From 9bdb28895eca0f76df7502d8e71fced294abac72 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Tue, 16 Oct 2018 13:54:54 +0200 Subject: [PATCH] small update on neural networks --- doc/pub/NeuralNet/html/._NeuralNet-bs000.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs001.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs002.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs003.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs004.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs005.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs006.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs007.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs008.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs009.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs010.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs011.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs012.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs013.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs014.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs015.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs016.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs017.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs018.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs019.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs020.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs021.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs022.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs023.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs024.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs025.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs026.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs027.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs028.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs029.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs030.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs031.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs032.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs033.html | 132 +++---- doc/pub/NeuralNet/html/._NeuralNet-bs034.html | 164 ++++----- doc/pub/NeuralNet/html/._NeuralNet-bs035.html | 233 +++++-------- doc/pub/NeuralNet/html/._NeuralNet-bs036.html | 240 +++++++------ doc/pub/NeuralNet/html/._NeuralNet-bs037.html | 195 ++++++----- doc/pub/NeuralNet/html/._NeuralNet-bs038.html | 199 +++++------ doc/pub/NeuralNet/html/._NeuralNet-bs039.html | 189 +++++----- doc/pub/NeuralNet/html/._NeuralNet-bs040.html | 171 ++++----- doc/pub/NeuralNet/html/._NeuralNet-bs041.html | 204 +++++------ doc/pub/NeuralNet/html/._NeuralNet-bs042.html | 210 +++++++----- doc/pub/NeuralNet/html/._NeuralNet-bs043.html | 175 +++++----- doc/pub/NeuralNet/html/._NeuralNet-bs044.html | 179 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doc/pub/NeuralNet/html/._NeuralNet-bs060.html | 170 ++++----- doc/pub/NeuralNet/html/._NeuralNet-bs061.html | 179 +++++----- doc/pub/NeuralNet/html/._NeuralNet-bs062.html | 174 +++++----- doc/pub/NeuralNet/html/._NeuralNet-bs063.html | 162 +++++---- doc/pub/NeuralNet/html/._NeuralNet-bs064.html | 179 +++++----- doc/pub/NeuralNet/html/._NeuralNet-bs065.html | 183 +++++----- doc/pub/NeuralNet/html/._NeuralNet-bs066.html | 323 +++++++++++++++++ doc/pub/NeuralNet/html/NeuralNet-bs.html | 132 +++---- doc/pub/NeuralNet/html/NeuralNet-reveal.html | 92 +++-- .../NeuralNet/html/NeuralNet-solarized.html | 151 ++++---- doc/pub/NeuralNet/html/NeuralNet.html | 151 ++++---- doc/pub/NeuralNet/ipynb/NeuralNet.ipynb | 52 +++ .../ipynb/ipynb-NeuralNet-src.tar.gz | Bin 210 -> 211 bytes doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf | Bin 469564 -> 470872 bytes doc/src/NeuralNet/NeuralNet.do.txt | 23 ++ 75 files changed, 6318 insertions(+), 5703 deletions(-) create mode 100644 doc/pub/NeuralNet/html/._NeuralNet-bs066.html diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs000.html b/doc/pub/NeuralNet/html/._NeuralNet-bs000.html index aeabd57ad..55accbec3 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs000.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs000.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose 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('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
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  • Full object-oriented implementation
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  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
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  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -300,7 +302,7 @@ MathJax.Hub.Config({
  • 9
  • 10
  • ...
    • -
  • 66
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  • Defining the cost function
  • Example: binary classification problem
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  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Evaluate model performance on test data
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
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  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
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  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
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  • Visualization
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    • +
  • Tensorflow
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  • Collect and pre-process data
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  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -285,7 +287,7 @@ a weight variable.
  • 10
  • 11
  • ...
    • -
  • 66
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  • Example: binary classification problem
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  • Developing a code for doing neural networks with back propagation
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    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • The RELU function family
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  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -335,7 +337,7 @@ humanities to life science and medicine.
  • 11
  • 12
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs003.html b/doc/pub/NeuralNet/html/._NeuralNet-bs003.html index dc57c22b0..209d7fcaa 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs003.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs003.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
    • -
  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
    • -
  • Matrix multiplication
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  • Improving performance
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  • Building neural networks in Tensorflow and Keras
    • -
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  • Using TensorFlow backend
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  • Optimizing and using gradient descent
    • -
  • Using Keras
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  • Which activation function should I use?
    • -
  • Is the Logistic activation function (Sigmoid) our choice?
    • -
  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -300,7 +302,7 @@ methods we discussed earlier.
  • 12
  • 13
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs004.html b/doc/pub/NeuralNet/html/._NeuralNet-bs004.html index 567102e7a..2a5f5c51d 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs004.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs004.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • -
  • Train and test datasets
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  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
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  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -292,7 +294,7 @@ to all nodes in the subsequent layer, making this a so-called
  • 13
  • 14
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs005.html b/doc/pub/NeuralNet/html/._NeuralNet-bs005.html index 873e374bd..1a6e2b047 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs005.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs005.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
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  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • +
  • Train and test datasets
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  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -301,7 +303,7 @@ recognition.
  • 14
  • 15
  • ...
    • -
  • 66
    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs006.html b/doc/pub/NeuralNet/html/._NeuralNet-bs006.html index 3f1994444..6831a5e64 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs006.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs006.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
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  • Train and test datasets
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Adjust hyperparameters
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  • scikit-learn implementation
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  • Building neural networks in Tensorflow and Keras
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  • Optimizing and using gradient descent
    • -
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  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -293,7 +295,7 @@ especially well-suited for handwriting and speech recognition.
  • 15
  • 16
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs007.html b/doc/pub/NeuralNet/html/._NeuralNet-bs007.html index a31b5dce4..7477f3bbb 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs007.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs007.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Building neural networks in Tensorflow and Keras
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    • -
  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -294,7 +296,7 @@ type of NN due the unusual activation functions.
  • 16
  • 17
  • ...
    • -
  • 66
    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs008.html b/doc/pub/NeuralNet/html/._NeuralNet-bs008.html index 2fae7fb2c..389112241 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs008.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs008.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -292,7 +294,7 @@ Such networks are often called multilayer perceptrons (MLPs).
  • 17
  • 18
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs009.html b/doc/pub/NeuralNet/html/._NeuralNet-bs009.html index d6be8593d..6db617982 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs009.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs009.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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    • @@ -297,7 +299,7 @@ as to not restrict the range of output values.
  • 18
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    • -
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    • @@ -298,7 +300,7 @@ of the outputs of all neurons in the previous layer.
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    • @@ -327,7 +329,7 @@ is obtained.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs013.html b/doc/pub/NeuralNet/html/._NeuralNet-bs013.html index 598a73d5d..20788e1fb 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs013.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs013.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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  • Train and test datasets
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  • Matrix multiplications
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  • Optimizing the cost function
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  • scikit-learn implementation
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  • Optimizing and using gradient descent
    • -
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  • Is the Logistic activation function (Sigmoid) our choice?
    • -
  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -300,7 +302,7 @@ variables are the input values \( x_n \).
  • 22
  • 23
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs014.html b/doc/pub/NeuralNet/html/._NeuralNet-bs014.html index 89f79e50f..12aebdafd 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs014.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs014.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
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  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
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    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
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    • +
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  • Visualization
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  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • @@ -309,7 +311,7 @@ flexibility of a neural network.
  • 23
  • 24
  • ...
    • -
  • 66
    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs015.html b/doc/pub/NeuralNet/html/._NeuralNet-bs015.html index c00812ea2..75f455d95 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs015.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs015.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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    • -
  • Train and test datasets
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  • Define model and architecture
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  • Optimizing the cost function
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  • Which activation function should we use?
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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    • +
  • Limitations of supervised learning with deep networks
    • @@ -319,7 +321,7 @@ $$
  • 24
  • 25
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs016.html b/doc/pub/NeuralNet/html/._NeuralNet-bs016.html index 662103b8a..20ab2d14a 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs016.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs016.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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    • -
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
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  • The Softmax function
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  • Developing a code for doing neural networks with back propagation
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  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
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  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
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  • Improving performance
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  • Full object-oriented implementation
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    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -303,7 +305,7 @@ used as input to the activation functions. For each operation
  • 25
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs017.html b/doc/pub/NeuralNet/html/._NeuralNet-bs017.html index 4a4434ae1..1dfbd2c6a 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs017.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs017.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
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  • Developing a code for doing neural networks with back propagation
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  • Matrix multiplications
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  • Optimizing the cost function
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  • The RELU function family
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  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Optimizing and using gradient descent
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  • Using Keras
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • The RELU function family
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  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
    • @@ -297,7 +299,7 @@ for a FFNN to fulfill the universal approximation theorem
  • 26
  • 27
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html index 73bbfd2a3..2fba6eb34 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
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  • Developing a code for doing neural networks with back propagation
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  • Optimizing the cost function
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  • Which activation function should we use?
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  • The RELU function family
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  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
    • @@ -305,7 +307,7 @@ $$
  • 27
  • 28
  • ...
    • -
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    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs019.html b/doc/pub/NeuralNet/html/._NeuralNet-bs019.html index 0bd555817..4eebc4d90 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs019.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs019.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • Defining the cost function
  • Example: binary classification problem
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  • Developing a code for doing neural networks with back propagation
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  • Train and test datasets
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  • Define model and architecture
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Regularization
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  • Limitations of supervised learning with deep networks
    • @@ -367,7 +369,7 @@ plt.show()
  • 28
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs020.html b/doc/pub/NeuralNet/html/._NeuralNet-bs020.html index 41fba85a8..316612137 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs020.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs020.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • The RELU function family
    • -
  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
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  • Collect and pre-process data
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  • Define model and architecture
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  • Weights and biases
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
    • +
  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
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  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
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  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • @@ -318,7 +320,7 @@ like logistic regression or linear regression and their modifications on the oth
  • 29
  • 30
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs021.html b/doc/pub/NeuralNet/html/._NeuralNet-bs021.html index ff17bac08..e32373ef4 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs021.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs021.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
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  • Developing a code for doing neural networks with back propagation
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    • -
  • Train and test datasets
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  • Which activation function should we use?
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  • Choose cost function and optimizer
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  • Collect and pre-process data
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    • +
  • Using Keras
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  • Limitations of supervised learning with deep networks
    • @@ -308,7 +310,7 @@ the potential of being universal approximators.
  • 30
  • 31
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs022.html b/doc/pub/NeuralNet/html/._NeuralNet-bs022.html index 588834449..f443fdbe5 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs022.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs022.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • Developing a code for doing neural networks with back propagation
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  • Limitations of supervised learning with deep networks
    • @@ -314,7 +316,7 @@ classes.
  • 31
  • 32
  • ...
    • -
  • 66
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs023.html b/doc/pub/NeuralNet/html/._NeuralNet-bs023.html index 00261c1b3..a151443c8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs023.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs023.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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    • @@ -318,7 +320,7 @@ $$
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    • @@ -302,7 +304,7 @@ $$
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    • @@ -305,7 +307,7 @@ $$
  • 34
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs026.html b/doc/pub/NeuralNet/html/._NeuralNet-bs026.html index 823d0411c..8dce9b6dd 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs026.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs026.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs027.html b/doc/pub/NeuralNet/html/._NeuralNet-bs027.html index e998bcea1..566cdd8b4 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs027.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs027.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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  • Improving performance
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
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  • Define model and architecture
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  • Layers
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
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  • Improving performance
    • +
  • Full object-oriented implementation
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  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
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  • Visualization
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  • Building neural networks in Tensorflow and Keras
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  • Collect and pre-process data
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  • Optimizing and using gradient descent
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • +
  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
    • @@ -297,7 +299,7 @@ That is, the error \( \delta_j^L \) is exactly equal to the rate of change of th
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs028.html b/doc/pub/NeuralNet/html/._NeuralNet-bs028.html index 181635ec9..c73902140 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs028.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs028.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • The RELU function family
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  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
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  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
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  • Define model and architecture
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Evaluate model performance on test data
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  • Adjust hyperparameters
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  • Visualization
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  • Using TensorFlow backend
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  • The derivative of the Logistic funtion
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  • The RELU function family
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  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • @@ -342,7 +344,7 @@ one \( L-1 \) in terms of the errors in the final output layer.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs029.html b/doc/pub/NeuralNet/html/._NeuralNet-bs029.html index a77376e91..f73bea6ea 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs029.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs029.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
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  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
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  • Define model and architecture
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Improving performance
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  • Adjust hyperparameters
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  • Visualization
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  • The derivative of the Logistic funtion
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  • The RELU function family
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  • Which activation function should we use?
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  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -312,7 +314,7 @@ We are now ready to set up the algorithm for back propagation and learning the w
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs030.html b/doc/pub/NeuralNet/html/._NeuralNet-bs030.html index 34cf226ff..dc50afb97 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs030.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs030.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
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  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
    • -
  • Train and test datasets
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  • Define model and architecture
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  • Layers
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
    • -
  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
    • -
  • Regularization
    • -
  • Matrix multiplication
    • -
  • Improving performance
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  • Evaluate model performance on test data
    • -
  • Adjust hyperparameters
    • -
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  • scikit-learn implementation
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  • Building neural networks in Tensorflow and Keras
    • -
  • Tensorflow
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  • Collect and pre-process data
    • -
  • Using TensorFlow backend
    • -
  • Optimizing and using gradient descent
    • -
  • Using Keras
    • -
  • Which activation function should I use?
    • -
  • Is the Logistic activation function (Sigmoid) our choice?
    • -
  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -356,7 +358,7 @@ Here it is convenient to use stochastic gradient descent (see the examples below
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs031.html b/doc/pub/NeuralNet/html/._NeuralNet-bs031.html index e930b5526..4925bb7b7 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs031.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs031.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
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    • -
  • Train and test datasets
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  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
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  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
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  • Train and test datasets
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  • Define model and architecture
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
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  • Regularization
    • +
  • Matrix multiplication
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  • Improving performance
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  • Full object-oriented implementation
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  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
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  • Visualization
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  • Visualization
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  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
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  • Collect and pre-process data
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  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -318,7 +320,7 @@ of our network.
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  • ...
    • -
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs032.html b/doc/pub/NeuralNet/html/._NeuralNet-bs032.html index 418007b57..e8c29bd5d 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs032.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs032.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
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  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
    • -
  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
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  • Regularization
    • -
  • Matrix multiplication
    • -
  • Improving performance
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  • Full object-oriented implementation
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  • Evaluate model performance on test data
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  • The derivative of the Logistic funtion
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  • The RELU function family
    • -
  • Which activation function should we use?
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  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
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  • The Softmax function
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  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
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  • Train and test datasets
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  • Define model and architecture
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  • Layers
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  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Full object-oriented implementation
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  • Evaluate model performance on test data
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  • Adjust hyperparameters
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  • Visualization
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  • scikit-learn implementation
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  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -340,7 +342,7 @@ The back propagation equations need now only a small change, namely the definiti
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs033.html b/doc/pub/NeuralNet/html/._NeuralNet-bs033.html index c949a9d21..433191a0c 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs033.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs033.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
    • -
  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
    • -
  • Regularization
    • -
  • Matrix multiplication
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  • Improving performance
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  • The RELU function family
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  • Which activation function should we use?
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  • A top-down perspective on Neural networks
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  • Limitations of supervised learning with deep networks
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  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
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  • Collect and pre-process data
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  • Train and test datasets
    • +
  • Define model and architecture
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  • Layers
    • +
  • Weights and biases
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  • Feed-forward pass
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  • Matrix multiplications
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  • Choose cost function and optimizer
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  • Optimizing the cost function
    • +
  • Regularization
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  • Matrix multiplication
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  • Improving performance
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  • Full object-oriented implementation
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  • Evaluate model performance on test data
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  • Adjust hyperparameters
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  • Visualization
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  • scikit-learn implementation
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  • Visualization
    • +
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  • Collect and pre-process data
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  • Using TensorFlow backend
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  • Using Keras
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  • Which activation function should I use?
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  • Is the Logistic activation function (Sigmoid) our choice?
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  • The derivative of the Logistic funtion
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  • The RELU function family
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  • Which activation function should we use?
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    • +
  • Limitations of supervised learning with deep networks
    • @@ -328,7 +330,7 @@ In case we use another activation function than the logistic one, we need to eva
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs034.html b/doc/pub/NeuralNet/html/._NeuralNet-bs034.html index c5af01771..acd32b1a3 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs034.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs034.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
    • -
  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
    • -
  • Regularization
    • -
  • Matrix multiplication
    • -
  • Improving performance
    • -
  • Full object-oriented implementation
    • -
  • Evaluate model performance on test data
    • -
  • Adjust hyperparameters
    • -
  • Visualization
    • -
  • scikit-learn implementation
    • -
  • Visualization
    • -
  • Building neural networks in Tensorflow and Keras
    • -
  • Tensorflow
    • -
  • Collect and pre-process data
    • -
  • Using TensorFlow backend
    • -
  • Optimizing and using gradient descent
    • -
  • Using Keras
    • -
  • Which activation function should I use?
    • -
  • Is the Logistic activation function (Sigmoid) our choice?
    • -
  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -255,22 +257,28 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Developing a code for doing neural networks with back propagation

    +

    The Softmax function

    +In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need +$$ +\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = +\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}. +$$ + +For the Softmax function we have +$$ +f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}. +$$ + +Its derivative with respect to \( z_j^l \) gives +$$ +\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right), +$$ + +which in case of the simply binary model reduces to having \( i=j \).

    -One can identify a set of key steps when using neural networks to solve supervised learning problems: - -

      -
    1. Collect and pre-process data
      2. -
    2. Define model and architecture
      3. -
    3. Choose cost function and optimizer
      4. -
    4. Train the model
      5. -
    5. Evaluate model performance on test data
      6. -
    6. Adjust hyperparameters (if necessary, network architecture)
      7. -
    -

    diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs035.html b/doc/pub/NeuralNet/html/._NeuralNet-bs035.html index c031fd78d..1382acec1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs035.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs035.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
    • -
  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
    • -
  • Regularization
    • -
  • Matrix multiplication
    • -
  • Improving performance
    • -
  • Full object-oriented implementation
    • -
  • Evaluate model performance on test data
    • -
  • Adjust hyperparameters
    • -
  • Visualization
    • -
  • scikit-learn implementation
    • -
  • Visualization
    • -
  • Building neural networks in Tensorflow and Keras
    • -
  • Tensorflow
    • -
  • Collect and pre-process data
    • -
  • Using TensorFlow backend
    • -
  • Optimizing and using gradient descent
    • -
  • Using Keras
    • -
  • Which activation function should I use?
    • -
  • Is the Logistic activation function (Sigmoid) our choice?
    • -
  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -255,101 +257,22 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Collect and pre-process data

    +

    Developing a code for doing neural networks with back propagation

    -Here we will be using the MNIST dataset, which is readily available through the scikit-learn -package. You may also find it for example here. -The MNIST (Modified National Institute of Standards and Technology) database is a large database -of handwritten digits that is commonly used for training various image processing systems. -The MNIST dataset consists of 70 000 images of size 28x28 pixels, each labeled from 0 to 9. -The scikit-learn dataset we will use consists of a selection of 1797 images of size \( 8\times 8 \) collected and processed from this database. +One can identify a set of key steps when using neural networks to solve supervised learning problems: -

    -To feed data into a feed-forward neural network we need to represent -the inputs as a feature matrix \( X = (n_{inputs}, n_{features}) \). Each -row represents an input, in this case a handwritten digit, and -each column represents a feature, in this case a pixel. The -correct answers, also known as labels or targets are -represented as a 1D array of integers -\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \). +

      +
    1. Collect and pre-process data
      2. +
    2. Define model and architecture
      3. +
    3. Choose cost function and optimizer
      4. +
    4. Train the model
      5. +
    5. Evaluate model performance on test data
      6. +
    6. Adjust hyperparameters (if necessary, network architecture)
      7. +
    -

    -As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from -measurements of height (in m) -and weight (in kg). If we have measurements of 5 people the feature matrix could be for example: - -$$ X = \begin{bmatrix} -1.85 & 81\\ -1.71 & 65\\ -1.95 & 103\\ -1.55 & 42\\ -1.63 & 56 -\end{bmatrix} ,$$ - -

    -and the targets would be: - -$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ - -

    -Since each input image is a 2D matrix, we need to flatten the image -(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a -feature matrix. This means we lose all spatial information in the -image, such as locality and translational invariance. More complicated -architectures such as Convolutional Neural Networks can take advantage -of such information, and are most commonly applied when analyzing -images. - -

    - - -

    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# flatten the image
-# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
-n_inputs = len(inputs)
-inputs = inputs.reshape(n_inputs, -1)
-print("X = (n_inputs, n_features) = " + str(inputs.shape))
-
-
-# choose some random images to display
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
    -

    diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html index b82e20c62..f1c2fc395 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html @@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source '___sec30'), ('Defining the cost function', 2, None, '___sec31'), ('Example: binary classification problem', 2, None, '___sec32'), + ('The Softmax function', 2, None, '___sec33'), ('Developing a code for doing neural networks with back ' 'propagation', 2, None, - '___sec33'), - ('Collect and pre-process data', 2, None, '___sec34'), - ('Train and test datasets', 2, None, '___sec35'), - ('Define model and architecture', 2, None, '___sec36'), - ('Layers', 2, None, '___sec37'), - ('Weights and biases', 2, None, '___sec38'), - ('Feed-forward pass', 2, None, '___sec39'), - ('Matrix multiplications', 2, None, '___sec40'), - ('Choose cost function and optimizer', 2, None, '___sec41'), - ('Optimizing the cost function', 2, None, '___sec42'), - ('Regularization', 2, None, '___sec43'), - ('Matrix multiplication', 2, None, '___sec44'), - ('Improving performance', 2, None, '___sec45'), - ('Full object-oriented implementation', 2, None, '___sec46'), - ('Evaluate model performance on test data', 2, None, '___sec47'), - ('Adjust hyperparameters', 2, None, '___sec48'), - ('Visualization', 2, None, '___sec49'), - ('scikit-learn implementation', 2, None, '___sec50'), - ('Visualization', 2, None, '___sec51'), + '___sec34'), + ('Collect and pre-process data', 2, None, '___sec35'), + ('Train and test datasets', 2, None, '___sec36'), + ('Define model and architecture', 2, None, '___sec37'), + ('Layers', 2, None, '___sec38'), + ('Weights and biases', 2, None, '___sec39'), + ('Feed-forward pass', 2, None, '___sec40'), + ('Matrix multiplications', 2, None, '___sec41'), + ('Choose cost function and optimizer', 2, None, '___sec42'), + ('Optimizing the cost function', 2, None, '___sec43'), + ('Regularization', 2, None, '___sec44'), + ('Matrix multiplication', 2, None, '___sec45'), + ('Improving performance', 2, None, '___sec46'), + ('Full object-oriented implementation', 2, None, '___sec47'), + ('Evaluate model performance on test data', 2, None, '___sec48'), + ('Adjust hyperparameters', 2, None, '___sec49'), + ('Visualization', 2, None, '___sec50'), + ('scikit-learn implementation', 2, None, '___sec51'), + ('Visualization', 2, None, '___sec52'), ('Building neural networks in Tensorflow and Keras', 2, None, - '___sec52'), - ('Tensorflow', 2, None, '___sec53'), - ('Collect and pre-process data', 2, None, '___sec54'), - ('Using TensorFlow backend', 2, None, '___sec55'), - ('Optimizing and using gradient descent', 2, None, '___sec56'), - ('Using Keras', 2, None, '___sec57'), - ('Which activation function should I use?', 2, None, '___sec58'), + '___sec53'), + ('Tensorflow', 2, None, '___sec54'), + ('Collect and pre-process data', 2, None, '___sec55'), + ('Using TensorFlow backend', 2, None, '___sec56'), + ('Optimizing and using gradient descent', 2, None, '___sec57'), + ('Using Keras', 2, None, '___sec58'), + ('Which activation function should I use?', 2, None, '___sec59'), ('Is the Logistic activation function (Sigmoid) our choice?', 2, None, - '___sec59'), - ('The derivative of the Logistic funtion', 2, None, '___sec60'), - ('The RELU function family', 2, None, '___sec61'), - ('Which activation function should we use?', 2, None, '___sec62'), + '___sec60'), + ('The derivative of the Logistic funtion', 2, None, '___sec61'), + ('The RELU function family', 2, None, '___sec62'), + ('Which activation function should we use?', 2, None, '___sec63'), ('A top-down perspective on Neural networks', 2, None, - '___sec63'), + '___sec64'), ('Limitations of supervised learning with deep networks', 2, None, - '___sec64')]} + '___sec65')]} end of tocinfo --> @@ -210,38 +211,39 @@ MathJax.Hub.Config({
  • Setting up a Multi-layer perceptron model for classification
  • Defining the cost function
  • Example: binary classification problem
    • -
  • Developing a code for doing neural networks with back propagation
    • -
  • Collect and pre-process data
    • -
  • Train and test datasets
    • -
  • Define model and architecture
    • -
  • Layers
    • -
  • Weights and biases
    • -
  • Feed-forward pass
    • -
  • Matrix multiplications
    • -
  • Choose cost function and optimizer
    • -
  • Optimizing the cost function
    • -
  • Regularization
    • -
  • Matrix multiplication
    • -
  • Improving performance
    • -
  • Full object-oriented implementation
    • -
  • Evaluate model performance on test data
    • -
  • Adjust hyperparameters
    • -
  • Visualization
    • -
  • scikit-learn implementation
    • -
  • Visualization
    • -
  • Building neural networks in Tensorflow and Keras
    • -
  • Tensorflow
    • -
  • Collect and pre-process data
    • -
  • Using TensorFlow backend
    • -
  • Optimizing and using gradient descent
    • -
  • Using Keras
    • -
  • Which activation function should I use?
    • -
  • Is the Logistic activation function (Sigmoid) our choice?
    • -
  • The derivative of the Logistic funtion
    • -
  • The RELU function family
    • -
  • Which activation function should we use?
    • -
  • A top-down perspective on Neural networks
    • -
  • Limitations of supervised learning with deep networks
    • +
  • The Softmax function
    • +
  • Developing a code for doing neural networks with back propagation
    • +
  • Collect and pre-process data
    • +
  • Train and test datasets
    • +
  • Define model and architecture
    • +
  • Layers
    • +
  • Weights and biases
    • +
  • Feed-forward pass
    • +
  • Matrix multiplications
    • +
  • Choose cost function and optimizer
    • +
  • Optimizing the cost function
    • +
  • Regularization
    • +
  • Matrix multiplication
    • +
  • Improving performance
    • +
  • Full object-oriented implementation
    • +
  • Evaluate model performance on test data
    • +
  • Adjust hyperparameters
    • +
  • Visualization
    • +
  • scikit-learn implementation
    • +
  • Visualization
    • +
  • Building neural networks in Tensorflow and Keras
    • +
  • Tensorflow
    • +
  • Collect and pre-process data
    • +
  • Using TensorFlow backend
    • +
  • Optimizing and using gradient descent
    • +
  • Using Keras
    • +
  • Which activation function should I use?
    • +
  • Is the Logistic activation function (Sigmoid) our choice?
    • +
  • The derivative of the Logistic funtion
    • +
  • The RELU function family
    • +
  • Which activation function should we use?
    • +
  • A top-down perspective on Neural networks
    • +
  • Limitations of supervised learning with deep networks
    • @@ -257,51 +259,97 @@ MathJax.Hub.Config({ -

    Train and test datasets

    +

    Collect and pre-process data

    -Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions. +Here we will be using the MNIST dataset, which is readily available through the scikit-learn +package. You may also find it for example here. +The MNIST (Modified National Institute of Standards and Technology) database is a large database +of handwritten digits that is commonly used for training various image processing systems. +The MNIST dataset consists of 70 000 images of size 28x28 pixels, each labeled from 0 to 9. +The scikit-learn dataset we will use consists of a selection of 1797 images of size \( 8\times 8 \) collected and processed from this database.

    -We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing. +To feed data into a feed-forward neural network we need to represent +the inputs as a feature matrix \( X = (n_{inputs}, n_{features}) \). Each +row represents an input, in this case a handwritten digit, and +each column represents a feature, in this case a pixel. The +correct answers, also known as labels or targets are +represented as a 1D array of integers +\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \).

    -It is important that the train and test datasets are drawn randomly from our dataset, to ensure -no bias in the sampling. -Say you are taking measurements of weather data to predict the weather in the coming 5 days. -You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data -collected from 12.00 to 24.00. +As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from +measurements of height (in m) +and weight (in kg). If we have measurements of 5 people the feature matrix could be for example: + +$$ X = \begin{bmatrix} +1.85 & 81\\ +1.71 & 65\\ +1.95 & 103\\ +1.55 & 42\\ +1.63 & 56 +\end{bmatrix} ,$$ + +

    +and the targets would be: + +$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ + +

    +Since each input image is a 2D matrix, we need to flatten the image +(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a +feature matrix. This means we lose all spatial information in the +image, such as locality and translational invariance. More complicated +architectures such as Convolutional Neural Networks can take advantage +of such information, and are most commonly applied when analyzing +images.

    -

    from sklearn.model_selection import train_test_split
+
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
 
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
 
-# equivalently in numpy
-def train_test_split_numpy(inputs, labels, train_size, test_size):
-    n_inputs = len(inputs)
-    inputs_shuffled = inputs.copy()
-    labels_shuffled = labels.copy()
-    
-    np.random.shuffle(inputs_shuffled)
-    np.random.shuffle(labels_shuffled)
-    
-    train_end = int(n_inputs*train_size)
-    X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
-    Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
-    
-    return X_train, X_test, Y_train, Y_test
+# ensure the same random numbers appear every time
+np.random.seed(0)
 
-#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
 
-print("Number of training images: " + str(len(X_train)))
-print("Number of test images: " + str(len(X_test)))
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# flatten the image
+# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
+n_inputs = len(inputs)
+inputs = inputs.reshape(n_inputs, -1)
+print("X = (n_inputs, n_features) = " + str(inputs.shape))
+
+
+# choose some random images to display
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
 
    
 
    
 
    
@@ -329,7 +377,7 @@ X_train, X_test, Y_train, Y_test = train_tes
   
  • 45
    • 
   
  • 46
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs037.html b/doc/pub/NeuralNet/html/._NeuralNet-bs037.html
index 74ef76c31..dd46359f6 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs037.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs037.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,49 +259,52 @@ MathJax.Hub.Config({
 
 
 
-
    Define model and architecture 
    
+
    Train and test datasets 
    
 
 
    
-Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have 
-
-$$ z = \sum_{i=1}^n w_i a_i ,$$
-
-$$ y = f(z) ,$$
+Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
 
 
    
-where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer
-and \( w_i \) is the weight to input \( i \).  
-The activation of the neurons in the input layer is just the features (e.g. a pixel value).
+We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
 
 
    
-The simplest activation function for a neuron is the Heaviside function:
-
-$$ f(z) = 
-\begin{cases}
-1,  &  z > 0\\
-0,  & \text{otherwise}
-\end{cases}
-$$
+It is important that the train and test datasets are drawn randomly from our dataset, to ensure
+no bias in the sampling.  
+Say you are taking measurements of weather data to predict the weather in the coming 5 days.
+You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data
+collected from 12.00 to 24.00.
 
 
    
-A feed-forward neural network with this activation is known as a perceptron.  
-For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.  
-This activation can be generalized to \( k \) classes (using e.g. the one-against-all strategy), 
-and we call these architectures multiclass perceptrons.
 
-
    
-However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and  
-Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.
+
+
    from sklearn.model_selection import train_test_split
 
-
    
-Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).  
-We will be using the sigmoid function \( \sigma(x) \):  
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
 
-$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$
+# equivalently in numpy
+def train_test_split_numpy(inputs, labels, train_size, test_size):
+    n_inputs = len(inputs)
+    inputs_shuffled = inputs.copy()
+    labels_shuffled = labels.copy()
+    
+    np.random.shuffle(inputs_shuffled)
+    np.random.shuffle(labels_shuffled)
+    
+    train_end = int(n_inputs*train_size)
+    X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
+    Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
+    
+    return X_train, X_test, Y_train, Y_test
 
-
    
-which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.
+#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
 
+print("Number of training images: " + str(len(X_train)))
+print("Number of test images: " + str(len(X_test)))
+
    
 
    
 
    
 
@@ -326,7 +331,7 @@ which is inspired by probability theory (see logistic regression) and was most c
   
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    • 
   
  • 47
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs038.html b/doc/pub/NeuralNet/html/._NeuralNet-bs038.html
index fe1f53db9..669d2bb60 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs038.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs038.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,49 +257,50 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Layers 
    
-
-
      
-
    •  Input
      •  
-
    
-
-Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.  
-
-
      
-
    •  Hidden layer
      • 
-
    
-
-We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.  
-Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.  
-
-
      
-
    •  Output
      • 
-
    
-
-If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
-which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.
+
    Define model and architecture 
    
 
 
    
-For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.
+Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have 
+
+$$ z = \sum_{i=1}^n w_i a_i ,$$
+
+$$ y = f(z) ,$$
 
 
    
-Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the softmax function:  
-
-$$ P(\text{class \( j \)} \mid \text{input \( \hat{a} \)}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}}
-{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,$$
+where \( f \) is the activation function, \( a_i \) represents input from neuron \( i \) in the preceding layer
+and \( w_i \) is the weight to input \( i \).  
+The activation of the neurons in the input layer is just the features (e.g. a pixel value).
 
 
    
-i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \hat{a} \), with \( \hat{w}_j \) the weights of neuron \( j \) to the inputs.  
-The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.  
-The exponent is just the weighted sum of inputs as before:  
+The simplest activation function for a neuron is the Heaviside function:
 
-$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$
+$$ f(z) = 
+\begin{cases}
+1,  &  z > 0\\
+0,  & \text{otherwise}
+\end{cases}
+$$
 
 
    
-Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
-weights to the output layer.
+A feed-forward neural network with this activation is known as a perceptron.  
+For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.  
+This activation can be generalized to \( k \) classes (using e.g. the one-against-all strategy), 
+and we call these architectures multiclass perceptrons.
+
+
    
+However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and  
+Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.
+
+
    
+Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).  
+We will be using the sigmoid function \( \sigma(x) \):  
+
+$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$
+
+
    
+which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.
 
 
    
 
    
@@ -325,7 +328,7 @@ weights to the output layer.
   
  • 47
    • 
   
  • 48
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs039.html b/doc/pub/NeuralNet/html/._NeuralNet-bs039.html
index 2f21e4b4f..d0a8591d9 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs039.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs039.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,39 +259,48 @@ MathJax.Hub.Config({
 
 
 
-
    Weights and biases 
    
+
    Layers 
    
+
+
      
+
    •  Input
      •  
+
    
+
+Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.  
+
+
      
+
    •  Hidden layer
      • 
+
    
+
+We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.  
+Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.  
+
+
      
+
    •  Output
      • 
+
    
+
+If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,
+which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.
 
 
    
-Typically weights are initialized with small values distributed around zero, drawn from a uniform
-or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.
+For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.
 
 
    
-Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
-of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \( j \), \( b_j \):  
+Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \( j = 0,1,...,9 \). The activation of each output neuron \( j \) will be according to the softmax function:  
 
-$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$
+$$ P(\text{class \( j \)} \mid \text{input \( \hat{a} \)}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}}
+{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,$$
 
 
    
-The bias weights \( \hat{b} \) are often initialized to zero, but a small value like \( 0.01 \) ensures all neurons have some output which can be backpropagated in the first training cycle.
+i.e. each neuron \( j \) outputs the probability of being in class \( j \) given an input from the hidden layer \( \hat{a} \), with \( \hat{w}_j \) the weights of neuron \( j \) to the inputs.  
+The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.  
+The exponent is just the weighted sum of inputs as before:  
+
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$
+
 
    
+Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500
+weights to the output layer.
 
-
-
    # building our neural network
-
-n_inputs, n_features = X_train.shape
-n_hidden_neurons = 50
-n_categories = 10
-
-# we make the weights normally distributed using numpy.random.randn
-
-# weights and bias in the hidden layer
-hidden_weights = np.random.randn(n_features, n_hidden_neurons)
-hidden_bias = np.zeros(n_hidden_neurons) + 0.01
-
-# weights and bias in the output layer
-output_weights = np.random.randn(n_hidden_neurons, n_categories)
-output_bias = np.zeros(n_categories) + 0.01
-
    
 
    
 
    
 
@@ -316,7 +327,7 @@ output_bias = np48
   
  • 49
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs040.html b/doc/pub/NeuralNet/html/._NeuralNet-bs040.html
index ee0727ce9..842f99d27 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs040.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs040.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,32 +257,41 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Feed-forward pass 
    
+
    Weights and biases 
    
 
 
    
-Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.  
-For each input image we calculate a weighted sum of input features (pixel values) to each neuron \( j \) in the hidden layer \( l \):  
-
-$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
+Typically weights are initialized with small values distributed around zero, drawn from a uniform
+or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.
 
 
    
-this is then passed through our activation function  
+Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range
+of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \( j \), \( b_j \):  
 
-$$ a_{j}^{l} = f(z_{j}^{l}) .$$
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$
 
 
    
-We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:  
-
-$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
-
+The bias weights \( \hat{b} \) are often initialized to zero, but a small value like \( 0.01 \) ensures all neurons have some output which can be backpropagated in the first training cycle.
 
    
-Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:  
 
-$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$
+
+
    # building our neural network
 
+n_inputs, n_features = X_train.shape
+n_hidden_neurons = 50
+n_categories = 10
+
+# we make the weights normally distributed using numpy.random.randn
+
+# weights and bias in the hidden layer
+hidden_weights = np.random.randn(n_features, n_hidden_neurons)
+hidden_bias = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+output_weights = np.random.randn(n_hidden_neurons, n_categories)
+output_bias = np.zeros(n_categories) + 0.01
+
    
 
    
 
    
 
@@ -307,7 +318,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
   
  • 49
    • 
   
  • 50
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs041.html b/doc/pub/NeuralNet/html/._NeuralNet-bs041.html
index d01d443c9..156e55d98 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs041.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs041.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,80 +257,32 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Matrix multiplications 
    
+
    Feed-forward pass 
    
 
 
    
-Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
-layer have the dimensions  
-\( W_{hidden} = (n_{features}, n_{hidden}) \),
-we can easily feed the network all our training data in one go by taking the matrix product  
+Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.  
+For each input image we calculate a weighted sum of input features (pixel values) to each neuron \( j \) in the hidden layer \( l \):  
 
-$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
+$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
 
 
    
-and obtain a matrix that holds the weighted sum of inputs to the hidden layer
-for each input image and each hidden neuron.    
-We also add the bias to obtain a matrix of weighted sums to the hidden layer \( Z^{h} \):  
+this is then passed through our activation function  
 
-$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$
+$$ a_{j}^{l} = f(z_{j}^{l}) .$$
 
 
    
-meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.  
-This is then passed through the activation:  
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:  
 
-$$ \hat{a}^{l} = f(\hat{z}^l) .$$
+$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
 
 
    
-This is fed to the output layer:  
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:  
 
-$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$
+$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
+{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$
 
-
    
-Finally we receive our output values for each image and each category by passing it through the softmax function:  
-
-$$ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$
-
-
    
-
-
-
    # setup the feed-forward pass, subscript h = hidden layer
-
-def sigmoid(x):
-    return 1/(1 + np.exp(-x))
-
-def feed_forward(X):
-    # weighted sum of inputs to the hidden layer
-    z_h = np.matmul(X, hidden_weights) + hidden_bias
-    # activation in the hidden layer
-    a_h = sigmoid(z_h)
-    
-    # weighted sum of inputs to the output layer
-    z_o = np.matmul(a_h, output_weights) + output_bias
-    # softmax output
-    # axis 0 holds each input and axis 1 the probabilities of each category
-    exp_term = np.exp(z_o)
-    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-    
-    return probabilities
-
-probabilities = feed_forward(X_train)
-print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
-print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
-print("probabilities sum up to: " + str(probabilities[0].sum()))
-print()
-
-# we obtain a prediction by taking the class with the highest likelihood
-def predict(X):
-    probabilities = feed_forward(X)
-    return np.argmax(probabilities, axis=1)
-
-predictions = predict(X_train)
-print("predictions = (n_inputs) = " + str(predictions.shape))
-print("prediction for image 0: " + str(predictions[0]))
-print("correct label for image 0: " + str(Y_train[0]))
-
    
 
    
 
    
 
@@ -355,7 +309,7 @@ predictions = predict(X_train)
   
  • 50
    • 
   
  • 51
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs042.html b/doc/pub/NeuralNet/html/._NeuralNet-bs042.html
index c51a3bae6..044950df5 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs042.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs042.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,36 +257,80 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Choose cost function and optimizer 
    
+
    Matrix multiplications 
    
 
 
    
-To measure how well our neural network is doing we need to introduce a cost function.  
-We will call the function that gives the error of a single sample output the loss function, and the function
-that gives the total error of our network across all samples the cost function.
-A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood.
+Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
+layer have the dimensions  
+\( W_{hidden} = (n_{features}, n_{hidden}) \),
+we can easily feed the network all our training data in one go by taking the matrix product  
+
+$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
 
 
    
-In multiclass classification it is common to treat each integer label as a so called one-hot vector:  
+and obtain a matrix that holds the weighted sum of inputs to the hidden layer
+for each input image and each hidden neuron.    
+We also add the bias to obtain a matrix of weighted sums to the hidden layer \( Z^{h} \):  
 
-$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$  
-
-$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
+$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$
 
 
    
-i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
+meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.  
+This is then passed through the activation:  
+
+$$ \hat{a}^{l} = f(\hat{z}^l) .$$
 
 
    
-Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector.  
-We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \hat{x}_i \) in the dataset.
+This is fed to the output layer:  
+
+$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$
 
 
    
-In the one-hot representation only one of the terms in the loss function is non-zero, namely the
-probability of the correct category \( c' \)  
-(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong
-you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \hat{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
+Finally we receive our output values for each image and each category by passing it through the softmax function:  
 
+$$ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$
+
+
    
+
+
+
    # setup the feed-forward pass, subscript h = hidden layer
+
+def sigmoid(x):
+    return 1/(1 + np.exp(-x))
+
+def feed_forward(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    exp_term = np.exp(z_o)
+    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+    
+    return probabilities
+
+probabilities = feed_forward(X_train)
+print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
+print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
+print("probabilities sum up to: " + str(probabilities[0].sum()))
+print()
+
+# we obtain a prediction by taking the class with the highest likelihood
+def predict(X):
+    probabilities = feed_forward(X)
+    return np.argmax(probabilities, axis=1)
+
+predictions = predict(X_train)
+print("predictions = (n_inputs) = " + str(predictions.shape))
+print("prediction for image 0: " + str(predictions[0]))
+print("correct label for image 0: " + str(Y_train[0]))
+
    
 
    
 
    
 
@@ -311,7 +357,7 @@ you got the correct label. The probability of category \( c \) is given by the s
   
  • 51
    • 
   
  • 52
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs043.html b/doc/pub/NeuralNet/html/._NeuralNet-bs043.html
index c8a2b2793..504ed3872 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs043.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs043.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,42 +259,33 @@ MathJax.Hub.Config({
 
 
 
-
    Optimizing the cost function 
    
+
    Choose cost function and optimizer 
    
 
 
    
-The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
-is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function.  
-Each parameter \( \theta \) is iteratively adjusted according to the rule  
-
-$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
+To measure how well our neural network is doing we need to introduce a cost function.  
+We will call the function that gives the error of a single sample output the loss function, and the function
+that gives the total error of our network across all samples the cost function.
+A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood.
 
 
    
-where \( \eta \) is known as the learning rate, which controls how big a step we take towards the minimum.  
-This update can be repeated for any number of iterations, or until we are satisfied with the result.
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:  
+
+$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$  
+
+$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
 
 
    
-A simple and effective improvement is a variant called Batch Gradient Descent.  
-Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
-on a subset of the data called a minibatch.  
-If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches
-is \( N/M \).  
-We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes:  
-
-$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
 
 
    
-i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
+Let \( y_{ic} \) denote the \( c \)-th component of the \( i \)-th one-hot vector.  
+We define the cost function \( \mathcal{C} \) as a sum over the cross-entropy loss for each point \( \hat{x}_i \) in the dataset.
 
 
    
-This has two important benefits:  
-
-
      
-
    1.  Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.
      2.   
-
    2.  It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.
      3.   
-
    
-
-The various optmization  methods, with codes and algorithms,  are discussed in our lectures on Gradient descent approaches.
+In the one-hot representation only one of the terms in the loss function is non-zero, namely the
+probability of the correct category \( c' \)  
+(i.e. the category \( c' \) such that \( y_{ic'} = 1 \)). This means that the cross entropy loss only punishes you for how wrong
+you got the correct label. The probability of category \( c \) is given by the softmax function. The vector \( \hat{\theta} \) represents the parameters of our network, i.e. all the weights and biases.
 
 
    
 
    
@@ -320,7 +313,7 @@ The various optmization  methods, with codes and algorithms,  are discussed in o
   
  • 52
    • 
   
  • 53
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs044.html b/doc/pub/NeuralNet/html/._NeuralNet-bs044.html
index 734d0f29b..e2fb902a3 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs044.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs044.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,37 +257,44 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Regularization 
    
+
    Optimizing the cost function 
    
 
 
    
-It is common to add an extra term to the cost function, proportional
-to the size of the weights.  This is equivalent to constraining the
-size of the weights, so that they do not grow out of control.
-Constraining the size of the weights means that the weights cannot
-grow arbitrarily large to fit the training data, and in this way
-reduces overfitting.
+The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
+is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function.  
+Each parameter \( \theta \) is iteratively adjusted according to the rule  
+
+$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
 
 
    
-We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:  
+where \( \eta \) is known as the learning rate, which controls how big a step we take towards the minimum.  
+This update can be repeated for any number of iterations, or until we are satisfied with the result.
+
+
    
+A simple and effective improvement is a variant called Batch Gradient Descent.  
+Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient
+on a subset of the data called a minibatch.  
+If there are \( N \) data points and we have a minibatch size of \( M \), the total number of batches
+is \( N/M \).  
+We denote each minibatch \( B_k \), with \( k = 1, 2,...,N/M \). The gradient then becomes:  
 
 $$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2 
-= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
+\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
 
 
    
-i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
 
 
    
-In order to train the model, we need to calculate the derivative of
-the cost function with respect to every bias and weight in the
-network.  In total our network has \( (64 + 1)\times 50=3250 \) weights in
-the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output
-layer (\( +1 \) for the bias), and the gradient must be calculated for
-every parameter.  We use the backpropagation algorithm discussed
-above. This is a clever use of the chain rule that allows us to
-calculate the gradient efficently.
+This has two important benefits:  
+
+
      
+
    1.  Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.
      2.   
+
    2.  It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.
      3.   
+
    
+
+The various optmization  methods, with codes and algorithms,  are discussed in our lectures on Gradient descent approaches.
 
 
    
 
    
@@ -313,7 +322,7 @@ calculate the gradient efficently.
   
  • 53
    • 
   
  • 54
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs045.html b/doc/pub/NeuralNet/html/._NeuralNet-bs045.html
index 1331dd56d..e1b7f70bd 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs045.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs045.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,119 +257,38 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Matrix  multiplication 
    
+
    Regularization 
    
 
 
    
-To more efficently train our network these equations are implemented using matrix operations.  
-The error in the output layer is calculated simply as, with \( \hat{t} \) being our targets,  
-
-$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
+It is common to add an extra term to the cost function, proportional
+to the size of the weights.  This is equivalent to constraining the
+size of the weights, so that they do not grow out of control.
+Constraining the size of the weights means that the weights cannot
+grow arbitrarily large to fit the training data, and in this way
+reduces overfitting.
 
 
    
-The gradient for the output weights is calculated as  
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:  
 
-$$ \nabla W_{L} = \hat{a}^T \delta_L   = (n_{hidden}, n_{categories}) ,$$
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2 
+= \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
 
 
    
-where \( \hat{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input.  
-Since we are going backwards we have to transpose the activation matrix.
+i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
 
 
    
-The gradient with respect to the output bias is then  
+In order to train the model, we need to calculate the derivative of
+the cost function with respect to every bias and weight in the
+network.  In total our network has \( (64 + 1)\times 50=3250 \) weights in
+the hidden layer and \( (50 + 1)\times 10=510 \) weights to the output
+layer (\( +1 \) for the bias), and the gradient must be calculated for
+every parameter.  We use the backpropagation algorithm discussed
+above. This is a clever use of the chain rule that allows us to
+calculate the gradient efficently.
 
-$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
-
-
    
-The error in the hidden layer is  
-
-$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$
-
-
    
-where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean
-that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes
-the Hadamard product, meaning element-wise multiplication.
-
-
    
-This again gives us the gradients in the hidden layer:  
-
-$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$  
-
-$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$
-
-
    
-
-
-
    # to categorical turns our integer vector into a onehot representation
-from sklearn.metrics import accuracy_score
-
-# one-hot in numpy
-def to_categorical_numpy(integer_vector):
-    n_inputs = len(integer_vector)
-    n_categories = np.max(integer_vector) + 1
-    onehot_vector = np.zeros((n_inputs, n_categories))
-    onehot_vector[range(n_inputs), integer_vector] = 1
-    
-    return onehot_vector
-
-#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
-Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
-
-def feed_forward_train(X):
-    # weighted sum of inputs to the hidden layer
-    z_h = np.matmul(X, hidden_weights) + hidden_bias
-    # activation in the hidden layer
-    a_h = sigmoid(z_h)
-    
-    # weighted sum of inputs to the output layer
-    z_o = np.matmul(a_h, output_weights) + output_bias
-    # softmax output
-    # axis 0 holds each input and axis 1 the probabilities of each category
-    exp_term = np.exp(z_o)
-    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-    
-    # for backpropagation need activations in hidden and output layers
-    return a_h, probabilities
-
-def backpropagation(X, Y):
-    a_h, probabilities = feed_forward_train(X)
-    
-    # error in the output layer
-    error_output = probabilities - Y
-    # error in the hidden layer
-    error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
-    
-    # gradients for the output layer
-    output_weights_gradient = np.matmul(a_h.T, error_output)
-    output_bias_gradient = np.sum(error_output, axis=0)
-    
-    # gradient for the hidden layer
-    hidden_weights_gradient = np.matmul(X.T, error_hidden)
-    hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
-    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
-
-print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
-eta = 0.01
-lmbd = 0.01
-for i in range(1000):
-    # calculate gradients
-    dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
-    
-    # regularization term gradients
-    dWo += lmbd * output_weights
-    dWh += lmbd * hidden_weights
-    
-    # update weights and biases
-    output_weights -= eta * dWo
-    output_bias -= eta * dBo
-    hidden_weights -= eta * dWh
-    hidden_bias -= eta * dBh
-
-print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
    
 
    
 
    
 
@@ -394,7 +315,7 @@ lmbd = 0.0154
   
  • 55
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs046.html b/doc/pub/NeuralNet/html/._NeuralNet-bs046.html
index 762f31a53..f382128e0 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs046.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs046.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,23 +259,117 @@ MathJax.Hub.Config({
 
 
 
-
    Improving performance 
    
+
    Matrix  multiplication 
    
 
 
    
-As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.  
-In order to obtain a network that does something useful, we will have to do a bit more work.
+To more efficently train our network these equations are implemented using matrix operations.  
+The error in the output layer is calculated simply as, with \( \hat{t} \) being our targets,  
+
+$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
 
 
    
-The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).
+The gradient for the output weights is calculated as  
+
+$$ \nabla W_{L} = \hat{a}^T \delta_L   = (n_{hidden}, n_{categories}) ,$$
 
 
    
-Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period
-going through the entire dataset (\( n/M \) batches) an epoch.
+where \( \hat{a} = (n_{inputs}, n_{hidden}) \). This simply means that we are summing up the gradients for each input.  
+Since we are going backwards we have to transpose the activation matrix.
 
 
    
-If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.  
-Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here.
+The gradient with respect to the output bias is then  
 
+$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
+
+
    
+The error in the hidden layer is  
+
+$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$
+
+
    
+where \( f'(a_{h}) \) is the derivative of the activation in the hidden layer. The matrix products mean
+that we are summing up the products for each neuron in the output layer. The symbol \( \circ \) denotes
+the Hadamard product, meaning element-wise multiplication.
+
+
    
+This again gives us the gradients in the hidden layer:  
+
+$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$  
+
+$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$
+
+
    
+
+
+
    # to categorical turns our integer vector into a onehot representation
+from sklearn.metrics import accuracy_score
+
+# one-hot in numpy
+def to_categorical_numpy(integer_vector):
+    n_inputs = len(integer_vector)
+    n_categories = np.max(integer_vector) + 1
+    onehot_vector = np.zeros((n_inputs, n_categories))
+    onehot_vector[range(n_inputs), integer_vector] = 1
+    
+    return onehot_vector
+
+#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
+Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
+
+def feed_forward_train(X):
+    # weighted sum of inputs to the hidden layer
+    z_h = np.matmul(X, hidden_weights) + hidden_bias
+    # activation in the hidden layer
+    a_h = sigmoid(z_h)
+    
+    # weighted sum of inputs to the output layer
+    z_o = np.matmul(a_h, output_weights) + output_bias
+    # softmax output
+    # axis 0 holds each input and axis 1 the probabilities of each category
+    exp_term = np.exp(z_o)
+    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+    
+    # for backpropagation need activations in hidden and output layers
+    return a_h, probabilities
+
+def backpropagation(X, Y):
+    a_h, probabilities = feed_forward_train(X)
+    
+    # error in the output layer
+    error_output = probabilities - Y
+    # error in the hidden layer
+    error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
+    
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_h.T, error_output)
+    output_bias_gradient = np.sum(error_output, axis=0)
+    
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(X.T, error_hidden)
+    hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
+eta = 0.01
+lmbd = 0.01
+for i in range(1000):
+    # calculate gradients
+    dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
+    
+    # regularization term gradients
+    dWo += lmbd * output_weights
+    dWh += lmbd * hidden_weights
+    
+    # update weights and biases
+    output_weights -= eta * dWo
+    output_bias -= eta * dBo
+    hidden_weights -= eta * dWh
+    hidden_bias -= eta * dBh
+
+print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
+
    
 
    
 
    
 
@@ -300,7 +396,7 @@ Andrew Ng goes through some of these considerations in this 55
   
  • 56
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs047.html b/doc/pub/NeuralNet/html/._NeuralNet-bs047.html
index 90bba8b4c..f37757f07 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs047.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs047.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,116 +259,23 @@ MathJax.Hub.Config({
 
 
 
-
    Full object-oriented implementation 
    
+
    Improving performance 
    
 
 
    
-It is very natural to think of the network as an object, with specific instances of the network
-being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.
+As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.  
+In order to obtain a network that does something useful, we will have to do a bit more work.
 
 
    
+The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \( \eta = 10^{-6}, 10^{-5},...,10^{-1} \) with different regularization parameters \( \lambda = 10^{-6},...,10^{-0} \).
 
-
-
    class NeuralNetwork:
-    def __init__(
-        self,
-        X_data,
-        Y_data,
-        n_hidden_neurons=50,
-        n_categories=10,
-        epochs=10,
-        batch_size=100,
-        eta=0.1,
-        lmbd=0.0,
+
    
+Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period
+going through the entire dataset (\( n/M \) batches) an epoch.
 
-    ):
-        self.X_data_full = X_data
-        self.Y_data_full = Y_data
+
    
+If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.  
+Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here.
 
-        self.n_inputs = X_data.shape[0]
-        self.n_features = X_data.shape[1]
-        self.n_hidden_neurons = n_hidden_neurons
-        self.n_categories = n_categories
-
-        self.epochs = epochs
-        self.batch_size = batch_size
-        self.iterations = self.n_inputs // self.batch_size
-        self.eta = eta
-        self.lmbd = lmbd
-
-        self.create_biases_and_weights()
-
-    def create_biases_and_weights(self):
-        self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
-        self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
-
-        self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
-        self.output_bias = np.zeros(self.n_categories) + 0.01
-
-    def feed_forward(self):
-        # feed-forward for training
-        self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
-        self.a_h = sigmoid(self.z_h)
-
-        self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
-
-        exp_term = np.exp(self.z_o)
-        self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
-    def feed_forward_out(self, X):
-        # feed-forward for output
-        z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
-        a_h = sigmoid(z_h)
-
-        z_o = np.matmul(a_h, self.output_weights) + self.output_bias
-        
-        exp_term = np.exp(z_o)
-        probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-        return probabilities
-
-    def backpropagation(self):
-        error_output = self.probabilities - self.Y_data
-        error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
-
-        self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
-        self.output_bias_gradient = np.sum(error_output, axis=0)
-
-        self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
-        self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
-        if self.lmbd > 0.0:
-            self.output_weights_gradient += self.lmbd * self.output_weights
-            self.hidden_weights_gradient += self.lmbd * self.hidden_weights
-
-        self.output_weights -= self.eta * self.output_weights_gradient
-        self.output_bias -= self.eta * self.output_bias_gradient
-        self.hidden_weights -= self.eta * self.hidden_weights_gradient
-        self.hidden_bias -= self.eta * self.hidden_bias_gradient
-
-    def predict(self, X):
-        probabilities = self.feed_forward_out(X)
-        return np.argmax(probabilities, axis=1)
-
-    def predict_probabilities(self, X):
-        probabilities = self.feed_forward_out(X)
-        return probabilities
-
-    def train(self):
-        data_indices = np.arange(self.n_inputs)
-
-        for i in range(self.epochs):
-            for j in range(self.iterations):
-                # pick datapoints with replacement
-                chosen_datapoints = np.random.choice(
-                    data_indices, size=self.batch_size, replace=False
-                )
-
-                # minibatch training data
-                self.X_data = self.X_data_full[chosen_datapoints]
-                self.Y_data = self.Y_data_full[chosen_datapoints]
-
-                self.feed_forward()
-                self.backpropagation()
-
    
 
    
 
    
 
@@ -393,7 +302,7 @@ being realizations of this object with different hyperparameters. An implementat
   
  • 56
    • 
   
  • 57
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs048.html b/doc/pub/NeuralNet/html/._NeuralNet-bs048.html
index 77963059e..e9264ade4 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs048.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs048.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,37 +259,115 @@ MathJax.Hub.Config({
 
 
 
-
    Evaluate model performance on test data 
    
+
    Full object-oriented implementation 
    
 
 
    
-To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.  
-We measure the performance of the network using the accuracy score.  
-The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \).  
-
-$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,$$
-
-
    
-where \( I \) is the indicator function, \( 1 \) if \( \hat{y}_i = y_i \) and \( 0 \) otherwise.
+It is very natural to think of the network as an object, with specific instances of the network
+being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.
 
 
    
 
 
-
    epochs = 100
-batch_size = 100
+
    class NeuralNetwork:
+    def __init__(
+        self,
+        X_data,
+        Y_data,
+        n_hidden_neurons=50,
+        n_categories=10,
+        epochs=10,
+        batch_size=100,
+        eta=0.1,
+        lmbd=0.0,
 
-dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
-                    n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
-dnn.train()
-test_predict = dnn.predict(X_test)
+    ):
+        self.X_data_full = X_data
+        self.Y_data_full = Y_data
 
-# accuracy score from scikit library
-print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+        self.n_inputs = X_data.shape[0]
+        self.n_features = X_data.shape[1]
+        self.n_hidden_neurons = n_hidden_neurons
+        self.n_categories = n_categories
 
-# equivalent in numpy
-def accuracy_score_numpy(Y_test, Y_pred):
-    return np.sum(Y_test == Y_pred) / len(Y_test)
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
 
-#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
+        self.create_biases_and_weights()
+
+    def create_biases_and_weights(self):
+        self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
+        self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
+
+        self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
+        self.output_bias = np.zeros(self.n_categories) + 0.01
+
+    def feed_forward(self):
+        # feed-forward for training
+        self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
+        self.a_h = sigmoid(self.z_h)
+
+        self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
+
+        exp_term = np.exp(self.z_o)
+        self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
+    def feed_forward_out(self, X):
+        # feed-forward for output
+        z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
+        a_h = sigmoid(z_h)
+
+        z_o = np.matmul(a_h, self.output_weights) + self.output_bias
+        
+        exp_term = np.exp(z_o)
+        probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+        return probabilities
+
+    def backpropagation(self):
+        error_output = self.probabilities - self.Y_data
+        error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
+
+        self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
+        self.output_bias_gradient = np.sum(error_output, axis=0)
+
+        self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
+        self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
+
+        if self.lmbd > 0.0:
+            self.output_weights_gradient += self.lmbd * self.output_weights
+            self.hidden_weights_gradient += self.lmbd * self.hidden_weights
+
+        self.output_weights -= self.eta * self.output_weights_gradient
+        self.output_bias -= self.eta * self.output_bias_gradient
+        self.hidden_weights -= self.eta * self.hidden_weights_gradient
+        self.hidden_bias -= self.eta * self.hidden_bias_gradient
+
+    def predict(self, X):
+        probabilities = self.feed_forward_out(X)
+        return np.argmax(probabilities, axis=1)
+
+    def predict_probabilities(self, X):
+        probabilities = self.feed_forward_out(X)
+        return probabilities
+
+    def train(self):
+        data_indices = np.arange(self.n_inputs)
+
+        for i in range(self.epochs):
+            for j in range(self.iterations):
+                # pick datapoints with replacement
+                chosen_datapoints = np.random.choice(
+                    data_indices, size=self.batch_size, replace=False
+                )
+
+                # minibatch training data
+                self.X_data = self.X_data_full[chosen_datapoints]
+                self.Y_data = self.Y_data_full[chosen_datapoints]
+
+                self.feed_forward()
+                self.backpropagation()
 
    
 
    
 
    
@@ -315,7 +395,7 @@ test_predict = dnn57
   
  • 58
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs049.html b/doc/pub/NeuralNet/html/._NeuralNet-bs049.html
index 3815d10a1..aa4744012 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs049.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs049.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,35 +259,37 @@ MathJax.Hub.Config({
 
 
 
-
    Adjust hyperparameters 
    
+
    Evaluate model performance on test data 
    
 
 
    
-We now perform a grid search to find the optimal hyperparameters for the network.  
-Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate).
+To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.  
+We measure the performance of the network using the accuracy score.  
+The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \( 1 \).  
+
+$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,$$
+
+
    
+where \( I \) is the indicator function, \( 1 \) if \( \hat{y}_i = y_i \) and \( 0 \) otherwise.
 
 
    
 
 
-
    eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-# store the models for later use
-DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
    epochs = 100
+batch_size = 100
 
-# grid search
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
-                            n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
-        dnn.train()
-        
-        DNN_numpy[i][j] = dnn
-        
-        test_predict = dnn.predict(X_test)
-        
-        print("Learning rate  = ", eta)
-        print("Lambda = ", lmbd)
-        print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
-        print()
+dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+                    n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+dnn.train()
+test_predict = dnn.predict(X_test)
+
+# accuracy score from scikit library
+print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+
+# equivalent in numpy
+def accuracy_score_numpy(Y_test, Y_pred):
+    return np.sum(Y_test == Y_pred) / len(Y_test)
+
+#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
 
    
 
    
 
    
@@ -313,7 +317,7 @@ DNN_numpy = np.
   
  • 58
    • 
   
  • 59
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs050.html b/doc/pub/NeuralNet/html/._NeuralNet-bs050.html
index 47b18cf03..ff0e388c6 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs050.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs050.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,44 +259,35 @@ MathJax.Hub.Config({
 
 
 
-
    Visualization 
    
+
    Adjust hyperparameters 
    
+
+
    
+We now perform a grid search to find the optimal hyperparameters for the network.  
+Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \( 98\% \) (\( 2\% \) error rate).
 
 
    
 
 
-
    # visual representation of grid search
-# uses seaborn heatmap, you can also do this with matplotlib imshow
-import seaborn as sns
+
    eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
 
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        dnn = DNN_numpy[i][j]
+# grid search
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
+                            n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
+        dnn.train()
         
-        train_pred = dnn.predict(X_train) 
-        test_pred = dnn.predict(X_test)
-
-        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
-        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
-
+        DNN_numpy[i][j] = dnn
         
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+        test_predict = dnn.predict(X_test)
+        
+        print("Learning rate  = ", eta)
+        print("Lambda = ", lmbd)
+        print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
+        print()
 
    
 
    
 
    
@@ -322,7 +315,7 @@ plt.show()
   
  • 59
    • 
   
  • 60
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs051.html b/doc/pub/NeuralNet/html/._NeuralNet-bs051.html
index 669324e88..e1876b12b 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs051.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs051.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,41 +259,44 @@ MathJax.Hub.Config({
 
 
 
-
    scikit-learn implementation 
    
-
-
    
-scikit-learn focuses more
-on traditional machine learning methods, such as regression,
-clustering, decision trees, etc. As such, it has only two types of
-neural networks: Multi Layer Perceptron outputting continuous values,
-MPLRegressor, and Multi Layer Perceptron outputting labels,
-MLPClassifier. We will see how simple it is to use these classes.
-
-
    
-scikit-learn implements a few improvements from our neural network,
-such as early stopping, a varying learning rate, different
-optimization methods, etc. We would therefore expect a better
-performance overall.
+
    Visualization 
    
 
 
    
 
 
-
    from sklearn.neural_network import MLPClassifier
-# store models for later use
-DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
    # visual representation of grid search
+# uses seaborn heatmap, you can also do this with matplotlib imshow
+import seaborn as sns
 
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
-                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
-        dnn.fit(X_train, Y_train)
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        dnn = DNN_numpy[i][j]
         
-        DNN_scikit[i][j] = dnn
+        train_pred = dnn.predict(X_train) 
+        test_pred = dnn.predict(X_test)
+
+        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
+
         
-        print("Learning rate  = ", eta)
-        print("Lambda = ", lmbd)
-        print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
-        print()
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
 
    
 
    
 
    
@@ -319,7 +324,7 @@ DNN_scikit = np
   
  • 60
    • 
   
  • 61
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs052.html b/doc/pub/NeuralNet/html/._NeuralNet-bs052.html
index bddb22bf1..c8112b974 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs052.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs052.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,44 +259,41 @@ MathJax.Hub.Config({
 
 
 
-
    Visualization 
    
+
    scikit-learn implementation 
    
+
+
    
+scikit-learn focuses more
+on traditional machine learning methods, such as regression,
+clustering, decision trees, etc. As such, it has only two types of
+neural networks: Multi Layer Perceptron outputting continuous values,
+MPLRegressor, and Multi Layer Perceptron outputting labels,
+MLPClassifier. We will see how simple it is to use these classes.
+
+
    
+scikit-learn implements a few improvements from our neural network,
+such as early stopping, a varying learning rate, different
+optimization methods, etc. We would therefore expect a better
+performance overall.
+
 
    
 
 
-
    # optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
+
    from sklearn.neural_network import MLPClassifier
+# store models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
 
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        dnn = DNN_scikit[i][j]
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X_train, Y_train)
         
-        train_pred = dnn.predict(X_train) 
-        test_pred = dnn.predict(X_test)
-
-        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
-        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
-
+        DNN_scikit[i][j] = dnn
         
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+        print("Learning rate  = ", eta)
+        print("Lambda = ", lmbd)
+        print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
+        print()
 
    
 
    
 
    
@@ -322,7 +321,7 @@ plt.show()
   
  • 61
    • 
   
  • 62
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs053.html b/doc/pub/NeuralNet/html/._NeuralNet-bs053.html
index 063cc6412..f86c23aec 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs053.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs053.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,18 +259,45 @@ MathJax.Hub.Config({
 
 
 
-
    Building neural networks in Tensorflow and Keras 
    
-
+
    Visualization 
    
 
    
-Now we want  to build on the experience gained from our neural network implementation in NumPy and scikit-learn
-and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
-and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.
 
-
    
-In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
-clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
-NumPy arrays.
+
+
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
 
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        dnn = DNN_scikit[i][j]
+        
+        train_pred = dnn.predict(X_train) 
+        test_pred = dnn.predict(X_test)
+
+        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
+        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    
 
    
 
    
 
@@ -295,7 +324,7 @@ NumPy arrays.
   
  • 62
    • 
   
  • 63
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs054.html b/doc/pub/NeuralNet/html/._NeuralNet-bs054.html
index 3a00235c3..1b3bd2377 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs054.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs054.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,48 +259,18 @@ MathJax.Hub.Config({
 
 
 
-
    Tensorflow 
    
+
    Building neural networks in Tensorflow and Keras 
    
 
 
    
-Tensorflow is an open source library machine learning library
-developed by the Google Brain team for internal use. It was released
-under the Apache 2.0 open source license in November 9, 2015.
+Now we want  to build on the experience gained from our neural network implementation in NumPy and scikit-learn
+and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy
+and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.
 
 
    
-Tensorflow is a computational framework that allows you to construct
-machine learning models at different levels of abstraction, from
-high-level, object-oriented APIs like Keras, down to the C++ kernels
-that Tensorflow is built upon. The higher levels of abstraction are
-simpler to use, but less flexible, and our choice of implementation
-should reflect the problems we are trying to solve.
+In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite
+clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or
+NumPy arrays.
 
-
    
-Tensorflow uses so-called graphs to represent your computation
-in terms of the dependencies between individual operations, such that you first build a Tensorflow graph
-to represent your model, and then create a Tensorflow session to run the graph.
-
-
    
-In this guide we will analyze the same data as we did in our NumPy and
-scikit-learn tutorial, gathered from the MNIST database of images. We
-will give an introduction to the lower level Python Application
-Program Interfaces (APIs), and see how we use them to build our graph.
-Then we will build (effectively) the same graph in Keras, to see just
-how simple solving a machine learning problem can be.
-
-
    
-To install tensorflow on Unix/Linux systems, use pip as
-
    
-
-
-
    pip3 install tensorflow
-
    
-
    
-and/or if you use anaconda, just write (or install from the graphical user interface)
-
    
-
-
-
    conda install tensorflow
-
    
 
    
 
    
 
@@ -325,7 +297,7 @@ and/or if you use anaconda, just write (or install from the graphical use
   
  • 63
    • 
   
  • 64
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs055.html b/doc/pub/NeuralNet/html/._NeuralNet-bs055.html
index d3b4f78e5..07238066a 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs055.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs055.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,68 +259,47 @@ MathJax.Hub.Config({
 
 
 
-
    Collect and pre-process data 
    
+
    Tensorflow 
    
 
+
    
+Tensorflow is an open source library machine learning library
+developed by the Google Brain team for internal use. It was released
+under the Apache 2.0 open source license in November 9, 2015.
+
+
    
+Tensorflow is a computational framework that allows you to construct
+machine learning models at different levels of abstraction, from
+high-level, object-oriented APIs like Keras, down to the C++ kernels
+that Tensorflow is built upon. The higher levels of abstraction are
+simpler to use, but less flexible, and our choice of implementation
+should reflect the problems we are trying to solve.
+
+
    
+Tensorflow uses so-called graphs to represent your computation
+in terms of the dependencies between individual operations, such that you first build a Tensorflow graph
+to represent your model, and then create a Tensorflow session to run the graph.
+
+
    
+In this guide we will analyze the same data as we did in our NumPy and
+scikit-learn tutorial, gathered from the MNIST database of images. We
+will give an introduction to the lower level Python Application
+Program Interfaces (APIs), and see how we use them to build our graph.
+Then we will build (effectively) the same graph in Keras, to see just
+how simple solving a machine learning problem can be.
+
+
    
+To install tensorflow on Unix/Linux systems, use pip as
 
    
 
 
-
    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# flatten the image
-# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
-n_inputs = len(inputs)
-inputs = inputs.reshape(n_inputs, -1)
-print("X = (n_inputs, n_features) = " + str(inputs.shape))
-
-
-# choose some random images to display
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
+
    pip3 install tensorflow
 
    
 
    
+and/or if you use anaconda, just write (or install from the graphical user interface)
+
    
 
 
-
    from keras.utils import to_categorical
-from sklearn.model_selection import train_test_split
-
-# one-hot representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
+
    conda install tensorflow
 
    
 
    
 
    
@@ -346,7 +327,7 @@ X_train, X_test, Y_train, Y_test = train_tes
   
  • 64
    • 
   
  • 65
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs056.html b/doc/pub/NeuralNet/html/._NeuralNet-bs056.html
index d5471dc7e..22fc4edd5 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs056.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs056.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,147 +259,68 @@ MathJax.Hub.Config({
 
 
 
-
    Using TensorFlow backend 
    
-
-
      
-
    1.  Define model and architecture
      2. 
-
    2.  Choose cost function and optimizer
      3. 
-
    
+
    Collect and pre-process data 
    
 
 
    
 
 
-
    import tensorflow as tf
+
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
 
-class NeuralNetworkTensorflow:
-    def __init__(
-        self,
-        X_train,
-        Y_train,
-        X_test,
-        Y_test,
-        n_neurons_layer1=100,
-        n_neurons_layer2=50,
-        n_categories=2,
-        epochs=10,
-        batch_size=100,
-        eta=0.1,
-        lmbd=0.0,
-    ):
-        
-        # keep track of number of steps
-        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
-        
-        self.X_train = X_train
-        self.Y_train = Y_train
-        self.X_test = X_test
-        self.Y_test = Y_test
-        
-        self.n_inputs = X_train.shape[0]
-        self.n_features = X_train.shape[1]
-        self.n_neurons_layer1 = n_neurons_layer1
-        self.n_neurons_layer2 = n_neurons_layer2
-        self.n_categories = n_categories
-        
-        self.epochs = epochs
-        self.batch_size = batch_size
-        self.iterations = self.n_inputs // self.batch_size
-        self.eta = eta
-        self.lmbd = lmbd
-        
-        # build network piece by piece
-        # name scopes (with) are used to enforce creation of new variables
-        # https://www.tensorflow.org/guide/variables
-        self.create_placeholders()
-        self.create_DNN()
-        self.create_loss()
-        self.create_optimiser()
-        self.create_accuracy()
-    
-    def create_placeholders(self):
-        # placeholders are fine here, but "Datasets" are the preferred method
-        # of streaming data into a model
-        with tf.name_scope('data'):
-            self.X = tf.placeholder(tf.float32, shape=(None, self.n_features), name='X_data')
-            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
-    
-    def create_DNN(self):
-        with tf.name_scope('DNN'):
-            # the weights are stored to calculate regularization loss later
-            
-            # Fully connected layer 1
-            self.W_fc1 = self.weight_variable([self.n_features, self.n_neurons_layer1], name='fc1', dtype=tf.float32)
-            b_fc1 = self.bias_variable([self.n_neurons_layer1], name='fc1', dtype=tf.float32)
-            a_fc1 = tf.nn.sigmoid(tf.matmul(self.X, self.W_fc1) + b_fc1)
-            
-            # Fully connected layer 2
-            self.W_fc2 = self.weight_variable([self.n_neurons_layer1, self.n_neurons_layer2], name='fc2', dtype=tf.float32)
-            b_fc2 = self.bias_variable([self.n_neurons_layer2], name='fc2', dtype=tf.float32)
-            a_fc2 = tf.nn.sigmoid(tf.matmul(a_fc1, self.W_fc2) + b_fc2)
-            
-            # Output layer
-            self.W_out = self.weight_variable([self.n_neurons_layer2, self.n_categories], name='out', dtype=tf.float32)
-            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
-            self.z_out = tf.matmul(a_fc2, self.W_out) + b_out
-    
-    def create_loss(self):
-        with tf.name_scope('loss'):
-            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
-            
-            regularizer_loss_fc1 = tf.nn.l2_loss(self.W_fc1)
-            regularizer_loss_fc2 = tf.nn.l2_loss(self.W_fc2)
-            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
-            regularizer_loss = self.lmbd*(regularizer_loss_fc1 + regularizer_loss_fc2 + regularizer_loss_out)
-            
-            self.loss = softmax_loss + regularizer_loss
 
-    def create_accuracy(self):
-        with tf.name_scope('accuracy'):
-            probabilities = tf.nn.softmax(self.z_out)
-            predictions = tf.argmax(probabilities, axis=1)
-            labels = tf.argmax(self.Y, axis=1)
-            
-            correct_predictions = tf.equal(predictions, labels)
-            correct_predictions = tf.cast(correct_predictions, tf.float32)
-            self.accuracy = tf.reduce_mean(correct_predictions)
-    
-    def create_optimiser(self):
-        with tf.name_scope('optimizer'):
-            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
-            
-    def weight_variable(self, shape, name='', dtype=tf.float32):
-        initial = tf.truncated_normal(shape, stddev=0.1)
-        return tf.Variable(initial, name=name, dtype=dtype)
-    
-    def bias_variable(self, shape, name='', dtype=tf.float32):
-        initial = tf.constant(0.1, shape=shape)
-        return tf.Variable(initial, name=name, dtype=dtype)
-    
-    def fit(self):
-        data_indices = np.arange(self.n_inputs)
+# ensure the same random numbers appear every time
+np.random.seed(0)
 
-        with tf.Session() as sess:
-            sess.run(tf.global_variables_initializer())
-            for i in range(self.epochs):
-                for j in range(self.iterations):
-                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
-                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
-            
-                    sess.run([DNN.loss, DNN.optimizer],
-                        feed_dict={DNN.X: batch_X,
-                                   DNN.Y: batch_Y})
-                    accuracy = sess.run(DNN.accuracy,
-                        feed_dict={DNN.X: batch_X,
-                                   DNN.Y: batch_Y})
-                    step = sess.run(DNN.global_step)
-    
-            self.train_loss, self.train_accuracy = sess.run([DNN.loss, DNN.accuracy],
-                feed_dict={DNN.X: self.X_train,
-                           DNN.Y: self.Y_train})
-        
-            self.test_loss, self.test_accuracy = sess.run([DNN.loss, DNN.accuracy],
-                feed_dict={DNN.X: self.X_test,
-                           DNN.Y: self.Y_test})
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# flatten the image
+# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
+n_inputs = len(inputs)
+inputs = inputs.reshape(n_inputs, -1)
+print("X = (n_inputs, n_features) = " + str(inputs.shape))
+
+
+# choose some random images to display
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
    
+
    
+
+
+
    from keras.utils import to_categorical
+from sklearn.model_selection import train_test_split
+
+# one-hot representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
 
    
 
    
 
    
@@ -424,6 +347,8 @@ MathJax.Hub.Config({
   
  • 64
    • 
   
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    • 
   
  • 66
    • 
+  
  • ...
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs057.html b/doc/pub/NeuralNet/html/._NeuralNet-bs057.html
index 049c7039c..008c096cc 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs057.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs057.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,80 +259,147 @@ MathJax.Hub.Config({
 
 
 
-
    Optimizing and using gradient descent 
    
+
    Using TensorFlow backend 
    
+
+
      
+
    1.  Define model and architecture
      2. 
+
    2.  Choose cost function and optimizer
      3. 
+
    
 
 
    
 
 
-
    epochs = 100
-batch_size = 100
-n_neurons_layer1 = 100
-n_neurons_layer2 = 50
-n_categories = 10
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-
    
-
    
+
    import tensorflow as tf
 
-
-
    DNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+class NeuralNetworkTensorflow:
+    def __init__(
+        self,
+        X_train,
+        Y_train,
+        X_test,
+        Y_test,
+        n_neurons_layer1=100,
+        n_neurons_layer2=50,
+        n_categories=2,
+        epochs=10,
+        batch_size=100,
+        eta=0.1,
+        lmbd=0.0,
+    ):
         
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        DNN = NeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
-                                      n_neurons_layer1, n_neurons_layer2, n_categories,
-                                      epochs=epochs, batch_size=batch_size, eta=eta, lmbd=lmbd)
-        DNN.fit()
+        # keep track of number of steps
+        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
         
-        DNN_tf[i][j] = DNN
+        self.X_train = X_train
+        self.Y_train = Y_train
+        self.X_test = X_test
+        self.Y_test = Y_test
         
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % DNN.test_accuracy)
-        print()
-
    
-
    
-
-
-
    # optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        DNN = DNN_tf[i][j]
-
-        train_accuracy[i][j] = DNN.train_accuracy
-        test_accuracy[i][j] = DNN.test_accuracy
-
+        self.n_inputs = X_train.shape[0]
+        self.n_features = X_train.shape[1]
+        self.n_neurons_layer1 = n_neurons_layer1
+        self.n_neurons_layer2 = n_neurons_layer2
+        self.n_categories = n_categories
         
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
+        
+        # build network piece by piece
+        # name scopes (with) are used to enforce creation of new variables
+        # https://www.tensorflow.org/guide/variables
+        self.create_placeholders()
+        self.create_DNN()
+        self.create_loss()
+        self.create_optimiser()
+        self.create_accuracy()
+    
+    def create_placeholders(self):
+        # placeholders are fine here, but "Datasets" are the preferred method
+        # of streaming data into a model
+        with tf.name_scope('data'):
+            self.X = tf.placeholder(tf.float32, shape=(None, self.n_features), name='X_data')
+            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
+    
+    def create_DNN(self):
+        with tf.name_scope('DNN'):
+            # the weights are stored to calculate regularization loss later
+            
+            # Fully connected layer 1
+            self.W_fc1 = self.weight_variable([self.n_features, self.n_neurons_layer1], name='fc1', dtype=tf.float32)
+            b_fc1 = self.bias_variable([self.n_neurons_layer1], name='fc1', dtype=tf.float32)
+            a_fc1 = tf.nn.sigmoid(tf.matmul(self.X, self.W_fc1) + b_fc1)
+            
+            # Fully connected layer 2
+            self.W_fc2 = self.weight_variable([self.n_neurons_layer1, self.n_neurons_layer2], name='fc2', dtype=tf.float32)
+            b_fc2 = self.bias_variable([self.n_neurons_layer2], name='fc2', dtype=tf.float32)
+            a_fc2 = tf.nn.sigmoid(tf.matmul(a_fc1, self.W_fc2) + b_fc2)
+            
+            # Output layer
+            self.W_out = self.weight_variable([self.n_neurons_layer2, self.n_categories], name='out', dtype=tf.float32)
+            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
+            self.z_out = tf.matmul(a_fc2, self.W_out) + b_out
+    
+    def create_loss(self):
+        with tf.name_scope('loss'):
+            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
+            
+            regularizer_loss_fc1 = tf.nn.l2_loss(self.W_fc1)
+            regularizer_loss_fc2 = tf.nn.l2_loss(self.W_fc2)
+            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
+            regularizer_loss = self.lmbd*(regularizer_loss_fc1 + regularizer_loss_fc2 + regularizer_loss_out)
+            
+            self.loss = softmax_loss + regularizer_loss
 
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
    
-
    
+    def create_accuracy(self):
+        with tf.name_scope('accuracy'):
+            probabilities = tf.nn.softmax(self.z_out)
+            predictions = tf.argmax(probabilities, axis=1)
+            labels = tf.argmax(self.Y, axis=1)
+            
+            correct_predictions = tf.equal(predictions, labels)
+            correct_predictions = tf.cast(correct_predictions, tf.float32)
+            self.accuracy = tf.reduce_mean(correct_predictions)
+    
+    def create_optimiser(self):
+        with tf.name_scope('optimizer'):
+            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
+            
+    def weight_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.truncated_normal(shape, stddev=0.1)
+        return tf.Variable(initial, name=name, dtype=dtype)
+    
+    def bias_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.constant(0.1, shape=shape)
+        return tf.Variable(initial, name=name, dtype=dtype)
+    
+    def fit(self):
+        data_indices = np.arange(self.n_inputs)
 
-
-
    # optional
-# we can use log files to visualize our graph in Tensorboard
-writer = tf.summary.FileWriter('logs/')
-writer.add_graph(tf.get_default_graph())
+        with tf.Session() as sess:
+            sess.run(tf.global_variables_initializer())
+            for i in range(self.epochs):
+                for j in range(self.iterations):
+                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
+                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
+            
+                    sess.run([DNN.loss, DNN.optimizer],
+                        feed_dict={DNN.X: batch_X,
+                                   DNN.Y: batch_Y})
+                    accuracy = sess.run(DNN.accuracy,
+                        feed_dict={DNN.X: batch_X,
+                                   DNN.Y: batch_Y})
+                    step = sess.run(DNN.global_step)
+    
+            self.train_loss, self.train_accuracy = sess.run([DNN.loss, DNN.accuracy],
+                feed_dict={DNN.X: self.X_train,
+                           DNN.Y: self.Y_train})
+        
+            self.test_loss, self.test_accuracy = sess.run([DNN.loss, DNN.accuracy],
+                feed_dict={DNN.X: self.X_test,
+                           DNN.Y: self.Y_test})
 
    
 
    
 
    
@@ -356,6 +425,7 @@ writer.add_graph(tf64
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs058.html b/doc/pub/NeuralNet/html/._NeuralNet-bs058.html
index 2da05d923..fd4d44b40 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs058.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs058.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,65 +259,36 @@ MathJax.Hub.Config({
 
 
 
-
    Using Keras 
    
-
-
    
-Keras is a high level neural network
-that supports Tensorflow, CTNK and Theano as backends.  
-If you have Tensorflow installed Keras is available through the tf.keras module.  
-If you have Anaconda installed you may run the following command
-
    
-
-
-
    conda install keras
-
    
-
    
-Alternatively, if you have Tensorflow or one of the other supported backends install you may use the pip package manager:
+
    Optimizing and using gradient descent 
    
 
 
    
 
 
-
    pip3 install keras
-
    
-
    
-or look up the instructions here.
-
-
    
-
-
-
    from keras.models import Sequential
-from keras.layers import Dense
-from keras.regularizers import l2
-from keras.optimizers import SGD
-
-def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
-    model = Sequential()
-    model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=l2(lmbd)))
-    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=l2(lmbd)))
-    model.add(Dense(n_categories, activation='softmax'))
-    
-    sgd = SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-    
-    return model
+
    epochs = 100
+batch_size = 100
+n_neurons_layer1 = 100
+n_neurons_layer2 = 50
+n_categories = 10
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
 
    
 
    
 
 
-
    DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
    DNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
         
 for i, eta in enumerate(eta_vals):
     for j, lmbd in enumerate(lmbd_vals):
-        DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
-                                         eta=eta, lmbd=lmbd)
-        DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = DNN.evaluate(X_test, Y_test)
+        DNN = NeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
+                                      n_neurons_layer1, n_neurons_layer2, n_categories,
+                                      epochs=epochs, batch_size=batch_size, eta=eta, lmbd=lmbd)
+        DNN.fit()
         
-        DNN_keras[i][j] = DNN
+        DNN_tf[i][j] = DNN
         
         print("Learning rate = ", eta)
         print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
+        print("Test accuracy: %.3f" % DNN.test_accuracy)
         print()
 
    
 
    
@@ -333,10 +306,10 @@ test_accuracy = npfor i in range(len(eta_vals)):
     for j in range(len(lmbd_vals)):
-        DNN = DNN_keras[i][j]
+        DNN = DNN_tf[i][j]
 
-        train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+        train_accuracy[i][j] = DNN.train_accuracy
+        test_accuracy[i][j] = DNN.test_accuracy
 
         
 fig, ax = plt.subplots(figsize = (10, 10))
@@ -354,6 +327,14 @@ ax.set_xlabel(&
 plt.show()
 
    
 
    
+
+
+
    # optional
+# we can use log files to visualize our graph in Tensorboard
+writer = tf.summary.FileWriter('logs/')
+writer.add_graph(tf.get_default_graph())
+
    
+
    
 
    
 
 
    
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html
index 13d0e4d5e..28d76760e 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,34 +257,104 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Which activation function should I use? 
    
+
    Using Keras 
    
 
 
    
-The Back propagation algorithm we derived above works by going from
-the output layer to the input layer, propagating the error gradient on
-the way. Once the algorithm has computed the gradient of the cost
-function with regards to each parameter in the network, it uses these
-gradients to update each parameter with a Gradient Descent (GD) step.
+Keras is a high level neural network
+that supports Tensorflow, CTNK and Theano as backends.  
+If you have Tensorflow installed Keras is available through the tf.keras module.  
+If you have Anaconda installed you may run the following command
+
    
+
+
+
    conda install keras
+
    
+
    
+Alternatively, if you have Tensorflow or one of the other supported backends install you may use the pip package manager:
 
 
    
-Unfortunately for us, the gradients often get smaller and smaller as the
-algorithm progresses down to the first hidden layers. As a result, the
-GD update leaves the lower layer connection weights
-virtually unchanged, and training never converges to a good
-solution. This is known in the literature as 
-the vanishing gradients problem.
+
+
+
    pip3 install keras
+
    
+
    
+or look up the instructions here.
 
 
    
-In other cases, the opposite can happen, namely the the gradients can grow bigger and
-bigger. The result is that many of the layers get large updates of the 
-weights the
-algorithm diverges. This is the exploding gradients problem, which is
-mostly encountered in recurrent neural networks. More generally, deep
-neural networks suffer from unstable gradients, different layers may
-learn at widely different speeds
 
+
+
    from keras.models import Sequential
+from keras.layers import Dense
+from keras.regularizers import l2
+from keras.optimizers import SGD
+
+def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
+    model = Sequential()
+    model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=l2(lmbd)))
+    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax'))
+    
+    sgd = SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
    
+
    
+
+
+
    DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
+                                         eta=eta, lmbd=lmbd)
+        DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = DNN.evaluate(X_test, Y_test)
+        
+        DNN_keras[i][j] = DNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    
+
    
+
+
+
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        DNN = DNN_keras[i][j]
+
+        train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    
 
    
 
    
 
@@ -305,6 +377,7 @@ learn at widely different speeds
   
  • 64
    • 
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs060.html b/doc/pub/NeuralNet/html/._NeuralNet-bs060.html
index 4e8a17850..e08ced460 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs060.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs060.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,32 +259,31 @@ MathJax.Hub.Config({
 
 
 
-
    Is the Logistic activation function (Sigmoid)  our choice? 
    
+
    Which activation function should I use? 
    
 
 
    
-Although this unfortunate behavior has been empirically observed for
-quite a while (it was one of the reasons why deep neural networks were
-mostly abandoned for a long time), it is only around 2010 that
-significant progress was made in understanding it.
+The Back propagation algorithm we derived above works by going from
+the output layer to the input layer, propagating the error gradient on
+the way. Once the algorithm has computed the gradient of the cost
+function with regards to each parameter in the network, it uses these
+gradients to update each parameter with a Gradient Descent (GD) step.
 
 
    
-A paper titled Understanding the Difficulty of Training Deep
-Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that
-the problems with the popular logistic
-sigmoid activation function and the weight initialization technique
-that was most popular at the time, namely random initialization using
-a normal distribution with a mean of 0 and a standard deviation of
-1.
+Unfortunately for us, the gradients often get smaller and smaller as the
+algorithm progresses down to the first hidden layers. As a result, the
+GD update leaves the lower layer connection weights
+virtually unchanged, and training never converges to a good
+solution. This is known in the literature as 
+the vanishing gradients problem.
 
 
    
-They showed that with this activation function and this
-initialization scheme, the variance of the outputs of each layer is
-much greater than the variance of its inputs. Going forward in the
-network, the variance keeps increasing after each layer until the
-activation function saturates at the top layers. This is actually made
-worse by the fact that the logistic function has a mean of 0.5, not 0
-(the hyperbolic tangent function has a mean of 0 and behaves slightly
-better than the logistic function in deep networks).
+In other cases, the opposite can happen, namely the the gradients can grow bigger and
+bigger. The result is that many of the layers get large updates of the 
+weights the
+algorithm diverges. This is the exploding gradients problem, which is
+mostly encountered in recurrent neural networks. More generally, deep
+neural networks suffer from unstable gradients, different layers may
+learn at widely different speeds
 
 
    
 
    
@@ -305,6 +306,7 @@ better than the logistic function in deep networks).
   
  • 64
    • 
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs061.html b/doc/pub/NeuralNet/html/._NeuralNet-bs061.html
index 4419076ee..e878faabd 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs061.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs061.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,40 +257,34 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    The derivative of the Logistic funtion 
    
+
    Is the Logistic activation function (Sigmoid)  our choice? 
    
 
 
    
-Looking at the logistic activation function, when inputs become large
-(negative or positive), the function saturates at 0 or 1, with a
-derivative extremely close to 0. Thus when backpropagation kicks in,
-it has virtually no gradient to propagate back through the network,
-and what little gradient exists keeps getting diluted as
-backpropagation progresses down through the top layers, so there is
-really nothing left for the lower layers.
+Although this unfortunate behavior has been empirically observed for
+quite a while (it was one of the reasons why deep neural networks were
+mostly abandoned for a long time), it is only around 2010 that
+significant progress was made in understanding it.
 
 
    
-In their paper, Glorot and Bengio propose a way to significantly
-alleviate this problem. We need the signal to flow properly in both
-directions: in the forward direction when making predictions, and in
-the reverse direction when backpropagating gradients. We don’t want
-the signal to die out, nor do we want it to explode and saturate. For
-the signal to flow properly, the authors argue that we need the
-variance of the outputs of each layer to be equal to the variance of
-its inputs, and we also need the gradients to have equal variance
-before and after flowing through a layer in the reverse direction.
+A paper titled Understanding the Difficulty of Training Deep
+Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that
+the problems with the popular logistic
+sigmoid activation function and the weight initialization technique
+that was most popular at the time, namely random initialization using
+a normal distribution with a mean of 0 and a standard deviation of
+1.
 
 
    
-One of the insights in the 2010 paper by Glorot and Bengio was that
-the vanishing/exploding gradients problems were in part due to a poor
-choice of activation function. Until then most people had assumed that
-if Nature had chosen to use roughly sigmoid activation functions in
-biological neurons, they must be an excellent choice. But it turns out
-that other activation functions behave much better in deep neural
-networks, in particular the ReLU activation function, mostly because
-it does not saturate for positive values (and also because it is quite
-fast to compute).
+They showed that with this activation function and this
+initialization scheme, the variance of the outputs of each layer is
+much greater than the variance of its inputs. Going forward in the
+network, the variance keeps increasing after each layer until the
+activation function saturates at the top layers. This is actually made
+worse by the fact that the logistic function has a mean of 0.5, not 0
+(the hyperbolic tangent function has a mean of 0 and behaves slightly
+better than the logistic function in deep networks).
 
 
    
 
    
@@ -310,6 +306,7 @@ fast to compute).
   
  • 64
    • 
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs062.html b/doc/pub/NeuralNet/html/._NeuralNet-bs062.html
index 40b629114..3a0f23bae 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs062.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs062.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,29 +259,38 @@ MathJax.Hub.Config({
 
 
 
-
    The RELU function family 
    
+
    The derivative of the Logistic funtion 
    
 
 
    
-The ReLU activation function suffers from a problem known as the dying
-ReLUs: during training, some neurons effectively die, meaning they
-stop outputting anything other than 0.
+Looking at the logistic activation function, when inputs become large
+(negative or positive), the function saturates at 0 or 1, with a
+derivative extremely close to 0. Thus when backpropagation kicks in,
+it has virtually no gradient to propagate back through the network,
+and what little gradient exists keeps getting diluted as
+backpropagation progresses down through the top layers, so there is
+really nothing left for the lower layers.
 
 
    
-In some cases, you may find that half of your network’s neurons are
-dead, especially if you used a large learning rate. During training,
-if a neuron’s weights get updated such that the weighted sum of the
-neuron’s inputs is negative, it will start outputting 0. When this
-happen, the neuron is unlikely to come back to life since the gradient
-of the ReLU function is 0 when its input is negative.
+In their paper, Glorot and Bengio propose a way to significantly
+alleviate this problem. We need the signal to flow properly in both
+directions: in the forward direction when making predictions, and in
+the reverse direction when backpropagating gradients. We don’t want
+the signal to die out, nor do we want it to explode and saturate. For
+the signal to flow properly, the authors argue that we need the
+variance of the outputs of each layer to be equal to the variance of
+its inputs, and we also need the gradients to have equal variance
+before and after flowing through a layer in the reverse direction.
 
 
    
-To solve this problem, nowadays practitioners use a  variant of the ReLU
-function, such as the leaky ReLU discussed above or the so-called
-exponential linear unit (ELU) function
-
-$$
-ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\  z & z \ge 0.\end{array}\right. 
-$$
+One of the insights in the 2010 paper by Glorot and Bengio was that
+the vanishing/exploding gradients problems were in part due to a poor
+choice of activation function. Until then most people had assumed that
+if Nature had chosen to use roughly sigmoid activation functions in
+biological neurons, they must be an excellent choice. But it turns out
+that other activation functions behave much better in deep neural
+networks, in particular the ReLU activation function, mostly because
+it does not saturate for positive values (and also because it is quite
+fast to compute).
 
 
    
 
    
@@ -300,6 +311,7 @@ $$
   
  • 64
    • 
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs063.html b/doc/pub/NeuralNet/html/._NeuralNet-bs063.html
index 8441353dd..a684f9a05 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs063.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs063.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -257,22 +259,29 @@ MathJax.Hub.Config({
 
 
 
-
    Which activation function should we use? 
    
+
    The RELU function family 
    
 
 
    
-In general it seems that the ELU activation function is better than
-the leaky ReLU function (and its variants), which is better than
-ReLU. ReLU performs better than \( \tanh \) which in turn performs better
-than the logistic function.
+The ReLU activation function suffers from a problem known as the dying
+ReLUs: during training, some neurons effectively die, meaning they
+stop outputting anything other than 0.
 
 
    
-If runtime
-performance is an issue, then you may opt for the  leaky ReLU function  over the 
-ELU function If you don’t
-want to tweak yet another hyperparameter, you may just use the default
-\( \alpha \) of \( 0.01 \) for the leaky ReLU, and \( 1 \) for ELU. If you have
-spare time and computing power, you can use cross-validation or
-bootstrap to evaluate other activation functions.
+In some cases, you may find that half of your network’s neurons are
+dead, especially if you used a large learning rate. During training,
+if a neuron’s weights get updated such that the weighted sum of the
+neuron’s inputs is negative, it will start outputting 0. When this
+happen, the neuron is unlikely to come back to life since the gradient
+of the ReLU function is 0 when its input is negative.
+
+
    
+To solve this problem, nowadays practitioners use a  variant of the ReLU
+function, such as the leaky ReLU discussed above or the so-called
+exponential linear unit (ELU) function
+
+$$
+ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\  z & z \ge 0.\end{array}\right. 
+$$
 
 
    
 
    
@@ -292,6 +301,7 @@ bootstrap to evaluate other activation functions.
   
  • 64
    • 
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs064.html b/doc/pub/NeuralNet/html/._NeuralNet-bs064.html
index eaae2cf1e..6d03ac994 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs064.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs064.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,46 +257,24 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    A top-down perspective on Neural networks 
    
+
    Which activation function should we use? 
    
 
 
    
-The first thing we would like to do is divide the data into two or three
-parts. A training set, a validation or dev (development) set, and a
-test set. The test set is the data on which we want to make
-predictions. The dev set is a subset of the training data we use to
-check how well we are doing out-of-sample, after training the model on
-the training dataset. We use the validation error as a proxy for the
-test error in order to make tweaks to our model. It is crucial that we
-do not use any of the test data to train the algorithm. This is a
-cardinal sin in ML. Then:
-
-
      
-
    •  Estimate optimal error rate
      • 
-
    •  Minimize underfitting (bias) on training data set.
      • 
-
    •  Make sure you are not overfitting.
      • 
-
    
-
-If the validation and test sets are drawn from the same distributions,
-then a good performance on the validation set should lead to similarly
-good performance on the test set.
+In general it seems that the ELU activation function is better than
+the leaky ReLU function (and its variants), which is better than
+ReLU. ReLU performs better than \( \tanh \) which in turn performs better
+than the logistic function.
 
 
    
-However, sometimes
-the training data and test data differ in subtle ways because, for
-example, they are collected using slightly different methods, or
-because it is cheaper to collect data in one way versus another. In
-this case, there can be a mismatch between the training and test
-data. This can lead to the neural network overfitting these small
-differences between the test and training sets, and a poor performance
-on the test set despite having a good performance on the validation
-set. To rectify this, Andrew Ng suggests making two validation or dev
-sets, one constructed from the training data and one constructed from
-the test data. The difference between the performance of the algorithm
-on these two validation sets quantifies the train-test mismatch. This
-can serve as another important diagnostic when using DNNs for
-supervised learning.
+If runtime
+performance is an issue, then you may opt for the  leaky ReLU function  over the 
+ELU function If you don’t
+want to tweak yet another hyperparameter, you may just use the default
+\( \alpha \) of \( 0.01 \) for the leaky ReLU, and \( 1 \) for ELU. If you have
+spare time and computing power, you can use cross-validation or
+bootstrap to evaluate other activation functions.
 
 
    
 
    
@@ -313,6 +293,7 @@ supervised learning.
   
  • 64
    • 
   
  • 65
    • 
   
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs065.html b/doc/pub/NeuralNet/html/._NeuralNet-bs065.html
index fc43897ec..4cdd73ae6 100644
--- a/doc/pub/NeuralNet/html/._NeuralNet-bs065.html
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs065.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -255,31 +257,48 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Limitations of supervised learning with deep networks 
    
+
    A top-down perspective on Neural networks 
    
 
 
    
-Like all statistical methods, supervised learning using neural
-networks has important limitations. This is especially important when
-one seeks to apply these methods, especially to physics problems. Like
-all tools, DNNs are not a universal solution. Often, the same or
-better performance on a task can be achieved by using a few
-hand-engineered features (or even a collection of random
-features).
-
-
    
-Here we list some of the important limitations of supervised neural network based models. 
+The first thing we would like to do is divide the data into two or three
+parts. A training set, a validation or dev (development) set, and a
+test set. The test set is the data on which we want to make
+predictions. The dev set is a subset of the training data we use to
+check how well we are doing out-of-sample, after training the model on
+the training dataset. We use the validation error as a proxy for the
+test error in order to make tweaks to our model. It is crucial that we
+do not use any of the test data to train the algorithm. This is a
+cardinal sin in ML. Then:
 
 
      
-
    •  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).
      • 
-
    •  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.
      • 
-
    •  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.
      • 
-
    •  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.
      • 
+
    •  Estimate optimal error rate
      • 
+
    •  Minimize underfitting (bias) on training data set.
      • 
+
    •  Make sure you are not overfitting.
      • 
 
    
 
-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 circumnavigate these problems.
+If the validation and test sets are drawn from the same distributions,
+then a good performance on the validation set should lead to similarly
+good performance on the test set.
 
+
    
+However, sometimes
+the training data and test data differ in subtle ways because, for
+example, they are collected using slightly different methods, or
+because it is cheaper to collect data in one way versus another. In
+this case, there can be a mismatch between the training and test
+data. This can lead to the neural network overfitting these small
+differences between the test and training sets, and a poor performance
+on the test set despite having a good performance on the validation
+set. To rectify this, Andrew Ng suggests making two validation or dev
+sets, one constructed from the training data and one constructed from
+the test data. The difference between the performance of the algorithm
+on these two validation sets quantifies the train-test mismatch. This
+can serve as another important diagnostic when using DNNs for
+supervised learning.
+
+
    
 
    
 
 
      
@@ -295,6 +314,8 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
   
    • 64
      • 
   
    • 65
      • 
   
    • 66
      • 
+  
    • 67
      • 
+  
    • »
      • 
 
    
 
 
diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs066.html b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html
new file mode 100644
index 000000000..e1b566889
--- /dev/null
+++ b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html
@@ -0,0 +1,323 @@
+
+
+
+
+
+
+
+Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    
+
+
    
+  
    
+    
+    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
+  
    
+
+  
    
+    
+  
    
+
    
+
     
+
+
    
+
+
     
     
     
     
+
+
+
+
+
    Limitations of supervised learning with deep networks 
    
+
+
    
+Like all statistical methods, supervised learning using neural
+networks has important limitations. This is especially important when
+one seeks to apply these methods, especially to physics problems. Like
+all tools, DNNs are not a universal solution. Often, the same or
+better performance on a task can be achieved by using a few
+hand-engineered features (or even a collection of random
+features).
+
+
    
+Here we list some of the important limitations of supervised neural network based models. 
+
+
      
+
    •  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).
      • 
+
    •  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.
      • 
+
    •  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.
      • 
+
    •  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.
      • 
+
    
+
+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 circumnavigate these problems.
+
+
    
+
+
    
+
+
+
      
+
+
+
+
+
+
+
+
    
+
+
    
+
+
+
+
+    
+
diff --git a/doc/pub/NeuralNet/html/NeuralNet-bs.html b/doc/pub/NeuralNet/html/NeuralNet-bs.html
index aeabd57ad..55accbec3 100644
--- a/doc/pub/NeuralNet/html/NeuralNet-bs.html
+++ b/doc/pub/NeuralNet/html/NeuralNet-bs.html
@@ -92,54 +92,55 @@ Automatically generated HTML file from DocOnce source
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -210,38 +211,39 @@ MathJax.Hub.Config({
       
  • Setting up a Multi-layer perceptron model for classification
    • 
       
  • Defining the cost function
    • 
       
  • Example: binary classification problem
    • 
-      
  • Developing a code for doing neural networks with back propagation
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Train and test datasets
    • 
-      
  • Define model and architecture
    • 
-      
  • Layers
    • 
-      
  • Weights and biases
    • 
-      
  • Feed-forward pass
    • 
-      
  • Matrix multiplications
    • 
-      
  • Choose cost function and optimizer
    • 
-      
  • Optimizing the cost function
    • 
-      
  • Regularization
    • 
-      
  • Matrix  multiplication
    • 
-      
  • Improving performance
    • 
-      
  • Full object-oriented implementation
    • 
-      
  • Evaluate model performance on test data
    • 
-      
  • Adjust hyperparameters
    • 
-      
  • Visualization
    • 
-      
  • scikit-learn implementation
    • 
-      
  • Visualization
    • 
-      
  • Building neural networks in Tensorflow and Keras
    • 
-      
  • Tensorflow
    • 
-      
  • Collect and pre-process data
    • 
-      
  • Using TensorFlow backend
    • 
-      
  • Optimizing and using gradient descent
    • 
-      
  • Using Keras
    • 
-      
  • Which activation function should I use?
    • 
-      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
-      
  • The derivative of the Logistic funtion
    • 
-      
  • The RELU function family
    • 
-      
  • Which activation function should we use?
    • 
-      
  • A top-down perspective on Neural networks
    • 
-      
  • Limitations of supervised learning with deep networks
    • 
+      
  • The Softmax function
    • 
+      
  • Developing a code for doing neural networks with back propagation
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Train and test datasets
    • 
+      
  • Define model and architecture
    • 
+      
  • Layers
    • 
+      
  • Weights and biases
    • 
+      
  • Feed-forward pass
    • 
+      
  • Matrix multiplications
    • 
+      
  • Choose cost function and optimizer
    • 
+      
  • Optimizing the cost function
    • 
+      
  • Regularization
    • 
+      
  • Matrix  multiplication
    • 
+      
  • Improving performance
    • 
+      
  • Full object-oriented implementation
    • 
+      
  • Evaluate model performance on test data
    • 
+      
  • Adjust hyperparameters
    • 
+      
  • Visualization
    • 
+      
  • scikit-learn implementation
    • 
+      
  • Visualization
    • 
+      
  • Building neural networks in Tensorflow and Keras
    • 
+      
  • Tensorflow
    • 
+      
  • Collect and pre-process data
    • 
+      
  • Using TensorFlow backend
    • 
+      
  • Optimizing and using gradient descent
    • 
+      
  • Using Keras
    • 
+      
  • Which activation function should I use?
    • 
+      
  • Is the Logistic activation function (Sigmoid)  our choice?
    • 
+      
  • The derivative of the Logistic funtion
    • 
+      
  • The RELU function family
    • 
+      
  • Which activation function should we use?
    • 
+      
  • A top-down perspective on Neural networks
    • 
+      
  • Limitations of supervised learning with deep networks
    • 
 
         
       
@@ -300,7 +302,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 66
    • 
+  
  • 67
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/NeuralNet/html/NeuralNet-reveal.html b/doc/pub/NeuralNet/html/NeuralNet-reveal.html
index 59ebbaf6a..861a02477 100644
--- a/doc/pub/NeuralNet/html/NeuralNet-reveal.html
+++ b/doc/pub/NeuralNet/html/NeuralNet-reveal.html
@@ -1383,7 +1383,35 @@ In case we use another activation function than the logistic one, we need to eva
 
 
 
    
-
    Developing a code for doing neural networks with back propagation 
    
+
    The Softmax function 
    
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
+
     
    
+$$
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+$$
+
     
    
+
+For the Softmax function we have
+
     
    
+$$
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+$$
+
     
    
+
+Its derivative with respect to \( z_j^l \) gives 
+
     
    
+$$
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right), 
+$$
+
     
    
+
+which in case of the simply binary model reduces to  having \( i=j \).
+
    
+
+
+
    
+
    Developing a code for doing neural networks with back propagation 
    
 
 
    
 One can identify a set of key steps when using neural networks to solve supervised learning problems:  
@@ -1405,7 +1433,7 @@ One can identify a set of key steps when using neural networks to solve supervis
 
 
 
    
-
    Collect and pre-process data 
    
+
    Collect and pre-process data 
    
 
 
    
 Here we will be using the MNIST dataset, which is readily available through the scikit-learn
@@ -1505,7 +1533,7 @@ plt.show()
 
 
 
    
-
    Train and test datasets 
    
+
    Train and test datasets 
    
 
 
    
 Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
@@ -1555,7 +1583,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
 
 
 
    
-
    Define model and architecture 
    
+
    Define model and architecture 
    
 
 
    
 Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have
@@ -1609,7 +1637,7 @@ which is inspired by probability theory (see logistic regression) and was most c
 
 
 
    
-
    Layers 
    
+
    Layers 
    
 
 
      
 
    •  Input
      •  
@@ -1661,7 +1689,7 @@ weights to the output layer.
 
 
 
      
-
      Weights and biases 
      
+
      Weights and biases 
      
 
 
      
 Typically weights are initialized with small values distributed around zero, drawn from a uniform
@@ -1700,7 +1728,7 @@ output_bias = np.zeros(n_categories) + 0.01
 
 
 
      
-
      Feed-forward pass 
      
+
      Feed-forward pass 
      
 
 
      
 Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.  
@@ -1735,7 +1763,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
 
 
 
      
-
      Matrix multiplications 
      
+
      Matrix multiplications 
      
 
 
      
 Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
@@ -1821,7 +1849,7 @@ predictions = predict(X_train)
 
 
 
      
-
      Choose cost function and optimizer 
      
+
      Choose cost function and optimizer 
      
 
 
      
 To measure how well our neural network is doing we need to introduce a cost function.  
@@ -1856,7 +1884,7 @@ you got the correct label. The probability of category \( c \) is given by the s
 
 
 
      
-
      Optimizing the cost function 
      
+
      Optimizing the cost function 
      
 
 
      
 The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
@@ -1902,7 +1930,7 @@ The various optmization  methods, with codes and algorithms,  are discussed in o
 
 
 
      
-
      Regularization 
      
+
      Regularization 
      
 
 
      
 It is common to add an extra term to the cost function, proportional
@@ -1937,7 +1965,7 @@ calculate the gradient efficently.
 
 
 
      
-
      Matrix  multiplication 
      
+
      Matrix  multiplication 
      
 
 
      
 To more efficently train our network these equations are implemented using matrix operations.  
@@ -2064,7 +2092,7 @@ lmbd = 0.01
 
 
 
      
-
      Improving performance 
      
+
      Improving performance 
      
 
 
      
 As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.  
@@ -2084,7 +2112,7 @@ Andrew Ng goes through some of these considerations in this Full object-oriented implementation 
+
      Full object-oriented implementation 
      
 
 
      
 It is very natural to think of the network as an object, with specific instances of the network
@@ -2198,7 +2226,7 @@ being realizations of this object with different hyperparameters. An implementat
 
 
 
      
-
      Evaluate model performance on test data 
      
+
      Evaluate model performance on test data 
      
 
 
      
 To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.  
@@ -2236,7 +2264,7 @@ test_predict = dnn.predict(X_test)
 
 
 
      
-
      Adjust hyperparameters 
      
+
      Adjust hyperparameters 
      
 
 
      
 We now perform a grid search to find the optimal hyperparameters for the network.  
@@ -2270,7 +2298,7 @@ DNN_numpy = np.zeros((len(eta_vals), 
-
      Visualization 
      
+
      Visualization 
      
 
 
      
 
@@ -2313,7 +2341,7 @@ plt.show()
 
 
 
      
-
      scikit-learn implementation 
      
+
      scikit-learn implementation 
      
 
 
      
 scikit-learn focuses more
@@ -2353,7 +2381,7 @@ DNN_scikit = np.zeros((len(eta_vals), 
-
      Visualization 
      
+
      Visualization 
      
 
      
 
 
@@ -2396,7 +2424,7 @@ plt.show()
 
 
 
      
-
      Building neural networks in Tensorflow and Keras 
      
+
      Building neural networks in Tensorflow and Keras 
      
 
 
      
 Now we want  to build on the experience gained from our neural network implementation in NumPy and scikit-learn
@@ -2411,7 +2439,7 @@ NumPy arrays.
 
 
 
      
-
      Tensorflow 
      
+
      Tensorflow 
      
 
 
      
 Tensorflow is an open source library machine learning library
@@ -2457,7 +2485,7 @@ and/or if you use anaconda, just write (or install from the graphical use
 
 
 
      
-
      Collect and pre-process data 
      
+
      Collect and pre-process data 
      
 
 
      
 
@@ -2524,7 +2552,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
 
 
 
      
-
      Using TensorFlow backend 
      
+
      Using TensorFlow backend 
      
 
 
        
 
      1.  Define model and architecture
        2. 
@@ -2669,7 +2697,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
 
 
 
        
-
        Optimizing and using gradient descent 
        
+
        Optimizing and using gradient descent 
        
 
 
        
 
@@ -2748,7 +2776,7 @@ writer.add_graph(tf.get_default_graph())
 
 
 
        
-
        Using Keras 
        
+
        Using Keras 
        
 
 
        
 Keras is a high level neural network
@@ -2848,7 +2876,7 @@ plt.show()
 
 
 
        
-
        Which activation function should I use? 
        
+
        Which activation function should I use? 
        
 
 
        
 The Back propagation algorithm we derived above works by going from
@@ -2877,7 +2905,7 @@ learn at widely different speeds
 
 
 
        
-
        Is the Logistic activation function (Sigmoid)  our choice? 
        
+
        Is the Logistic activation function (Sigmoid)  our choice? 
        
 
 
        
 Although this unfortunate behavior has been empirically observed for
@@ -2907,7 +2935,7 @@ better than the logistic function in deep networks).
 
 
 
        
-
        The derivative of the Logistic funtion 
        
+
        The derivative of the Logistic funtion 
        
 
 
        
 Looking at the logistic activation function, when inputs become large
@@ -2943,7 +2971,7 @@ fast to compute).
 
 
 
        
-
        The RELU function family 
        
+
        The RELU function family 
        
 
 
        
 The ReLU activation function suffers from a problem known as the dying
@@ -2972,7 +3000,7 @@ $$
 
 
 
        
-
        Which activation function should we use? 
        
+
        Which activation function should we use? 
        
 
 
        
 In general it seems that the ELU activation function is better than
@@ -2992,7 +3020,7 @@ bootstrap to evaluate other activation functions.
 
 
 
        
-
        A top-down perspective on Neural networks 
        
+
        A top-down perspective on Neural networks 
        
 
 
        
 The first thing we would like to do is divide the data into two or three
@@ -3035,7 +3063,7 @@ supervised learning.
 
 
 
        
-
        Limitations of supervised learning with deep networks 
        
+
        Limitations of supervised learning with deep networks 
        
 
 
        
 Like all statistical methods, supervised learning using neural
diff --git a/doc/pub/NeuralNet/html/NeuralNet-solarized.html b/doc/pub/NeuralNet/html/NeuralNet-solarized.html
index fa2aa10d3..e4faff3f1 100644
--- a/doc/pub/NeuralNet/html/NeuralNet-solarized.html
+++ b/doc/pub/NeuralNet/html/NeuralNet-solarized.html
@@ -112,54 +112,55 @@ div { text-align: justify; text-justify: inter-word; }
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -1313,10 +1314,32 @@ $$
 
 In case we use another activation function than the logistic one, we need to evaluate other derivatives.
 
+
        
+









        
+
+
        The Softmax function 
        
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
+$$
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+$$
+
+For the Softmax function we have
+$$
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+$$
+
+Its derivative with respect to \( z_j^l \) gives 
+$$
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right), 
+$$
+
+which in case of the simply binary model reduces to  having \( i=j \).
+
 
        
 
 
-
        Developing a code for doing neural networks with back propagation 
        
+
        Developing a code for doing neural networks with back propagation 
        
 
 
        
 One can identify a set of key steps when using neural networks to solve supervised learning problems:  
@@ -1332,7 +1355,7 @@ One can identify a set of key steps when using neural networks to solve supervis
 
 









        
 
-
        Collect and pre-process data 
        
+
        Collect and pre-process data 
        
 
 
        
 Here we will be using the MNIST dataset, which is readily available through the scikit-learn
@@ -1427,7 +1450,7 @@ plt.show()
 
        
 









        
 
-
        Train and test datasets 
        
+
        Train and test datasets 
        
 
 
        
 Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
@@ -1476,7 +1499,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
 
        
 









        
 
-
        Define model and architecture 
        
+
        Define model and architecture 
        
 
 
        
 Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have 
@@ -1522,7 +1545,7 @@ which is inspired by probability theory (see logistic regression) and was most c
 
        
 
 
-
        Layers 
        
+
        Layers 
        
 
 
          
 
        •  Input
          •  
@@ -1567,7 +1590,7 @@ weights to the output layer.
 
          
 
 
-
          Weights and biases 
          
+
          Weights and biases 
          
 
 
          
 Typically weights are initialized with small values distributed around zero, drawn from a uniform
@@ -1603,7 +1626,7 @@ output_bias = np.zeros(n_categories) + 0.01
 
          
 









          
 
-
          Feed-forward pass 
          
+
          Feed-forward pass 
          
 
 
          
 Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.  
@@ -1630,7 +1653,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
 
          
 
 
-
          Matrix multiplications 
          
+
          Matrix multiplications 
          
 
 
          
 Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
@@ -1705,7 +1728,7 @@ predictions = predict(X_train)
 
          
 









          
 
-
          Choose cost function and optimizer 
          
+
          Choose cost function and optimizer 
          
 
 
          
 To measure how well our neural network is doing we need to introduce a cost function.  
@@ -1736,7 +1759,7 @@ you got the correct label. The probability of category \( c \) is given by the s
 
          
 









          
 
-
          Optimizing the cost function 
          
+
          Optimizing the cost function 
          
 
 
          
 The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
@@ -1776,7 +1799,7 @@ The various optmization  methods, with codes and algorithms,  are discussed in o
 
          
 
 
-
          Regularization 
          
+
          Regularization 
          
 
 
          
 It is common to add an extra term to the cost function, proportional
@@ -1809,7 +1832,7 @@ calculate the gradient efficently.
 
          
 









          
 
-
          Matrix  multiplication 
          
+
          Matrix  multiplication 
          
 
 
          
 To more efficently train our network these equations are implemented using matrix operations.  
@@ -1923,7 +1946,7 @@ lmbd = 0.01
 
          
 









          
 
-
          Improving performance 
          
+
          Improving performance 
          
 
 
          
 As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.  
@@ -1943,7 +1966,7 @@ Andrew Ng goes through some of these considerations in this Full object-oriented implementation 
+
          Full object-oriented implementation 
          
 
 
          
 It is very natural to think of the network as an object, with specific instances of the network
@@ -2056,7 +2079,7 @@ being realizations of this object with different hyperparameters. An implementat
 
          
 









          
 
-
          Evaluate model performance on test data 
          
+
          Evaluate model performance on test data 
          
 
 
          
 To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.  
@@ -2091,7 +2114,7 @@ test_predict = dnn.predict(X_test)
 
          
 









          
 
-
          Adjust hyperparameters 
          
+
          Adjust hyperparameters 
          
 
 
          
 We now perform a grid search to find the optimal hyperparameters for the network.  
@@ -2124,7 +2147,7 @@ DNN_numpy = np.zeros((len(eta_vals), 
 









          
 
-
          Visualization 
          
+
          Visualization 
          
 
 
          
 
@@ -2166,7 +2189,7 @@ plt.show()
 
          
 









          
 
-
          scikit-learn implementation 
          
+
          scikit-learn implementation 
          
 
 
          
 scikit-learn focuses more
@@ -2205,7 +2228,7 @@ DNN_scikit = np.zeros((len(eta_vals), 
 









          
 
-
          Visualization 
          
+
          Visualization 
          
 
          
 
 
@@ -2247,7 +2270,7 @@ plt.show()
 
          
 









          
 
-
          Building neural networks in Tensorflow and Keras 
          
+
          Building neural networks in Tensorflow and Keras 
          
 
 
          
 Now we want  to build on the experience gained from our neural network implementation in NumPy and scikit-learn
@@ -2262,7 +2285,7 @@ NumPy arrays.
 
          
 









          
 
-
          Tensorflow 
          
+
          Tensorflow 
          
 
 
          
 Tensorflow is an open source library machine learning library
@@ -2307,7 +2330,7 @@ and/or if you use anaconda, just write (or install from the graphical use
 
          
 









          
 
-
          Collect and pre-process data 
          
+
          Collect and pre-process data 
          
 
 
          
 
@@ -2373,7 +2396,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
 
          
 









          
 
-
          Using TensorFlow backend 
          
+
          Using TensorFlow backend 
          
 
 
            
 
          1.  Define model and architecture
            2. 
@@ -2518,7 +2541,7 @@ X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=t
 
            
 









            
 
-
            Optimizing and using gradient descent 
            
+
            Optimizing and using gradient descent 
            
 
 
            
 
@@ -2596,7 +2619,7 @@ writer.add_graph(tf.get_default_graph())
 
            
 









            
 
-
            Using Keras 
            
+
            Using Keras 
            
 
 
            
 Keras is a high level neural network
@@ -2695,7 +2718,7 @@ plt.show()
 
            
 
 
-
            Which activation function should I use? 
            
+
            Which activation function should I use? 
            
 
 
            
 The Back propagation algorithm we derived above works by going from
@@ -2724,7 +2747,7 @@ learn at widely different speeds
 
            
 
 
-
            Is the Logistic activation function (Sigmoid)  our choice? 
            
+
            Is the Logistic activation function (Sigmoid)  our choice? 
            
 
 
            
 Although this unfortunate behavior has been empirically observed for
@@ -2754,7 +2777,7 @@ better than the logistic function in deep networks).
 
            
 









            
 
-
            The derivative of the Logistic funtion 
            
+
            The derivative of the Logistic funtion 
            
 
 
            
 Looking at the logistic activation function, when inputs become large
@@ -2790,7 +2813,7 @@ fast to compute).
 
            
 









            
 
-
            The RELU function family 
            
+
            The RELU function family 
            
 
 
            
 The ReLU activation function suffers from a problem known as the dying
@@ -2817,7 +2840,7 @@ $$
 
            
 









            
 
-
            Which activation function should we use? 
            
+
            Which activation function should we use? 
            
 
 
            
 In general it seems that the ELU activation function is better than
@@ -2837,7 +2860,7 @@ bootstrap to evaluate other activation functions.
 
            
 
 
-
            A top-down perspective on Neural networks 
            
+
            A top-down perspective on Neural networks 
            
 
 
            
 The first thing we would like to do is divide the data into two or three
@@ -2879,7 +2902,7 @@ supervised learning.
 
            
 









            
 
-
            Limitations of supervised learning with deep networks 
            
+
            Limitations of supervised learning with deep networks 
            
 
 
            
 Like all statistical methods, supervised learning using neural
diff --git a/doc/pub/NeuralNet/html/NeuralNet.html b/doc/pub/NeuralNet/html/NeuralNet.html
index b0a0523e3..eda365b42 100644
--- a/doc/pub/NeuralNet/html/NeuralNet.html
+++ b/doc/pub/NeuralNet/html/NeuralNet.html
@@ -117,54 +117,55 @@ div { text-align: justify; text-justify: inter-word; }
                '___sec30'),
               ('Defining the cost function', 2, None, '___sec31'),
               ('Example: binary classification problem', 2, None, '___sec32'),
+              ('The Softmax function', 2, None, '___sec33'),
               ('Developing a code for doing neural networks with back '
                'propagation',
                2,
                None,
-               '___sec33'),
-              ('Collect and pre-process data', 2, None, '___sec34'),
-              ('Train and test datasets', 2, None, '___sec35'),
-              ('Define model and architecture', 2, None, '___sec36'),
-              ('Layers', 2, None, '___sec37'),
-              ('Weights and biases', 2, None, '___sec38'),
-              ('Feed-forward pass', 2, None, '___sec39'),
-              ('Matrix multiplications', 2, None, '___sec40'),
-              ('Choose cost function and optimizer', 2, None, '___sec41'),
-              ('Optimizing the cost function', 2, None, '___sec42'),
-              ('Regularization', 2, None, '___sec43'),
-              ('Matrix  multiplication', 2, None, '___sec44'),
-              ('Improving performance', 2, None, '___sec45'),
-              ('Full object-oriented implementation', 2, None, '___sec46'),
-              ('Evaluate model performance on test data', 2, None, '___sec47'),
-              ('Adjust hyperparameters', 2, None, '___sec48'),
-              ('Visualization', 2, None, '___sec49'),
-              ('scikit-learn implementation', 2, None, '___sec50'),
-              ('Visualization', 2, None, '___sec51'),
+               '___sec34'),
+              ('Collect and pre-process data', 2, None, '___sec35'),
+              ('Train and test datasets', 2, None, '___sec36'),
+              ('Define model and architecture', 2, None, '___sec37'),
+              ('Layers', 2, None, '___sec38'),
+              ('Weights and biases', 2, None, '___sec39'),
+              ('Feed-forward pass', 2, None, '___sec40'),
+              ('Matrix multiplications', 2, None, '___sec41'),
+              ('Choose cost function and optimizer', 2, None, '___sec42'),
+              ('Optimizing the cost function', 2, None, '___sec43'),
+              ('Regularization', 2, None, '___sec44'),
+              ('Matrix  multiplication', 2, None, '___sec45'),
+              ('Improving performance', 2, None, '___sec46'),
+              ('Full object-oriented implementation', 2, None, '___sec47'),
+              ('Evaluate model performance on test data', 2, None, '___sec48'),
+              ('Adjust hyperparameters', 2, None, '___sec49'),
+              ('Visualization', 2, None, '___sec50'),
+              ('scikit-learn implementation', 2, None, '___sec51'),
+              ('Visualization', 2, None, '___sec52'),
               ('Building neural networks in Tensorflow and Keras',
                2,
                None,
-               '___sec52'),
-              ('Tensorflow', 2, None, '___sec53'),
-              ('Collect and pre-process data', 2, None, '___sec54'),
-              ('Using TensorFlow backend', 2, None, '___sec55'),
-              ('Optimizing and using gradient descent', 2, None, '___sec56'),
-              ('Using Keras', 2, None, '___sec57'),
-              ('Which activation function should I use?', 2, None, '___sec58'),
+               '___sec53'),
+              ('Tensorflow', 2, None, '___sec54'),
+              ('Collect and pre-process data', 2, None, '___sec55'),
+              ('Using TensorFlow backend', 2, None, '___sec56'),
+              ('Optimizing and using gradient descent', 2, None, '___sec57'),
+              ('Using Keras', 2, None, '___sec58'),
+              ('Which activation function should I use?', 2, None, '___sec59'),
               ('Is the Logistic activation function (Sigmoid)  our choice?',
                2,
                None,
-               '___sec59'),
-              ('The derivative of the Logistic funtion', 2, None, '___sec60'),
-              ('The RELU function family', 2, None, '___sec61'),
-              ('Which activation function should we use?', 2, None, '___sec62'),
+               '___sec60'),
+              ('The derivative of the Logistic funtion', 2, None, '___sec61'),
+              ('The RELU function family', 2, None, '___sec62'),
+              ('Which activation function should we use?', 2, None, '___sec63'),
               ('A top-down perspective on Neural networks',
                2,
                None,
-               '___sec63'),
+               '___sec64'),
               ('Limitations of supervised learning with deep networks',
                2,
                None,
-               '___sec64')]}
+               '___sec65')]}
 end of tocinfo -->
 
 
@@ -1318,10 +1319,32 @@ $$
 
 In case we use another activation function than the logistic one, we need to evaluate other derivatives.
 
+
            
+









            
+
+
            The Softmax function 
            
+In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \( z_i^l \), that is we need
+$$
+\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
+\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{-1}.
+$$
+
+For the Softmax function we have
+$$
+f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{k=1}^K\exp{(z_k^l}}.
+$$
+
+Its derivative with respect to \( z_j^l \) gives 
+$$
+\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_i^l)\right), 
+$$
+
+which in case of the simply binary model reduces to  having \( i=j \).
+
 
            
 
 
-
            Developing a code for doing neural networks with back propagation 
            
+
            Developing a code for doing neural networks with back propagation 
            
 
 
            
 One can identify a set of key steps when using neural networks to solve supervised learning problems:  
@@ -1337,7 +1360,7 @@ One can identify a set of key steps when using neural networks to solve supervis
 
 









            
 
-
            Collect and pre-process data 
            
+
            Collect and pre-process data 
            
 
 
            
 Here we will be using the MNIST dataset, which is readily available through the scikit-learn
@@ -1432,7 +1455,7 @@ plt.show()
 
            
 









            
 
-
            Train and test datasets 
            
+
            Train and test datasets 
            
 
 
            
 Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
@@ -1481,7 +1504,7 @@ X_train, X_test, Y_train, Y_test = train_tes
 
            
 









            
 
-
            Define model and architecture 
            
+
            Define model and architecture 
            
 
 
            
 Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \( y \) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have 
@@ -1527,7 +1550,7 @@ which is inspired by probability theory (see logistic regression) and was most c
 
            
 
 
-
            Layers 
            
+
            Layers 
            
 
 
              
 
            •  Input
              •  
@@ -1572,7 +1595,7 @@ weights to the output layer.
 
              
 
 
-
              Weights and biases 
              
+
              Weights and biases 
              
 
 
              
 Typically weights are initialized with small values distributed around zero, drawn from a uniform
@@ -1608,7 +1631,7 @@ output_bias = np
 









              
 
-
              Feed-forward pass 
              
+
              Feed-forward pass 
              
 
 
              
 Denote \( F \) the number of features, \( H \) the number of hidden neurons and \( C \) the number of categories.  
@@ -1635,7 +1658,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
 
              
 
 
-
              Matrix multiplications 
              
+
              Matrix multiplications 
              
 
 
              
 Since our data has the dimensions \( X = (n_{inputs}, n_{features}) \) and our weights to the hidden
@@ -1710,7 +1733,7 @@ predictions = predict(X_train)
 
              
 









              
 
-
              Choose cost function and optimizer 
              
+
              Choose cost function and optimizer 
              
 
 
              
 To measure how well our neural network is doing we need to introduce a cost function.  
@@ -1741,7 +1764,7 @@ you got the correct label. The probability of category \( c \) is given by the s
 
              
 









              
 
-
              Optimizing the cost function 
              
+
              Optimizing the cost function 
              
 
 
              
 The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent
@@ -1781,7 +1804,7 @@ The various optmization  methods, with codes and algorithms,  are discussed in o
 
              
 
 
-
              Regularization 
              
+
              Regularization 
              
 
 
              
 It is common to add an extra term to the cost function, proportional
@@ -1814,7 +1837,7 @@ calculate the gradient efficently.
 
              
 









              
 
-
              Matrix  multiplication 
              
+
              Matrix  multiplication 
              
 
 
              
 To more efficently train our network these equations are implemented using matrix operations.  
@@ -1928,7 +1951,7 @@ lmbd = 0.01
 









              
 
-
              Improving performance 
              
+
              Improving performance 
              
 
 
              
 As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.  
@@ -1948,7 +1971,7 @@ Andrew Ng goes through some of these considerations in this Full object-oriented implementation 
+
              Full object-oriented implementation 
              
 
 
              
 It is very natural to think of the network as an object, with specific instances of the network
@@ -2061,7 +2084,7 @@ being realizations of this object with different hyperparameters. An implementat
 
              
 









              
 
-
              Evaluate model performance on test data 
              
+
              Evaluate model performance on test data 
              
 
 
              
 To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.  
@@ -2096,7 +2119,7 @@ test_predict = dnnAdjust hyperparameters 
+
              Adjust hyperparameters 
              
 
 
              
 We now perform a grid search to find the optimal hyperparameters for the network.  
@@ -2129,7 +2152,7 @@ DNN_numpy = np.
 
              
 









              
 
-
              Visualization 
              
+
              Visualization 
              
 
 
              
 
@@ -2171,7 +2194,7 @@ plt.show()
 
              
 









              
 
-
              scikit-learn implementation 
              
+
              scikit-learn implementation 
              
 
 
              
 scikit-learn focuses more
@@ -2210,7 +2233,7 @@ DNN_scikit = np
 
              
 









              
 
-
              Visualization 
              
+
              Visualization 
              
 
              
 
 
@@ -2252,7 +2275,7 @@ plt.show()
 
              
 









              
 
-
              Building neural networks in Tensorflow and Keras 
              
+
              Building neural networks in Tensorflow and Keras 
              
 
 
              
 Now we want  to build on the experience gained from our neural network implementation in NumPy and scikit-learn
@@ -2267,7 +2290,7 @@ NumPy arrays.
 
              
 









              
 
-
              Tensorflow 
              
+
              Tensorflow 
              
 
 
              
 Tensorflow is an open source library machine learning library
@@ -2312,7 +2335,7 @@ and/or if you use anaconda, just write (or install from the graphical use
 
              
 









              
 
-
              Collect and pre-process data 
              
+
              Collect and pre-process data 
              
 
 
              
 
@@ -2378,7 +2401,7 @@ X_train, X_test, Y_train, Y_test = train_tes
 
              
 









              
 
-
              Using TensorFlow backend 
              
+
              Using TensorFlow backend 
              
 
 
                
 
              1.  Define model and architecture
                2. 
@@ -2523,7 +2546,7 @@ X_train, X_test, Y_train, Y_test = train_tes
 
                
 









                
 
-
                Optimizing and using gradient descent 
                
+
                Optimizing and using gradient descent 
                
 
 
                
 
@@ -2601,7 +2624,7 @@ writer.add_graph(tfUsing Keras 
+
                Using Keras 
                
 
 
                
 Keras is a high level neural network
@@ -2700,7 +2723,7 @@ plt.show()
 
                
 
 
-
                Which activation function should I use? 
                
+
                Which activation function should I use? 
                
 
 
                
 The Back propagation algorithm we derived above works by going from
@@ -2729,7 +2752,7 @@ learn at widely different speeds
 
                
 
 
-
                Is the Logistic activation function (Sigmoid)  our choice? 
                
+
                Is the Logistic activation function (Sigmoid)  our choice? 
                
 
 
                
 Although this unfortunate behavior has been empirically observed for
@@ -2759,7 +2782,7 @@ better than the logistic function in deep networks).
 
                
 









                
 
-
                The derivative of the Logistic funtion 
                
+
                The derivative of the Logistic funtion 
                
 
 
                
 Looking at the logistic activation function, when inputs become large
@@ -2795,7 +2818,7 @@ fast to compute).
 
                
 









                
 
-
                The RELU function family 
                
+
                The RELU function family 
                
 
 
                
 The ReLU activation function suffers from a problem known as the dying
@@ -2822,7 +2845,7 @@ $$
 
                
 









                
 
-
                Which activation function should we use? 
                
+
                Which activation function should we use? 
                
 
 
                
 In general it seems that the ELU activation function is better than
@@ -2842,7 +2865,7 @@ bootstrap to evaluate other activation functions.
 
                
 
 
-
                A top-down perspective on Neural networks 
                
+
                A top-down perspective on Neural networks 
                
 
 
                
 The first thing we would like to do is divide the data into two or three
@@ -2884,7 +2907,7 @@ supervised learning.
 
                
 









                
 
-
                Limitations of supervised learning with deep networks 
                
+
                Limitations of supervised learning with deep networks 
                
 
 
                
 Like all statistical methods, supervised learning using neural
diff --git a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
index c0c8483f6..3d3b3a872 100644
--- a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
+++ b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb
@@ -1576,6 +1576,58 @@
     "In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n",
     "\n",
     "\n",
+    "## The Softmax function\n",
+    "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\frac{\\partial f(z_i^l)}{\\partial w_{jk}^l} =\n",
+    "\\frac{\\partial f(z_i^l)}{\\partial z_j^l} \\frac{\\partial z_j^l}{\\partial w_{jk}^l}= \\frac{\\partial f(z_i^l)}{\\partial z_j^l}a_k^{-1}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "For the Softmax function we have"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "f(z_i^l) = \\frac{\\exp{(z_i^l)}}{\\sum_{k=1}^K\\exp{(z_k^l}}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Its derivative with respect to $z_j^l$ gives"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\frac{\\partial f(z_i^l)}{\\partial z_j^l}= f(z_i^l)\\left(\\delta_{ij}-f(z_i^l)\\right),\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "which in case of the simply binary model reduces to  having $i=j$. \n",
+    "\n",
     "\n",
     "## Developing a code for doing neural networks with back propagation\n",
     "\n",
diff --git a/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz b/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz
index 8fa98fbb32c06f645f84bae58ce0819cff00ae06..461b35ece952d7c9830d1dab4f5f2b2a8401f071 100644
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