hidden layers, in addition to\n", + "the input and output layers. Each layer $l$ has a set of parameters\n", + "$\\boldsymbol{\\Theta}^{(l)}=(\\boldsymbol{W}^{(l)},\\boldsymbol{b}^{(l)})$ which are related to the\n", + "parameters in other layers through a series of affine transformations,\n", + "for a standard NN these are matrix-matrix and matrix-vector\n", + "multiplications. For all layers we will simply use a collective variable $\\boldsymbol{\\Theta}$.\n", + "\n", + "It consist of two basic steps:\n", + "1. a feed forward stage which takes a given input and produces a final output which is compared with the target values through our cost/loss function.\n", + "\n", + "2. a back-propagation state where the unknown parameters $\\boldsymbol{\\Theta}$ are updated through the optimization of the their gradients. The expressions for the gradients are obtained via the chain rule, starting from the derivative of the cost/function.\n", + "\n", + "These two steps make up one iteration. This iterative process is continued till we reach an eventual stopping criterion." + ] + }, + { + "cell_type": "markdown", + "id": "1753ad30", + "metadata": {}, + "source": [ + "## Overarching view of a neural network\n", + "\n", + "The architecture of a neural network defines our model. This model\n", + "aims at describing some function $f(\\boldsymbol{x}$ which represents\n", + "some final result (outputs or tagrget values) given a specific inpput\n", + "$\\boldsymbol{x}$. Note that here $\\boldsymbol{y}$ and $\\boldsymbol{x}$ are not limited to be\n", + "vectors.\n", + "\n", + "The architecture consists of\n", + "1. An input and an output layer where the input layer is defined by the inputs $\\boldsymbol{x}$. The output layer produces the model ouput $\\boldsymbol{\\tilde{y}}$ which is compared with the target value $\\boldsymbol{y}$\n", + "\n", + "2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)\n", + "\n", + "3. A given activation function $\\sigma(\\boldsymbol{z})$ with arguments $\\boldsymbol{z}$ to be defined below. The activation functions may differ from layer to layer.\n", + "\n", + "4. The last layer, normally called **output** layer has normally an activation function tailored to the specific problem\n", + "\n", + "5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model." + ] + }, + { + "cell_type": "markdown", + "id": "177dab3c", + "metadata": {}, + "source": [ + "## The optimization problem\n", + "\n", + "The cost function is a function of the unknown parameters\n", + "$\\boldsymbol{\\Theta}$ where the latter is a container for all possible\n", + "parameters needed to define a neural network\n", + "\n", + "If we are dealing with a regression task a typical cost/loss function\n", + "is the mean squared error" + ] + }, + { + "cell_type": "markdown", + "id": "4881c1a1", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{\\Theta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8efe3ea8", + "metadata": {}, + "source": [ + "This function represents one of many possible ways to define\n", + "the so-called cost function. Note that here we have assumed a linear dependence in terms of the paramters $\\boldsymbol{\\Theta}$. This is in general not the case." + ] + }, + { + "cell_type": "markdown", + "id": "7301a507", + "metadata": {}, + "source": [ + "## Parameters of neural networks\n", + "For neural networks the parameters\n", + "$\\boldsymbol{\\Theta}$ are given by the so-called weights and biases (to be\n", + "defined below).\n", + "\n", + "The weights are given by matrix elements $w_{ij}^{(l)}$ where the\n", + "superscript indicates the layer number. The biases are typically given\n", + "by vector elements representing each single node of a given layer,\n", + "that is $b_j^{(l)}$." + ] + }, + { + "cell_type": "markdown", + "id": "f98bf567", + "metadata": {}, + "source": [ + "## Other ingredients of a neural network\n", + "\n", + "Having defined the architecture of a neural network, the optimization\n", + "of the cost function with respect to the parameters $\\boldsymbol{\\Theta}$,\n", + "involves the calculations of gradients and their optimization. The\n", + "gradients represent the derivatives of a multidimensional object and\n", + "are often approximated by various gradient methods, including\n", + "1. various quasi-Newton methods,\n", + "\n", + "2. plain gradient descent (GD) with a constant learning rate $\\eta$,\n", + "\n", + "3. GD with momentum and other approximations to the learning rates such as\n", + "\n", + " * Adapative gradient (ADAgrad)\n", + "\n", + " * Root mean-square propagation (RMSprop)\n", + "\n", + " * Adaptive gradient with momentum (ADAM) and many other\n", + "\n", + "4. Stochastic gradient descent and various families of learning rate approximations" + ] + }, + { + "cell_type": "markdown", + "id": "e85b2410", + "metadata": {}, + "source": [ + "## Other parameters\n", + "\n", + "In addition to the above, there are often additional hyperparamaters\n", + "which are included in the setup of a neural network. These will be\n", + "discussed below." + ] + }, + { + "cell_type": "markdown", + "id": "6f2ca4f7", + "metadata": {}, + "source": [ + "## Universal approximation theorem\n", + "\n", + "The universal approximation theorem plays a central role in deep\n", + "learning. [Cybenko (1989)](https://link.springer.com/article/10.1007/BF02551274) showed\n", + "the following:\n", + "\n", + "Let $\\sigma$ be any continuous sigmoidal function such that" + ] + }, + { + "cell_type": "markdown", + "id": "300fd9e5", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma(z) = \\left\\{\\begin{array}{cc} 1 & z\\rightarrow \\infty\\\\ 0 & z \\rightarrow -\\infty \\end{array}\\right.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "118e03ae", + "metadata": {}, + "source": [ + "Given a continuous and deterministic function $F(\\boldsymbol{x})$ on the unit\n", + "cube in $d$-dimensions $F\\in [0,1]^d$, $x\\in [0,1]^d$ and a parameter\n", + "$\\epsilon >0$, there is a one-layer (hidden) neural network\n", + "$f(\\boldsymbol{x};\\boldsymbol{\\Theta})$ with $\\boldsymbol{\\Theta}=(\\boldsymbol{W},\\boldsymbol{b})$ and $\\boldsymbol{W}\\in\n", + "\\mathbb{R}^{m\\times n}$ and $\\boldsymbol{b}\\in \\mathbb{R}^{n}$, for which" + ] + }, + { + "cell_type": "markdown", + "id": "6428c149", + "metadata": {}, + "source": [ + "$$\n", + "\\vert F(\\boldsymbol{x})-f(\\boldsymbol{x};\\boldsymbol{\\Theta})\\vert < \\epsilon \\hspace{0.1cm} \\forall \\boldsymbol{x}\\in[0,1]^d.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7d83601f", + "metadata": {}, + "source": [ + "## Some parallels from real analysis\n", + "\n", + "For those of you familiar with for example the [Stone-Weierstrass\n", + "theorem](https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem)\n", + "for polynomial approximations or the convergence criterion for Fourier\n", + "series, there are similarities in the derivation of the proof for\n", + "neural networks." + ] + }, + { + "cell_type": "markdown", + "id": "c47e5a62", + "metadata": {}, + "source": [ + "## The approximation theorem in words\n", + "\n", + "**Any continuous function $y=F(\\boldsymbol{x})$ supported on the unit cube in\n", + "$d$-dimensions can be approximated by a one-layer sigmoidal network to\n", + "arbitrary accuracy.**\n", + "\n", + "[Hornik (1991)](https://www.sciencedirect.com/science/article/abs/pii/089360809190009T) extended the theorem by letting any non-constant, bounded activation function to be included using that the expectation value" + ] + }, + { + "cell_type": "markdown", + "id": "1d2f653b", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}[\\vert F(\\boldsymbol{x})\\vert^2] =\\int_{\\boldsymbol{x}\\in D} \\vert F(\\boldsymbol{x})\\vert^2p(\\boldsymbol{x})d\\boldsymbol{x} < \\infty.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "dc3f8cb4", + "metadata": {}, + "source": [ + "Then we have" + ] + }, + { + "cell_type": "markdown", + "id": "28d329ee", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}[\\vert F(\\boldsymbol{x})-f(\\boldsymbol{x};\\boldsymbol{\\Theta})\\vert^2] =\\int_{\\boldsymbol{x}\\in D} \\vert F(\\boldsymbol{x})-f(\\boldsymbol{x};\\boldsymbol{\\Theta})\\vert^2p(\\boldsymbol{x})d\\boldsymbol{x} < \\epsilon.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "82b63881", + "metadata": {}, + "source": [ + "## More on the general approximation theorem\n", + "\n", + "None of the proofs give any insight into the relation between the\n", + "number of of hidden layers and nodes and the approximation error\n", + "$\\epsilon$, nor the magnitudes of $\\boldsymbol{W}$ and $\\boldsymbol{b}$.\n", + "\n", + "Neural networks (NNs) have what we may call a kind of universality no matter what function we want to compute.\n", + "\n", + "It does not mean that an NN can be used to exactly compute any function. Rather, we get an approximation that is as good as we want." + ] + }, + { + "cell_type": "markdown", + "id": "d0ee1ac5", + "metadata": {}, + "source": [ + "## Class of functions we can approximate\n", + "\n", + "The class of functions that can be approximated are the continuous ones.\n", + "If the function $F(\\boldsymbol{x})$ is discontinuous, it won't in general be possible to approximate it. However, an NN may still give an approximation even if we fail in some points." + ] + }, + { + "cell_type": "markdown", + "id": "c6a0e720", + "metadata": {}, + "source": [ + "## Setting up the equations for a neural network\n", + "\n", + "The questions we want to ask are how do changes in the biases and the\n", + "weights in our network change the cost function and how can we use the\n", + "final output to modify the weights and biases?\n", + "\n", + "To derive these equations let us start with a plain regression problem\n", + "and define our cost function as" + ] + }, + { + "cell_type": "markdown", + "id": "2b6eecfa", + "metadata": {}, + "source": [ + "$$\n", + "{\\cal C}(\\boldsymbol{\\Theta}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - \\tilde{y}_i\\right)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b3738b9e", + "metadata": {}, + "source": [ + "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 $\\boldsymbol{x}$ are given by $\\boldsymbol{\\tilde{y}}_i$." + ] + }, + { + "cell_type": "markdown", + "id": "4028a89a", + "metadata": {}, + "source": [ + "## Layout of a neural network with three hidden layers\n", + "\n", + "\n", + "\n", + "\n", + "

    Figure 1:

    \n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "98e7592b", + "metadata": {}, + "source": [ + "## Definitions\n", + "\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", + "$\\hat{a}^{l-1}$ from the previous layer as" + ] + }, + { + "cell_type": "markdown", + "id": "9ad9bc5c", + "metadata": {}, + "source": [ + "$$\n", + "z_j^l = \\sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "447185c0", + "metadata": {}, + "source": [ + "where $b_k^l$ are the biases from layer $l$. Here $M_{l-1}$\n", + "represents the total number of nodes/neurons/units of layer $l-1$. The\n", + "figure in the whiteboard notes illustrates this equation. We can rewrite this in a more\n", + "compact form as the matrix-vector products we discussed earlier," + ] + }, + { + "cell_type": "markdown", + "id": "1bcbad43", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{z}^l = \\left(\\hat{W}^l\\right)^T\\hat{a}^{l-1}+\\hat{b}^l.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "62050d17", + "metadata": {}, + "source": [ + "## Inputs to the 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" + ] + }, + { + "cell_type": "markdown", + "id": "71278918", + "metadata": {}, + "source": [ + "$$\n", + "a_j^l = \\sigma(z_j^l) = \\frac{1}{1+\\exp{-(z_j^l)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1e73edcc", + "metadata": {}, + "source": [ + "## Derivatives and the chain rule\n", + "\n", + "From the definition of the activation $z_j^l$ we have" + ] + }, + { + "cell_type": "markdown", + "id": "ab5df9a5", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial z_j^l}{\\partial w_{ij}^l} = a_i^{l-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b4389fd7", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "4a950750", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial z_j^l}{\\partial a_i^{l-1}} = w_{ji}^l.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "868c60ff", + "metadata": {}, + "source": [ + "With our definition of the activation function we have that (note that this function depends only on $z_j^l$)" + ] + }, + { + "cell_type": "markdown", + "id": "ee1e3021", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial a_j^l}{\\partial z_j^{l}} = a_j^l(1-a_j^l)=\\sigma(z_j^l)(1-\\sigma(z_j^l)).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "56ea1322", + "metadata": {}, + "source": [ + "## Derivative of the cost function\n", + "\n", + "With these definitions we can now compute the derivative of the cost function in terms of the weights.\n", + "\n", + "Let us specialize to the output layer $l=L$. Our cost function is" + ] + }, + { + "cell_type": "markdown", + "id": "ae095881", + "metadata": {}, + "source": [ + "$$\n", + "{\\cal C}(\\boldsymbol{\\Theta}^L) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - \\tilde{y}_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - y_i\\right)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8fe5367d", + "metadata": {}, + "source": [ + "The derivative of this function with respect to the weights is" + ] + }, + { + "cell_type": "markdown", + "id": "1374c254", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial{\\cal C}(\\boldsymbol{\\Theta}^L)}{\\partial w_{jk}^L} = \\left(a_j^L - y_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "03f0501f", + "metadata": {}, + "source": [ + "The last partial derivative can easily be computed and reads (by applying the chain rule)" + ] + }, + { + "cell_type": "markdown", + "id": "2d9c19ee", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "dc1121f4", + "metadata": {}, + "source": [ + "## Simpler examples first, and automatic differentiation\n", + "\n", + "In order to understand the back propagation algorithm and its\n", + "derivation (an implementation of the chain rule), let us first digress\n", + "with some simple examples. These examples are also meant to motivate\n", + "the link with back propagation and [automatic differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation)." + ] + }, + { + "cell_type": "markdown", + "id": "ac618fa0", + "metadata": {}, + "source": [ + "## Reminder on the chain rule and gradients\n", + "\n", + "If we have a multivariate function $f(x,y)$ where $x=x(t)$ and $y=y(t)$ are functions of a variable $t$, we have that the gradient of $f$ with respect to $t$ (without the explicit unit vector components)" + ] + }, + { + "cell_type": "markdown", + "id": "43405ea8", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{dt} = \\begin{bmatrix}\\frac{\\partial f}{\\partial x} & \\frac{\\partial f}{\\partial y} \\end{bmatrix} \\begin{bmatrix}\\frac{\\partial x}{\\partial t} \\\\ \\frac{\\partial y}{\\partial t} \\end{bmatrix}=\\frac{\\partial f}{\\partial x} \\frac{\\partial x}{\\partial t} +\\frac{\\partial f}{\\partial y} \\frac{\\partial y}{\\partial t}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f5582370", + "metadata": {}, + "source": [ + "## Multivariable functions\n", + "\n", + "If we have a multivariate function $f(x,y)$ where $x=x(t,s)$ and $y=y(t,s)$ are functions of the variables $t$ and $s$, we have that the partial derivatives" + ] + }, + { + "cell_type": "markdown", + "id": "dfd5ae6c", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial s}=\\frac{\\partial f}{\\partial x}\\frac{\\partial x}{\\partial s}+\\frac{\\partial f}{\\partial y}\\frac{\\partial y}{\\partial s},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4f86cb27", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "e17ccd72", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial t}=\\frac{\\partial f}{\\partial x}\\frac{\\partial x}{\\partial t}+\\frac{\\partial f}{\\partial y}\\frac{\\partial y}{\\partial t}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e963806e", + "metadata": {}, + "source": [ + "the gradient of $f$ with respect to $t$ and $s$ (without the explicit unit vector components)" + ] + }, + { + "cell_type": "markdown", + "id": "a5237a7e", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{d(s,t)} = \\begin{bmatrix}\\frac{\\partial f}{\\partial x} & \\frac{\\partial f}{\\partial y} \\end{bmatrix} \\begin{bmatrix}\\frac{\\partial x}{\\partial s} &\\frac{\\partial x}{\\partial t} \\\\ \\frac{\\partial y}{\\partial s} & \\frac{\\partial y}{\\partial t} \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "12c58997", + "metadata": {}, + "source": [ + "## Automatic differentiation through examples\n", + "\n", + "A great introduction to automatic differentiation is given by Baydin et al., see .\n", + "\n", + "Automatic differentiation is a represented by a repeated application\n", + "of the chain rule on well-known functions and allows for the\n", + "calculation of derivatives to numerical precision. It is not the same\n", + "as the calculation of symbolic derivatives via for example SymPy, nor\n", + "does it use approximative formulae based on Taylor-expansions of a\n", + "function around a given value. The latter are error prone due to\n", + "truncation errors and values of the step size $\\Delta$." + ] + }, + { + "cell_type": "markdown", + "id": "d0a63af2", + "metadata": {}, + "source": [ + "## Simple example\n", + "\n", + "Our first example is rather simple," + ] + }, + { + "cell_type": "markdown", + "id": "dab248c4", + "metadata": {}, + "source": [ + "$$\n", + "f(x) =\\exp{x^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2cb4fb05", + "metadata": {}, + "source": [ + "with derivative" + ] + }, + { + "cell_type": "markdown", + "id": "2a298e4b", + "metadata": {}, + "source": [ + "$$\n", + "f'(x) =2x\\exp{x^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6dc4acbf", + "metadata": {}, + "source": [ + "We can use SymPy to extract the pertinent lines of Python code through the following simple example" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "cca0b96b", + "metadata": {}, + "outputs": [], + "source": [ + "from __future__ import division\n", + "from sympy import *\n", + "x = symbols('x')\n", + "expr = exp(x*x)\n", + "simplify(expr)\n", + "derivative = diff(expr,x)\n", + "print(python(expr))\n", + "print(python(derivative))" + ] + }, + { + "cell_type": "markdown", + "id": "bb9f566d", + "metadata": {}, + "source": [ + "## Smarter way of evaluating the above function\n", + "If we study this function, we note that we can reduce the number of operations by introducing an intermediate variable" + ] + }, + { + "cell_type": "markdown", + "id": "e68602cb", + "metadata": {}, + "source": [ + "$$\n", + "a = x^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "da2ab16d", + "metadata": {}, + "source": [ + "leading to" + ] + }, + { + "cell_type": "markdown", + "id": "5e2a24f9", + "metadata": {}, + "source": [ + "$$\n", + "f(x) = f(a(x)) = b= \\exp{a}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c931b292", + "metadata": {}, + "source": [ + "We now assume that all operations can be counted in terms of equal\n", + "floating point operations. This means that in order to calculate\n", + "$f(x)$ we need first to square $x$ and then compute the exponential. We\n", + "have thus two floating point operations only." + ] + }, + { + "cell_type": "markdown", + "id": "17b3056b", + "metadata": {}, + "source": [ + "## Reducing the number of operations\n", + "\n", + "With the introduction of a precalculated quantity $a$ and thereby $f(x)$ we have that the derivative can be written as" + ] + }, + { + "cell_type": "markdown", + "id": "ee5a7a9d", + "metadata": {}, + "source": [ + "$$\n", + "f'(x) = 2xb,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6cd2da78", + "metadata": {}, + "source": [ + "which reduces the number of operations from four in the orginal\n", + "expression to two. This means that if we need to compute $f(x)$ and\n", + "its derivative (a common task in optimizations), we have reduced the\n", + "number of operations from six to four in total.\n", + "\n", + "**Note** that the usage of a symbolic software like SymPy does not\n", + "include such simplifications and the calculations of the function and\n", + "the derivatives yield in general more floating point operations." + ] + }, + { + "cell_type": "markdown", + "id": "8e06a657", + "metadata": {}, + "source": [ + "## Chain rule, forward and reverse modes\n", + "\n", + "In the above example we have introduced the variables $a$ and $b$, and our function is" + ] + }, + { + "cell_type": "markdown", + "id": "46b00956", + "metadata": {}, + "source": [ + "$$\n", + "f(x) = f(a(x)) = b= \\exp{a},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "605894c4", + "metadata": {}, + "source": [ + "with $a=x^2$. We can decompose the derivative of $f$ with respect to $x$ as" + ] + }, + { + "cell_type": "markdown", + "id": "a62bf386", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{dx}=\\frac{df}{db}\\frac{db}{da}\\frac{da}{dx}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "de356218", + "metadata": {}, + "source": [ + "We note that since $b=f(x)$ that" + ] + }, + { + "cell_type": "markdown", + "id": "d5a1f929", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{db}=1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2249c76d", + "metadata": {}, + "source": [ + "leading to" + ] + }, + { + "cell_type": "markdown", + "id": "973586ae", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{dx}=\\frac{db}{da}\\frac{da}{dx}=2x\\exp{x^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d2b9bd9b", + "metadata": {}, + "source": [ + "as before." + ] + }, + { + "cell_type": "markdown", + "id": "4c445e81", + "metadata": {}, + "source": [ + "## Forward and reverse modes\n", + "\n", + "We have that" + ] + }, + { + "cell_type": "markdown", + "id": "9c86d400", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{dx}=\\frac{df}{db}\\frac{db}{da}\\frac{da}{dx},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5fc81fb2", + "metadata": {}, + "source": [ + "which we can rewrite either as" + ] + }, + { + "cell_type": "markdown", + "id": "03703809", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{dx}=\\left[\\frac{df}{db}\\frac{db}{da}\\right]\\frac{da}{dx},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6c1d033a", + "metadata": {}, + "source": [ + "or" + ] + }, + { + "cell_type": "markdown", + "id": "cf65ac5c", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{df}{dx}=\\frac{df}{db}\\left[\\frac{db}{da}\\frac{da}{dx}\\right].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0ad9942c", + "metadata": {}, + "source": [ + "The first expression is called reverse mode (or back propagation)\n", + "since we start by evaluating the derivatives at the end point and then\n", + "propagate backwards. This is the standard way of evaluating\n", + "derivatives (gradients) when optimizing the parameters of a neural\n", + "network. In the context of deep learning this is computationally\n", + "more efficient since the output of a neural network consists of either\n", + "one or some few other output variables.\n", + "\n", + "The second equation defines the so-called **forward mode**." + ] + }, + { + "cell_type": "markdown", + "id": "5b425fad", + "metadata": {}, + "source": [ + "## More complicated function\n", + "\n", + "We increase our ambitions and introduce a slightly more complicated function" + ] + }, + { + "cell_type": "markdown", + "id": "662ed332", + "metadata": {}, + "source": [ + "$$\n", + "f(x) =\\sqrt{x^2+exp{x^2}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "86f3ff9b", + "metadata": {}, + "source": [ + "with derivative" + ] + }, + { + "cell_type": "markdown", + "id": "8fe92564", + "metadata": {}, + "source": [ + "$$\n", + "f'(x) =\\frac{x(1+\\exp{x^2})}{\\sqrt{x^2+exp{x^2}}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5a914c32", + "metadata": {}, + "source": [ + "The corresponding SymPy code reads" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "64ddb47b", + "metadata": {}, + "outputs": [], + "source": [ + "from __future__ import division\n", + "from sympy import *\n", + "x = symbols('x')\n", + "expr = sqrt(x*x+exp(x*x))\n", + "simplify(expr)\n", + "derivative = diff(expr,x)\n", + "print(python(expr))\n", + "print(python(derivative))" + ] + }, + { + "cell_type": "markdown", + "id": "66d9c66e", + "metadata": {}, + "source": [ + "## Counting the number of floating point operations\n", + "\n", + "A simple count of operations shows that we need five operations for\n", + "the function itself and ten for the derivative. Fifteen operations in total if we wish to proceed with the above codes.\n", + "\n", + "Can we reduce this to\n", + "say half the number of operations?" + ] + }, + { + "cell_type": "markdown", + "id": "fbc1f092", + "metadata": {}, + "source": [ + "## Defining intermediate operations\n", + "\n", + "We can indeed reduce the number of operation to half of those listed in the brute force approach above.\n", + "We define the following quantities" + ] + }, + { + "cell_type": "markdown", + "id": "621baea0", + "metadata": {}, + "source": [ + "$$\n", + "a = x^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ebf674b6", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "87c1f6d9", + "metadata": {}, + "source": [ + "$$\n", + "b = \\exp{x^2} = \\exp{a},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5d2b47fe", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "545340e8", + "metadata": {}, + "source": [ + "$$\n", + "c= a+b,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "264638b7", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "f7875bc1", + "metadata": {}, + "source": [ + "$$\n", + "d=f(x)=\\sqrt{c}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4331cb65", + "metadata": {}, + "source": [ + "## New expression for the derivative\n", + "\n", + "With these definitions we obtain the following partial derivatives" + ] + }, + { + "cell_type": "markdown", + "id": "3cf0da90", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial a}{\\partial x} = 2x,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e8c7a997", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "c13d59ac", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial b}{\\partial a} = \\exp{a},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c8e3c278", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "8d68820e", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial c}{\\partial a} = 1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "58e8617c", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "9df9f1cd", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial c}{\\partial b} = 1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a5c1e218", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "ff44cfc4", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial d}{\\partial c} = \\frac{1}{2\\sqrt{c}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4c2a5d35", + "metadata": {}, + "source": [ + "and finally" + ] + }, + { + "cell_type": "markdown", + "id": "17abb6e0", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial d} = 1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7d2973dc", + "metadata": {}, + "source": [ + "## Final derivatives\n", + "Our final derivatives are thus" + ] + }, + { + "cell_type": "markdown", + "id": "a93bb4b2", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial c} = \\frac{\\partial f}{\\partial d} \\frac{\\partial d}{\\partial c} = \\frac{1}{2\\sqrt{c}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "677f30e9", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial b} = \\frac{\\partial f}{\\partial c} \\frac{\\partial c}{\\partial b} = \\frac{1}{2\\sqrt{c}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "43df8648", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial a} = \\frac{\\partial f}{\\partial c} \\frac{\\partial c}{\\partial a}+\n", + "\\frac{\\partial f}{\\partial b} \\frac{\\partial b}{\\partial a} = \\frac{1+\\exp{a}}{2\\sqrt{c}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3e630739", + "metadata": {}, + "source": [ + "and finally" + ] + }, + { + "cell_type": "markdown", + "id": "3283b158", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial x} = \\frac{\\partial f}{\\partial a} \\frac{\\partial a}{\\partial x} = \\frac{x(1+\\exp{a})}{\\sqrt{c}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "fff99695", + "metadata": {}, + "source": [ + "which is just" + ] + }, + { + "cell_type": "markdown", + "id": "d1d4beab", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial x} = \\frac{x(1+b)}{d},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3afd9dc2", + "metadata": {}, + "source": [ + "and requires only three operations if we can reuse all intermediate variables." + ] + }, + { + "cell_type": "markdown", + "id": "c4beed1f", + "metadata": {}, + "source": [ + "## In general not this simple\n", + "\n", + "In general, see the generalization below, unless we can obtain simple\n", + "analytical expressions which we can simplify further, the final\n", + "implementation of automatic differentiation involves repeated\n", + "calculations (and thereby operations) of derivatives of elementary\n", + "functions." + ] + }, + { + "cell_type": "markdown", + "id": "76f39e7a", + "metadata": {}, + "source": [ + "## Automatic differentiation\n", + "\n", + "We can make this example more formal. Automatic differentiation is a\n", + "formalization of the previous example (see graph).\n", + "\n", + "We define $\\boldsymbol{x}\\in x_1,\\dots, x_l$ input variables to a given function $f(\\boldsymbol{x})$ and $x_{l+1},\\dots, x_L$ intermediate variables.\n", + "\n", + "In the above example we have only one input variable, $l=1$ and four intermediate variables, that is" + ] + }, + { + "cell_type": "markdown", + "id": "22cf8332", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{bmatrix} x_1=x & x_2 = x^2=a & x_3 =\\exp{a}= b & x_4=c=a+b & x_5 = \\sqrt{c}=d \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3c3a2261", + "metadata": {}, + "source": [ + "Furthemore, for $i=l+1, \\dots, L$ (here $i=2,3,4,5$ and $f=x_L=d$), we\n", + "define the elementary functions $g_i(x_{Pa(x_i)})$ where $x_{Pa(x_i)}$ are the parent nodes of the variable $x_i$.\n", + "\n", + "In our case, we have for example for $x_3=g_3(x_{Pa(x_i)})=\\exp{a}$, that $g_3=\\exp{()}$ and $x_{Pa(x_3)}=a$." + ] + }, + { + "cell_type": "markdown", + "id": "840cbfe7", + "metadata": {}, + "source": [ + "## Chain rule\n", + "\n", + "We can now compute the gradients by back-propagating the derivatives using the chain rule.\n", + "We have defined" + ] + }, + { + "cell_type": "markdown", + "id": "e53e9f3c", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial x_L} = 1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4e3f9558", + "metadata": {}, + "source": [ + "which allows us to find the derivatives of the various variables $x_i$ as" + ] + }, + { + "cell_type": "markdown", + "id": "5b0958c4", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial f}{\\partial x_i} = \\sum_{x_j:x_i\\in Pa(x_j)}\\frac{\\partial f}{\\partial x_j} \\frac{\\partial x_j}{\\partial x_i}=\\sum_{x_j:x_i\\in Pa(x_j)}\\frac{\\partial f}{\\partial x_j} \\frac{\\partial g_j}{\\partial x_i}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "27c6d568", + "metadata": {}, + "source": [ + "Whenever we have a function which can be expressed as a computation\n", + "graph and the various functions can be expressed in terms of\n", + "elementary functions that are differentiable, then automatic\n", + "differentiation works. The functions may not need to be elementary\n", + "functions, they could also be computer programs, although not all\n", + "programs can be automatically differentiated." + ] + }, + { + "cell_type": "markdown", + "id": "d7eecc2c", + "metadata": {}, + "source": [ + "## First network example, simple percepetron with one input\n", + "\n", + "As yet another example we define now a simple perceptron model with all quantities given by scalars. We consider only one input variable $x$ and one target value $y$. We define an activation function $\\sigma_1$ which takes as input" + ] + }, + { + "cell_type": "markdown", + "id": "0f2ddd67", + "metadata": {}, + "source": [ + "$$\n", + "z_1 = w_1x+b_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "56f5f523", + "metadata": {}, + "source": [ + "where $w_1$ is the weight and $b_1$ is the bias. These are the\n", + "parameters we want to optimize. The output is $a_1=\\sigma(z_1)$ (see\n", + "graph from whiteboard notes). This output is then fed into the\n", + "**cost/loss** function, which we here for the sake of simplicity just\n", + "define as the squared error" + ] + }, + { + "cell_type": "markdown", + "id": "45472101", + "metadata": {}, + "source": [ + "$$\n", + "C(x;w_1,b_1)=\\frac{1}{2}(a_1-y)^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "baf7b0bd", + "metadata": {}, + "source": [ + "## Optimizing the parameters\n", + "\n", + "In setting up the feed forward and back propagation parts of the\n", + "algorithm, we need now the derivative of the various variables we want\n", + "to train.\n", + "\n", + "We need" + ] + }, + { + "cell_type": "markdown", + "id": "ce972a0c", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial w_1} \\hspace{0.1cm}\\mathrm{and}\\hspace{0.1cm}\\frac{\\partial C}{\\partial b_1}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2155ccd9", + "metadata": {}, + "source": [ + "Using the chain rule we find" + ] + }, + { + "cell_type": "markdown", + "id": "c071d63f", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial w_1}=\\frac{\\partial C}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial w_1}=(a_1-y)\\sigma_1'x,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "58dc2700", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "804bb5c2", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial b_1}=\\frac{\\partial C}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial b_1}=(a_1-y)\\sigma_1',\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "094d7f3f", + "metadata": {}, + "source": [ + "which we later will just define as" + ] + }, + { + "cell_type": "markdown", + "id": "4dc8cbb0", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}=\\delta_1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8d0b215e", + "metadata": {}, + "source": [ + "## Adding a hidden layer\n", + "\n", + "We change our simple model to a (see graph on whiteboard notes)\n", + "network with just one hidden layer but with scalar variables only.\n", + "\n", + "Our output variable changes to $a_2$ and $a_1$ is now the output from the hidden node and $a_0=x$.\n", + "We have then" + ] + }, + { + "cell_type": "markdown", + "id": "11e48362", + "metadata": {}, + "source": [ + "$$\n", + "z_1 = w_1a_0+b_1 \\hspace{0.1cm} \\wedge a_1 = \\sigma_1(z_1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2e32b0e5", + "metadata": {}, + "source": [ + "$$\n", + "z_2 = w_2a_1+b_2 \\hspace{0.1cm} \\wedge a_2 = \\sigma_2(z_2),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c46e1b63", + "metadata": {}, + "source": [ + "and the cost function" + ] + }, + { + "cell_type": "markdown", + "id": "020e7033", + "metadata": {}, + "source": [ + "$$\n", + "C(x;\\boldsymbol{\\Theta})=\\frac{1}{2}(a_2-y)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "52ccc8e3", + "metadata": {}, + "source": [ + "with $\\boldsymbol{\\Theta}=[w_1,w_2,b_1,b_2]$." + ] + }, + { + "cell_type": "markdown", + "id": "428f04f1", + "metadata": {}, + "source": [ + "## The derivatives\n", + "\n", + "The derivatives are now, using the chain rule again" + ] + }, + { + "cell_type": "markdown", + "id": "c68531b0", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial w_2}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial w_2}=(a_2-y)\\sigma_2'a_1=\\delta_2a_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bbb26b9f", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial b_2}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial b_2}=(a_2-y)\\sigma_2'=\\delta_2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d4320790", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial w_1}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial w_1}=(a_2-y)\\sigma_2'a_1\\sigma_1'a_0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "85de71a5", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial b_1}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial b_1}=(a_2-y)\\sigma_2'\\sigma_1'=\\delta_1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0cff7c34", + "metadata": {}, + "source": [ + "Can you generalize this to more than one hidden layer?" + ] + }, + { + "cell_type": "markdown", + "id": "48280819", + "metadata": {}, + "source": [ + "## Important observations\n", + "\n", + "From the above equations we see that the derivatives of the activation\n", + "functions play a central role. If they vanish, the training may\n", + "stop. This is called the vanishing gradient problem. If they become\n", + "large, the parameters $w_i$ and $b_i$ may simply go to infinity. This\n", + "is, pretty obvious, called the exploding gradient problem." + ] + }, + { + "cell_type": "markdown", + "id": "59662ad7", + "metadata": {}, + "source": [ + "## The training\n", + "\n", + "The training of the parameters is done through various gradient descent approximations with" + ] + }, + { + "cell_type": "markdown", + "id": "7bd57568", + "metadata": {}, + "source": [ + "$$\n", + "w_{i}\\leftarrow w_{i}- \\eta \\delta_i a_{i-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d29bd5d8", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "3cac6c38", + "metadata": {}, + "source": [ + "$$\n", + "b_i \\leftarrow b_i-\\eta \\delta_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5f9b3a9e", + "metadata": {}, + "source": [ + "with $\\eta$ is the learning rate.\n", + "\n", + "One iteration consists of one feed forward step and one back propagation step. Each back propagation step does one update of the parameters $\\boldsymbol{\\Theta}$.\n", + "\n", + "For the first hidden layer $a_{i-1}=a_0=x$ for this simple model." + ] + }, + { + "cell_type": "markdown", + "id": "79f167df", + "metadata": {}, + "source": [ + "## Code examples for the simple models" + ] + }, + { + "cell_type": "markdown", + "id": "3924f537", + "metadata": {}, + "source": [ + "## Simple neural network and the back propagation equations\n", + "\n", + "Let us now try to increase our level of ambitions and attempt to set up the equations for a neural network with two input nodes, one hidden layer with two hidden nodes and one utput layers.\n", + "\n", + "We need to define the following parameters and variables\n", + "**input layer (layer $(0)$).**\n", + "\n", + "Input nodes $x_0$ and $x_1$)" + ] + }, + { + "cell_type": "markdown", + "id": "1fd3642e", + "metadata": {}, + "source": [ + "$$\n", + "x_0 = a_0^{(0)} \\wedge x_1 = a_1^{(0)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5d4bb336", + "metadata": {}, + "source": [ + "**hidden layer (layer $(1)$).**\n", + "\n", + "with hidden nodes yielding the outputs $a_0^{(1)}$ and $a_1^{(1)}$) and weights $\\boldsymbol{w}$ and biases $\\boldsymbol{b}$" + ] + }, + { + "cell_type": "markdown", + "id": "98e1951b", + "metadata": {}, + "source": [ + "$$\n", + "w_{ij}^{(1)}=\\left\\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\\right\\} \\wedge b^{(1)}=\\left{ b_0^{(1)},b_1^{(1)}\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7db04ef3", + "metadata": {}, + "source": [ + "## The ouput layer\n", + "\n", + "Finally, we have the ouput layer given by layer label $(2)$ with output $a^{(2)}$ and weights and biases" + ] + }, + { + "cell_type": "markdown", + "id": "19f8654b", + "metadata": {}, + "source": [ + "$$\n", + "w_{i}^{(2)}=\\left\\{w_{0}^{(2)},w_{1}^{(2)}\\right\\} \\wedge b^{(2)}\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a44da164", + "metadata": {}, + "source": [ + "Our output is $\\tilde{y}=a^{(2)}$ and we define a generic cost function $C(a^{(2)},y;\\boldsymbol{\\Theta})$ where $y$ is the target value (a scalar here).\n", + "The parameters we need to optimize are given by" + ] + }, + { + "cell_type": "markdown", + "id": "c3f0e829", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\Theta}=\\left\\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "73c5f94f", + "metadata": {}, + "source": [ + "## Compact expressions\n", + "\n", + "We can define the inputs to the activation functions for the various layers in terms of various matrix-vector multiplications and vector additions.\n", + "The inputs to the first hidden layer are" + ] + }, + { + "cell_type": "markdown", + "id": "f0f9347e", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{bmatrix}z_0^{(1)} \\\\ z_1^{(1)} \\end{bmatrix}=\\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\\\ w_{10}^{(1)} &w_{11}^{(1)} \\end{bmatrix}\\begin{bmatrix}a_0^{(0)} \\\\ a_1^{(0)} \\end{bmatrix}+\\begin{bmatrix}b_0^{(1)} \\\\ b_1^{(1)} \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9e4ca185", + "metadata": {}, + "source": [ + "with outputs" + ] + }, + { + "cell_type": "markdown", + "id": "3d408f6a", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{bmatrix}a_0^{(1)} \\\\ a_1^{(1)} \\end{bmatrix}=\\begin{bmatrix}\\sigma^{(1)}(z_0^{(1)}) \\\\ \\sigma^{(1)}(z_1^{(1)}) \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2920f924", + "metadata": {}, + "source": [ + "## For the output layer" + ] + }, + { + "cell_type": "markdown", + "id": "7958efd8", + "metadata": {}, + "source": [ + "## Explicit derivatives\n", + "\n", + "Since we have only nine quantities we want to optimize for this system, we can for pedagogical reasons spell all out all of them.\n", + "Using the chain rule (or just the back propagation algorithm) we can find all derivatives. For the output and hidden layer we have thus" + ] + }, + { + "cell_type": "markdown", + "id": "78b950c8", + "metadata": {}, + "source": [ + "## Setting up the equations for the optimization\n", + "\n", + "For th" + ] + }, + { + "cell_type": "markdown", + "id": "1dbbdee5", + "metadata": {}, + "source": [ + "## Getting serious, the back propagation equations for a neural network\n", + "\n", + "Now it is time to move away from one node in each layer only. Our inputs are also represented either by several inputs.\n", + "\n", + "We have thus" + ] + }, + { + "cell_type": "markdown", + "id": "984b7fd6", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial{\\cal C}((\\boldsymbol{\\Theta}^L)}{\\partial w_{jk}^L} = \\left(a_j^L - y_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b09c2e59", + "metadata": {}, + "source": [ + "Defining" + ] + }, + { + "cell_type": "markdown", + "id": "c8d7a4e6", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - y_j\\right) = \\sigma'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4a813cea", + "metadata": {}, + "source": [ + "and using the Hadamard product of two vectors we can write this as" + ] + }, + { + "cell_type": "markdown", + "id": "0e09a1e1", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\delta}^L = \\sigma'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\boldsymbol{a}^L)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7ec6af7c", + "metadata": {}, + "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", + "much on a particular output node $j$, then $\\delta_j^L$ will be small,\n", + "which is what we would expect. The first term on the right, measures\n", + "how fast the activation function $f$ is changing at a given activation\n", + "value $z_j^L$." + ] + }, + { + "cell_type": "markdown", + "id": "17b74b7f", + "metadata": {}, + "source": [ + "## More considerations\n", + "\n", + "Notice that everything in the above equations is easily computed. In\n", + "particular, we compute $z_j^L$ while computing the behaviour of the\n", + "network, and it is only a small additional overhead to compute\n", + "$\\sigma'(z^L_j)$. The exact form of the derivative with respect to the\n", + "output depends on the form of the cost function.\n", + "However, provided the cost function is known there should be little\n", + "trouble in calculating" + ] + }, + { + "cell_type": "markdown", + "id": "6a3d1a6c", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5158f4ec", + "metadata": {}, + "source": [ + "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" + ] + }, + { + "cell_type": "markdown", + "id": "c923a350", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial{\\cal C}}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "74edad6b", + "metadata": {}, + "source": [ + "## Derivatives in terms of $z_j^L$\n", + "\n", + "It is also easy to see that our previous equation can be written as" + ] + }, + { + "cell_type": "markdown", + "id": "75fde721", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^L =\\frac{\\partial {\\cal C}}{\\partial z_j^L}= \\frac{\\partial {\\cal C}}{\\partial a_j^L}\\frac{\\partial a_j^L}{\\partial z_j^L},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7c5e74c3", + "metadata": {}, + "source": [ + "which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely" + ] + }, + { + "cell_type": "markdown", + "id": "84c9f004", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L}\\frac{\\partial b_j^L}{\\partial z_j^L}=\\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b1b4555c", + "metadata": {}, + "source": [ + "That is, the error $\\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias." + ] + }, + { + "cell_type": "markdown", + "id": "2966fca6", + "metadata": {}, + "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" + ] + }, + { + "cell_type": "markdown", + "id": "7e7314de", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1},\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e39830a4", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "92d3a3e3", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\delta_j^L = \\sigma'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8838ccc8", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "12eb944c", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", + "\\label{_auto3} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6fec0af3", + "metadata": {}, + "source": [ + "## Final back propagating equation\n", + "\n", + "We have that (replacing $L$ with a general layer $l$)" + ] + }, + { + "cell_type": "markdown", + "id": "d04f8f3f", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^l =\\frac{\\partial {\\cal C}}{\\partial z_j^l}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9d59178e", + "metadata": {}, + "source": [ + "We want to express this in terms of the equations for layer $l+1$." + ] + }, + { + "cell_type": "markdown", + "id": "99774079", + "metadata": {}, + "source": [ + "## Using the chain rule and summing over all $k$ entries\n", + "\n", + "We obtain" + ] + }, + { + "cell_type": "markdown", + "id": "140c731a", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^l =\\sum_k \\frac{\\partial {\\cal C}}{\\partial z_k^{l+1}}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}}=\\sum_k \\delta_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6ad40c12", + "metadata": {}, + "source": [ + "and recalling that" + ] + }, + { + "cell_type": "markdown", + "id": "d0b94459", + "metadata": {}, + "source": [ + "$$\n", + "z_j^{l+1} = \\sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "08e8ea4c", + "metadata": {}, + "source": [ + "with $M_l$ being the number of nodes in layer $l$, we obtain" + ] + }, + { + "cell_type": "markdown", + "id": "9e66e116", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^l =\\sum_k \\delta_k^{l+1}w_{kj}^{l+1}\\sigma'(z_j^l),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "64e1c072", + "metadata": {}, + "source": [ + "This is our final equation.\n", + "\n", + "We are now ready to set up the algorithm for back propagation and learning the weights and biases." + ] + }, + { + "cell_type": "markdown", + "id": "883f67ff", + "metadata": {}, + "source": [ + "## 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", + "$\\hat{z}_1$ 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", + "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", + "$l=1,2,3,\\dots,L$.\n", + "\n", + "**Notation**: The first hidden layer has $l=1$ as label and the final output layer has $l=L$." + ] + }, + { + "cell_type": "markdown", + "id": "ea192358", + "metadata": {}, + "source": [ + "## 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": "c8789d0e", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^L = \\sigma'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d6332cf3", + "metadata": {}, + "source": [ + "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,1$ as" + ] + }, + { + "cell_type": "markdown", + "id": "9988c609", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}\\sigma'(z_j^l).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3d5d076e", + "metadata": {}, + "source": [ + "## Setting up the Back propagation algorithm, part 3\n", + "\n", + "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": "29cf31df", + "metadata": {}, + "source": [ + "$$\n", + "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a4eb8d03", + "metadata": {}, + "source": [ + "$$\n", + "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "490f29c9", + "metadata": {}, + "source": [ + "with $\\eta$ being the learning rate." + ] + }, + { + "cell_type": "markdown", + "id": "e37488c0", + "metadata": {}, + "source": [ + "## Updating the gradients\n", + "\n", + "With the back propagate error for each $l=L-1,L-2,\\dots,1$ as" + ] + }, + { + "cell_type": "markdown", + "id": "769b6083", + "metadata": {}, + "source": [ + "$$\n", + "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}sigma'(z_j^l),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a802e409", + "metadata": {}, + "source": [ + "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": "03be34ec", + "metadata": {}, + "source": [ + "$$\n", + "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e6a80871", + "metadata": {}, + "source": [ + "$$\n", + "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6e1bf4b6", + "metadata": {}, + "source": [ + "## Fine-tuning neural network hyperparameters\n", + "\n", + "The flexibility of neural networks is also one of their main\n", + "drawbacks: there are many hyperparameters to tweak. Not only can you\n", + "use any imaginable network topology (how neurons/nodes are\n", + "interconnected), but even in a simple FFNN you can change the number\n", + "of layers, the number of neurons per layer, the type of activation\n", + "function to use in each layer, the weight initialization logic, the\n", + "stochastic gradient optmized and much more. How do you know what\n", + "combination of hyperparameters is the best for your task?\n", + "\n", + "* You can use grid search with cross-validation to find the right hyperparameters.\n", + "\n", + "However,since there are many hyperparameters to tune, and since\n", + "training a neural network on a large dataset takes a lot of time, you\n", + "will only be able to explore a tiny part of the hyperparameter space.\n", + "\n", + "* You can use randomized search.\n", + "\n", + "* Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly." + ] + }, + { + "cell_type": "markdown", + "id": "2da8c7a2", + "metadata": {}, + "source": [ + "## Hidden layers\n", + "\n", + "For many problems you can start with just one or two hidden layers and\n", + "it will work just fine. For the MNIST data set you ca easily get a\n", + "high accuracy using just one hidden layer with a few hundred neurons.\n", + "You can reach for this data set above 98% accuracy using two hidden\n", + "layers with the same total amount of neurons, in roughly the same\n", + "amount of training time.\n", + "\n", + "For more complex problems, you can gradually ramp up the number of\n", + "hidden layers, until you start overfitting the training set. Very\n", + "complex tasks, such as large image classification or speech\n", + "recognition, typically require networks with dozens of layers and they\n", + "need a huge amount of training data. However, you will rarely have to\n", + "train such networks from scratch: it is much more common to reuse\n", + "parts of a pretrained state-of-the-art network that performs a similar\n", + "task." + ] + }, + { + "cell_type": "markdown", + "id": "3dc9c620", + "metadata": {}, + "source": [ + "## Which activation function should I use?\n", + "\n", + "The Back propagation algorithm we derived above works by going from\n", + "the output layer to the input layer, propagating the error gradient on\n", + "the way. Once the algorithm has computed the gradient of the cost\n", + "function with regards to each parameter in the network, it uses these\n", + "gradients to update each parameter with a Gradient Descent (GD) step.\n", + "\n", + "Unfortunately for us, the gradients often get smaller and smaller as\n", + "the algorithm progresses down to the first hidden layers. As a result,\n", + "the GD update leaves the lower layer connection weights virtually\n", + "unchanged, and training never converges to a good solution. This is\n", + "known in the literature as **the vanishing gradients problem**.\n", + "\n", + "In other cases, the opposite can happen, namely the the gradients can\n", + "grow bigger and bigger. The result is that many of the layers get\n", + "large updates of the weights the algorithm diverges. This is the\n", + "**exploding gradients problem**, which is mostly encountered in\n", + "recurrent neural networks. More generally, deep neural networks suffer\n", + "from unstable gradients, different layers may learn at widely\n", + "different speeds" + ] + }, + { + "cell_type": "markdown", + "id": "68544785", + "metadata": {}, + "source": [ + "## Is the Logistic activation function (Sigmoid) our choice?\n", + "\n", + "Although this unfortunate behavior has been empirically observed for\n", + "quite a while (it was one of the reasons why deep neural networks were\n", + "mostly abandoned for a long time), it is only around 2010 that\n", + "significant progress was made in understanding it.\n", + "\n", + "A paper titled [Understanding the Difficulty of Training Deep\n", + "Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio](http://proceedings.mlr.press/v9/glorot10a.html) found that\n", + "the problems with the popular logistic\n", + "sigmoid activation function and the weight initialization technique\n", + "that was most popular at the time, namely random initialization using\n", + "a normal distribution with a mean of 0 and a standard deviation of\n", + "1. \n", + "\n", + "They showed that with this activation function and this\n", + "initialization scheme, the variance of the outputs of each layer is\n", + "much greater than the variance of its inputs. Going forward in the\n", + "network, the variance keeps increasing after each layer until the\n", + "activation function saturates at the top layers. This is actually made\n", + "worse by the fact that the logistic function has a mean of 0.5, not 0\n", + "(the hyperbolic tangent function has a mean of 0 and behaves slightly\n", + "better than the logistic function in deep networks)." + ] + }, + { + "cell_type": "markdown", + "id": "4df2cf86", + "metadata": {}, + "source": [ + "## The derivative of the Logistic funtion\n", + "\n", + "Looking at the logistic activation function, when inputs become large\n", + "(negative or positive), the function saturates at 0 or 1, with a\n", + "derivative extremely close to 0. Thus when backpropagation kicks in,\n", + "it has virtually no gradient to propagate back through the network,\n", + "and what little gradient exists keeps getting diluted as\n", + "backpropagation progresses down through the top layers, so there is\n", + "really nothing left for the lower layers.\n", + "\n", + "In their paper, Glorot and Bengio propose a way to significantly\n", + "alleviate this problem. We need the signal to flow properly in both\n", + "directions: in the forward direction when making predictions, and in\n", + "the reverse direction when backpropagating gradients. We don’t want\n", + "the signal to die out, nor do we want it to explode and saturate. For\n", + "the signal to flow properly, the authors argue that we need the\n", + "variance of the outputs of each layer to be equal to the variance of\n", + "its inputs, and we also need the gradients to have equal variance\n", + "before and after flowing through a layer in the reverse direction." + ] + }, + { + "cell_type": "markdown", + "id": "f992cd00", + "metadata": {}, + "source": [ + "## Insights from the paper by Glorot and Bengio\n", + "\n", + "One of the insights in the 2010 paper by Glorot and Bengio was that\n", + "the vanishing/exploding gradients problems were in part due to a poor\n", + "choice of activation function. Until then most people had assumed that\n", + "if Nature had chosen to use roughly sigmoid activation functions in\n", + "biological neurons, they must be an excellent choice. But it turns out\n", + "that other activation functions behave much better in deep neural\n", + "networks, in particular the ReLU activation function, mostly because\n", + "it does not saturate for positive values (and also because it is quite\n", + "fast to compute)." + ] + }, + { + "cell_type": "markdown", + "id": "93a1c57b", + "metadata": {}, + "source": [ + "## The RELU function family\n", + "\n", + "The ReLU activation function suffers from a problem known as the dying\n", + "ReLUs: during training, some neurons effectively die, meaning they\n", + "stop outputting anything other than 0.\n", + "\n", + "In some cases, you may find that half of your network’s neurons are\n", + "dead, especially if you used a large learning rate. During training,\n", + "if a neuron’s weights get updated such that the weighted sum of the\n", + "neuron’s inputs is negative, it will start outputting 0. When this\n", + "happen, the neuron is unlikely to come back to life since the gradient\n", + "of the ReLU function is 0 when its input is negative.\n", + "\n", + "To solve this problem, nowadays practitioners use a variant of the\n", + "ReLU function, such as the leaky ReLU discussed above or the so-called\n", + "exponential linear unit (ELU) function" + ] + }, + { + "cell_type": "markdown", + "id": "1ced80ba", + "metadata": {}, + "source": [ + "$$\n", + "ELU(z) = \\left\\{\\begin{array}{cc} \\alpha\\left( \\exp{(z)}-1\\right) & z < 0,\\\\ z & z \\ge 0.\\end{array}\\right.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "32022612", + "metadata": {}, + "source": [ + "## Which activation function should we use?\n", + "\n", + "In general it seems that the ELU activation function is better than\n", + "the leaky ReLU function (and its variants), which is better than\n", + "ReLU. ReLU performs better than $\\tanh$ which in turn performs better\n", + "than the logistic function.\n", + "\n", + "If runtime performance is an issue, then you may opt for the leaky\n", + "ReLU function over the ELU function If you don’t want to tweak yet\n", + "another hyperparameter, you may just use the default $\\alpha$ of\n", + "$0.01$ for the leaky ReLU, and $1$ for ELU. If you have spare time and\n", + "computing power, you can use cross-validation or bootstrap to evaluate\n", + "other activation functions." + ] + }, + { + "cell_type": "markdown", + "id": "41ef7fa9", + "metadata": {}, + "source": [ + "## More on activation functions, output layers\n", + "\n", + "In most cases you can use the ReLU activation function in the hidden\n", + "layers (or one of its variants).\n", + "\n", + "It is a bit faster to compute than other activation functions, and the\n", + "gradient descent optimization does in general not get stuck.\n", + "\n", + "**For the output layer:**\n", + "\n", + "* For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).\n", + "\n", + "* For regression tasks, you can simply use no activation function at all." + ] + }, + { + "cell_type": "markdown", + "id": "ac06391c", + "metadata": {}, + "source": [ + "## Batch Normalization\n", + "\n", + "Batch Normalization aims to address the vanishing/exploding gradients\n", + "problems, and more generally the problem that the distribution of each\n", + "layer’s inputs changes during training, as the parameters of the\n", + "previous layers change.\n", + "\n", + "The technique consists of adding an operation in the model just before\n", + "the activation function of each layer, simply zero-centering and\n", + "normalizing the inputs, then scaling and shifting the result using two\n", + "new parameters per layer (one for scaling, the other for shifting). In\n", + "other words, this operation lets the model learn the optimal scale and\n", + "mean of the inputs for each layer. In order to zero-center and\n", + "normalize the inputs, the algorithm needs to estimate the inputs’ mean\n", + "and standard deviation. It does so by evaluating the mean and standard\n", + "deviation of the inputs over the current mini-batch, from this the\n", + "name batch normalization." + ] + }, + { + "cell_type": "markdown", + "id": "c28b92f7", + "metadata": {}, + "source": [ + "## Dropout\n", + "\n", + "It is a fairly simple algorithm: at every training step, every neuron\n", + "(including the input neurons but excluding the output neurons) has a\n", + "probability $p$ of being temporarily dropped out, meaning it will be\n", + "entirely ignored during this training step, but it may be active\n", + "during the next step.\n", + "\n", + "The hyperparameter $p$ is called the dropout rate, and it is typically\n", + "set to 50%. After training, the neurons are not dropped anymore. It\n", + "is viewed as one of the most popular regularization techniques." + ] + }, + { + "cell_type": "markdown", + "id": "9614d3b5", + "metadata": {}, + "source": [ + "## Gradient Clipping\n", + "\n", + "A popular technique to lessen the exploding gradients problem is to\n", + "simply clip the gradients during backpropagation so that they never\n", + "exceed some threshold (this is mostly useful for recurrent neural\n", + "networks).\n", + "\n", + "This technique is called Gradient Clipping.\n", + "\n", + "In general however, Batch\n", + "Normalization is preferred." + ] + }, + { + "cell_type": "markdown", + "id": "5c0cd7be", + "metadata": {}, + "source": [ + "## A top-down perspective on Neural networks\n", + "\n", + "The first thing we would like to do is divide the data into two or\n", + "three parts. A training set, a validation or dev (development) set,\n", + "and a test set. The test set is the data on which we want to make\n", + "predictions. The dev set is a subset of the training data we use to\n", + "check how well we are doing out-of-sample, after training the model on\n", + "the training dataset. We use the validation error as a proxy for the\n", + "test error in order to make tweaks to our model. It is crucial that we\n", + "do not use any of the test data to train the algorithm. This is a\n", + "cardinal sin in ML. Then:\n", + "\n", + "* Estimate optimal error rate\n", + "\n", + "* Minimize underfitting (bias) on training data set.\n", + "\n", + "* Make sure you are not overfitting." + ] + }, + { + "cell_type": "markdown", + "id": "7ece19dd", + "metadata": {}, + "source": [ + "## More top-down perspectives\n", + "\n", + "If the validation and test sets are drawn from the same distributions,\n", + "then a good performance on the validation set should lead to similarly\n", + "good performance on the test set. \n", + "\n", + "However, sometimes\n", + "the training data and test data differ in subtle ways because, for\n", + "example, they are collected using slightly different methods, or\n", + "because it is cheaper to collect data in one way versus another. In\n", + "this case, there can be a mismatch between the training and test\n", + "data. This can lead to the neural network overfitting these small\n", + "differences between the test and training sets, and a poor performance\n", + "on the test set despite having a good performance on the validation\n", + "set. To rectify this, Andrew Ng suggests making two validation or dev\n", + "sets, one constructed from the training data and one constructed from\n", + "the test data. The difference between the performance of the algorithm\n", + "on these two validation sets quantifies the train-test mismatch. This\n", + "can serve as another important diagnostic when using DNNs for\n", + "supervised learning." + ] + }, + { + "cell_type": "markdown", + "id": "67c68979", + "metadata": {}, + "source": [ + "## Limitations of supervised learning with deep networks\n", + "\n", + "Like all statistical methods, supervised learning using neural\n", + "networks has important limitations. This is especially important when\n", + "one seeks to apply these methods, especially to physics problems. Like\n", + "all tools, DNNs are not a universal solution. Often, the same or\n", + "better performance on a task can be achieved by using a few\n", + "hand-engineered features (or even a collection of random\n", + "features)." + ] + }, + { + "cell_type": "markdown", + "id": "1742515c", + "metadata": {}, + "source": [ + "## Limitations of NNs\n", + "\n", + "Here we list some of the important limitations of supervised neural network based models. \n", + "\n", + "* **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).\n", + "\n", + "* **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs." + ] + }, + { + "cell_type": "markdown", + "id": "79aadc50", + "metadata": {}, + "source": [ + "## Homogeneous data\n", + "\n", + "* **Homogeneous data.** Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types." + ] + }, + { + "cell_type": "markdown", + "id": "9a4578b9", + "metadata": {}, + "source": [ + "## More limitations\n", + "\n", + "* **Many problems are not about prediction.** In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a *wrong* model. The model might or might not be useful for understanding the underlying science.\n", + "\n", + "Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems." + ] + }, + { + "cell_type": "markdown", + "id": "151aa1d0", + "metadata": {}, + "source": [ + "## Building a neural network code\n", + "\n", + "Here we present a flexible object oriented codebase\n", + "for a feed forward neural network, along with a demonstration of how\n", + "to use it. Before we get into the details of the neural network, we\n", + "will first present some implementations of various schedulers, cost\n", + "functions and activation functions that can be used together with the\n", + "neural network.\n", + "\n", + "The codes here were developed by Eric Reber and Gregor Kajda during spring 2023.\n", + "After these codes we present the TensorFlow inplementation. Pytorch will be discussed next week." + ] + }, + { + "cell_type": "markdown", + "id": "b4eba603", + "metadata": {}, + "source": [ + "## Learning rate methods\n", + "\n", + "The code below shows object oriented implementations of the Constant,\n", + "Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All\n", + "of the classes belong to the shared abstract **Scheduler class**, and\n", + "share the update$\\_$change() and reset() methods allowing for any of the\n", + "schedulers to be seamlessly used during the training stage, as will\n", + "later be shown in the fit() method of the neural\n", + "network. Update$\\_$change() only has one parameter, the gradient, and returns the change which will be subtracted\n", + "from the weights. The reset() function takes no parameters, and resets\n", + "the desired variables. For Constant and Momentum, reset does nothing." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "a6928ae9", + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "\n", + "class Scheduler:\n", + " \"\"\"\n", + " Abstract class for Schedulers\n", + " \"\"\"\n", + "\n", + " def __init__(self, eta):\n", + " self.eta = eta\n", + "\n", + " # should be overwritten\n", + " def update_change(self, gradient):\n", + " raise NotImplementedError\n", + "\n", + " # overwritten if needed\n", + " def reset(self):\n", + " pass\n", + "\n", + "\n", + "class Constant(Scheduler):\n", + " def __init__(self, eta):\n", + " super().__init__(eta)\n", + "\n", + " def update_change(self, gradient):\n", + " return self.eta * gradient\n", + " \n", + " def reset(self):\n", + " pass\n", + "\n", + "\n", + "class Momentum(Scheduler):\n", + " def __init__(self, eta: float, momentum: float):\n", + " super().__init__(eta)\n", + " self.momentum = momentum\n", + " self.change = 0\n", + "\n", + " def update_change(self, gradient):\n", + " self.change = self.momentum * self.change + self.eta * gradient\n", + " return self.change\n", + "\n", + " def reset(self):\n", + " pass\n", + "\n", + "\n", + "class Adagrad(Scheduler):\n", + " def __init__(self, eta):\n", + " super().__init__(eta)\n", + " self.G_t = None\n", + "\n", + " def update_change(self, gradient):\n", + " delta = 1e-8 # avoid division ny zero\n", + "\n", + " if self.G_t is None:\n", + " self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))\n", + "\n", + " self.G_t += gradient @ gradient.T\n", + "\n", + " G_t_inverse = 1 / (\n", + " delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))\n", + " )\n", + " return self.eta * gradient * G_t_inverse\n", + "\n", + " def reset(self):\n", + " self.G_t = None\n", + "\n", + "\n", + "class AdagradMomentum(Scheduler):\n", + " def __init__(self, eta, momentum):\n", + " super().__init__(eta)\n", + " self.G_t = None\n", + " self.momentum = momentum\n", + " self.change = 0\n", + "\n", + " def update_change(self, gradient):\n", + " delta = 1e-8 # avoid division ny zero\n", + "\n", + " if self.G_t is None:\n", + " self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))\n", + "\n", + " self.G_t += gradient @ gradient.T\n", + "\n", + " G_t_inverse = 1 / (\n", + " delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))\n", + " )\n", + " self.change = self.change * self.momentum + self.eta * gradient * G_t_inverse\n", + " return self.change\n", + "\n", + " def reset(self):\n", + " self.G_t = None\n", + "\n", + "\n", + "class RMS_prop(Scheduler):\n", + " def __init__(self, eta, rho):\n", + " super().__init__(eta)\n", + " self.rho = rho\n", + " self.second = 0.0\n", + "\n", + " def update_change(self, gradient):\n", + " delta = 1e-8 # avoid division ny zero\n", + " self.second = self.rho * self.second + (1 - self.rho) * gradient * gradient\n", + " return self.eta * gradient / (np.sqrt(self.second + delta))\n", + "\n", + " def reset(self):\n", + " self.second = 0.0\n", + "\n", + "\n", + "class Adam(Scheduler):\n", + " def __init__(self, eta, rho, rho2):\n", + " super().__init__(eta)\n", + " self.rho = rho\n", + " self.rho2 = rho2\n", + " self.moment = 0\n", + " self.second = 0\n", + " self.n_epochs = 1\n", + "\n", + " def update_change(self, gradient):\n", + " delta = 1e-8 # avoid division ny zero\n", + "\n", + " self.moment = self.rho * self.moment + (1 - self.rho) * gradient\n", + " self.second = self.rho2 * self.second + (1 - self.rho2) * gradient * gradient\n", + "\n", + " moment_corrected = self.moment / (1 - self.rho**self.n_epochs)\n", + " second_corrected = self.second / (1 - self.rho2**self.n_epochs)\n", + "\n", + " return self.eta * moment_corrected / (np.sqrt(second_corrected + delta))\n", + "\n", + " def reset(self):\n", + " self.n_epochs += 1\n", + " self.moment = 0\n", + " self.second = 0" + ] + }, + { + "cell_type": "markdown", + "id": "5d4c8f91", + "metadata": {}, + "source": [ + "## Usage of the above learning rate schedulers\n", + "\n", + "To initalize a scheduler, simply create the object and pass in the\n", + "necessary parameters such as the learning rate and the momentum as\n", + "shown below. As the Scheduler class is an abstract class it should not\n", + "called directly, and will raise an error upon usage." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "9c3ffda8", + "metadata": {}, + "outputs": [], + "source": [ + "momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)\n", + "adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)" + ] + }, + { + "cell_type": "markdown", + "id": "a107dbbd", + "metadata": {}, + "source": [ + "Here is a small example for how a segment of code using schedulers\n", + "could look. Switching out the schedulers is simple." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "281e616b", + "metadata": {}, + "outputs": [], + "source": [ + "weights = np.ones((3,3))\n", + "print(f\"Before scheduler:\\n{weights=}\")\n", + "\n", + "epochs = 10\n", + "for e in range(epochs):\n", + " gradient = np.random.rand(3, 3)\n", + " change = adam_scheduler.update_change(gradient)\n", + " weights = weights - change\n", + " adam_scheduler.reset()\n", + "\n", + "print(f\"\\nAfter scheduler:\\n{weights=}\")" + ] + }, + { + "cell_type": "markdown", + "id": "e39f085c", + "metadata": {}, + "source": [ + "## Cost functions\n", + "\n", + "Here we discuss cost functions that can be used when creating the\n", + "neural network. Every cost function takes the target vector as its\n", + "parameter, and returns a function valued only at $x$ such that it may\n", + "easily be differentiated." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "2ee26200", + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "\n", + "def CostOLS(target):\n", + " \n", + " def func(X):\n", + " return (1.0 / target.shape[0]) * np.sum((target - X) ** 2)\n", + "\n", + " return func\n", + "\n", + "\n", + "def CostLogReg(target):\n", + "\n", + " def func(X):\n", + " \n", + " return -(1.0 / target.shape[0]) * np.sum(\n", + " (target * np.log(X + 10e-10)) + ((1 - target) * np.log(1 - X + 10e-10))\n", + " )\n", + "\n", + " return func\n", + "\n", + "\n", + "def CostCrossEntropy(target):\n", + " \n", + " def func(X):\n", + " return -(1.0 / target.size) * np.sum(target * np.log(X + 10e-10))\n", + "\n", + " return func" + ] + }, + { + "cell_type": "markdown", + "id": "972d889d", + "metadata": {}, + "source": [ + "Below we give a short example of how these cost function may be used\n", + "to obtain results if you wish to test them out on your own using\n", + "AutoGrad's automatics differentiation." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "8f26e796", + "metadata": {}, + "outputs": [], + "source": [ + "from autograd import grad\n", + "\n", + "target = np.array([[1, 2, 3]]).T\n", + "a = np.array([[4, 5, 6]]).T\n", + "\n", + "cost_func = CostCrossEntropy\n", + "cost_func_derivative = grad(cost_func(target))\n", + "\n", + "valued_at_a = cost_func_derivative(a)\n", + "print(f\"Derivative of cost function {cost_func.__name__} valued at a:\\n{valued_at_a}\")" + ] + }, + { + "cell_type": "markdown", + "id": "d53f05b5", + "metadata": {}, + "source": [ + "## Activation functions\n", + "\n", + "Finally, before we look at the neural network, we will look at the\n", + "activation functions which can be specified between the hidden layers\n", + "and as the output function. Each function can be valued for any given\n", + "vector or matrix X, and can be differentiated via derivate()." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "b3fa8e1f", + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import elementwise_grad\n", + "\n", + "def identity(X):\n", + " return X\n", + "\n", + "\n", + "def sigmoid(X):\n", + " try:\n", + " return 1.0 / (1 + np.exp(-X))\n", + " except FloatingPointError:\n", + " return np.where(X > np.zeros(X.shape), np.ones(X.shape), np.zeros(X.shape))\n", + "\n", + "\n", + "def softmax(X):\n", + " X = X - np.max(X, axis=-1, keepdims=True)\n", + " delta = 10e-10\n", + " return np.exp(X) / (np.sum(np.exp(X), axis=-1, keepdims=True) + delta)\n", + "\n", + "\n", + "def RELU(X):\n", + " return np.where(X > np.zeros(X.shape), X, np.zeros(X.shape))\n", + "\n", + "\n", + "def LRELU(X):\n", + " delta = 10e-4\n", + " return np.where(X > np.zeros(X.shape), X, delta * X)\n", + "\n", + "\n", + "def derivate(func):\n", + " if func.__name__ == \"RELU\":\n", + "\n", + " def func(X):\n", + " return np.where(X > 0, 1, 0)\n", + "\n", + " return func\n", + "\n", + " elif func.__name__ == \"LRELU\":\n", + "\n", + " def func(X):\n", + " delta = 10e-4\n", + " return np.where(X > 0, 1, delta)\n", + "\n", + " return func\n", + "\n", + " else:\n", + " return elementwise_grad(func)" + ] + }, + { + "cell_type": "markdown", + "id": "beb758fc", + "metadata": {}, + "source": [ + "Below follows a short demonstration of how to use an activation\n", + "function. The derivative of the activation function will be important\n", + "when calculating the output delta term during backpropagation. Note\n", + "that derivate() can also be used for cost functions for a more\n", + "generalized approach." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "ae884966", + "metadata": {}, + "outputs": [], + "source": [ + "z = np.array([[4, 5, 6]]).T\n", + "print(f\"Input to activation function:\\n{z}\")\n", + "\n", + "act_func = sigmoid\n", + "a = act_func(z)\n", + "print(f\"\\nOutput from {act_func.__name__} activation function:\\n{a}\")\n", + "\n", + "act_func_derivative = derivate(act_func)\n", + "valued_at_z = act_func_derivative(a)\n", + "print(f\"\\nDerivative of {act_func.__name__} activation function valued at z:\\n{valued_at_z}\")" + ] + }, + { + "cell_type": "markdown", + "id": "5e9280ca", + "metadata": {}, + "source": [ + "## The Neural Network\n", + "\n", + "Now that we have gotten a good understanding of the implementation of\n", + "some important components, we can take a look at an object oriented\n", + "implementation of a feed forward neural network. The feed forward\n", + "neural network has been implemented as a class named FFNN, which can\n", + "be initiated as a regressor or classifier dependant on the choice of\n", + "cost function. The FFNN can have any number of input nodes, hidden\n", + "layers with any amount of hidden nodes, and any amount of output nodes\n", + "meaning it can perform multiclass classification as well as binary\n", + "classification and regression problems. Although there is a lot of\n", + "code present, it makes for an easy to use and generalizeable interface\n", + "for creating many types of neural networks as will be demonstrated\n", + "below." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "53e7f332", + "metadata": {}, + "outputs": [], + "source": [ + "import math\n", + "import autograd.numpy as np\n", + "import sys\n", + "import warnings\n", + "from autograd import grad, elementwise_grad\n", + "from random import random, seed\n", + "from copy import deepcopy, copy\n", + "from typing import Tuple, Callable\n", + "from sklearn.utils import resample\n", + "\n", + "warnings.simplefilter(\"error\")\n", + "\n", + "\n", + "class FFNN:\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Feed Forward Neural Network with interface enabling flexible design of a\n", + " nerual networks architecture and the specification of activation function\n", + " in the hidden layers and output layer respectively. This model can be used\n", + " for both regression and classification problems, depending on the output function.\n", + "\n", + " Attributes:\n", + " ------------\n", + " I dimensions (tuple[int]): A list of positive integers, which specifies the\n", + " number of nodes in each of the networks layers. The first integer in the array\n", + " defines the number of nodes in the input layer, the second integer defines number\n", + " of nodes in the first hidden layer and so on until the last number, which\n", + " specifies the number of nodes in the output layer.\n", + " II hidden_func (Callable): The activation function for the hidden layers\n", + " III output_func (Callable): The activation function for the output layer\n", + " IV cost_func (Callable): Our cost function\n", + " V seed (int): Sets random seed, makes results reproducible\n", + " \"\"\"\n", + "\n", + " def __init__(\n", + " self,\n", + " dimensions: tuple[int],\n", + " hidden_func: Callable = sigmoid,\n", + " output_func: Callable = lambda x: x,\n", + " cost_func: Callable = CostOLS,\n", + " seed: int = None,\n", + " ):\n", + " self.dimensions = dimensions\n", + " self.hidden_func = hidden_func\n", + " self.output_func = output_func\n", + " self.cost_func = cost_func\n", + " self.seed = seed\n", + " self.weights = list()\n", + " self.schedulers_weight = list()\n", + " self.schedulers_bias = list()\n", + " self.a_matrices = list()\n", + " self.z_matrices = list()\n", + " self.classification = None\n", + "\n", + " self.reset_weights()\n", + " self._set_classification()\n", + "\n", + " def fit(\n", + " self,\n", + " X: np.ndarray,\n", + " t: np.ndarray,\n", + " scheduler: Scheduler,\n", + " batches: int = 1,\n", + " epochs: int = 100,\n", + " lam: float = 0,\n", + " X_val: np.ndarray = None,\n", + " t_val: np.ndarray = None,\n", + " ):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " This function performs the training the neural network by performing the feedforward and backpropagation\n", + " algorithm to update the networks weights.\n", + "\n", + " Parameters:\n", + " ------------\n", + " I X (np.ndarray) : training data\n", + " II t (np.ndarray) : target data\n", + " III scheduler (Scheduler) : specified scheduler (algorithm for optimization of gradient descent)\n", + " IV scheduler_args (list[int]) : list of all arguments necessary for scheduler\n", + "\n", + " Optional Parameters:\n", + " ------------\n", + " V batches (int) : number of batches the datasets are split into, default equal to 1\n", + " VI epochs (int) : number of iterations used to train the network, default equal to 100\n", + " VII lam (float) : regularization hyperparameter lambda\n", + " VIII X_val (np.ndarray) : validation set\n", + " IX t_val (np.ndarray) : validation target set\n", + "\n", + " Returns:\n", + " ------------\n", + " I scores (dict) : A dictionary containing the performance metrics of the model.\n", + " The number of the metrics depends on the parameters passed to the fit-function.\n", + "\n", + " \"\"\"\n", + "\n", + " # setup \n", + " if self.seed is not None:\n", + " np.random.seed(self.seed)\n", + "\n", + " val_set = False\n", + " if X_val is not None and t_val is not None:\n", + " val_set = True\n", + "\n", + " # creating arrays for score metrics\n", + " train_errors = np.empty(epochs)\n", + " train_errors.fill(np.nan)\n", + " val_errors = np.empty(epochs)\n", + " val_errors.fill(np.nan)\n", + "\n", + " train_accs = np.empty(epochs)\n", + " train_accs.fill(np.nan)\n", + " val_accs = np.empty(epochs)\n", + " val_accs.fill(np.nan)\n", + "\n", + " self.schedulers_weight = list()\n", + " self.schedulers_bias = list()\n", + "\n", + " batch_size = X.shape[0] // batches\n", + "\n", + " X, t = resample(X, t)\n", + "\n", + " # this function returns a function valued only at X\n", + " cost_function_train = self.cost_func(t)\n", + " if val_set:\n", + " cost_function_val = self.cost_func(t_val)\n", + "\n", + " # create schedulers for each weight matrix\n", + " for i in range(len(self.weights)):\n", + " self.schedulers_weight.append(copy(scheduler))\n", + " self.schedulers_bias.append(copy(scheduler))\n", + "\n", + " print(f\"{scheduler.__class__.__name__}: Eta={scheduler.eta}, Lambda={lam}\")\n", + "\n", + " try:\n", + " for e in range(epochs):\n", + " for i in range(batches):\n", + " # allows for minibatch gradient descent\n", + " if i == batches - 1:\n", + " # If the for loop has reached the last batch, take all thats left\n", + " X_batch = X[i * batch_size :, :]\n", + " t_batch = t[i * batch_size :, :]\n", + " else:\n", + " X_batch = X[i * batch_size : (i + 1) * batch_size, :]\n", + " t_batch = t[i * batch_size : (i + 1) * batch_size, :]\n", + "\n", + " self._feedforward(X_batch)\n", + " self._backpropagate(X_batch, t_batch, lam)\n", + "\n", + " # reset schedulers for each epoch (some schedulers pass in this call)\n", + " for scheduler in self.schedulers_weight:\n", + " scheduler.reset()\n", + "\n", + " for scheduler in self.schedulers_bias:\n", + " scheduler.reset()\n", + "\n", + " # computing performance metrics\n", + " pred_train = self.predict(X)\n", + " train_error = cost_function_train(pred_train)\n", + "\n", + " train_errors[e] = train_error\n", + " if val_set:\n", + " \n", + " pred_val = self.predict(X_val)\n", + " val_error = cost_function_val(pred_val)\n", + " val_errors[e] = val_error\n", + "\n", + " if self.classification:\n", + " train_acc = self._accuracy(self.predict(X), t)\n", + " train_accs[e] = train_acc\n", + " if val_set:\n", + " val_acc = self._accuracy(pred_val, t_val)\n", + " val_accs[e] = val_acc\n", + "\n", + " # printing progress bar\n", + " progression = e / epochs\n", + " print_length = self._progress_bar(\n", + " progression,\n", + " train_error=train_errors[e],\n", + " train_acc=train_accs[e],\n", + " val_error=val_errors[e],\n", + " val_acc=val_accs[e],\n", + " )\n", + " except KeyboardInterrupt:\n", + " # allows for stopping training at any point and seeing the result\n", + " pass\n", + "\n", + " # visualization of training progression (similiar to tensorflow progression bar)\n", + " sys.stdout.write(\"\\r\" + \" \" * print_length)\n", + " sys.stdout.flush()\n", + " self._progress_bar(\n", + " 1,\n", + " train_error=train_errors[e],\n", + " train_acc=train_accs[e],\n", + " val_error=val_errors[e],\n", + " val_acc=val_accs[e],\n", + " )\n", + " sys.stdout.write(\"\")\n", + "\n", + " # return performance metrics for the entire run\n", + " scores = dict()\n", + "\n", + " scores[\"train_errors\"] = train_errors\n", + "\n", + " if val_set:\n", + " scores[\"val_errors\"] = val_errors\n", + "\n", + " if self.classification:\n", + " scores[\"train_accs\"] = train_accs\n", + "\n", + " if val_set:\n", + " scores[\"val_accs\"] = val_accs\n", + "\n", + " return scores\n", + "\n", + " def predict(self, X: np.ndarray, *, threshold=0.5):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Performs prediction after training of the network has been finished.\n", + "\n", + " Parameters:\n", + " ------------\n", + " I X (np.ndarray): The design matrix, with n rows of p features each\n", + "\n", + " Optional Parameters:\n", + " ------------\n", + " II threshold (float) : sets minimal value for a prediction to be predicted as the positive class\n", + " in classification problems\n", + "\n", + " Returns:\n", + " ------------\n", + " I z (np.ndarray): A prediction vector (row) for each row in our design matrix\n", + " This vector is thresholded if regression=False, meaning that classification results\n", + " in a vector of 1s and 0s, while regressions in an array of decimal numbers\n", + "\n", + " \"\"\"\n", + "\n", + " predict = self._feedforward(X)\n", + "\n", + " if self.classification:\n", + " return np.where(predict > threshold, 1, 0)\n", + " else:\n", + " return predict\n", + "\n", + " def reset_weights(self):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Resets/Reinitializes the weights in order to train the network for a new problem.\n", + "\n", + " \"\"\"\n", + " if self.seed is not None:\n", + " np.random.seed(self.seed)\n", + "\n", + " self.weights = list()\n", + " for i in range(len(self.dimensions) - 1):\n", + " weight_array = np.random.randn(\n", + " self.dimensions[i] + 1, self.dimensions[i + 1]\n", + " )\n", + " weight_array[0, :] = np.random.randn(self.dimensions[i + 1]) * 0.01\n", + "\n", + " self.weights.append(weight_array)\n", + "\n", + " def _feedforward(self, X: np.ndarray):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Calculates the activation of each layer starting at the input and ending at the output.\n", + " Each following activation is calculated from a weighted sum of each of the preceeding\n", + " activations (except in the case of the input layer).\n", + "\n", + " Parameters:\n", + " ------------\n", + " I X (np.ndarray): The design matrix, with n rows of p features each\n", + "\n", + " Returns:\n", + " ------------\n", + " I z (np.ndarray): A prediction vector (row) for each row in our design matrix\n", + " \"\"\"\n", + "\n", + " # reset matrices\n", + " self.a_matrices = list()\n", + " self.z_matrices = list()\n", + "\n", + " # if X is just a vector, make it into a matrix\n", + " if len(X.shape) == 1:\n", + " X = X.reshape((1, X.shape[0]))\n", + "\n", + " # Add a coloumn of zeros as the first coloumn of the design matrix, in order\n", + " # to add bias to our data\n", + " bias = np.ones((X.shape[0], 1)) * 0.01\n", + " X = np.hstack([bias, X])\n", + "\n", + " # a^0, the nodes in the input layer (one a^0 for each row in X - where the\n", + " # exponent indicates layer number).\n", + " a = X\n", + " self.a_matrices.append(a)\n", + " self.z_matrices.append(a)\n", + "\n", + " # The feed forward algorithm\n", + " for i in range(len(self.weights)):\n", + " if i < len(self.weights) - 1:\n", + " z = a @ self.weights[i]\n", + " self.z_matrices.append(z)\n", + " a = self.hidden_func(z)\n", + " # bias column again added to the data here\n", + " bias = np.ones((a.shape[0], 1)) * 0.01\n", + " a = np.hstack([bias, a])\n", + " self.a_matrices.append(a)\n", + " else:\n", + " try:\n", + " # a^L, the nodes in our output layers\n", + " z = a @ self.weights[i]\n", + " a = self.output_func(z)\n", + " self.a_matrices.append(a)\n", + " self.z_matrices.append(z)\n", + " except Exception as OverflowError:\n", + " print(\n", + " \"OverflowError in fit() in FFNN\\nHOW TO DEBUG ERROR: Consider lowering your learning rate or scheduler specific parameters such as momentum, or check if your input values need scaling\"\n", + " )\n", + "\n", + " # this will be a^L\n", + " return a\n", + "\n", + " def _backpropagate(self, X, t, lam):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Performs the backpropagation algorithm. In other words, this method\n", + " calculates the gradient of all the layers starting at the\n", + " output layer, and moving from right to left accumulates the gradient until\n", + " the input layer is reached. Each layers respective weights are updated while\n", + " the algorithm propagates backwards from the output layer (auto-differentation in reverse mode).\n", + "\n", + " Parameters:\n", + " ------------\n", + " I X (np.ndarray): The design matrix, with n rows of p features each.\n", + " II t (np.ndarray): The target vector, with n rows of p targets.\n", + " III lam (float32): regularization parameter used to punish the weights in case of overfitting\n", + "\n", + " Returns:\n", + " ------------\n", + " No return value.\n", + "\n", + " \"\"\"\n", + " out_derivative = derivate(self.output_func)\n", + " hidden_derivative = derivate(self.hidden_func)\n", + "\n", + " for i in range(len(self.weights) - 1, -1, -1):\n", + " # delta terms for output\n", + " if i == len(self.weights) - 1:\n", + " # for multi-class classification\n", + " if (\n", + " self.output_func.__name__ == \"softmax\"\n", + " ):\n", + " delta_matrix = self.a_matrices[i + 1] - t\n", + " # for single class classification\n", + " else:\n", + " cost_func_derivative = grad(self.cost_func(t))\n", + " delta_matrix = out_derivative(\n", + " self.z_matrices[i + 1]\n", + " ) * cost_func_derivative(self.a_matrices[i + 1])\n", + "\n", + " # delta terms for hidden layer\n", + " else:\n", + " delta_matrix = (\n", + " self.weights[i + 1][1:, :] @ delta_matrix.T\n", + " ).T * hidden_derivative(self.z_matrices[i + 1])\n", + "\n", + " # calculate gradient\n", + " gradient_weights = self.a_matrices[i][:, 1:].T @ delta_matrix\n", + " gradient_bias = np.sum(delta_matrix, axis=0).reshape(\n", + " 1, delta_matrix.shape[1]\n", + " )\n", + "\n", + " # regularization term\n", + " gradient_weights += self.weights[i][1:, :] * lam\n", + "\n", + " # use scheduler\n", + " update_matrix = np.vstack(\n", + " [\n", + " self.schedulers_bias[i].update_change(gradient_bias),\n", + " self.schedulers_weight[i].update_change(gradient_weights),\n", + " ]\n", + " )\n", + "\n", + " # update weights and bias\n", + " self.weights[i] -= update_matrix\n", + "\n", + " def _accuracy(self, prediction: np.ndarray, target: np.ndarray):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Calculates accuracy of given prediction to target\n", + "\n", + " Parameters:\n", + " ------------\n", + " I prediction (np.ndarray): vector of predicitons output network\n", + " (1s and 0s in case of classification, and real numbers in case of regression)\n", + " II target (np.ndarray): vector of true values (What the network ideally should predict)\n", + "\n", + " Returns:\n", + " ------------\n", + " A floating point number representing the percentage of correctly classified instances.\n", + " \"\"\"\n", + " assert prediction.size == target.size\n", + " return np.average((target == prediction))\n", + " def _set_classification(self):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Decides if FFNN acts as classifier (True) og regressor (False),\n", + " sets self.classification during init()\n", + " \"\"\"\n", + " self.classification = False\n", + " if (\n", + " self.cost_func.__name__ == \"CostLogReg\"\n", + " or self.cost_func.__name__ == \"CostCrossEntropy\"\n", + " ):\n", + " self.classification = True\n", + "\n", + " def _progress_bar(self, progression, **kwargs):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Displays progress of training\n", + " \"\"\"\n", + " print_length = 40\n", + " num_equals = int(progression * print_length)\n", + " num_not = print_length - num_equals\n", + " arrow = \">\" if num_equals > 0 else \"\"\n", + " bar = \"[\" + \"=\" * (num_equals - 1) + arrow + \"-\" * num_not + \"]\"\n", + " perc_print = self._format(progression * 100, decimals=5)\n", + " line = f\" {bar} {perc_print}% \"\n", + "\n", + " for key in kwargs:\n", + " if not np.isnan(kwargs[key]):\n", + " value = self._format(kwargs[key], decimals=4)\n", + " line += f\"| {key}: {value} \"\n", + " sys.stdout.write(\"\\r\" + line)\n", + " sys.stdout.flush()\n", + " return len(line)\n", + "\n", + " def _format(self, value, decimals=4):\n", + " \"\"\"\n", + " Description:\n", + " ------------\n", + " Formats decimal numbers for progress bar\n", + " \"\"\"\n", + " if value > 0:\n", + " v = value\n", + " elif value < 0:\n", + " v = -10 * value\n", + " else:\n", + " v = 1\n", + " n = 1 + math.floor(math.log10(v))\n", + " if n >= decimals - 1:\n", + " return str(round(value))\n", + " return f\"{value:.{decimals-n-1}f}\"" + ] + }, + { + "cell_type": "markdown", + "id": "3d122aa0", + "metadata": {}, + "source": [ + "Before we make a model, we will quickly generate a dataset we can use\n", + "for our linear regression problem as shown below" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "f2e26ec9", + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from sklearn.model_selection import train_test_split\n", + "\n", + "def SkrankeFunction(x, y):\n", + " return np.ravel(0 + 1*x + 2*y + 3*x**2 + 4*x*y + 5*y**2)\n", + "\n", + "def create_X(x, y, n):\n", + " if len(x.shape) > 1:\n", + " x = np.ravel(x)\n", + " y = np.ravel(y)\n", + "\n", + " N = len(x)\n", + " l = int((n + 1) * (n + 2) / 2) # Number of elements in beta\n", + " X = np.ones((N, l))\n", + "\n", + " for i in range(1, n + 1):\n", + " q = int((i) * (i + 1) / 2)\n", + " for k in range(i + 1):\n", + " X[:, q + k] = (x ** (i - k)) * (y**k)\n", + "\n", + " return X\n", + "\n", + "step=0.5\n", + "x = np.arange(0, 1, step)\n", + "y = np.arange(0, 1, step)\n", + "x, y = np.meshgrid(x, y)\n", + "target = SkrankeFunction(x, y)\n", + "target = target.reshape(target.shape[0], 1)\n", + "\n", + "poly_degree=3\n", + "X = create_X(x, y, poly_degree)\n", + "\n", + "X_train, X_test, t_train, t_test = train_test_split(X, target)" + ] + }, + { + "cell_type": "markdown", + "id": "50193282", + "metadata": {}, + "source": [ + "Now that we have our dataset ready for the regression, we can create\n", + "our regressor. Note that with the seed parameter, we can make sure our\n", + "results stay the same every time we run the neural network. For\n", + "inititialization, we simply specify the dimensions (we wish the amount\n", + "of input nodes to be equal to the datapoints, and the output to\n", + "predict one value)." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "72e089fd", + "metadata": {}, + "outputs": [], + "source": [ + "input_nodes = X_train.shape[1]\n", + "output_nodes = 1\n", + "\n", + "linear_regression = FFNN((input_nodes, output_nodes), output_func=identity, cost_func=CostOLS, seed=2023)" + ] + }, + { + "cell_type": "markdown", + "id": "36d0292f", + "metadata": {}, + "source": [ + "We then fit our model with our training data using the scheduler of our choice." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "f57f585c", + "metadata": {}, + "outputs": [], + "source": [ + "linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", + "\n", + "scheduler = Constant(eta=1e-3)\n", + "scores = linear_regression.fit(X_train, t_train, scheduler)" + ] + }, + { + "cell_type": "markdown", + "id": "2cb5c961", + "metadata": {}, + "source": [ + "Due to the progress bar we can see the MSE (train$\\_$error) throughout\n", + "the FFNN's training. Note that the fit() function has some optional\n", + "parameters with defualt arguments. For example, the regularization\n", + "hyperparameter can be left ignored if not needed, and equally the FFNN\n", + "will by default run for 100 epochs. These can easily be changed, such\n", + "as for example:" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "2ca466da", + "metadata": {}, + "outputs": [], + "source": [ + "linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", + "\n", + "scores = linear_regression.fit(X_train, t_train, scheduler, lam=1e-4, epochs=1000)" + ] + }, + { + "cell_type": "markdown", + "id": "7aa6b5cf", + "metadata": {}, + "source": [ + "We see that given more epochs to train on, the regressor reaches a lower MSE.\n", + "\n", + "Let us then switch to a binary classification. We use a binary\n", + "classification dataset, and follow a similar setup to the regression\n", + "case." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "14ef27bb", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.datasets import load_breast_cancer\n", + "from sklearn.preprocessing import MinMaxScaler\n", + "\n", + "wisconsin = load_breast_cancer()\n", + "X = wisconsin.data\n", + "target = wisconsin.target\n", + "target = target.reshape(target.shape[0], 1)\n", + "\n", + "X_train, X_val, t_train, t_val = train_test_split(X, target)\n", + "\n", + "scaler = MinMaxScaler()\n", + "scaler.fit(X_train)\n", + "X_train = scaler.transform(X_train)\n", + "X_val = scaler.transform(X_val)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "60ed9af1", + "metadata": {}, + "outputs": [], + "source": [ + "input_nodes = X_train.shape[1]\n", + "output_nodes = 1\n", + "\n", + "logistic_regression = FFNN((input_nodes, output_nodes), output_func=sigmoid, cost_func=CostLogReg, seed=2023)" + ] + }, + { + "cell_type": "markdown", + "id": "7581648e", + "metadata": {}, + "source": [ + "We will now make use of our validation data by passing it into our fit function as a keyword argument" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "5b60e340", + "metadata": {}, + "outputs": [], + "source": [ + "logistic_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", + "\n", + "scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)\n", + "scores = logistic_regression.fit(X_train, t_train, scheduler, epochs=1000, X_val=X_val, t_val=t_val)" + ] + }, + { + "cell_type": "markdown", + "id": "60943cb4", + "metadata": {}, + "source": [ + "Finally, we will create a neural network with 2 hidden layers with activation functions." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "81064b40", + "metadata": {}, + "outputs": [], + "source": [ + "input_nodes = X_train.shape[1]\n", + "hidden_nodes1 = 100\n", + "hidden_nodes2 = 30\n", + "output_nodes = 1\n", + "\n", + "dims = (input_nodes, hidden_nodes1, hidden_nodes2, output_nodes)\n", + "\n", + "neural_network = FFNN(dims, hidden_func=RELU, output_func=sigmoid, cost_func=CostLogReg, seed=2023)" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "01255d33", + "metadata": {}, + "outputs": [], + "source": [ + "neural_network.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", + "\n", + "scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999)\n", + "scores = neural_network.fit(X_train, t_train, scheduler, epochs=1000, X_val=X_val, t_val=t_val)" + ] + }, + { + "cell_type": "markdown", + "id": "9b945ba1", + "metadata": {}, + "source": [ + "## Multiclass classification\n", + "\n", + "Finally, we will demonstrate the use case of multiclass classification\n", + "using our FFNN with the famous MNIST dataset, which contain images of\n", + "digits between the range of 0 to 9." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "2255dcf3", + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.datasets import load_digits\n", + "\n", + "def onehot(target: np.ndarray):\n", + " onehot = np.zeros((target.size, target.max() + 1))\n", + " onehot[np.arange(target.size), target] = 1\n", + " return onehot\n", + "\n", + "digits = load_digits()\n", + "\n", + "X = digits.data\n", + "target = digits.target\n", + "target = onehot(target)\n", + "\n", + "input_nodes = 64\n", + "hidden_nodes1 = 100\n", + "hidden_nodes2 = 30\n", + "output_nodes = 10\n", + "\n", + "dims = (input_nodes, hidden_nodes1, hidden_nodes2, output_nodes)\n", + "\n", + "multiclass = FFNN(dims, hidden_func=LRELU, output_func=softmax, cost_func=CostCrossEntropy)\n", + "\n", + "multiclass.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", + "\n", + "scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999)\n", + "scores = multiclass.fit(X, target, scheduler, epochs=1000)" + ] + }, + { + "cell_type": "markdown", + "id": "b67c5d21", + "metadata": {}, + "source": [ + "## Testing the XOR gate and other gates\n", + "\n", + "Let us now use our code to test the XOR gate." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "id": "0a838e23", + "metadata": {}, + "outputs": [], + "source": [ + "X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)\n", + "\n", + "# The XOR gate\n", + "yXOR = np.array( [[ 0], [1] ,[1], [0]])\n", + "\n", + "input_nodes = X.shape[1]\n", + "output_nodes = 1\n", + "\n", + "logistic_regression = FFNN((input_nodes, output_nodes), output_func=sigmoid, cost_func=CostLogReg, seed=2023)\n", + "logistic_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", + "scheduler = Adam(eta=1e-1, rho=0.9, rho2=0.999)\n", + "scores = logistic_regression.fit(X, yXOR, scheduler, epochs=1000)" + ] + }, + { + "cell_type": "markdown", + "id": "d60f0d48", + "metadata": {}, + "source": [ + "Not bad, but the results depend strongly on the learning reate. Try different learning rates." + ] + }, + { + "cell_type": "markdown", + "id": "0e7733c6", + "metadata": {}, + "source": [ + "## Building neural networks in Tensorflow and Keras\n", + "\n", + "Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn\n", + "and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy\n", + "and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer. \n", + "\n", + "In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite\n", + "clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or\n", + "NumPy arrays." + ] + }, + { + "cell_type": "markdown", + "id": "d6235cfb", + "metadata": {}, + "source": [ + "## Tensorflow\n", + "\n", + "Tensorflow is an open source library machine learning library\n", + "developed by the Google Brain team for internal use. It was released\n", + "under the Apache 2.0 open source license in November 9, 2015.\n", + "\n", + "Tensorflow is a computational framework that allows you to construct\n", + "machine learning models at different levels of abstraction, from\n", + "high-level, object-oriented APIs like Keras, down to the C++ kernels\n", + "that Tensorflow is built upon. The higher levels of abstraction are\n", + "simpler to use, but less flexible, and our choice of implementation\n", + "should reflect the problems we are trying to solve.\n", + "\n", + "[Tensorflow uses](https://www.tensorflow.org/guide/graphs) so-called graphs to represent your computation\n", + "in terms of the dependencies between individual operations, such that you first build a Tensorflow *graph*\n", + "to represent your model, and then create a Tensorflow *session* to run the graph.\n", + "\n", + "In this guide we will analyze the same data as we did in our NumPy and\n", + "scikit-learn tutorial, gathered from the MNIST database of images. We\n", + "will give an introduction to the lower level Python Application\n", + "Program Interfaces (APIs), and see how we use them to build our graph.\n", + "Then we will build (effectively) the same graph in Keras, to see just\n", + "how simple solving a machine learning problem can be.\n", + "\n", + "To install tensorflow on Unix/Linux systems, use pip as" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "d9a957fa", + "metadata": {}, + "outputs": [], + "source": [ + "pip3 install tensorflow" + ] + }, + { + "cell_type": "markdown", + "id": "d5816511", + "metadata": {}, + "source": [ + "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", + "(current release of CPU-only TensorFlow)" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "4beaf713", + "metadata": {}, + "outputs": [], + "source": [ + "conda create -n tf tensorflow\n", + "conda activate tf" + ] + }, + { + "cell_type": "markdown", + "id": "5f5c9603", + "metadata": {}, + "source": [ + "To install the current release of GPU TensorFlow" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "1910be62", + "metadata": {}, + "outputs": [], + "source": [ + "conda create -n tf-gpu tensorflow-gpu\n", + "conda activate tf-gpu" + ] + }, + { + "cell_type": "markdown", + "id": "db1c0682", + "metadata": {}, + "source": [ + "## Using Keras\n", + "\n", + "Keras is a high level [neural network](https://en.wikipedia.org/wiki/Application_programming_interface)\n", + "that supports Tensorflow, CTNK and Theano as backends. \n", + "If you have Anaconda installed you may run the following command" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "id": "dd2c291d", + "metadata": {}, + "outputs": [], + "source": [ + "conda install keras" + ] + }, + { + "cell_type": "markdown", + "id": "bcd3aa2d", + "metadata": {}, + "source": [ + "You can look up the [instructions here](https://keras.io/) for more information.\n", + "\n", + "We will to a large extent use **keras** in this course." + ] + }, + { + "cell_type": "markdown", + "id": "bfbd5190", + "metadata": {}, + "source": [ + "## Collect and pre-process data\n", + "\n", + "Let us look again at the MINST data set." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "a1257ae3", + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "# import necessary packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import tensorflow as tf\n", + "from sklearn import datasets\n", + "\n", + "\n", + "# ensure the same random numbers appear every time\n", + "np.random.seed(0)\n", + "\n", + "# display images in notebook\n", + "%matplotlib inline\n", + "plt.rcParams['figure.figsize'] = (12,12)\n", + "\n", + "\n", + "# download MNIST dataset\n", + "digits = datasets.load_digits()\n", + "\n", + "# define inputs and labels\n", + "inputs = digits.images\n", + "labels = digits.target\n", + "\n", + "print(\"inputs = (n_inputs, pixel_width, pixel_height) = \" + str(inputs.shape))\n", + "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", + "\n", + "\n", + "# flatten the image\n", + "# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64\n", + "n_inputs = len(inputs)\n", + "inputs = inputs.reshape(n_inputs, -1)\n", + "print(\"X = (n_inputs, n_features) = \" + str(inputs.shape))\n", + "\n", + "\n", + "# choose some random images to display\n", + "indices = np.arange(n_inputs)\n", + "random_indices = np.random.choice(indices, size=5)\n", + "\n", + "for i, image in enumerate(digits.images[random_indices]):\n", + " plt.subplot(1, 5, i+1)\n", + " plt.axis('off')\n", + " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", + " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "963f6b3c", + "metadata": {}, + "outputs": [], + "source": [ + "from tensorflow.keras.layers import Input\n", + "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", + "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", + "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", + "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", + "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", + "\n", + "from sklearn.model_selection import train_test_split\n", + "\n", + "# one-hot representation of labels\n", + "labels = to_categorical(labels)\n", + "\n", + "# split into train and test data\n", + "train_size = 0.8\n", + "test_size = 1 - train_size\n", + "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", + " test_size=test_size)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "c16ce821", + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "epochs = 100\n", + "batch_size = 100\n", + "n_neurons_layer1 = 100\n", + "n_neurons_layer2 = 50\n", + "n_categories = 10\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)\n", + "def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):\n", + " model = Sequential()\n", + " model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))\n", + " model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))\n", + " model.add(Dense(n_categories, activation='softmax'))\n", + " \n", + " sgd = optimizers.SGD(lr=eta)\n", + " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", + " \n", + " return model" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "id": "f1505f52", + "metadata": {}, + "outputs": [], + "source": [ + "DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + " \n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,\n", + " eta=eta, lmbd=lmbd)\n", + " DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", + " scores = DNN.evaluate(X_test, Y_test)\n", + " \n", + " DNN_keras[i][j] = DNN\n", + " \n", + " print(\"Learning rate = \", eta)\n", + " print(\"Lambda = \", lmbd)\n", + " print(\"Test accuracy: %.3f\" % scores[1])\n", + " print()" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "id": "83c3854e", + "metadata": {}, + "outputs": [], + "source": [ + "# optional\n", + "# visual representation of grid search\n", + "# uses seaborn heatmap, could probably do this in matplotlib\n", + "import seaborn as sns\n", + "\n", + "sns.set()\n", + "\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "\n", + "for i in range(len(eta_vals)):\n", + " for j in range(len(lmbd_vals)):\n", + " DNN = DNN_keras[i][j]\n", + "\n", + " train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]\n", + " test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]\n", + "\n", + " \n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Test Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ff59e2bd", + "metadata": {}, + "source": [ + "## And using PyTorch (more discussions to follow)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "id": "df278a5c", + "metadata": {}, + "outputs": [], + "source": [ + "# Simple neural-network (NN) code using pytorch\n", + "import numpy as np\n", + "import torch\n", + "import torch.nn as nn\n", + "import torch.optim as optim\n", + " \n", + "# load the dataset, split into input (X) and output (y) variables\n", + "dataset = np.loadtxt('yourdata.csv', delimiter=',')\n", + "X = dataset[:,0:8]\n", + "y = dataset[:,8]\n", + " \n", + "X = torch.tensor(X, dtype=torch.float32)\n", + "y = torch.tensor(y, dtype=torch.float32).reshape(-1, 1)\n", + "\n", + "# define the model\n", + "class NNClassifier(nn.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.hidden1 = nn.Linear(8, 12)\n", + " self.act1 = nn.ReLU()\n", + " self.hidden2 = nn.Linear(12, 8)\n", + " self.act2 = nn.ReLU()\n", + " self.output = nn.Linear(8, 1)\n", + " self.act_output = nn.Sigmoid()\n", + " \n", + " def forward(self, x):\n", + " x = self.act1(self.hidden1(x))\n", + " x = self.act2(self.hidden2(x))\n", + " x = self.act_output(self.output(x))\n", + " return x\n", + " \n", + "model = NNClassifier()\n", + "print(model)\n", + " \n", + "# train the model\n", + "loss_fn = nn.BCELoss() # binary cross entropy\n", + "optimizer = optim.Adam(model.parameters(), lr=0.001)\n", + "n_epochs = 100\n", + "batch_size = 10\n", + " \n", + "for epoch in range(n_epochs):\n", + " for i in range(0, len(X), batch_size):\n", + " Xbatch = X[i:i+batch_size]\n", + " y_pred = model(Xbatch)\n", + " ybatch = y[i:i+batch_size]\n", + " loss = loss_fn(y_pred, ybatch)\n", + " optimizer.zero_grad()\n", + " loss.backward()\n", + " optimizer.step()\n", + " \n", + "# compute accuracy\n", + "y_pred = model(X)\n", + "accuracy = (y_pred.round() == y).float().mean()\n", + "print(f\"Accuracy {accuracy}\")\n", + "\n", + "# make class predictions with the model\n", + "predictions = (model(X) > 0.5).int()\n", + "for i in range(5):\n", + " print('%s => %d (expected %d)' % (X[i].tolist(), predictions[i], y[i]))" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.10" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/src/week3/figures/simple.pdf b/doc/src/week3/figures/simple.pdf new file mode 100644 index 0000000000000000000000000000000000000000..65610969e6639838d4381e8d1e17974776fc11f2 GIT binary patch literal 1814058 zcmb5Vb97~2vo0Llc6Myrw%M`Mv2EM#q+=T$+fHZ4?%1|*`}dyn-gE9a_ZxS7|E#_D ztXi{XJ@w2QHEXX$p)4-Rz{1D|M=^A8dvI2Emp?N!0>=hm1~`~l!}0S2m?W)iUCo{U ze%czlnv0v8I+&RQnB>jvEnF=DtgLJRK|wedS7&o$J2+2JePtH=L1u)FkLWjHE7d}B 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