From 0ffba66be539d256a0b3178f5f6df64eaf451412 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 8 Oct 2020 21:09:16 +0200 Subject: [PATCH] updating week 41 --- doc/pub/week41/html/._week41-bs000.html | 186 ++- doc/pub/week41/html/._week41-bs001.html | 188 ++- doc/pub/week41/html/._week41-bs002.html | 257 ++-- doc/pub/week41/html/._week41-bs003.html | 274 ++-- doc/pub/week41/html/._week41-bs004.html | 264 ++-- doc/pub/week41/html/._week41-bs005.html | 258 ++-- doc/pub/week41/html/._week41-bs006.html | 228 ++- doc/pub/week41/html/._week41-bs007.html | 216 ++- doc/pub/week41/html/._week41-bs008.html | 285 ++-- doc/pub/week41/html/._week41-bs009.html | 292 ++-- doc/pub/week41/html/._week41-bs010.html | 247 ++- doc/pub/week41/html/._week41-bs011.html | 251 ++- doc/pub/week41/html/._week41-bs012.html | 241 ++- doc/pub/week41/html/._week41-bs013.html | 223 ++- doc/pub/week41/html/._week41-bs014.html | 256 ++-- doc/pub/week41/html/._week41-bs015.html | 262 ++-- doc/pub/week41/html/._week41-bs016.html | 227 +-- doc/pub/week41/html/._week41-bs017.html | 235 ++- doc/pub/week41/html/._week41-bs018.html | 307 ++-- doc/pub/week41/html/._week41-bs019.html | 294 ++-- doc/pub/week41/html/._week41-bs020.html | 296 ++-- doc/pub/week41/html/._week41-bs021.html | 303 ++-- doc/pub/week41/html/._week41-bs022.html | 230 ++- doc/pub/week41/html/._week41-bs023.html | 237 +-- doc/pub/week41/html/._week41-bs024.html | 243 ++- doc/pub/week41/html/._week41-bs025.html | 245 +-- doc/pub/week41/html/._week41-bs026.html | 229 ++- doc/pub/week41/html/._week41-bs027.html | 228 +-- doc/pub/week41/html/._week41-bs028.html | 271 ++-- doc/pub/week41/html/._week41-bs029.html | 372 +++-- doc/pub/week41/html/._week41-bs030.html | 376 ++--- doc/pub/week41/html/._week41-bs031.html | 260 ++-- doc/pub/week41/html/._week41-bs032.html | 412 ++--- doc/pub/week41/html/._week41-bs033.html | 373 ++--- doc/pub/week41/html/._week41-bs034.html | 223 ++- doc/pub/week41/html/._week41-bs035.html | 232 +-- doc/pub/week41/html/._week41-bs036.html | 227 ++- doc/pub/week41/html/._week41-bs037.html | 214 ++- doc/pub/week41/html/._week41-bs038.html | 231 +-- doc/pub/week41/html/._week41-bs039.html | 233 ++- doc/pub/week41/html/._week41-bs040.html | 228 +-- doc/pub/week41/html/._week41-bs041.html | 230 ++- doc/pub/week41/html/._week41-bs042.html | 225 +-- doc/pub/week41/html/._week41-bs043.html | 232 ++- doc/pub/week41/html/._week41-bs044.html | 213 ++- doc/pub/week41/html/._week41-bs045.html | 211 ++- doc/pub/week41/html/week41-bs.html | 186 ++- doc/pub/week41/html/week41-reveal.html | 105 +- doc/pub/week41/html/week41-solarized.html | 196 ++- doc/pub/week41/html/week41.html | 196 ++- doc/pub/week41/ipynb/ipynb-week41-src.tar.gz | Bin 87344 -> 87344 bytes doc/pub/week41/ipynb/week41.ipynb | 1429 ++---------------- doc/src/week41/week41.do.txt | 12 +- 53 files changed, 6277 insertions(+), 7612 deletions(-) diff --git a/doc/pub/week41/html/._week41-bs000.html b/doc/pub/week41/html/._week41-bs000.html index 63ba71fc4..045b39a7d 100644 --- a/doc/pub/week41/html/._week41-bs000.html +++ b/doc/pub/week41/html/._week41-bs000.html @@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source @@ -149,52 +148,51 @@ MathJax.Hub.Config({
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    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Oct 6, 2020

    +

    Oct 8, 2020


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    Plan for week 40

    +

    Plan for week 41

    @@ -237,7 +235,7 @@ extbooks/TensorflowML.pdf" target="_self">Aurelien Geron's chapters 10-11 an
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    Overview video for week 41

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    Setting up the Back propagation algorithm

    -"Overview Video, from Stochastic Gradient methods to Neural Networks":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20\ -/forelesningsvideoer/OverviewWeek41.mp4?vrtx=view-as-webpage" +The four equations derived last week provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm. + +

    +

    +
    +

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

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    + + +

    +

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

    +
    + + +

    +

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    +

    +Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all +$$ +\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. +$$ +

    +
    + + +

    +

    +
    +

    +Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as +$$ +\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). +$$ +

    +
    + + +

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    +Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules +$$ +w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, +$$ + + +$$ +b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, +$$ +

    +
    + + +

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

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    - + -

    Setting up the Back propagation algorithm

    +

    Setting up a Multi-layer perceptron model for classification

    -The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm. +We are now gong to develop an example based on the MNIST data +base. This is a classification problem and we need to use our +cross-entropy function we discussed in connection with logistic +regression. The cross-entropy defines our cost function for the +classificaton problems with neural networks.

    -

    -
    -

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

    -
    - +In binary classification with two classes \( (0, 1) \) we define the +logistic/sigmoid function as the probability that a particular input +is in class \( 0 \) or \( 1 \). This is possible because the logistic +function takes any input from the real numbers and inputs a number +between 0 and 1, and can therefore be interpreted as a probability. It +also has other nice properties, such as a derivative that is simple to +calculate.

    -

    -
    -

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

    -
    +For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \) +is in class 0 or 1 is just. We let \( \theta \) represent the unknown weights and biases to be adjusted by our equations). The variable \( x \) +represents our activation values \( z \). We have +$$ +P(y = 0 \mid \hat{x}, \hat{\theta}) = \frac{1}{1 + \exp{(- \hat{x}})} , +$$ +and +$$ +P(y = 1 \mid \hat{x}, \hat{\theta}) = 1 - P(y = 0 \mid \hat{x}, \hat{\theta}) , +$$

    -

    -
    -

    -Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all -$$ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -$$ -

    -
    - - -

    -

    -
    -

    -Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as -$$ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). -$$ -

    -
    - - -

    -

    -
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    -Finally, we update the weights and the biases using gradient descent for each \( l=L-1,L-2,\dots,2 \) and update the weights and biases according to the rules -$$ -w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, -$$ - - -$$ -b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, -$$ -

    -
    - - -

    -The parameter \( \eta \) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training. +where \( y \in \{0, 1\} \) and \( \hat{\theta} \) represents the weights and biases +of our network.

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    - + -

    Setting up a Multi-layer perceptron model for classification

    +

    Defining the cost function

    -We are now gong to develop an example based on the MNIST data -base. This is a classification problem and we need to use our -cross-entropy function we discussed in connection with logistic -regression. The cross-entropy defines our cost function for the -classificaton problems with neural networks. - -

    -In binary classification with two classes \( (0, 1) \) we define the -logistic/sigmoid function as the probability that a particular input -is in class \( 0 \) or \( 1 \). This is possible because the logistic -function takes any input from the real numbers and inputs a number -between 0 and 1, and can therefore be interpreted as a probability. It -also has other nice properties, such as a derivative that is simple to -calculate. - -

    -For an input \( \boldsymbol{a} \) from the hidden layer, the probability that the input \( \boldsymbol{x} \) -is in class 0 or 1 is just. We let \( \theta \) represent the unknown weights and biases to be adjusted by our equations). The variable \( x \) -represents our activation values \( z \). We have +Our cost function is given as (see the Logistic regression lectures) $$ -P(y = 0 \mid \hat{x}, \hat{\theta}) = \frac{1}{1 + \exp{(- \hat{x}})} , -$$ - -and -$$ -P(y = 1 \mid \hat{x}, \hat{\theta}) = 1 - P(y = 0 \mid \hat{x}, \hat{\theta}) , +\mathcal{C}(\hat{\theta}) = - \ln P(\mathcal{D} \mid \hat{\theta}) = - \sum_{i=1}^n +y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\hat{\theta}) . $$

    -where \( y \in \{0, 1\} \) and \( \hat{\theta} \) represents the weights and biases -of our network. +This last equality means that we can interpret our cost function as a sum over the loss function +for each point in the dataset \( \mathcal{L}_i(\hat{\theta}) \). +The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather +than maximizing a negative number. + +

    +In multiclass classification it is common to treat each integer label as a so called one-hot vector: + +

    +\( y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and + +

    +\( y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \) + +

    +i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \)).. + +

    +If \( \hat{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th +output vector \( \hat{y}_i \). +The probability of \( \hat{x}_i \) being in class \( c \) will be given by the softmax function: + +$$ +P(y_{ic} = 1 \mid \hat{x}_i, \hat{\theta}) = \frac{\exp{((\hat{a}_i^{hidden})^T \hat{w}_c)}} +{\sum_{c'=0}^{C-1} \exp{((\hat{a}_i^{hidden})^T \hat{w}_{c'})}} , +$$ + +

    +which reduces to the logistic function in the binary case. +The likelihood of this \( C \)-class classifier +is now given as: + +$$ +P(\mathcal{D} \mid \hat{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} . +$$ + +Again we take the negative log-likelihood to define our cost function: + +$$ +\mathcal{C}(\hat{\theta}) = - \log{P(\mathcal{D} \mid \hat{\theta})}. +$$ + +See the logistic regression lectures for a full definition of the cost function. + +

    +The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!

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    Defining the cost function

    +

    Example: binary classification problem

    -Our cost function is given as (see the Logistic regression lectures) +As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \( \beta \) as $$ -\mathcal{C}(\hat{\theta}) = - \ln P(\mathcal{D} \mid \hat{\theta}) = - \sum_{i=1}^n -y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\hat{\theta}) . +\mathcal{C}(\hat{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\hat{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\hat{\beta})}\right), $$ -

    -This last equality means that we can interpret our cost function as a sum over the loss function -for each point in the dataset \( \mathcal{L}_i(\hat{\theta}) \). -The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather -than maximizing a negative number. - -

    -In multiclass classification it is common to treat each integer label as a so called one-hot vector: - -

    -\( y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , \) and - -

    -\( y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) , \) - -

    -i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset (numbers from \( 0 \) to \( 9 \)).. - -

    -If \( \hat{x}_i \) is the \( i \)-th input (image), \( y_{ic} \) refers to the \( c \)-th component of the \( i \)-th -output vector \( \hat{y}_i \). -The probability of \( \hat{x}_i \) being in class \( c \) will be given by the softmax function: - +where we had defined the logistic (sigmoid) function $$ -P(y_{ic} = 1 \mid \hat{x}_i, \hat{\theta}) = \frac{\exp{((\hat{a}_i^{hidden})^T \hat{w}_c)}} -{\sum_{c'=0}^{C-1} \exp{((\hat{a}_i^{hidden})^T \hat{w}_{c'})}} , +p(y_i =1\vert x_i,\hat{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, $$ -

    -which reduces to the logistic function in the binary case. -The likelihood of this \( C \)-class classifier -is now given as: - +and $$ -P(\mathcal{D} \mid \hat{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} . +p(y_i =0\vert x_i,\hat{\beta})=1-p(y_i =1\vert x_i,\hat{\beta}). $$ -Again we take the negative log-likelihood to define our cost function: - -$$ -\mathcal{C}(\hat{\theta}) = - \log{P(\mathcal{D} \mid \hat{\theta})}. -$$ - -See the logistic regression lectures for a full definition of the cost function. +The parameters \( \hat{\beta} \) were defined using a minimization method like gradient descent or Newton-Raphson's method.

    -The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before! +Now we replace \( x_i \) with the activation \( z_i^l \) for a given layer \( l \) and the outputs as \( y_i=a_i^l=f(z_i^l) \), with \( z_i^l \) now being a function of the weights \( w_{ij}^l \) and biases \( b_i^l \). +We have then +$$ +a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, +$$ + +with +$$ +z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, +$$ + +where the superscript \( l-1 \) indicates that these are the outputs from layer \( l-1 \). +Our cost function at the final layer \( l=L \) is now +$$ +\mathcal{C}(\hat{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), +$$ + +where we have defined the targets \( t_i \). The derivatives of the cost function with respect to the output \( a_i^L \) are then easily calculated and we get +$$ +\frac{\partial \mathcal{C}(\hat{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. +$$ + +In case we use another activation function than the logistic one, we need to evaluate other derivatives.

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    Example: binary classification problem

    - -

    -As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \( \beta \) as +

    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 $$ -\mathcal{C}(\hat{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\hat{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\hat{\beta})}\right), +\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = +\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}. $$ -where we had defined the logistic (sigmoid) function +For the Softmax function we have $$ -p(y_i =1\vert x_i,\hat{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, +f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}. $$ -and +Its derivative with respect to \( z_j^l \) gives $$ -p(y_i =0\vert x_i,\hat{\beta})=1-p(y_i =1\vert x_i,\hat{\beta}). +\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), $$ -The parameters \( \hat{\beta} \) were defined using a minimization method like gradient descent or Newton-Raphson's method. - -

    -Now we replace \( x_i \) with the activation \( z_i^l \) for a given layer \( l \) and the outputs as \( y_i=a_i^l=f(z_i^l) \), with \( z_i^l \) now being a function of the weights \( w_{ij}^l \) and biases \( b_i^l \). -We have then -$$ -a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, -$$ - -with -$$ -z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, -$$ - -where the superscript \( l-1 \) indicates that these are the outputs from layer \( l-1 \). -Our cost function at the final layer \( l=L \) is now -$$ -\mathcal{C}(\hat{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), -$$ - -where we have defined the targets \( t_i \). The derivatives of the cost function with respect to the output \( a_i^L \) are then easily calculated and we get -$$ -\frac{\partial \mathcal{C}(\hat{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. -$$ - -In case we use another activation function than the logistic one, we need to evaluate other derivatives. +which in case of the simply binary model reduces to having \( i=j \).

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    - + -

    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 -$$ -\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = -\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}. -$$ - -For the Softmax function we have -$$ -f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}. -$$ - -Its derivative with respect to \( z_j^l \) gives -$$ -\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), -$$ - -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: + +

      +
    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/week41/html/._week41-bs008.html b/doc/pub/week41/html/._week41-bs008.html index 1ef420d7e..d6d3caf3d 100644 --- a/doc/pub/week41/html/._week41-bs008.html +++ b/doc/pub/week41/html/._week41-bs008.html @@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source @@ -149,52 +148,51 @@ MathJax.Hub.Config({
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    • @@ -208,22 +206,101 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Developing a code for doing neural networks with back propagation

    +

    Collect and pre-process data

    -One can identify a set of key steps when using neural networks to solve supervised learning problems: +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 \( 28\times 28 \) 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. -

      -
    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. -
    +

    +To feed data into a feed-forward neural network we need to represent +the inputs as a design/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 +\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \). +

    +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 design/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 +design/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/week41/html/._week41-bs009.html b/doc/pub/week41/html/._week41-bs009.html index c5943ced8..e972f1efa 100644 --- a/doc/pub/week41/html/._week41-bs009.html +++ b/doc/pub/week41/html/._week41-bs009.html @@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source @@ -149,52 +148,51 @@ MathJax.Hub.Config({
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    • @@ -210,97 +208,51 @@ MathJax.Hub.Config({ -

    Collect and pre-process data

    +

    Train and test datasets

    -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 \( 28\times 28 \) 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. +Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.

    -To feed data into a feed-forward neural network we need to represent -the inputs as a design/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 -\( Y = (n_{inputs}) = (5, 3, 1, 8,...) \). +We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.

    -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 design/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 -design/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. +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.

    -

    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
+
    from sklearn.model_selection import train_test_split
 
+# 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)
 
-# ensure the same random numbers appear every time
-np.random.seed(0)
+# 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
 
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
+#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
 
-
-# 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()
+print("Number of training images: " + str(len(X_train)))
+print("Number of test images: " + str(len(X_test)))
 
    
 
    
 
    
@@ -327,7 +279,7 @@ plt.show()
   
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diff --git a/doc/pub/week41/html/._week41-bs010.html b/doc/pub/week41/html/._week41-bs010.html
index 8c37923b7..86a0fe91b 100644
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-
    Train and test datasets 
    
+
    Define model and architecture 
    
 
 
    
-Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.
+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 
+
+$$ z = \sum_{i=1}^n w_i a_i ,$$
+
+$$ y = f(z) ,$$
 
 
    
-We will reserve \( 80 \% \) of our dataset for training and \( 20 \% \) for testing.
+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).
 
 
    
-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.
+The simplest activation function for a neuron is the Heaviside function:
+
+$$ f(z) = 
+\begin{cases}
+1,  &  z > 0\\
+0,  & \text{otherwise}
+\end{cases}
+$$
 
 
    
+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.
 
-
-
    from sklearn.model_selection import train_test_split
+
    
+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.
 
-# 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)
+
    
+Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).  
+We will be using the sigmoid function \( \sigma(x) \):  
 
-# 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
+$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$
 
-#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
+
    
+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.
 
-print("Number of training images: " + str(len(X_train)))
-print("Number of test images: " + str(len(X_test)))
-
    
 
    
 
    
 
@@ -282,7 +277,7 @@ X_train, X_test, Y_train, Y_test = train_tes
   
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diff --git a/doc/pub/week41/html/._week41-bs011.html b/doc/pub/week41/html/._week41-bs011.html
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+++ b/doc/pub/week41/html/._week41-bs011.html
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         Contents 
         
       
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-
+
 
-
    Define model and architecture 
    
+
    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.
 
 
    
-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 
-
-$$ z = \sum_{i=1}^n w_i a_i ,$$
-
-$$ y = f(z) ,$$
+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.
 
 
    
-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).
+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)}} ,$$
 
 
    
-The simplest activation function for a neuron is the Heaviside function:
+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:  
 
-$$ f(z) = 
-\begin{cases}
-1,  &  z > 0\\
-0,  & \text{otherwise}
-\end{cases}
-$$
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$
 
 
    
-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.
+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.
 
 
    
 
    
@@ -279,7 +276,7 @@ which is inspired by probability theory (see logistic regression) and was most c
   
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diff --git a/doc/pub/week41/html/._week41-bs012.html b/doc/pub/week41/html/._week41-bs012.html
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    • 
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-
    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.
+
    Weights and biases 
    
 
 
    
-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.
+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.
 
 
    
-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:  
+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 \( j \), \( b_j \):  
 
-$$ 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)}} ,$$
+$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$
 
 
    
-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:  
-
-$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$
-
+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.
 
    
-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
+
    
 
    
 
    
 
@@ -278,7 +267,7 @@ weights to the output layer.
   
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diff --git a/doc/pub/week41/html/._week41-bs013.html b/doc/pub/week41/html/._week41-bs013.html
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-
+
 
-
    Weights and biases 
    
+
    Feed-forward pass 
    
 
 
    
-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.
+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 \( l \):  
+
+$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
 
 
    
-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 \( j \), \( b_j \):  
+this is then passed through our activation function  
 
-$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$
+$$ a_{j}^{l} = f(z_{j}^{l}) .$$
 
 
    
-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.
+We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:  
+
+$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
+
 
    
+Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:  
 
-
-
    # building our neural network
+$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
+{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$
 
-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
-
    
 
    
 
    
 
@@ -269,7 +258,7 @@ output_bias = np22
   
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diff --git a/doc/pub/week41/html/._week41-bs014.html b/doc/pub/week41/html/._week41-bs014.html
index 36081dfe7..5e0fe8f6c 100644
--- a/doc/pub/week41/html/._week41-bs014.html
+++ b/doc/pub/week41/html/._week41-bs014.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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         Contents 
         
       
    • 
@@ -208,32 +206,80 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Feed-forward pass 
    
+
    Matrix multiplications 
    
 
 
    
-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 \( l \):  
+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  
 
-$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$
+$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
 
 
    
-this is then passed through our activation function  
+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 \( Z^{h} \):  
 
-$$ a_{j}^{l} = f(z_{j}^{l}) .$$
+$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$
 
 
    
-We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \( j \) in the output layer:  
+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:  
 
-$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$
+$$ \hat{a}^{l} = f(\hat{z}^l) .$$
 
 
    
-Finally we calculate the output of neuron \( j \) in the output layer using the softmax function:  
+This is fed to the output layer:  
 
-$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
-{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$
+$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$
 
+
    
+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]))
+
    
 
    
 
    
 
@@ -260,7 +306,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
   
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diff --git a/doc/pub/week41/html/._week41-bs015.html b/doc/pub/week41/html/._week41-bs015.html
index 7f970e2dd..593cd7abd 100644
--- a/doc/pub/week41/html/._week41-bs015.html
+++ b/doc/pub/week41/html/._week41-bs015.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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    • 
@@ -208,80 +206,36 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Matrix multiplications 
    
+
    Choose cost function and optimizer 
    
 
 
    
-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  
-
-$$ X W^{h} = (n_{inputs}, n_{hidden}),$$
+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.
 
 
    
-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 \( Z^{h} \):  
+In multiclass classification it is common to treat each integer label as a so called one-hot vector:  
 
-$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$
+$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$  
+
+$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
 
 
    
-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:  
-
-$$ \hat{a}^{l} = f(\hat{z}^l) .$$
+i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
 
 
    
-This is fed to the output layer:  
-
-$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$
+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.
 
 
    
-Finally we receive our output values for each image and each category by passing it through the softmax function:  
+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.
 
-$$ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$
-
-
    
-
-
-
    # 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]))
-
    
 
    
 
    
 
@@ -308,7 +262,7 @@ predictions = predict(X_train)
   
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diff --git a/doc/pub/week41/html/._week41-bs016.html b/doc/pub/week41/html/._week41-bs016.html
index 34d3776cf..a0aaead4e 100644
--- a/doc/pub/week41/html/._week41-bs016.html
+++ b/doc/pub/week41/html/._week41-bs016.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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@@ -210,33 +208,42 @@ MathJax.Hub.Config({
 
 
 
-
    Choose cost function and optimizer 
    
+
    Optimizing the cost function 
    
 
 
    
-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.
+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  
+
+$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
 
 
    
-In multiclass classification it is common to treat each integer label as a so called one-hot vector:  
-
-$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$  
-
-$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$
+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.
 
 
    
-i.e. a binary bit string of length \( C \), where \( C = 10 \) is the number of classes in the MNIST dataset.
+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:  
+
+$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$
 
 
    
-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.
+i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.
 
 
    
-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.
+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.
 
 
    
 
    
@@ -264,7 +271,7 @@ you got the correct label. The probability of category \( c \) is given by the s
   
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diff --git a/doc/pub/week41/html/._week41-bs017.html b/doc/pub/week41/html/._week41-bs017.html
index 8d53a40c3..29fa1c793 100644
--- a/doc/pub/week41/html/._week41-bs017.html
+++ b/doc/pub/week41/html/._week41-bs017.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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@@ -208,44 +206,37 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Optimizing the cost function 
    
+
    Regularization 
    
 
 
    
-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  
-
-$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$
+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.
 
 
    
-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.
+We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:  
+
+$$  \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
+\frac{1}{N} \sum_{i=1}^N  \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2 
+= \frac{1}{N} \sum_{i=1}^N  \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
 
 
    
-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. 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.
-
-
    
-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 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.
 
 
    
 
    
@@ -273,7 +264,7 @@ The various optmization  methods, with codes and algorithms,  are discussed in o
   
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diff --git a/doc/pub/week41/html/._week41-bs018.html b/doc/pub/week41/html/._week41-bs018.html
index b0fa2bea8..e5b290858 100644
--- a/doc/pub/week41/html/._week41-bs018.html
+++ b/doc/pub/week41/html/._week41-bs018.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
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-
+
 
-
    Regularization 
    
+
    Matrix  multiplication 
    
 
 
    
-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.
+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,  
+
+$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
 
 
    
-We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:  
+The gradient for the output weights is calculated as  
 
-$$  \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad
-\frac{1}{N} \sum_{i=1}^N  \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2 
-= \frac{1}{N} \sum_{i=1}^N  \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$
+$$ \nabla W_{L} = \hat{a}^T \delta_L   = (n_{hidden}, n_{categories}) ,$$
 
 
    
-i.e. we sum up all the weights squared. The factor \( \lambda \) is known as a regularization parameter.
+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.
 
 
    
-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.
+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)))
+
    
 
    
 
    
 
@@ -266,7 +345,7 @@ calculate the gradient efficently.
   
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diff --git a/doc/pub/week41/html/._week41-bs019.html b/doc/pub/week41/html/._week41-bs019.html
index 90d3e975a..d8a1b5cb3 100644
--- a/doc/pub/week41/html/._week41-bs019.html
+++ b/doc/pub/week41/html/._week41-bs019.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,117 +208,23 @@ MathJax.Hub.Config({
 
 
 
-
    Matrix  multiplication 
    
+
    Improving performance 
    
 
 
    
-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,  
-
-$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$
+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 gradient for the output weights is calculated as  
-
-$$ \nabla W_{L} = \hat{a}^T \delta_L   = (n_{hidden}, n_{categories}) ,$$
+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 \( \lambda = 10^{-6},...,10^{-0} \).
 
 
    
-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.
+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.
 
 
    
-The gradient with respect to the output bias is then  
+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.
 
-$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$
-
-
    
-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)))
-
    
 
    
 
    
 
@@ -347,7 +251,7 @@ lmbd = 0.0128
   
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diff --git a/doc/pub/week41/html/._week41-bs020.html b/doc/pub/week41/html/._week41-bs020.html
index 34afc6338..ce7cea4a6 100644
--- a/doc/pub/week41/html/._week41-bs020.html
+++ b/doc/pub/week41/html/._week41-bs020.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,23 +208,115 @@ MathJax.Hub.Config({
 
 
 
-
    Improving performance 
    
+
    Full object-oriented implementation 
    
 
 
    
-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.
+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.
 
 
    
-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 \( \lambda = 10^{-6},...,10^{-0} \).
 
-
    
-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.
+
+
    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):
 
-
    
-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.X_data_full = X_data
+        self.Y_data_full = Y_data
 
+        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()
+
    
 
    
 
    
 
@@ -253,7 +343,7 @@ Andrew Ng goes through some of these considerations in this 29
   
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diff --git a/doc/pub/week41/html/._week41-bs021.html b/doc/pub/week41/html/._week41-bs021.html
index e83e31548..f0fa21428 100644
--- a/doc/pub/week41/html/._week41-bs021.html
+++ b/doc/pub/week41/html/._week41-bs021.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,114 +208,37 @@ MathJax.Hub.Config({
 
 
 
-
    Full object-oriented implementation 
    
+
    Evaluate model performance on test data 
    
 
 
    
-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.
+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.
 
 
    
 
 
-
    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):
+
    epochs = 100
+batch_size = 100
 
-        self.X_data_full = X_data
-        self.Y_data_full = Y_data
+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.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
+# accuracy score from scikit library
+print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
 
-        self.epochs = epochs
-        self.batch_size = batch_size
-        self.iterations = self.n_inputs // self.batch_size
-        self.eta = eta
-        self.lmbd = lmbd
+# equivalent in numpy
+def accuracy_score_numpy(Y_test, Y_pred):
+    return np.sum(Y_test == Y_pred) / len(Y_test)
 
-        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()
+#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
 
    
 
    
 
    
@@ -345,7 +266,7 @@ being realizations of this object with different hyperparameters. An implementat
   
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diff --git a/doc/pub/week41/html/._week41-bs022.html b/doc/pub/week41/html/._week41-bs022.html
index 93bd88e91..53f0575ac 100644
--- a/doc/pub/week41/html/._week41-bs022.html
+++ b/doc/pub/week41/html/._week41-bs022.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,37 +208,35 @@ MathJax.Hub.Config({
 
 
 
-
    Evaluate model performance on test data 
    
+
    Adjust hyperparameters 
    
 
 
    
-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.
+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).
 
 
    
 
 
-
    epochs = 100
-batch_size = 100
+
    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)
 
-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))
+# 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()
 
    
 
    
 
    
@@ -268,7 +264,7 @@ test_predict = dnn31
   
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diff --git a/doc/pub/week41/html/._week41-bs023.html b/doc/pub/week41/html/._week41-bs023.html
index 14a32ae48..583224d05 100644
--- a/doc/pub/week41/html/._week41-bs023.html
+++ b/doc/pub/week41/html/._week41-bs023.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,35 +208,44 @@ MathJax.Hub.Config({
 
 
 
-
    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).
+
    Visualization 
    
 
 
    
 
 
-
    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)
+
    # visual representation of grid search
+# uses seaborn heatmap, you can also do this with matplotlib imshow
+import seaborn as sns
 
-# 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()
+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_numpy[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)
+
         
-        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()
+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()
 
    
 
    
 
    
@@ -266,7 +273,7 @@ DNN_numpy = np.
   
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diff --git a/doc/pub/week41/html/._week41-bs024.html b/doc/pub/week41/html/._week41-bs024.html
index baabd08c6..1204b5d13 100644
--- a/doc/pub/week41/html/._week41-bs024.html
+++ b/doc/pub/week41/html/._week41-bs024.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,44 +208,41 @@ 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.
 
 
    
 
 
-
    # visual representation of grid search
-# uses seaborn heatmap, you can also do this with matplotlib imshow
-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_numpy[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()
 
    
 
    
 
    
@@ -275,7 +270,7 @@ plt.show()
   
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diff --git a/doc/pub/week41/html/._week41-bs025.html b/doc/pub/week41/html/._week41-bs025.html
index 8e57c4b3c..1a992141c 100644
--- a/doc/pub/week41/html/._week41-bs025.html
+++ b/doc/pub/week41/html/._week41-bs025.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,41 +208,44 @@ 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)
+
    # optional
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+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_scikit[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()
 
    
 
    
 
    
@@ -272,7 +273,7 @@ DNN_scikit = np
   
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diff --git a/doc/pub/week41/html/._week41-bs026.html b/doc/pub/week41/html/._week41-bs026.html
index e9066c7c6..dde4abd53 100644
--- a/doc/pub/week41/html/._week41-bs026.html
+++ b/doc/pub/week41/html/._week41-bs026.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,45 +208,18 @@ MathJax.Hub.Config({
 
 
 
-
    Visualization 
    
+
    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
+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.
 
-
-
    # optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
+
    
+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.
 
-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()
-
    
 
    
 
    
 
@@ -275,7 +246,7 @@ plt.show()
   
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diff --git a/doc/pub/week41/html/._week41-bs027.html b/doc/pub/week41/html/._week41-bs027.html
index a887402a8..ba7e13d54 100644
--- a/doc/pub/week41/html/._week41-bs027.html
+++ b/doc/pub/week41/html/._week41-bs027.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,18 +208,48 @@ MathJax.Hub.Config({
 
 
 
-
    Building neural networks in Tensorflow and Keras 
    
+
    Tensorflow 
    
 
 
    
-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 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.
 
 
    
-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 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
+
    
+
+
+
    pip3 install tensorflow
+
    
+
    
+and/or if you use anaconda, just write (or install from the graphical user interface)
+
    
+
+
+
    conda install tensorflow
+
    
 
    
 
    
 
@@ -248,7 +276,7 @@ NumPy arrays.
   
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diff --git a/doc/pub/week41/html/._week41-bs028.html b/doc/pub/week41/html/._week41-bs028.html
index edeb39806..4599537ae 100644
--- a/doc/pub/week41/html/._week41-bs028.html
+++ b/doc/pub/week41/html/._week41-bs028.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,47 +208,68 @@ MathJax.Hub.Config({
 
 
 
-
    Tensorflow 
    
+
    Collect and pre-process data 
    
 
-
    
-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
 
    
 
 
-
    pip3 install tensorflow
+
    # 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()
 
    
 
    
-and/or if you use anaconda, just write (or install from the graphical user interface)
-
    
 
 
-
    conda install tensorflow
+
    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)
 
    
 
    
 
    
@@ -278,7 +297,7 @@ and/or if you use anaconda, just write (or install from the graphical use
   
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diff --git a/doc/pub/week41/html/._week41-bs029.html b/doc/pub/week41/html/._week41-bs029.html
index b2bc01b7d..19dae5624 100644
--- a/doc/pub/week41/html/._week41-bs029.html
+++ b/doc/pub/week41/html/._week41-bs029.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
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    • 
@@ -210,68 +208,146 @@ MathJax.Hub.Config({
 
 
 
-
    Collect and pre-process data 
    
+
    Using TensorFlow backend 
    
+
+
      
+
    1.  Define model and architecture
      2. 
+
    2.  Choose cost function and optimizer
      3. 
+
    
 
 
    
 
 
-
    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
+
    import tensorflow as tf
 
+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
 
-# ensure the same random numbers appear every time
-np.random.seed(0)
+    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)
 
-# 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)
+        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})
 
    
 
    
 
    
@@ -299,7 +375,7 @@ X_train, X_test, Y_train, Y_test = train_tes
   
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diff --git a/doc/pub/week41/html/._week41-bs030.html b/doc/pub/week41/html/._week41-bs030.html
index a5f51fd80..34a1449be 100644
--- a/doc/pub/week41/html/._week41-bs030.html
+++ b/doc/pub/week41/html/._week41-bs030.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,146 +208,80 @@ MathJax.Hub.Config({
 
 
 
-
    Using TensorFlow backend 
    
-
-
      
-
    1.  Define model and architecture
      2. 
-
    2.  Choose cost function and optimizer
      3. 
-
    
+
    Optimizing and using gradient descent 
    
 
 
    
 
 
-
    import tensorflow as tf
+
    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)
+
    
+
    
 
-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):
+
+
    DNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
         
-        # keep track of number of steps
-        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
+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()
         
-        self.X_train = X_train
-        self.Y_train = Y_train
-        self.X_test = X_test
-        self.Y_test = Y_test
+        DNN_tf[i][j] = DNN
         
-        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
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % DNN.test_accuracy)
+        print()
+
    
+
    
 
-    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
+# 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
 
-        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})
+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()
+
    
+
    
+
+
+
    # optional
+# we can use log files to visualize our graph in Tensorboard
+writer = tf.summary.FileWriter('logs/')
+writer.add_graph(tf.get_default_graph())
 
    
 
    
 
    
@@ -377,7 +309,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week41/html/._week41-bs031.html b/doc/pub/week41/html/._week41-bs031.html
index 56ede3ad9..a7841dc59 100644
--- a/doc/pub/week41/html/._week41-bs031.html
+++ b/doc/pub/week41/html/._week41-bs031.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,36 +208,68 @@ MathJax.Hub.Config({
 
 
 
-
    Optimizing and using gradient descent 
    
+
    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:
 
 
    
 
 
-
    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)
+
    pip install keras
+
    
+
    
+or look up the instructions here.
+
+
    
+
+
+
    import tensorflow as tf
+from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
+
+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=regularizers.l2(lmbd)))
+    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax'))
+    
+    sgd = optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
 
    
 
    
 
 
-
    DNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
    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 = 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 = 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_tf[i][j] = DNN
+        DNN_keras[i][j] = DNN
         
         print("Learning rate = ", eta)
         print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % DNN.test_accuracy)
+        print("Test accuracy: %.3f" % scores[1])
         print()
 
    
 
    
@@ -257,10 +287,10 @@ test_accuracy = npfor i in range(len(eta_vals)):
     for j in range(len(lmbd_vals)):
-        DNN = DNN_tf[i][j]
+        DNN = DNN_keras[i][j]
 
-        train_accuracy[i][j] = DNN.train_accuracy
-        test_accuracy[i][j] = DNN.test_accuracy
+        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))
@@ -278,14 +308,6 @@ 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())
-
    
-
    
 
    
 
 
      
@@ -311,7 +333,7 @@ writer.add_graph(tf40
   
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diff --git a/doc/pub/week41/html/._week41-bs032.html b/doc/pub/week41/html/._week41-bs032.html
index 329845044..c389e464b 100644
--- a/doc/pub/week41/html/._week41-bs032.html
+++ b/doc/pub/week41/html/._week41-bs032.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,28 +208,7 @@ 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:
-
-
    
-
-
-
    pip install keras
-
    
-
    
-or look up the instructions here.
+
    The Breast Cancer Data, now with Keras 
    
 
 
    
 
@@ -243,71 +220,164 @@ or look up the instructions here.
 from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
 from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
 from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-
-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=regularizers.l2(lmbd)))
-    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(Dense(n_categories, activation='softmax'))
-    
-    sgd = optimizers.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 numpy as np
+import matplotlib.pyplot as plt
 import seaborn as sns
+from sklearn.model_selection import train_test_split as splitter
+from sklearn.datasets import load_breast_cancer
+import pickle
+import os 
 
-sns.set()
 
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+"""Load breast cancer dataset"""
 
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        DNN = DNN_keras[i][j]
+np.random.seed(0)        #create same seed for random number every time
 
-        train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
+cancer=load_breast_cancer()      #Download breast cancer dataset
 
-        
-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("$\lambda$")
+inputs=cancer.data                     #Feature matrix of 569 rows (samples) and 30 columns (parameters)
+outputs=cancer.target                  #Label array of 569 rows (0 for benign and 1 for malignant)
+labels=cancer.feature_names[0:30]
+
+print('The content of the breast cancer dataset is:')      #Print information about the datasets
+print(labels)
+print('-------------------------')
+print("inputs =  " + str(inputs.shape))
+print("outputs =  " + str(outputs.shape))
+print("labels =  "+ str(labels.shape))
+
+x=inputs      #Reassign the Feature and Label matrices to other variables
+y=outputs
+
+#%% 
+
+# Visualisation of dataset (for correlation analysis)
+
+plt.figure()
+plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean radius',fontweight='bold')
+plt.ylabel('Mean perimeter',fontweight='bold')
 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.figure()
+plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)
+plt.xlabel('Mean compactness',fontweight='bold')
+plt.ylabel('Mean concavity',fontweight='bold')
 plt.show()
+
+
+plt.figure()
+plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean radius',fontweight='bold')
+plt.ylabel('Mean texture',fontweight='bold')
+plt.show()
+
+plt.figure()
+plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
+plt.xlabel('Mean perimeter',fontweight='bold')
+plt.ylabel('Mean compactness',fontweight='bold')
+plt.show()
+
+
+# Generate training and testing datasets
+
+#Select features relevant to classification (texture,perimeter,compactness and symmetery) 
+#and add to input matrix
+
+temp1=np.reshape(x[:,1],(len(x[:,1]),1))
+temp2=np.reshape(x[:,2],(len(x[:,2]),1))
+X=np.hstack((temp1,temp2))      
+temp=np.reshape(x[:,5],(len(x[:,5]),1))
+X=np.hstack((X,temp))       
+temp=np.reshape(x[:,8],(len(x[:,8]),1))
+X=np.hstack((X,temp))       
+
+X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1)   #Split datasets into training and testing
+
+y_train=to_categorical(y_train)     #Convert labels to categorical when using categorical cross entropy
+y_test=to_categorical(y_test)
+
+del temp1,temp2,temp
+
+# %%
+
+# Define tunable parameters"
+
+eta=np.logspace(-3,-1,3)                    #Define vector of learning rates (parameter to SGD optimiser)
+lamda=0.01                                  #Define hyperparameter
+n_layers=2                                  #Define number of hidden layers in the model
+n_neuron=np.logspace(0,3,4,dtype=int)       #Define number of neurons per layer
+epochs=100                                   #Number of reiterations over the input data
+batch_size=100                              #Number of samples per gradient update
+
+# %%
+
+"""Define function to return Deep Neural Network model"""
+
+def NN_model(inputsize,n_layers,n_neuron,eta,lamda):
+    model=Sequential()      
+    for i in range(n_layers):       #Run loop to add hidden layers to the model
+        if (i==0):                  #First layer requires input dimensions
+            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))
+        else:                       #Subsequent layers are capable of automatic shape inferencing
+            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))
+    model.add(Dense(2,activation='softmax'))  #2 outputs - ordered and disordered (softmax for prob)
+    sgd=optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])
+    return model
+
+    
+Train_accuracy=np.zeros((len(n_neuron),len(eta)))      #Define matrices to store accuracy scores as a function
+Test_accuracy=np.zeros((len(n_neuron),len(eta)))       #of learning rate and number of hidden neurons for 
+
+for i in range(len(n_neuron)):     #run loops over hidden neurons and learning rates to calculate 
+    for j in range(len(eta)):      #accuracy scores 
+        DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)
+        DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)
+        Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]
+        Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]
+               
+
+def plot_data(x,y,data,title=None):
+
+    # plot results
+    fontsize=16
+
+
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)
+    
+    cbar=fig.colorbar(cax)
+    cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)
+    cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])
+    cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])
+
+    # put text on matrix elements
+    for i, x_val in enumerate(np.arange(len(x))):
+        for j, y_val in enumerate(np.arange(len(y))):
+            c = "${0:.1f}\\%$".format( 100*data[j,i])  
+            ax.text(x_val, y_val, c, va='center', ha='center')
+
+    # convert axis vaues to to string labels
+    x=[str(i) for i in x]
+    y=[str(i) for i in y]
+
+
+    ax.set_xticklabels(['']+x)
+    ax.set_yticklabels(['']+y)
+
+    ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize)
+    ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize)
+    if title is not None:
+        ax.set_title(title)
+
+    plt.tight_layout()
+
+    plt.show()
+    
+plot_data(eta,n_neuron,Train_accuracy, 'training')
+plot_data(eta,n_neuron,Test_accuracy, 'testing')
 
    
 
    
 
    
@@ -335,7 +405,7 @@ plt.show()
   
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diff --git a/doc/pub/week41/html/._week41-bs033.html b/doc/pub/week41/html/._week41-bs033.html
index fdb6f0693..f1eccedc3 100644
--- a/doc/pub/week41/html/._week41-bs033.html
+++ b/doc/pub/week41/html/._week41-bs033.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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         Contents 
         
       
    • 
@@ -208,179 +206,34 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    The Breast Cancer Data, now with Keras 
    
+
    Which activation function should I use? 
    
 
 
    
+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.
 
-
-
    import tensorflow as tf
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-import numpy as np
-import matplotlib.pyplot as plt
-import seaborn as sns
-from sklearn.model_selection import train_test_split as splitter
-from sklearn.datasets import load_breast_cancer
-import pickle
-import os 
+
    
+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.
 
+
    
+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
 
-"""Load breast cancer dataset"""
-
-np.random.seed(0)        #create same seed for random number every time
-
-cancer=load_breast_cancer()      #Download breast cancer dataset
-
-inputs=cancer.data                     #Feature matrix of 569 rows (samples) and 30 columns (parameters)
-outputs=cancer.target                  #Label array of 569 rows (0 for benign and 1 for malignant)
-labels=cancer.feature_names[0:30]
-
-print('The content of the breast cancer dataset is:')      #Print information about the datasets
-print(labels)
-print('-------------------------')
-print("inputs =  " + str(inputs.shape))
-print("outputs =  " + str(outputs.shape))
-print("labels =  "+ str(labels.shape))
-
-x=inputs      #Reassign the Feature and Label matrices to other variables
-y=outputs
-
-#%% 
-
-# Visualisation of dataset (for correlation analysis)
-
-plt.figure()
-plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean radius',fontweight='bold')
-plt.ylabel('Mean perimeter',fontweight='bold')
-plt.show()
-
-plt.figure()
-plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)
-plt.xlabel('Mean compactness',fontweight='bold')
-plt.ylabel('Mean concavity',fontweight='bold')
-plt.show()
-
-
-plt.figure()
-plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean radius',fontweight='bold')
-plt.ylabel('Mean texture',fontweight='bold')
-plt.show()
-
-plt.figure()
-plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean perimeter',fontweight='bold')
-plt.ylabel('Mean compactness',fontweight='bold')
-plt.show()
-
-
-# Generate training and testing datasets
-
-#Select features relevant to classification (texture,perimeter,compactness and symmetery) 
-#and add to input matrix
-
-temp1=np.reshape(x[:,1],(len(x[:,1]),1))
-temp2=np.reshape(x[:,2],(len(x[:,2]),1))
-X=np.hstack((temp1,temp2))      
-temp=np.reshape(x[:,5],(len(x[:,5]),1))
-X=np.hstack((X,temp))       
-temp=np.reshape(x[:,8],(len(x[:,8]),1))
-X=np.hstack((X,temp))       
-
-X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1)   #Split datasets into training and testing
-
-y_train=to_categorical(y_train)     #Convert labels to categorical when using categorical cross entropy
-y_test=to_categorical(y_test)
-
-del temp1,temp2,temp
-
-# %%
-
-# Define tunable parameters"
-
-eta=np.logspace(-3,-1,3)                    #Define vector of learning rates (parameter to SGD optimiser)
-lamda=0.01                                  #Define hyperparameter
-n_layers=2                                  #Define number of hidden layers in the model
-n_neuron=np.logspace(0,3,4,dtype=int)       #Define number of neurons per layer
-epochs=100                                   #Number of reiterations over the input data
-batch_size=100                              #Number of samples per gradient update
-
-# %%
-
-"""Define function to return Deep Neural Network model"""
-
-def NN_model(inputsize,n_layers,n_neuron,eta,lamda):
-    model=Sequential()      
-    for i in range(n_layers):       #Run loop to add hidden layers to the model
-        if (i==0):                  #First layer requires input dimensions
-            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))
-        else:                       #Subsequent layers are capable of automatic shape inferencing
-            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))
-    model.add(Dense(2,activation='softmax'))  #2 outputs - ordered and disordered (softmax for prob)
-    sgd=optimizers.SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])
-    return model
-
-    
-Train_accuracy=np.zeros((len(n_neuron),len(eta)))      #Define matrices to store accuracy scores as a function
-Test_accuracy=np.zeros((len(n_neuron),len(eta)))       #of learning rate and number of hidden neurons for 
-
-for i in range(len(n_neuron)):     #run loops over hidden neurons and learning rates to calculate 
-    for j in range(len(eta)):      #accuracy scores 
-        DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)
-        DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)
-        Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]
-        Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]
-               
-
-def plot_data(x,y,data,title=None):
-
-    # plot results
-    fontsize=16
-
-
-    fig = plt.figure()
-    ax = fig.add_subplot(111)
-    cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)
-    
-    cbar=fig.colorbar(cax)
-    cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)
-    cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])
-    cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])
-
-    # put text on matrix elements
-    for i, x_val in enumerate(np.arange(len(x))):
-        for j, y_val in enumerate(np.arange(len(y))):
-            c = "${0:.1f}\\%$".format( 100*data[j,i])  
-            ax.text(x_val, y_val, c, va='center', ha='center')
-
-    # convert axis vaues to to string labels
-    x=[str(i) for i in x]
-    y=[str(i) for i in y]
-
-
-    ax.set_xticklabels(['']+x)
-    ax.set_yticklabels(['']+y)
-
-    ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize)
-    ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize)
-    if title is not None:
-        ax.set_title(title)
-
-    plt.tight_layout()
-
-    plt.show()
-    
-plot_data(eta,n_neuron,Train_accuracy, 'training')
-plot_data(eta,n_neuron,Test_accuracy, 'testing')
-
    
 
    
 
    
 
@@ -407,7 +260,7 @@ plot_data(eta,n_neuron,Test_accuracy, 'testing&
   
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diff --git a/doc/pub/week41/html/._week41-bs034.html b/doc/pub/week41/html/._week41-bs034.html
index c1bf94a05..62f905f8b 100644
--- a/doc/pub/week41/html/._week41-bs034.html
+++ b/doc/pub/week41/html/._week41-bs034.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -210,31 +208,32 @@ MathJax.Hub.Config({
 
 
 
-
    Which activation function should I use? 
    
+
    Is the Logistic activation function (Sigmoid)  our choice? 
    
 
 
    
-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.
+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.
 
 
    
-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.
+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.
 
 
    
-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
+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).
 
 
    
 
    
@@ -262,7 +261,7 @@ learn at widely different speeds
   
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diff --git a/doc/pub/week41/html/._week41-bs035.html b/doc/pub/week41/html/._week41-bs035.html
index aa0b1c558..1b36d8f28 100644
--- a/doc/pub/week41/html/._week41-bs035.html
+++ b/doc/pub/week41/html/._week41-bs035.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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         Contents 
         
       
    • 
@@ -208,34 +206,40 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Is the Logistic activation function (Sigmoid)  our choice? 
    
+
    The derivative of the Logistic funtion 
    
 
 
    
-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.
+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.
 
 
    
-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.
+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.
 
 
    
-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).
+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).
 
 
    
 
    
@@ -263,7 +267,7 @@ better than the logistic function in deep networks).
   
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diff --git a/doc/pub/week41/html/._week41-bs036.html b/doc/pub/week41/html/._week41-bs036.html
index 8435b22c6..b0b85507f 100644
--- a/doc/pub/week41/html/._week41-bs036.html
+++ b/doc/pub/week41/html/._week41-bs036.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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    • 
@@ -210,38 +208,29 @@ MathJax.Hub.Config({
 
 
 
-
    The derivative of the Logistic funtion 
    
+
    The RELU function family 
    
 
 
    
-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.
+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.
 
 
    
-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.
+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.
 
 
    
-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).
+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
+
+$$
+ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\  z & z \ge 0.\end{array}\right. 
+$$
 
 
    
 
    
@@ -268,8 +257,6 @@ fast to compute).
   
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diff --git a/doc/pub/week41/html/._week41-bs037.html b/doc/pub/week41/html/._week41-bs037.html
index 1659fc38a..e7f838e61 100644
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+++ b/doc/pub/week41/html/._week41-bs037.html
@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
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-
    The RELU function family 
    
+
    Which activation function should we use? 
    
 
 
    
-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.
+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.
 
 
    
-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
-
-$$
-ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\  z & z \ge 0.\end{array}\right. 
-$$
+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.
 
 
    
 
    
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-
+
 
-
    Which activation function should we use? 
    
+
    A top-down perspective on Neural networks 
    
 
 
    
-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 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.
 
 
    
-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.
+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.
 
 
    
 
    
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diff --git a/doc/pub/week41/html/._week41-bs039.html b/doc/pub/week41/html/._week41-bs039.html
index 6a515a03b..ca8b4f3d2 100644
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@@ -208,46 +206,30 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    A top-down perspective on Neural networks 
    
+
    Limitations of supervised learning with deep networks 
    
 
 
    
-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:
+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. 
 
 
      
-
    •  Estimate optimal error rate
      • 
-
    •  Minimize underfitting (bias) on training data set.
      • 
-
    •  Make sure you are not overfitting.
      • 
+
    •  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.
      • 
 
    
 
-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.
+Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.
 
 
    
 
    
@@ -271,7 +253,6 @@ supervised learning.
   
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diff --git a/doc/pub/week41/html/._week41-bs040.html b/doc/pub/week41/html/._week41-bs040.html
index 983038dc1..150f15632 100644
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-
    Limitations of supervised learning with deep networks 
    
+
    Convolutional Neural Networks (recognizing images) 
    
 
 
    
-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).
+Convolutional neural networks (CNNs) were developed during the last
+decade of the previous century, with a focus on character recognition
+tasks. Nowadays, CNNs are a central element in the spectacular success
+of deep learning methods. The success in for example image
+classifications have made them a central tool for most machine
+learning practitioners.
 
 
    
-Here we list some of the important limitations of supervised neural network based models. 
+CNNs are very similar to ordinary Neural Networks.
+They are made up of neurons that have learnable weights and
+biases. Each neuron receives some inputs, performs a dot product and
+optionally follows it with a non-linearity. The whole network still
+expresses a single differentiable score function: from the raw image
+pixels on one end to class scores at the other. And they still have a
+loss function (for example Softmax) on the last (fully-connected) layer
+and all the tips/tricks we developed for learning regular Neural
+Networks still apply (back propagation, gradient descent etc etc).
 
-
      
-
    •  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.
      • 
-
    
+
    
+What is the difference? CNN architectures make the explicit assumption that
+the inputs are images, which allows us to encode certain properties
+into the architecture. These then make the forward function more
+efficient to implement and vastly reduce the amount of parameters in
+the network.
 
-Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.
+
    
+Here we provide only a superficial overview, for the more interested, we recommend highly the course
+IN5400 – Machine Learning for Image Analysis
+and the slides of CS231.
+
+
    
+Another good read is the article here https://arxiv.org/pdf/1603.07285.pdf.
 
 
    
 
    
@@ -254,7 +265,6 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
   
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diff --git a/doc/pub/week41/html/._week41-bs041.html b/doc/pub/week41/html/._week41-bs041.html
index aa478ca4a..a0c67248e 100644
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-
    Convolutional Neural Networks (recognizing images) 
    
+
    Regular NNs don’t scale well to full images 
    
 
 
    
-Convolutional neural networks (CNNs) were developed during the last
-decade of the previous century, with a focus on character recognition
-tasks. Nowadays, CNNs are a central element in the spectacular success
-of deep learning methods. The success in for example image
-classifications have made them a central tool for most machine
-learning practitioners.
+As an example, consider
+an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a
+single fully-connected neuron in a first hidden layer of a regular
+Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still
+seems manageable, but clearly this fully-connected structure does not
+scale to larger images. For example, an image of more respectable
+size, say \( 200\times 200\times 3 \), would lead to neurons that have 
+\( 200\times 200\times 3 = 120,000 \) weights.
 
 
    
-CNNs are very similar to ordinary Neural Networks.
-They are made up of neurons that have learnable weights and
-biases. Each neuron receives some inputs, performs a dot product and
-optionally follows it with a non-linearity. The whole network still
-expresses a single differentiable score function: from the raw image
-pixels on one end to class scores at the other. And they still have a
-loss function (for example Softmax) on the last (fully-connected) layer
-and all the tips/tricks we developed for learning regular Neural
-Networks still apply (back propagation, gradient descent etc etc).
+We could have
+several such neurons, and the parameters would add up quickly! Clearly,
+this full connectivity is wasteful and the huge number of parameters
+would quickly lead to possible overfitting.
 
 
    
-What is the difference? CNN architectures make the explicit assumption that
-the inputs are images, which allows us to encode certain properties
-into the architecture. These then make the forward function more
-efficient to implement and vastly reduce the amount of parameters in
-the network.
-
-
    
-Here we provide only a superficial overview, for the more interested, we recommend highly the course
-IN5400 – Machine Learning for Image Analysis
-and the slides of CS231.
-
-
    
-Another good read is the article here https://arxiv.org/pdf/1603.07285.pdf.
+
      
+
    
+
    Figure 1:  A regular 3-layer Neural Network. 
    
+
    
+
    
 
 
    
 
    
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-
    Regular NNs don’t scale well to full images 
    
+
    3D volumes of neurons 
    
 
 
    
-As an example, consider
-an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a
-single fully-connected neuron in a first hidden layer of a regular
-Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still
-seems manageable, but clearly this fully-connected structure does not
-scale to larger images. For example, an image of more respectable
-size, say \( 200\times 200\times 3 \), would lead to neurons that have 
-\( 200\times 200\times 3 = 120,000 \) weights.
+Convolutional Neural Networks take advantage of the fact that the
+input consists of images and they constrain the architecture in a more
+sensible way.
 
 
    
-We could have
-several such neurons, and the parameters would add up quickly! Clearly,
-this full connectivity is wasteful and the huge number of parameters
-would quickly lead to possible overfitting.
+In particular, unlike a regular Neural Network, the
+layers of a CNN have neurons arranged in 3 dimensions: width,
+height, depth. (Note that the word depth here refers to the third
+dimension of an activation volume, not to the depth of a full Neural
+Network, which can refer to the total number of layers in a network.)
+
+
    
+To understand it better, the above example of an image 
+with an input volume of
+activations has dimensions \( 32\times 32\times 3 \) (width, height,
+depth respectively).
+
+
    
+The neurons in a layer will
+only be connected to a small region of the layer before it, instead of
+all of the neurons in a fully-connected manner. Moreover, the final
+output layer could  for this specific image have dimensions \( 1\times 1 \times 10 \), 
+because by the
+end of the CNN architecture we will reduce the full image into a
+single vector of class scores, arranged along the depth
+dimension.
 
 
    
 
      
 
    
-
    Figure 1:  A regular 3-layer Neural Network. 
    
-
    
+
    Figure 2:  A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels). 
    
+
    
 
    
 
 
    
@@ -254,7 +264,6 @@ would quickly lead to possible overfitting.
   
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-
+
 
-
    3D volumes of neurons 
    
+
    Layers used to build CNNs 
    
 
 
    
-Convolutional Neural Networks take advantage of the fact that the
-input consists of images and they constrain the architecture in a more
-sensible way.
+A simple CNN is a sequence of layers, and every layer of a CNN
+transforms one volume of activations to another through a
+differentiable function. We use three main types of layers to build
+CNN architectures: Convolutional Layer, Pooling Layer, and
+Fully-Connected Layer (exactly as seen in regular Neural Networks). We
+will stack these layers to form a full CNN architecture.
 
 
    
-In particular, unlike a regular Neural Network, the
-layers of a CNN have neurons arranged in 3 dimensions: width,
-height, depth. (Note that the word depth here refers to the third
-dimension of an activation volume, not to the depth of a full Neural
-Network, which can refer to the total number of layers in a network.)
+A simple CNN for image classification could have the architecture:
 
-
    
-To understand it better, the above example of an image 
-with an input volume of
-activations has dimensions \( 32\times 32\times 3 \) (width, height,
-depth respectively).
+
      
+
    •  INPUT (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
      • 
+
    •  CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.
      • 
+
    •  RELU layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).
      • 
+
    •  POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).
      • 
+
    •  FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
      • 
+
    
 
-
    
-The neurons in a layer will
-only be connected to a small region of the layer before it, instead of
-all of the neurons in a fully-connected manner. Moreover, the final
-output layer could  for this specific image have dimensions \( 1\times 1 \times 10 \), 
-because by the
-end of the CNN architecture we will reduce the full image into a
-single vector of class scores, arranged along the depth
-dimension.
-
-
    
-
      
-
    
-
    Figure 2:  A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels). 
    
-
    
-
    
-
-
    
 
    
 
 
    
 
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-
+
 
-
    Layers used to build CNNs 
    
+
    Transforming images 
    
 
 
    
-A simple CNN is a sequence of layers, and every layer of a CNN
-transforms one volume of activations to another through a
-differentiable function. We use three main types of layers to build
-CNN architectures: Convolutional Layer, Pooling Layer, and
-Fully-Connected Layer (exactly as seen in regular Neural Networks). We
-will stack these layers to form a full CNN architecture.
+CNNs transform the original image layer by layer from the original
+pixel values to the final class scores.
 
 
    
-A simple CNN for image classification could have the architecture:
-
-
      
-
    •  INPUT (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
      • 
-
    •  CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.
      • 
-
    •  RELU layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).
      • 
-
    •  POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).
      • 
-
    •  FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
      • 
-
    
+Observe that some layers contain
+parameters and other don’t. In particular, the CNN layers perform
+transformations that are a function of not only the activations in the
+input volume, but also of the parameters (the weights and biases of
+the neurons). On the other hand, the RELU/POOL layers will implement a
+fixed function. The parameters in the CONV/FC layers will be trained
+with gradient descent so that the class scores that the CNN computes
+are consistent with the labels in the training set for each image.
 
+
    
 
    
 
 
      
@@ -247,7 +241,6 @@ A simple CNN for image classification could have the architecture:
   
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-
    Transforming images 
    
+
    CNNs in brief 
    
 
 
    
-CNNs transform the original image layer by layer from the original
-pixel values to the final class scores.
+In summary:
+
+
      
+
    •  A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
      • 
+
    •  There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)
      • 
+
    •  Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function
      • 
+
    •  Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)
      • 
+
    •  Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
      • 
+
    
+
+For more material on convolutional networks, we strongly recommend
+the course
+IN5400 – Machine Learning for Image Analysis
+and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs.
 
 
    
-Observe that some layers contain
-parameters and other don’t. In particular, the CNN layers perform
-transformations that are a function of not only the activations in the
-input volume, but also of the parameters (the weights and biases of
-the neurons). On the other hand, the RELU/POOL layers will implement a
-fixed function. The parameters in the CONV/FC layers will be trained
-with gradient descent so that the class scores that the CNN computes
-are consistent with the labels in the training set for each image.
 
-
    
 
    
 
 
      
@@ -242,8 +243,6 @@ are consistent with the labels in the training set for each image.
   
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diff --git a/doc/pub/week41/html/week41-bs.html b/doc/pub/week41/html/week41-bs.html
index 63ba71fc4..045b39a7d 100644
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@@ -41,77 +41,76 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -149,52 +148,51 @@ MathJax.Hub.Config({
       
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         Contents 
         
       
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@@ -229,7 +227,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Oct 6, 2020
     
+
    Oct 8, 2020
     
 
    
 
    
 
@@ -253,7 +251,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week41/html/week41-reveal.html b/doc/pub/week41/html/week41-reveal.html
index da63c4200..05e3442c3 100644
--- a/doc/pub/week41/html/week41-reveal.html
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@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
     
    
-
    Oct 6, 2020
     
+
    Oct 8, 2020
     
 
    
 
    
 
@@ -159,10 +159,10 @@ MathJax.Hub.Config({
 
 
 
    
-
    Plan for week 40 
    
+
    Plan for week 41 
    
 
 
      
-
    •  Thursday: Building our own Feed-forward Neural Network
      • 
+
    •  Thursday: Building our own Feed-forward Neural Network. Video of Lecture
      • 
 
    •  Friday: Playing around with our own Feed-forward Neural Network and introduction to TensorFlow. Start convolutional Neural Networks (CNN).
      • 
 
    
 
    
@@ -173,19 +173,10 @@ extbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapters 10-11 a
 
 
 
    
-
    Overview video for week 41 
    
+
    Setting up the Back propagation algorithm 
    
 
 
    
-"Overview Video, from Stochastic Gradient methods to Neural Networks":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20\
-/forelesningsvideoer/OverviewWeek41.mp4?vrtx=view-as-webpage"
-
    
-
-
-
    
-
    Setting up the Back propagation algorithm 
    
-
-
    
-The four equations  provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
+The four equations  derived last week provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
 
 
    
 
    
@@ -255,7 +246,7 @@ Here it is convenient to use stochastic gradient descent (see the examples below
 
 
 
    
-
    Setting up a Multi-layer perceptron model for classification  
    
+
    Setting up a Multi-layer perceptron model for classification  
    
 
 
    
 We are now gong to develop an example based on the MNIST data
@@ -297,7 +288,7 @@ of our network.
 
 
 
    
-
    Defining the cost function 
    
+
    Defining the cost function 
    
 
 
    
 Our cost function is given as (see the Logistic regression lectures)
@@ -365,7 +356,7 @@ The back propagation equations need now only a small change, namely the definiti
 
 
 
    
-
    Example: binary classification problem 
    
+
    Example: binary classification problem 
    
 
 
    
 As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \( \beta \) as
@@ -427,7 +418,7 @@ In case we use another activation function than the logistic one, we need to eva
 
 
 
    
-
    The Softmax function 
    
+
    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
 
     
    
 $$
@@ -455,7 +446,7 @@ 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:  
@@ -477,7 +468,7 @@ 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
@@ -577,7 +568,7 @@ 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.
@@ -627,7 +618,7 @@ 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
@@ -681,7 +672,7 @@ which is inspired by probability theory (see logistic regression) and was most c
 
 
 
    
-
    Layers 
    
+
    Layers