From afdd0edde7ecb0197175b3360304b6f3a8de3271 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Thu, 7 Oct 2021 08:39:04 +0200 Subject: [PATCH] update --- doc/pub/week40/html/._week40-bs000.html | 127 +++++----- doc/pub/week40/html/._week40-bs001.html | 125 +++++----- doc/pub/week40/html/._week40-bs002.html | 125 +++++----- doc/pub/week40/html/._week40-bs003.html | 146 ++++++------ doc/pub/week40/html/._week40-bs004.html | 159 +++++++------ doc/pub/week40/html/._week40-bs005.html | 151 ++++++------ doc/pub/week40/html/._week40-bs006.html | 144 ++++++------ doc/pub/week40/html/._week40-bs007.html | 167 ++++++------- doc/pub/week40/html/._week40-bs008.html | 149 ++++++------ doc/pub/week40/html/._week40-bs009.html | 165 ++++++------- doc/pub/week40/html/._week40-bs010.html | 206 ++++++---------- doc/pub/week40/html/._week40-bs011.html | 190 ++++++++------- doc/pub/week40/html/._week40-bs012.html | 208 ++++++++++------ doc/pub/week40/html/._week40-bs013.html | 192 +++++++-------- doc/pub/week40/html/._week40-bs014.html | 167 ++++++------- doc/pub/week40/html/._week40-bs015.html | 186 +++++++++------ doc/pub/week40/html/._week40-bs016.html | 186 +++++++-------- doc/pub/week40/html/._week40-bs017.html | 158 +++++++------ doc/pub/week40/html/._week40-bs018.html | 235 ++++++++----------- doc/pub/week40/html/._week40-bs019.html | 162 ++++++------- doc/pub/week40/html/._week40-bs020.html | 218 ++++++++++------- doc/pub/week40/html/._week40-bs021.html | 164 ++++++------- doc/pub/week40/html/._week40-bs022.html | 173 ++++++++------ doc/pub/week40/html/._week40-bs023.html | 156 ++++++------ doc/pub/week40/html/._week40-bs024.html | 169 ++++++------- doc/pub/week40/html/._week40-bs025.html | 160 ++++++------- doc/pub/week40/html/._week40-bs026.html | 167 +++++++------ doc/pub/week40/html/._week40-bs027.html | 177 +++++++------- doc/pub/week40/html/._week40-bs028.html | 150 +++++++----- doc/pub/week40/html/._week40-bs029.html | 164 ++++++++----- doc/pub/week40/html/._week40-bs030.html | 142 +++++------ doc/pub/week40/html/._week40-bs031.html | 183 ++++++--------- doc/pub/week40/html/._week40-bs032.html | 154 ++++++------ doc/pub/week40/html/._week40-bs033.html | 190 +++++++++------ doc/pub/week40/html/._week40-bs034.html | 162 +++++++------ doc/pub/week40/html/._week40-bs035.html | 148 ++++++------ doc/pub/week40/html/._week40-bs036.html | 156 ++++++------ doc/pub/week40/html/._week40-bs037.html | 144 ++++++------ doc/pub/week40/html/._week40-bs038.html | 148 ++++++------ doc/pub/week40/html/._week40-bs039.html | 139 ++++++----- doc/pub/week40/html/._week40-bs040.html | 147 ++++++------ doc/pub/week40/html/._week40-bs041.html | 172 ++++++-------- doc/pub/week40/html/._week40-bs042.html | 152 ++++++------ doc/pub/week40/html/._week40-bs043.html | 168 +++++++------ doc/pub/week40/html/._week40-bs044.html | 161 +++++++------ doc/pub/week40/html/._week40-bs045.html | 166 ++++++------- doc/pub/week40/html/._week40-bs046.html | 151 ++++++------ doc/pub/week40/html/._week40-bs047.html | 169 +++++++------ doc/pub/week40/html/._week40-bs048.html | 157 +++++++------ doc/pub/week40/html/._week40-bs049.html | 217 ++++++----------- doc/pub/week40/html/._week40-bs050.html | 168 +++++++------ doc/pub/week40/html/._week40-bs051.html | 222 ++++++++++++------ doc/pub/week40/html/._week40-bs052.html | 171 ++++++++------ doc/pub/week40/html/._week40-bs053.html | 171 +++++++------- doc/pub/week40/html/._week40-bs054.html | 156 ++++++------ doc/pub/week40/html/._week40-bs055.html | 160 +++++++------ doc/pub/week40/html/._week40-bs056.html | 173 ++++++-------- doc/pub/week40/html/._week40-bs057.html | 145 +++++++----- doc/pub/week40/html/._week40-bs058.html | 199 ++++++++-------- doc/pub/week40/html/._week40-bs059.html | 152 ++++++------ doc/pub/week40/html/._week40-bs060.html | 219 +++++++++-------- doc/pub/week40/html/week40-bs.html | 127 +++++----- doc/pub/week40/html/week40-reveal.html | 57 ++++- doc/pub/week40/html/week40-solarized.html | 63 ++++- doc/pub/week40/html/week40.html | 63 ++++- doc/pub/week40/ipynb/ipynb-week40-src.tar.gz | Bin 33808 -> 33808 bytes doc/pub/week40/ipynb/week40.ipynb | 45 +++- doc/src/week40/week40.do.txt | 44 +++- 68 files changed, 5622 insertions(+), 4985 deletions(-) diff --git a/doc/pub/week40/html/._week40-bs000.html b/doc/pub/week40/html/._week40-bs000.html index 1e16f8b76..a8251bd75 100644 --- a/doc/pub/week40/html/._week40-bs000.html +++ b/doc/pub/week40/html/._week40-bs000.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -319,7 +326,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA

-

Oct 5, 2021

+

Oct 7, 2021


@@ -343,7 +350,7 @@ MathJax.Hub.Config({

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    • diff --git a/doc/pub/week40/html/._week40-bs001.html b/doc/pub/week40/html/._week40-bs001.html index 5781d2a57..199a4b18c 100644 --- a/doc/pub/week40/html/._week40-bs001.html +++ b/doc/pub/week40/html/._week40-bs001.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -328,7 +335,7 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.
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    • diff --git a/doc/pub/week40/html/._week40-bs002.html b/doc/pub/week40/html/._week40-bs002.html index b0552d05c..25d431b93 100644 --- a/doc/pub/week40/html/._week40-bs002.html +++ b/doc/pub/week40/html/._week40-bs002.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -323,7 +330,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week40/html/._week40-bs003.html b/doc/pub/week40/html/._week40-bs003.html index 82665cfc0..2453cae7a 100644 --- a/doc/pub/week40/html/._week40-bs003.html +++ b/doc/pub/week40/html/._week40-bs003.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,20 +307,21 @@ MathJax.Hub.Config({ -

    Stochastic Gradient Descent

    +

    Batches and mini-batches

    -Stochastic gradient descent (SGD) and variants thereof address some of -the shortcomings of the Gradient descent method discussed above. +In gradient descent we compute the cost function and its gradient for all data points we have.

    -The underlying idea of SGD comes from the observation that the cost -function, which we want to minimize, can almost always be written as a -sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \), -$$ -C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, -\mathbf{\beta}). -$$ +In large-scale applications such as the ILSVRC challenge, the +training data can have on order of millions of examples. Hence, it +seems wasteful to compute the full cost function over the entire +training set in order to perform only a single parameter update. A +very common approach to addressing this challenge is to compute the +gradient over batches of the training data. For example, in current +a typical batch could contain some thousand examples from +an entire training set of several millions. This batch is then used to +perform a parameter update.

    @@ -334,7 +342,7 @@ $$

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    • diff --git a/doc/pub/week40/html/._week40-bs004.html b/doc/pub/week40/html/._week40-bs004.html index f8884d7f8..7bf10cca7 100644 --- a/doc/pub/week40/html/._week40-bs004.html +++ b/doc/pub/week40/html/._week40-bs004.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,22 +307,32 @@ MathJax.Hub.Config({ -

    Computation of gradients

    +

    Stochastic Gradient Descent (SGD)

    -This in turn means that the gradient can be -computed as a sum over \( i \)-gradients -$$ -\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}). -$$ +In stochastic gradient descent, the extreme case is the case where we +have only one batch, that is we include the whole data set.

    -Stochasticity/randomness is introduced by only taking the -gradient on a subset of the data called minibatches. If there are \( n \) -data points and the size of each minibatch is \( M \), there will be \( n/M \) -minibatches. We denote these minibatches by \( B_k \) where -\( k=1,\cdots,n/M \). +This process is called Stochastic Gradient +Descent (SGD) (or also sometimes on-line gradient descent). This is +relatively less common to see because in practice due to vectorized +code optimizations it can be computationally much more efficient to +evaluate the gradient for 100 examples, than the gradient for one +example 100 times. Even though SGD technically refers to using a +single example at a time to evaluate the gradient, you will hear +people use the term SGD even when referring to mini-batch gradient +descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +for “Batch gradient descent” are rare to see), where it is usually +assumed that mini-batches are used. The size of the mini-batch is a +hyperparameter but it is not very common to cross-validate or bootstrap it. It is +usually based on memory constraints (if any), or set to some value, +e.g. 32, 64 or 128. We use powers of 2 in practice because many +vectorized operation implementations work faster when their inputs are +sized in powers of 2. + +

    +In our notes with SGD we mean stochastic gradient descent with mini-batches.

    @@ -337,7 +354,7 @@ minibatches. We denote these minibatches by \( B_k \) where

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    • diff --git a/doc/pub/week40/html/._week40-bs005.html b/doc/pub/week40/html/._week40-bs005.html index 7bb96eaab..92cd6b45c 100644 --- a/doc/pub/week40/html/._week40-bs005.html +++ b/doc/pub/week40/html/._week40-bs005.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,25 +307,19 @@ MathJax.Hub.Config({ -

    SGD example

    -As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) -and we choose to have \( M=5 \) minibathces, -then each minibatch contains two data points. In particular we have -\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = -(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you -have only a single batch with all data points and on the other extreme, -you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e -\( B_k = \mathbf{x}_k \). +

    Stochastic Gradient Descent

    -The idea is now to approximate the gradient by replacing the sum over -all data points with a sum over the data points in one the minibatches -picked at random in each gradient descent step +Stochastic gradient descent (SGD) and variants thereof address some of +the shortcomings of the Gradient descent method discussed above. + +

    +The underlying idea of SGD comes from the observation that the cost +function, which we want to minimize, can almost always be written as a +sum over \( n \) data points \( \{\mathbf{x}_i\}_{i=1}^n \), $$ -\nabla_{\beta} -C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta -c_i(\mathbf{x}_i, \mathbf{\beta}). +C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, +\mathbf{\beta}). $$

    @@ -342,7 +343,7 @@ $$

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    • diff --git a/doc/pub/week40/html/._week40-bs006.html b/doc/pub/week40/html/._week40-bs006.html index e860abc6e..2effe4689 100644 --- a/doc/pub/week40/html/._week40-bs006.html +++ b/doc/pub/week40/html/._week40-bs006.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,21 +307,22 @@ MathJax.Hub.Config({ -

    The gradient step

    +

    Computation of gradients

    -Thus a gradient descent step now looks like +This in turn means that the gradient can be +computed as a sum over \( i \)-gradients $$ -\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) +\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}). $$

    -where \( k \) is picked at random with equal -probability from \( [1,n/M] \). An iteration over the number of -minibathces (n/M) is commonly referred to as an epoch. Thus it is -typical to choose a number of epochs and for each epoch iterate over -the number of minibatches, as exemplified in the code below. +Stochasticity/randomness is introduced by only taking the +gradient on a subset of the data called minibatches. If there are \( n \) +data points and the size of each minibatch is \( M \), there will be \( n/M \) +minibatches. We denote these minibatches by \( B_k \) where +\( k=1,\cdots,n/M \).

    @@ -338,7 +346,7 @@ the number of minibatches, as exemplified in the code below.

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    • diff --git a/doc/pub/week40/html/._week40-bs007.html b/doc/pub/week40/html/._week40-bs007.html index c42d8cb74..0d9689cf0 100644 --- a/doc/pub/week40/html/._week40-bs007.html +++ b/doc/pub/week40/html/._week40-bs007.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,34 +307,28 @@ MathJax.Hub.Config({ -

    Simple example code

    +

    SGD example

    +As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) +and we choose to have \( M=5 \) minibathces, +then each minibatch contains two data points. In particular we have +\( B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = +(\mathbf{x}_9,\mathbf{x}_{10}) \). Note that if you choose \( M=1 \) you +have only a single batch with all data points and on the other extreme, +you may choose \( M=n \) resulting in a minibatch for each datapoint, i.e +\( B_k = \mathbf{x}_k \). - -

    import numpy as np 
-
-n = 100 #100 datapoints 
-M = 5   #size of each minibatch
-m = int(n/M) #number of minibatches
-n_epochs = 10 #number of epochs
-
-j = 0
-for epoch in range(1,n_epochs+1):
-    for i in range(m):
-        k = np.random.randint(m) #Pick the k-th minibatch at random
-        #Compute the gradient using the data in minibatch Bk
-        #Compute new suggestion for 
-        j += 1
-

    -Taking the gradient only on a subset of the data has two important -benefits. First, it introduces randomness which decreases the chance -that our opmization scheme gets stuck in a local minima. Second, if -the size of the minibatches are small relative to the number of -datapoints (\( M < n \)), the computation of the gradient is much -cheaper since we sum over the datapoints in the \( k-th \) minibatch and not -all \( n \) datapoints. +The idea is now to approximate the gradient by replacing the sum over +all data points with a sum over the data points in one the minibatches +picked at random in each gradient descent step +$$ +\nabla_{\beta} +C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta +c_i(\mathbf{x}_i, \mathbf{\beta}). +$$

    @@ -352,7 +353,7 @@ all \( n \) datapoints.

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    • diff --git a/doc/pub/week40/html/._week40-bs008.html b/doc/pub/week40/html/._week40-bs008.html index 487f9dc8a..462f59083 100644 --- a/doc/pub/week40/html/._week40-bs008.html +++ b/doc/pub/week40/html/._week40-bs008.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,19 +307,21 @@ MathJax.Hub.Config({ -

    When do we stop?

    +

    The gradient step

    -A natural question is when do we stop the search for a new minimum? -One possibility is to compute the full gradient after a given number -of epochs and check if the norm of the gradient is smaller than some -threshold and stop if true. However, the condition that the gradient -is zero is valid also for local minima, so this would only tell us -that we are close to a local/global minimum. However, we could also -evaluate the cost function at this point, store the result and -continue the search. If the test kicks in at a later stage we can -compare the values of the cost function and keep the \( \beta \) that -gave the lowest value. +Thus a gradient descent step now looks like +$$ +\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, +\mathbf{\beta}) +$$ + +

    +where \( k \) is picked at random with equal +probability from \( [1,n/M] \). An iteration over the number of +minibathces (n/M) is commonly referred to as an epoch. Thus it is +typical to choose a number of epochs and for each epoch iterate over +the number of minibatches, as exemplified in the code below.

    @@ -338,7 +347,7 @@ gave the lowest value.

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    • diff --git a/doc/pub/week40/html/._week40-bs009.html b/doc/pub/week40/html/._week40-bs009.html index 331e91e2b..027146370 100644 --- a/doc/pub/week40/html/._week40-bs009.html +++ b/doc/pub/week40/html/._week40-bs009.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,51 +307,35 @@ MathJax.Hub.Config({ -

    Slightly different approach

    - -

    -Another approach is to let the step length \( \gamma_j \) depend on the -number of epochs in such a way that it becomes very small after a -reasonable time such that we do not move at all. - -

    -As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in time \( t \). - -

    -In this way we can fix the number of epochs, compute \( \beta \) and -evaluate the cost function at the end. Repeating the computation will -give a different result since the scheme is random by design. Then we -pick the final \( \beta \) that gives the lowest value of the cost -function. +

    Simple example code 

    import numpy as np 
 
-def step_length(t,t0,t1):
-    return t0/(t+t1)
-
 n = 100 #100 datapoints 
 M = 5   #size of each minibatch
 m = int(n/M) #number of minibatches
-n_epochs = 500 #number of epochs
-t0 = 1.0
-t1 = 10
+n_epochs = 10 #number of epochs
 
-gamma_j = t0/t1
 j = 0
 for epoch in range(1,n_epochs+1):
     for i in range(m):
         k = np.random.randint(m) #Pick the k-th minibatch at random
         #Compute the gradient using the data in minibatch Bk
-        #Compute new suggestion for beta
-        t = epoch*m+i
-        gamma_j = step_length(t,t0,t1)
+        #Compute new suggestion for 
         j += 1
-
-print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
    +

    +Taking the gradient only on a subset of the data has two important +benefits. First, it introduces randomness which decreases the chance +that our opmization scheme gets stuck in a local minima. Second, if +the size of the minibatches are small relative to the number of +datapoints (\( M < n \)), the computation of the gradient is much +cheaper since we sum over the datapoints in the \( k-th \) minibatch and not +all \( n \) datapoints. +

    @@ -370,7 +361,7 @@ j = 0

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    Program for stochastic gradient

    +

    When do we stop?

    - - -

    # Importing various packages
-from math import exp, sqrt
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
-
-m = 100
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
-
-X = np.c_[np.ones((m,1)), x]
-theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
-print("Own inversion")
-print(theta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print("sgdreg from scikit")
-print(sgdreg.intercept_, sgdreg.coef_)
-
-
-theta = np.random.randn(2,1)
-eta = 0.1
-Niterations = 1000
-
-
-for iter in range(Niterations):
-    gradients = 2.0/m*X.T @ ((X @ theta)-y)
-    theta -= eta*gradients
-print("theta from own gd")
-print(theta)
-
-xnew = np.array([[0],[2]])
-Xnew = np.c_[np.ones((2,1)), xnew]
-ypredict = Xnew.dot(theta)
-ypredict2 = Xnew.dot(theta_linreg)
-
-
-n_epochs = 50
-t0, t1 = 5, 50
-def learning_schedule(t):
-    return t0/(t+t1)
-
-theta = np.random.randn(2,1)
-
-for epoch in range(n_epochs):
-    for i in range(m):
-        random_index = np.random.randint(m)
-        xi = X[random_index:random_index+1]
-        yi = y[random_index:random_index+1]
-        gradients = 2 * xi.T @ ((xi @ theta)-yi)
-        eta = learning_schedule(epoch*m+i)
-        theta = theta - eta*gradients
-print("theta from own sdg")
-print(theta)
-
-plt.plot(xnew, ypredict, "r-")
-plt.plot(xnew, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Random numbers ')
-plt.show()
-
    -

    -Challenge: try to write a similar code for a Logistic Regression case. +A natural question is when do we stop the search for a new minimum? +One possibility is to compute the full gradient after a given number +of epochs and check if the norm of the gradient is smaller than some +threshold and stop if true. However, the condition that the gradient +is zero is valid also for local minima, so this would only tell us +that we are close to a local/global minimum. However, we could also +evaluate the cost function at this point, store the result and +continue the search. If the test kicks in at a later stage we can +compare the values of the cost function and keep the \( \beta \) that +gave the lowest value.

    @@ -399,7 +347,7 @@ plt.show()

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    • diff --git a/doc/pub/week40/html/._week40-bs011.html b/doc/pub/week40/html/._week40-bs011.html index b6a1b2698..1e0c6c9d1 100644 --- a/doc/pub/week40/html/._week40-bs011.html +++ b/doc/pub/week40/html/._week40-bs011.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,40 +307,51 @@ MathJax.Hub.Config({ -

    Momentum based GD

    +

    Slightly different approach

    -The stochastic gradient descent (SGD) is almost always used with a -momentum or inertia term that serves as a memory of the direction we -are moving in parameter space. This is typically implemented as -follows - -$$ -\begin{align} -\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ -\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, -\tag{1} -\end{align} -$$ +Another approach is to let the step length \( \gamma_j \) depend on the +number of epochs in such a way that it becomes very small after a +reasonable time such that we do not move at all.

    -where we have introduced a momentum parameter \( \gamma \), with -\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to -indicate the gradient is to be taken over a different mini-batch at -each step. We call this algorithm gradient descent with momentum -(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a -running average of recently encountered gradients and -\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory -used in the averaging procedure. Consistent with this, when -\( \gamma=0 \), this just reduces down to ordinary SGD as discussed -earlier. An equivalent way of writing the updates is +As an example, let \( e = 0,1,2,3,\cdots \) denote the current epoch and let \( t_0, t_1 > 0 \) be two fixed numbers. Furthermore, let \( t = e \cdot m + i \) where \( m \) is the number of minibatches and \( i=0,\cdots,m-1 \). Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length \( \gamma_j (0; t_0, t_1) = t_0/t_1 \) which decays in time \( t \). -$$ -\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t), -$$ +

    +In this way we can fix the number of epochs, compute \( \beta \) and +evaluate the cost function at the end. Repeating the computation will +give a different result since the scheme is random by design. Then we +pick the final \( \beta \) that gives the lowest value of the cost +function. -where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \). +

    + +

    import numpy as np 
+
+def step_length(t,t0,t1):
+    return t0/(t+t1)
+
+n = 100 #100 datapoints 
+M = 5   #size of each minibatch
+m = int(n/M) #number of minibatches
+n_epochs = 500 #number of epochs
+t0 = 1.0
+t1 = 10
+
+gamma_j = t0/t1
+j = 0
+for epoch in range(1,n_epochs+1):
+    for i in range(m):
+        k = np.random.randint(m) #Pick the k-th minibatch at random
+        #Compute the gradient using the data in minibatch Bk
+        #Compute new suggestion for beta
+        t = epoch*m+i
+        gamma_j = step_length(t,t0,t1)
+        j += 1
+
+print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
+

    @@ -360,7 +378,7 @@ where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\

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    • diff --git a/doc/pub/week40/html/._week40-bs012.html b/doc/pub/week40/html/._week40-bs012.html index fda3a1c84..d9ee67aa9 100644 --- a/doc/pub/week40/html/._week40-bs012.html +++ b/doc/pub/week40/html/._week40-bs012.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,33 +307,78 @@ MathJax.Hub.Config({ -

    More on momentum based approaches

    +

    Program for stochastic gradient

    -Let us try to get more intuition from these equations. It is helpful -to consider a simple physical analogy with a particle of mass \( m \) -moving in a viscous medium with drag coefficient \( \mu \) and potential -\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \), -then its motion is described by -$$ -m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). -$$ + +

    # Importing various packages
+from math import exp, sqrt
+from random import random, seed
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import SGDRegressor
 
-
    
-We can discretize this equation in the usual way to get
+m = 100
+x = 2*np.random.rand(m,1)
+y = 4+3*x+np.random.randn(m,1)
 
-$$
-m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}).
-$$
+X = np.c_[np.ones((m,1)), x]
+theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
+sgdreg.fit(x,y.ravel())
+print("sgdreg from scikit")
+print(sgdreg.intercept_, sgdreg.coef_)
 
-
    
-Rearranging this equation, we can rewrite this as
 
-$$
-\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t.
-$$
+theta = np.random.randn(2,1)
+eta = 0.1
+Niterations = 1000
 
+
+for iter in range(Niterations):
+    gradients = 2.0/m*X.T @ ((X @ theta)-y)
+    theta -= eta*gradients
+print("theta from own gd")
+print(theta)
+
+xnew = np.array([[0],[2]])
+Xnew = np.c_[np.ones((2,1)), xnew]
+ypredict = Xnew.dot(theta)
+ypredict2 = Xnew.dot(theta_linreg)
+
+
+n_epochs = 50
+M = 10   #size of each minibatch
+m = int(n/M) #number of minibatches
+t0, t1 = 5, 50
+def learning_schedule(t):
+    return t0/(t+t1)
+
+theta = np.random.randn(2,1)
+
+for epoch in range(n_epochs):
+    for i in range(m):
+        random_index = np.random.randint(m)
+        xi = X[random_index:random_index+1]
+        yi = y[random_index:random_index+1]
+        gradients = 2 * xi.T @ ((xi @ theta)-yi)
+        eta = learning_schedule(epoch*m+i)
+        theta = theta - eta*gradients
+print("theta from own sdg")
+print(theta)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(xnew, ypredict2, "b-")
+plt.plot(x, y ,'ro')
+plt.axis([0,2.0,0, 15.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Random numbers ')
+plt.show()
+

    @@ -353,7 +405,7 @@ $$

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    • diff --git a/doc/pub/week40/html/._week40-bs013.html b/doc/pub/week40/html/._week40-bs013.html index 1a69bd13e..1567ee16f 100644 --- a/doc/pub/week40/html/._week40-bs013.html +++ b/doc/pub/week40/html/._week40-bs013.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,58 +307,39 @@ MathJax.Hub.Config({ -

    Momentum parameter

    +

    Momentum based GD

    -Notice that this equation is identical to previous one if we identify -the position of the particle, \( \mathbf{w} \), with the parameters -\( \boldsymbol{\theta} \). This allows us to identify the momentum -parameter and learning rate with the mass of the particle and the -viscous drag as: - -$$ -\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}. -$$ - -

    -Thus, as the name suggests, the momentum parameter is proportional to -the mass of the particle and effectively provides inertia. -Furthermore, in the large viscosity/small learning rate limit, our -memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \). - -

    -Why is momentum useful? SGD momentum helps the gradient descent -algorithm gain speed in directions with persistent but small gradients -even in the presence of stochasticity, while suppressing oscillations -in high-curvature directions. This becomes especially important in -situations where the landscape is shallow and flat in some directions -and narrow and steep in others. It has been argued that first-order -methods (with appropriate initial conditions) can perform comparable -to more expensive second order methods, especially in the context of -complex deep learning models. - -

    -These beneficial properties of momentum can sometimes become even more -pronounced by using a slight modification of the classical momentum -algorithm called Nesterov Accelerated Gradient (NAG). - -

    -In the NAG algorithm, rather than calculating the gradient at the -current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one -calculates the gradient at the expected value of the parameters given -our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma -\mathbf{v}_{t-1}) \). This yields the NAG update rule +The stochastic gradient descent (SGD) is almost always used with a +momentum or inertia term that serves as a memory of the direction we +are moving in parameter space. This is typically implemented as +follows $$ \begin{align} -\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ -\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. -\tag{2} +\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t) \nonumber \\ +\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}, +\tag{1} \end{align} $$

    -One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \). +where we have introduced a momentum parameter \( \gamma \), with +\( 0\le\gamma\le 1 \), and for brevity we dropped the explicit notation to +indicate the gradient is to be taken over a different mini-batch at +each step. We call this algorithm gradient descent with momentum +(GDM). From these equations, it is clear that \( \mathbf{v}_t \) is a +running average of recently encountered gradients and +\( (1-\gamma)^{-1} \) sets the characteristic time scale for the memory +used in the averaging procedure. Consistent with this, when +\( \gamma=0 \), this just reduces down to ordinary SGD as discussed +earlier. An equivalent way of writing the updates is + +$$ +\Delta \boldsymbol{\theta}_{t+1} = \gamma \Delta \boldsymbol{\theta}_t -\ \eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t), +$$ + +where we have defined \( \Delta \boldsymbol{\theta}_{t}= \boldsymbol{\theta}_t-\boldsymbol{\theta}_{t-1} \).

    @@ -379,7 +367,7 @@ One of the major advantages of NAG is that it allows for the use of a larger lea

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    • diff --git a/doc/pub/week40/html/._week40-bs014.html b/doc/pub/week40/html/._week40-bs014.html index 17f1d5113..8cedc10d4 100644 --- a/doc/pub/week40/html/._week40-bs014.html +++ b/doc/pub/week40/html/._week40-bs014.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,30 +307,32 @@ MathJax.Hub.Config({ -

    Second moment of the gradient

    +

    More on momentum based approaches

    -In stochastic gradient descent, with and without momentum, we still -have to specify a schedule for tuning the learning rates \( \eta_t \) -as a function of time. As discussed in the context of Newton's -method, this presents a number of dilemmas. The learning rate is -limited by the steepest direction which can change depending on the -current position in the landscape. To circumvent this problem, ideally -our algorithm would keep track of curvature and take large steps in -shallow, flat directions and small steps in steep, narrow directions. -Second-order methods accomplish this by calculating or approximating -the Hessian and normalizing the learning rate by the -curvature. However, this is very computationally expensive for -extremely large models. Ideally, we would like to be able to -adaptively change the step size to match the landscape without paying -the steep computational price of calculating or approximating -Hessians. +Let us try to get more intuition from these equations. It is helpful +to consider a simple physical analogy with a particle of mass \( m \) +moving in a viscous medium with drag coefficient \( \mu \) and potential +\( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \), +then its motion is described by + +$$ +m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). +$$

    -Recently, a number of methods have been introduced that accomplish -this by tracking not only the gradient, but also the second moment of -the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and -ADAM. +We can discretize this equation in the usual way to get + +$$ +m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}). +$$ + +

    +Rearranging this equation, we can rewrite this as + +$$ +\Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t. +$$

    @@ -351,7 +360,7 @@ ADAM.

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    • diff --git a/doc/pub/week40/html/._week40-bs015.html b/doc/pub/week40/html/._week40-bs015.html index 7fd1663fe..fd52c4012 100644 --- a/doc/pub/week40/html/._week40-bs015.html +++ b/doc/pub/week40/html/._week40-bs015.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,33 +307,58 @@ MathJax.Hub.Config({ -

    RMS prop

    +

    Momentum parameter

    -In RMS prop, in addition to keeping a running average of the first -moment of the gradient, we also keep track of the second moment -denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule -for RMS prop is given by +Notice that this equation is identical to previous one if we identify +the position of the particle, \( \mathbf{w} \), with the parameters +\( \boldsymbol{\theta} \). This allows us to identify the momentum +parameter and learning rate with the mass of the particle and the +viscous drag as: + +$$ +\gamma= {m \over m +\mu \Delta t }, \qquad \eta = {(\Delta t)^2 \over m +\mu \Delta t}. +$$ + +

    +Thus, as the name suggests, the momentum parameter is proportional to +the mass of the particle and effectively provides inertia. +Furthermore, in the large viscosity/small learning rate limit, our +memory time scales as \( (1-\gamma)^{-1} \approx m/(\mu \Delta t) \). + +

    +Why is momentum useful? SGD momentum helps the gradient descent +algorithm gain speed in directions with persistent but small gradients +even in the presence of stochasticity, while suppressing oscillations +in high-curvature directions. This becomes especially important in +situations where the landscape is shallow and flat in some directions +and narrow and steep in others. It has been argued that first-order +methods (with appropriate initial conditions) can perform comparable +to more expensive second order methods, especially in the context of +complex deep learning models. + +

    +These beneficial properties of momentum can sometimes become even more +pronounced by using a slight modification of the classical momentum +algorithm called Nesterov Accelerated Gradient (NAG). + +

    +In the NAG algorithm, rather than calculating the gradient at the +current parameters, \( \nabla_\theta E(\boldsymbol{\theta}_t) \), one +calculates the gradient at the expected value of the parameters given +our current momentum, \( \nabla_\theta E(\boldsymbol{\theta}_t +\gamma +\mathbf{v}_{t-1}) \). This yields the NAG update rule $$ \begin{align} -\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) -\tag{3}\\ -\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\ -\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber +\mathbf{v}_{t}&=\gamma \mathbf{v}_{t-1}+\eta_{t}\nabla_\theta E(\boldsymbol{\theta}_t +\gamma \mathbf{v}_{t-1}) \nonumber \\ +\boldsymbol{\theta}_{t+1}&= \boldsymbol{\theta}_t -\mathbf{v}_{t}. +\tag{2} \end{align} $$

    -where \( \beta \) controls the averaging time of the second moment and is -typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate -typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8} \) is a -small regularization constant to prevent divergences. Multiplication -and division by vectors is understood as an element-wise operation. It -is clear from this formula that the learning rate is reduced in -directions where the norm of the gradient is consistently large. This -greatly speeds up the convergence by allowing us to use a larger -learning rate for flat directions. +One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of \( \gamma \).

    @@ -354,7 +386,7 @@ learning rate for flat directions.

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    • diff --git a/doc/pub/week40/html/._week40-bs016.html b/doc/pub/week40/html/._week40-bs016.html index 803dc2f30..a66b6dcbf 100644 --- a/doc/pub/week40/html/._week40-bs016.html +++ b/doc/pub/week40/html/._week40-bs016.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,51 +307,30 @@ MathJax.Hub.Config({ -

    ADAM optimizer

    +

    Second moment of the gradient

    -A related algorithm is the ADAM optimizer. In ADAM, we keep a running -average of both the first and second moment of the gradient and use -this information to adaptively change the learning rate for different -parameters. In addition to keeping a running average of the first and -second moments of the gradient -(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and -\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM -performs an additional bias correction to account for the fact that we -are estimating the first two moments of the gradient using a running -average (denoted by the hats in the update rule below). The update -rule for ADAM is given by (where multiplication and division are once -again understood to be element-wise operations below) - -$$ -\begin{align} -\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) -\tag{4}\\ -\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\ -\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\ -\boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\ -\boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\ -\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\ -\tag{5} -\end{align} -$$ +In stochastic gradient descent, with and without momentum, we still +have to specify a schedule for tuning the learning rates \( \eta_t \) +as a function of time. As discussed in the context of Newton's +method, this presents a number of dilemmas. The learning rate is +limited by the steepest direction which can change depending on the +current position in the landscape. To circumvent this problem, ideally +our algorithm would keep track of curvature and take large steps in +shallow, flat directions and small steps in steep, narrow directions. +Second-order methods accomplish this by calculating or approximating +the Hessian and normalizing the learning rate by the +curvature. However, this is very computationally expensive for +extremely large models. Ideally, we would like to be able to +adaptively change the step size to match the landscape without paying +the steep computational price of calculating or approximating +Hessians.

    -where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and -second moment and are typically taken to be \( 0.9 \) and \( 0.99 \) -respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop. - -

    -Like in RMSprop, the effective step size of a parameter depends on the -magnitude of its gradient squared. To understand this better, let us -rewrite this expression in terms of the variance -\( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t - -(\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The -update rule for this parameter is given by - -$$ -\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. -$$ +Recently, a number of methods have been introduced that accomplish +this by tracking not only the gradient, but also the second moment of +the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and +ADAM.

    @@ -372,7 +358,7 @@ $$

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    • diff --git a/doc/pub/week40/html/._week40-bs017.html b/doc/pub/week40/html/._week40-bs017.html index 1fda3c44e..10b615900 100644 --- a/doc/pub/week40/html/._week40-bs017.html +++ b/doc/pub/week40/html/._week40-bs017.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,16 +307,33 @@ MathJax.Hub.Config({ -

    Practical tips

    +

    RMS prop

    - +

    +In RMS prop, in addition to keeping a running average of the first +moment of the gradient, we also keep track of the second moment +denoted by \( \mathbf{s}_t=\mathbb{E}[\mathbf{g}_t^2] \). The update rule +for RMS prop is given by -Geron's text, see chapter 11, has several interesting discussions. +$$ +\begin{align} +\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) +\tag{3}\\ +\mathbf{s}_t &=\beta \mathbf{s}_{t-1} +(1-\beta)\mathbf{g}_t^2 \nonumber \\ +\boldsymbol{\theta}_{t+1}&=&\boldsymbol{\theta}_t - \eta_t { \mathbf{g}_t \over \sqrt{\mathbf{s}_t +\epsilon}}, \nonumber +\end{align} +$$ + +

    +where \( \beta \) controls the averaging time of the second moment and is +typically taken to be about \( \beta=0.9 \), \( \eta_t \) is a learning rate +typically chosen to be \( 10^{-3} \), and \( \epsilon\sim 10^{-8} \) is a +small regularization constant to prevent divergences. Multiplication +and division by vectors is understood as an element-wise operation. It +is clear from this formula that the learning rate is reduced in +directions where the norm of the gradient is consistently large. This +greatly speeds up the convergence by allowing us to use a larger +learning rate for flat directions.

    @@ -337,7 +361,7 @@ Geron's text, see chapter 11, has several interesting discussions.

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    • diff --git a/doc/pub/week40/html/._week40-bs018.html b/doc/pub/week40/html/._week40-bs018.html index f04c7fa9f..fcaba956b 100644 --- a/doc/pub/week40/html/._week40-bs018.html +++ b/doc/pub/week40/html/._week40-bs018.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,88 +307,52 @@ MathJax.Hub.Config({ -

    Automatic differentiation

    +

    ADAM optimizer

    -Automatic differentiation (AD), -also called algorithmic -differentiation or computational differentiation,is a set of -techniques to numerically evaluate the derivative of a function -specified by a computer program. AD exploits the fact that every -computer program, no matter how complicated, executes a sequence of -elementary arithmetic operations (addition, subtraction, -multiplication, division, etc.) and elementary functions (exp, log, -sin, cos, etc.). By applying the chain rule repeatedly to these -operations, derivatives of arbitrary order can be computed -automatically, accurately to working precision, and using at most a -small constant factor more arithmetic operations than the original -program. +A related algorithm is the ADAM optimizer. In ADAM, we keep a running +average of both the first and second moment of the gradient and use +this information to adaptively change the learning rate for different +parameters. In addition to keeping a running average of the first and +second moments of the gradient +(i.e. \( \mathbf{m}_t=\mathbb{E}[\mathbf{g}_t] \) and +\( \mathbf{s}_t=\mathbb{E}[\mathbf{g}^2_t] \), respectively), ADAM +performs an additional bias correction to account for the fact that we +are estimating the first two moments of the gradient using a running +average (denoted by the hats in the update rule below). The update +rule for ADAM is given by (where multiplication and division are once +again understood to be element-wise operations below) + +$$ +\begin{align} +\mathbf{g}_t &= \nabla_\theta E(\boldsymbol{\theta}) +\tag{4}\\ +\mathbf{m}_t &= \beta_1 \mathbf{m}_{t-1} + (1-\beta_1) \mathbf{g}_t \nonumber \\ +\mathbf{s}_t &=\beta_2 \mathbf{s}_{t-1} +(1-\beta_2)\mathbf{g}_t^2 \nonumber \\ +\boldsymbol{\mathbf{m}}_t&={\mathbf{m}_t \over 1-\beta_1^t} \nonumber \\ +\boldsymbol{\mathbf{s}}_t &={\mathbf{s}_t \over1-\beta_2^t} \nonumber \\ +\boldsymbol{\theta}_{t+1}&=\boldsymbol{\theta}_t - \eta_t { \boldsymbol{\mathbf{m}}_t \over \sqrt{\boldsymbol{\mathbf{s}}_t} +\epsilon}, \nonumber \\ +\tag{5} +\end{align} +$$

    -Automatic differentiation is neither: - -

    - -Symbolic differentiation can lead to inefficient code and faces the -difficulty of converting a computer program into a single expression, -while numerical differentiation can introduce round-off errors in the -discretization process and cancellation +where \( \beta_1 \) and \( \beta_2 \) set the memory lifetime of the first and +second moment and are typically taken to be \( 0.9 \) and \( 0.99 \) +respectively, and \( \eta \) and \( \epsilon \) are identical to RMSprop.

    -Python has tools for so-called automatic differentiation. -Consider the following example +Like in RMSprop, the effective step size of a parameter depends on the +magnitude of its gradient squared. To understand this better, let us +rewrite this expression in terms of the variance +\( \boldsymbol{\sigma}_t^2 = \boldsymbol{\mathbf{s}}_t - +(\boldsymbol{\mathbf{m}}_t)^2 \). Consider a single parameter \( \theta_t \). The +update rule for this parameter is given by + $$ -f(x) = \sin\left(2\pi x + x^2\right) +\Delta \theta_{t+1}= -\eta_t { \boldsymbol{m}_t \over \sqrt{\sigma_t^2 + m_t^2 }+\epsilon}. $$ -which has the following derivative -$$ -f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) -$$ - -Using autograd we have - -

    - - -

    import autograd.numpy as np
-
-# To do elementwise differentiation:
-from autograd import elementwise_grad as egrad 
-
-# To plot:
-import matplotlib.pyplot as plt 
-
-
-def f(x):
-    return np.sin(2*np.pi*x + x**2)
-
-def f_grad_analytic(x):
-    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)
-
-# Do the comparison:
-x = np.linspace(0,1,1000)
-
-f_grad = egrad(f)
-
-computed = f_grad(x)
-analytic = f_grad_analytic(x)
-
-plt.title('Derivative computed from Autograd compared with the analytical derivative')
-plt.plot(x,computed,label='autograd')
-plt.plot(x,analytic,label='analytic')
-
-plt.xlabel('x')
-plt.ylabel('y')
-plt.legend()
-
-plt.show()
-
-print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))
-

    @@ -408,7 +379,7 @@ plt.show()

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    • diff --git a/doc/pub/week40/html/._week40-bs019.html b/doc/pub/week40/html/._week40-bs019.html index 084ba787e..f1f14c2c6 100644 --- a/doc/pub/week40/html/._week40-bs019.html +++ b/doc/pub/week40/html/._week40-bs019.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -298,38 +305,19 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Using autograd

    +

    Practical tips

    -

    -Here we -experiment with what kind of functions Autograd is capable -of finding the gradient of. The following Python functions are just -meant to illustrate what Autograd can do, but please feel free to -experiment with other, possibly more complicated, functions as well. +

    -

    +Geron's text, see chapter 11, has several interesting discussions. - -

    import autograd.numpy as np
-from autograd import grad
-
-def f1(x):
-    return x**3 + 1
-
-f1_grad = grad(f1)
-
-# Remember to send in float as argument to the computed gradient from Autograd!
-a = 1.0
-
-# See the evaluated gradient at a using autograd:
-print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
-
-# Compare with the analytical derivative, that is f1'(x) = 3*x**2 
-grad_analytical = 3*a**2
-print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
-

    @@ -356,7 +344,7 @@ grad_analytical = 28

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    • diff --git a/doc/pub/week40/html/._week40-bs020.html b/doc/pub/week40/html/._week40-bs020.html index 17c8b2e5a..70733fdee 100644 --- a/doc/pub/week40/html/._week40-bs020.html +++ b/doc/pub/week40/html/._week40-bs020.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,53 +307,88 @@ MathJax.Hub.Config({ -

    Autograd with more complicated functions

    +

    Automatic differentiation

    -To differentiate with respect to two (or more) arguments of a Python -function, Autograd need to know at which variable the function if -being differentiated with respect to. +Automatic differentiation (AD), +also called algorithmic +differentiation or computational differentiation,is a set of +techniques to numerically evaluate the derivative of a function +specified by a computer program. AD exploits the fact that every +computer program, no matter how complicated, executes a sequence of +elementary arithmetic operations (addition, subtraction, +multiplication, division, etc.) and elementary functions (exp, log, +sin, cos, etc.). By applying the chain rule repeatedly to these +operations, derivatives of arbitrary order can be computed +automatically, accurately to working precision, and using at most a +small constant factor more arithmetic operations than the original +program. + +

    +Automatic differentiation is neither: + +

    + +Symbolic differentiation can lead to inefficient code and faces the +difficulty of converting a computer program into a single expression, +while numerical differentiation can introduce round-off errors in the +discretization process and cancellation + +

    +Python has tools for so-called automatic differentiation. +Consider the following example +$$ +f(x) = \sin\left(2\pi x + x^2\right) +$$ + +which has the following derivative +$$ +f'(x) = \cos\left(2\pi x + x^2\right)\left(2\pi + 2x\right) +$$ + +Using autograd we have 

    import autograd.numpy as np
-from autograd import grad
-def f2(x1,x2):
-    return 3*x1**3 + x2*(x1 - 5) + 1
 
-# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
-f2_grad_x1 = grad(f2,0)
+# To do elementwise differentiation:
+from autograd import elementwise_grad as egrad 
 
-# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
-f2_grad_x2 = grad(f2,1)
+# To plot:
+import matplotlib.pyplot as plt 
 
-x1 = 1.0
-x2 = 3.0 
 
-print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
-print("-"*30)
+def f(x):
+    return np.sin(2*np.pi*x + x**2)
 
-# Compare with the analytical derivatives:
+def f_grad_analytic(x):
+    return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)
 
-# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
-f2_grad_x1_analytical = 9*x1**2 + x2
+# Do the comparison:
+x = np.linspace(0,1,1000)
 
-# Derivative of f2 w.r.t x2 is: x1 - 5:
-f2_grad_x2_analytical = x1 - 5
+f_grad = egrad(f)
 
-# See the evaluated derivations:
-print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+computed = f_grad(x)
+analytic = f_grad_analytic(x)
 
-print()
+plt.title('Derivative computed from Autograd compared with the analytical derivative')
+plt.plot(x,computed,label='autograd')
+plt.plot(x,analytic,label='analytic')
 
-print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
-print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+plt.xlabel('x')
+plt.ylabel('y')
+plt.legend()
+
+plt.show()
+
+print("The max absolute difference is: %g"%(np.max(np.abs(computed - analytic))))
    -

    -Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. -

    @@ -373,7 +415,7 @@ Note that the grad function will not produce the true gradient of the function.

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    • diff --git a/doc/pub/week40/html/._week40-bs021.html b/doc/pub/week40/html/._week40-bs021.html index 3e52fc31a..5e4c7f556 100644 --- a/doc/pub/week40/html/._week40-bs021.html +++ b/doc/pub/week40/html/._week40-bs021.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -298,39 +305,38 @@ MathJax.Hub.Config({

     

     

     

    - + -

    More complicated functions using the elements of their arguments directly

    +

    Using autograd

    + +

    +Here we +experiment with what kind of functions Autograd is capable +of finding the gradient of. The following Python functions are just +meant to illustrate what Autograd can do, but please feel free to +experiment with other, possibly more complicated, functions as well. 

    import autograd.numpy as np
 from autograd import grad
-def f3(x): # Assumes x is an array of length 5 or higher
-    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
 
-f3_grad = grad(f3)
+def f1(x):
+    return x**3 + 1
 
-x = np.linspace(0,4,5)
+f1_grad = grad(f1)
 
-# Print the computed gradient:
-print("The computed gradient of f3 is: ", f3_grad(x))
+# Remember to send in float as argument to the computed gradient from Autograd!
+a = 1.0
 
-# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
-f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+# See the evaluated gradient at a using autograd:
+print("The gradient of f1 evaluated at a = %g using autograd is: %g"%(a,f1_grad(a)))
 
-# Print the analytical gradient:
-print("The analytical gradient of f3 is: ", f3_grad_analytical)
+# Compare with the analytical derivative, that is f1'(x) = 3*x**2 
+grad_analytical = 3*a**2
+print("The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g"%(a,grad_analytical))
    -

    -Note that in this case, when sending an array as input argument, the -output from Autograd is another array. This is the true gradient of -the function, as opposed to the function in the previous example. By -using arrays to represent the variables, the output from Autograd -might be easier to work with, as the output is closer to what one -could expect form a gradient-evaluting function. -

    @@ -357,7 +363,7 @@ could expect form a gradient-evaluting function.

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    • diff --git a/doc/pub/week40/html/._week40-bs022.html b/doc/pub/week40/html/._week40-bs022.html index 0a68dbd53..e303a2251 100644 --- a/doc/pub/week40/html/._week40-bs022.html +++ b/doc/pub/week40/html/._week40-bs022.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -298,31 +305,55 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Functions using mathematical functions from Numpy

    +

    Autograd with more complicated functions

    + +

    +To differentiate with respect to two (or more) arguments of a Python +function, Autograd need to know at which variable the function if +being differentiated with respect to. 

    import autograd.numpy as np
 from autograd import grad
-def f4(x):
-    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
+def f2(x1,x2):
+    return 3*x1**3 + x2*(x1 - 5) + 1
 
-f4_grad = grad(f4)
+# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1
+f2_grad_x1 = grad(f2,0)
 
-x = 2.7
+# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad
+f2_grad_x2 = grad(f2,1)
 
-# Print the computed derivative:
-print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
+x1 = 1.0
+x2 = 3.0 
 
-# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi
-f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi
+print("Evaluating at x1 = %g, x2 = %g"%(x1,x2))
+print("-"*30)
 
-# Print the analytical gradient:
-print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))
+# Compare with the analytical derivatives:
+
+# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:
+f2_grad_x1_analytical = 9*x1**2 + x2
+
+# Derivative of f2 w.r.t x2 is: x1 - 5:
+f2_grad_x2_analytical = x1 - 5
+
+# See the evaluated derivations:
+print("The derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x1: %g"%( f2_grad_x1(x1,x2) ))
+
+print()
+
+print("The derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
+print("The analytical derivative of f2 w.r.t x2: %g"%( f2_grad_x2(x1,x2) ))
    +

    +Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable. +

    @@ -349,7 +380,7 @@ f4_grad_analytical = x31

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    • diff --git a/doc/pub/week40/html/._week40-bs023.html b/doc/pub/week40/html/._week40-bs023.html index 8e782b9d3..683873c3f 100644 --- a/doc/pub/week40/html/._week40-bs023.html +++ b/doc/pub/week40/html/._week40-bs023.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,26 +307,37 @@ MathJax.Hub.Config({ -

    More autograd

    +

    More complicated functions using the elements of their arguments directly 

    import autograd.numpy as np
 from autograd import grad
-def f5(x):
-    if x >= 0:
-        return x**2
-    else:
-        return -3*x + 1
+def f3(x): # Assumes x is an array of length 5 or higher
+    return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2
 
-f5_grad = grad(f5)
+f3_grad = grad(f3)
 
-x = 2.7
+x = np.linspace(0,4,5)
 
-# Print the computed derivative:
-print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
+# Print the computed gradient:
+print("The computed gradient of f3 is: ", f3_grad(x))
+
+# The analytical gradient is: (2, 3, 5, 7, 22*x[4])
+f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])
+
+# Print the analytical gradient:
+print("The analytical gradient of f3 is: ", f3_grad_analytical)
    +

    +Note that in this case, when sending an array as input argument, the +output from Autograd is another array. This is the true gradient of +the function, as opposed to the function in the previous example. By +using arrays to represent the variables, the output from Autograd +might be easier to work with, as the output is closer to what one +could expect form a gradient-evaluting function. +

    @@ -346,7 +364,7 @@ x = 2.7

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    • diff --git a/doc/pub/week40/html/._week40-bs024.html b/doc/pub/week40/html/._week40-bs024.html index 9a8963d26..94eae7f24 100644 --- a/doc/pub/week40/html/._week40-bs024.html +++ b/doc/pub/week40/html/._week40-bs024.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -298,50 +305,30 @@ MathJax.Hub.Config({

     

     

     

    - + -

    And with loops

    +

    Functions using mathematical functions from Numpy 

    import autograd.numpy as np
 from autograd import grad
-def f6_for(x):
-    val = 0
-    for i in range(10):
-        val = val + x**i
-    return val
+def f4(x):
+    return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)
 
-def f6_while(x):
-    val = 0
-    i = 0
-    while i < 10:
-        val = val + x**i
-        i = i + 1
-    return val
+f4_grad = grad(f4)
 
-f6_for_grad = grad(f6_for)
-f6_while_grad = grad(f6_while)
+x = 2.7
 
-x = 0.5
+# Print the computed derivative:
+print("The computed derivative of f4 at x = %g is: %g"%(x,f4_grad(x)))
 
-# Print the computed derivaties of f6_for and f6_while
-print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
-print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))
-
    -

    +# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi +f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi - -

    import autograd.numpy as np
-from autograd import grad
-# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9
-# The analytical derivative is: sum(i*x**(i-1)) 
-f6_grad_analytical = 0
-for i in range(10):
-    f6_grad_analytical += i*x**(i-1)
-
-print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+# Print the analytical gradient:
+print("The analytical gradient of f4 at x = %g is: %g"%(x,f4_grad_analytical))

    @@ -369,7 +356,7 @@ f6_grad_analytical = 33

  • 34
  • ...
    • -
  • 61
    • +
  • 63
  • »
    • diff --git a/doc/pub/week40/html/._week40-bs025.html b/doc/pub/week40/html/._week40-bs025.html index 908238f38..beb7b281e 100644 --- a/doc/pub/week40/html/._week40-bs025.html +++ b/doc/pub/week40/html/._week40-bs025.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,41 +307,26 @@ MathJax.Hub.Config({ -

    Using recursion

    +

    More autograd

    + 

    import autograd.numpy as np
 from autograd import grad
-
-def f7(n): # Assume that n is an integer
-    if n == 1 or n == 0:
-        return 1
+def f5(x):
+    if x >= 0:
+        return x**2
     else:
-        return n*f7(n-1)
+        return -3*x + 1
 
-f7_grad = grad(f7)
+f5_grad = grad(f5)
 
-n = 2.0
+x = 2.7
 
-print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
-
-# The function f7 is an implementation of the factorial of n.
-# By using the product rule, one can find that the derivative is:
-
-f7_grad_analytical = 0
-for i in range(int(n)-1):
-    tmp = 1
-    for k in range(int(n)-1):
-        if k != i:
-            tmp *= (n - k)
-    f7_grad_analytical += tmp
-
-print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))
+# Print the computed derivative:
+print("The computed derivative of f5 at x = %g is: %g"%(x,f5_grad(x)))
    -

    -Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. -

    @@ -361,7 +353,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi

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  • ...
    • -
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    • +
  • 63
  • »
    • diff --git a/doc/pub/week40/html/._week40-bs026.html b/doc/pub/week40/html/._week40-bs026.html index a3234bce0..7ed761a4e 100644 --- a/doc/pub/week40/html/._week40-bs026.html +++ b/doc/pub/week40/html/._week40-bs026.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,29 +307,49 @@ MathJax.Hub.Config({ -

    Unsupported functions

    -Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd. +

    And with loops

    -

    -Assigning a value to the variable being differentiated with respect to 

    import autograd.numpy as np
 from autograd import grad
-def f8(x): # Assume x is an array
-    x[2] = 3
-    return x*2
+def f6_for(x):
+    val = 0
+    for i in range(10):
+        val = val + x**i
+    return val
 
-f8_grad = grad(f8)
+def f6_while(x):
+    val = 0
+    i = 0
+    while i < 10:
+        val = val + x**i
+        i = i + 1
+    return val
 
-x = 8.4
+f6_for_grad = grad(f6_for)
+f6_while_grad = grad(f6_while)
 
-print("The derivative of f8 is:",f8_grad(x))
+x = 0.5
+
+# Print the computed derivaties of f6_for and f6_while
+print("The computed derivative of f6_for at x = %g is: %g"%(x,f6_for_grad(x)))
+print("The computed derivative of f6_while at x = %g is: %g"%(x,f6_while_grad(x)))

    -Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible. + +

    import autograd.numpy as np
+from autograd import grad
+# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9
+# The analytical derivative is: sum(i*x**(i-1)) 
+f6_grad_analytical = 0
+for i in range(10):
+    f6_grad_analytical += i*x**(i-1)
+
+print("The analytical derivative of f6 at x = %g is: %g"%(x,f6_grad_analytical))
+

    @@ -349,7 +376,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The

  • 35
  • 36
  • ...
    • -
  • 61
    • +
  • 63
  • »
    • diff --git a/doc/pub/week40/html/._week40-bs027.html b/doc/pub/week40/html/._week40-bs027.html index 681091a9c..6735ab09f 100644 --- a/doc/pub/week40/html/._week40-bs027.html +++ b/doc/pub/week40/html/._week40-bs027.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,45 +307,41 @@ MathJax.Hub.Config({ -

    The syntax a.dot(b) when finding the dot product

    +

    Using recursion 

    import autograd.numpy as np
 from autograd import grad
-def f9(a): # Assume a is an array with 2 elements
-    b = np.array([1.0,2.0])
-    return a.dot(b)
 
-f9_grad = grad(f9)
+def f7(n): # Assume that n is an integer
+    if n == 1 or n == 0:
+        return 1
+    else:
+        return n*f7(n-1)
 
-x = np.array([1.0,0.0])
+f7_grad = grad(f7)
 
-print("The derivative of f9 is:",f9_grad(x))
+n = 2.0
+
+print("The computed derivative of f7 at n = %d is: %g"%(n,f7_grad(n)))
+
+# The function f7 is an implementation of the factorial of n.
+# By using the product rule, one can find that the derivative is:
+
+f7_grad_analytical = 0
+for i in range(int(n)-1):
+    tmp = 1
+    for k in range(int(n)-1):
+        if k != i:
+            tmp *= (n - k)
+    f7_grad_analytical += tmp
+
+print("The analytical derivative of f7 at n = %d is: %g"%(n,f7_grad_analytical))

    -Here we are told that the 'dot' function does not belong to Autograd's -version of a Numpy array. To overcome this, an alternative syntax -which also computed the dot product can be used: +Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input. -

    - - -

    import autograd.numpy as np
-from autograd import grad
-def f9_alternative(x): # Assume a is an array with 2 elements
-    b = np.array([1.0,2.0])
-    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2
-
-f9_alternative_grad = grad(f9_alternative)
-
-x = np.array([3.0,0.0])
-
-print("The gradient of f9 is:",f9_alternative_grad(x))
-
-# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively
-# w.r.t x is (b_1, b_2).
-

    @@ -365,7 +368,7 @@ x = np.a

  • 36
  • 37
  • ...
    • -
  • 61
    • +
  • 63
  • »
    • diff --git a/doc/pub/week40/html/._week40-bs028.html b/doc/pub/week40/html/._week40-bs028.html index 0630cc652..f2c9db7c1 100644 --- a/doc/pub/week40/html/._week40-bs028.html +++ b/doc/pub/week40/html/._week40-bs028.html @@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -227,64 +232,66 @@ MathJax.Hub.Config({ @@ -300,16 +307,29 @@ MathJax.Hub.Config({ - -The documentation recommends to avoid inplace operations such as +

    Unsupported functions

    +Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd. + +

    +Assigning a value to the variable being differentiated with respect to

    -

    a += b
-a -= b
-a*= b
-a /=b
+
    import autograd.numpy as np
+from autograd import grad
+def f8(x): # Assume x is an array
+    x[2] = 3
+    return x*2
+
+f8_grad = grad(f8)
+
+x = 8.4
+
+print("The derivative of f8 is:",f8_grad(x))
 
    
+
    
+Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.
+
 
    
 
    
 
@@ -336,7 +356,7 @@ a /=b
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs029.html b/doc/pub/week40/html/._week40-bs029.html
index 55af95cf8..b189849d4 100644
--- a/doc/pub/week40/html/._week40-bs029.html
+++ b/doc/pub/week40/html/._week40-bs029.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,14 +307,45 @@ MathJax.Hub.Config({
 
 
 
-
    Videos on Neural Networks 
    
+
    The syntax a.dot(b) when finding the dot product 
    
+
    
+
+
+
    import autograd.numpy as np
+from autograd import grad
+def f9(a): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return a.dot(b)
+
+f9_grad = grad(f9)
+
+x = np.array([1.0,0.0])
+
+print("The derivative of f9 is:",f9_grad(x))
+
    
+
    
+Here we are told that the 'dot' function does not belong to Autograd's
+version of a Numpy array.  To overcome this, an alternative syntax
+which also computed the dot product can be used:
 
 
    
-Neural Networks demystified
 
-
    
-Building Neural Networks from scratch
+
+
    import autograd.numpy as np
+from autograd import grad
+def f9_alternative(x): # Assume a is an array with 2 elements
+    b = np.array([1.0,2.0])
+    return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2
 
+f9_alternative_grad = grad(f9_alternative)
+
+x = np.array([3.0,0.0])
+
+print("The gradient of f9 is:",f9_alternative_grad(x))
+
+# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively
+# w.r.t x is (b_1, b_2).
+
    
 
    
 
    
 
@@ -334,7 +372,7 @@ MathJax.Hub.Config({
   
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    • 
   
  • ...
    • 
-  
  • 61
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+  
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diff --git a/doc/pub/week40/html/._week40-bs030.html b/doc/pub/week40/html/._week40-bs030.html
index 15684b55d..59c9a3ace 100644
--- a/doc/pub/week40/html/._week40-bs030.html
+++ b/doc/pub/week40/html/._week40-bs030.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,17 +307,16 @@ MathJax.Hub.Config({
 
 
 
-
    Neural networks 
    
-
+
+The documentation recommends to avoid inplace operations such as
 
    
-Artificial neural networks are computational systems that can learn to
-perform tasks by considering examples, generally without being
-programmed with any task-specific rules. It is supposed to mimic a
-biological system, wherein neurons interact by sending signals in the
-form of mathematical functions between layers. All layers can contain
-an arbitrary number of neurons, and each connection is represented by
-a weight variable.
 
+
+
    a += b
+a -= b
+a*= b
+a /=b
+
    
 
    
 
    
 
@@ -337,7 +343,7 @@ a weight variable.
   
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    • 
   
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    • 
   
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    • 
-  
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+  
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    • 
   
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diff --git a/doc/pub/week40/html/._week40-bs031.html b/doc/pub/week40/html/._week40-bs031.html
index af29e2475..21071f93c 100644
--- a/doc/pub/week40/html/._week40-bs031.html
+++ b/doc/pub/week40/html/._week40-bs031.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,65 +307,13 @@ MathJax.Hub.Config({
 
 
 
-
    Artificial neurons  
    
+
    Videos on Neural Networks 
    
 
 
    
-The field of artificial neural networks has a long history of
-development, and is closely connected with the advancement of computer
-science and computers in general. A model of artificial neurons was
-first developed by McCulloch and Pitts in 1943 to study signal
-processing in the brain and has later been refined by others. The
-general idea is to mimic neural networks in the human brain, which is
-composed of billions of neurons that communicate with each other by
-sending electrical signals.  Each neuron accumulates its incoming
-signals, which must exceed an activation threshold to yield an
-output. If the threshold is not overcome, the neuron remains inactive,
-i.e. has zero output.
+Neural Networks demystified
 
 
    
-This behaviour has inspired a simple mathematical model for an artificial neuron.
-
-$$
-\begin{equation}
- y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
-\tag{6}
-\end{equation}
-$$
-
-Here, the output \( y \) of the neuron is the value of its activation function, which have as input
-a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
-
-
    
-Conceptually, it is helpful to divide neural networks into four
-categories:
-
-
      
-
    1.  general purpose neural networks for supervised learning,
      2. 
-
    2.  neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),
      3. 
-
    3.  neural networks for sequential data such as Recurrent Neural Networks (RNNs), and
      4. 
-
    4.  neural networks for unsupervised learning such as Deep Boltzmann Machines.
      5. 
-
    
-
-In natural science, DNNs and CNNs have already found numerous
-applications. In statistical physics, they have been applied to detect
-phase transitions in 2D Ising and Potts models, lattice gauge
-theories, and different phases of polymers, or solving the
-Navier-Stokes equation in weather forecasting.  Deep learning has also
-found interesting applications in quantum physics. Various quantum
-phase transitions can be detected and studied using DNNs and CNNs,
-topological phases, and even non-equilibrium many-body
-localization. Representing quantum states as DNNs quantum state
-tomography are among some of the impressive achievements to reveal the
-potential of DNNs to facilitate the study of quantum systems.
-
-
    
-In quantum information theory, it has been shown that one can perform
-gate decompositions with the help of neural.
-
-
    
-The applications are not limited to the natural sciences. There is a
-plethora of applications in essentially all disciplines, from the
-humanities to life science and medicine.
+Building Neural Networks from scratch
 
 
    
 
    
@@ -386,7 +341,7 @@ humanities to life science and medicine.
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs032.html b/doc/pub/week40/html/._week40-bs032.html
index a6af02343..32867cb5f 100644
--- a/doc/pub/week40/html/._week40-bs032.html
+++ b/doc/pub/week40/html/._week40-bs032.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,29 +307,16 @@ MathJax.Hub.Config({
 
 
 
-
    Neural network types 
    
+
    Neural networks 
    
 
 
    
-An artificial neural network (ANN), is a computational model that
-consists of layers of connected neurons, or nodes or units.  We will
-refer to these interchangeably as units or nodes, and sometimes as
-neurons.
-
-
    
-It is supposed to mimic a biological nervous system by letting each
-neuron interact with other neurons by sending signals in the form of
-mathematical functions between layers.  A wide variety of different
-ANNs have been developed, but most of them consist of an input layer,
-an output layer and eventual layers in-between, called hidden
-layers. All layers can contain an arbitrary number of nodes, and each
-connection between two nodes is associated with a weight variable.
-
-
    
-Neural networks (also called neural nets) are neural-inspired
-nonlinear models for supervised learning.  As we will see, neural nets
-can be viewed as natural, more powerful extensions of supervised
-learning methods such as linear and logistic regression and soft-max
-methods we discussed earlier.
+Artificial neural networks are computational systems that can learn to
+perform tasks by considering examples, generally without being
+programmed with any task-specific rules. It is supposed to mimic a
+biological system, wherein neurons interact by sending signals in the
+form of mathematical functions between layers. All layers can contain
+an arbitrary number of neurons, and each connection is represented by
+a weight variable.
 
 
    
 
    
@@ -350,7 +344,7 @@ methods we discussed earlier.
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs033.html b/doc/pub/week40/html/._week40-bs033.html
index ea70798d7..bbe8e6bbb 100644
--- a/doc/pub/week40/html/._week40-bs033.html
+++ b/doc/pub/week40/html/._week40-bs033.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,20 +307,65 @@ MathJax.Hub.Config({
 
 
 
-
    Feed-forward neural networks 
    
+
    Artificial neurons  
    
 
 
    
-The feed-forward neural network (FFNN) was the first and simplest type
-of ANNs that were devised. In this network, the information moves in
-only one direction: forward through the layers.
+The field of artificial neural networks has a long history of
+development, and is closely connected with the advancement of computer
+science and computers in general. A model of artificial neurons was
+first developed by McCulloch and Pitts in 1943 to study signal
+processing in the brain and has later been refined by others. The
+general idea is to mimic neural networks in the human brain, which is
+composed of billions of neurons that communicate with each other by
+sending electrical signals.  Each neuron accumulates its incoming
+signals, which must exceed an activation threshold to yield an
+output. If the threshold is not overcome, the neuron remains inactive,
+i.e. has zero output.
 
 
    
-Nodes are represented by circles, while the arrows display the
-connections between the nodes, including the direction of information
-flow. Additionally, each arrow corresponds to a weight variable
-(figure to come).  We observe that each node in a layer is connected
-to all nodes in the subsequent layer, making this a so-called
-fully-connected FFNN.
+This behaviour has inspired a simple mathematical model for an artificial neuron.
+
+$$
+\begin{equation}
+ y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
+\tag{6}
+\end{equation}
+$$
+
+Here, the output \( y \) of the neuron is the value of its activation function, which have as input
+a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
+
+
    
+Conceptually, it is helpful to divide neural networks into four
+categories:
+
+
      
+
    1.  general purpose neural networks for supervised learning,
      2. 
+
    2.  neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),
      3. 
+
    3.  neural networks for sequential data such as Recurrent Neural Networks (RNNs), and
      4. 
+
    4.  neural networks for unsupervised learning such as Deep Boltzmann Machines.
      5. 
+
    
+
+In natural science, DNNs and CNNs have already found numerous
+applications. In statistical physics, they have been applied to detect
+phase transitions in 2D Ising and Potts models, lattice gauge
+theories, and different phases of polymers, or solving the
+Navier-Stokes equation in weather forecasting.  Deep learning has also
+found interesting applications in quantum physics. Various quantum
+phase transitions can be detected and studied using DNNs and CNNs,
+topological phases, and even non-equilibrium many-body
+localization. Representing quantum states as DNNs quantum state
+tomography are among some of the impressive achievements to reveal the
+potential of DNNs to facilitate the study of quantum systems.
+
+
    
+In quantum information theory, it has been shown that one can perform
+gate decompositions with the help of neural.
+
+
    
+The applications are not limited to the natural sciences. There is a
+plethora of applications in essentially all disciplines, from the
+humanities to life science and medicine.
 
 
    
 
    
@@ -341,7 +393,7 @@ to all nodes in the subsequent layer, making this a so-called
   
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diff --git a/doc/pub/week40/html/._week40-bs034.html b/doc/pub/week40/html/._week40-bs034.html
index 6d494ebcc..7f7d92c76 100644
--- a/doc/pub/week40/html/._week40-bs034.html
+++ b/doc/pub/week40/html/._week40-bs034.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,28 +307,29 @@ MathJax.Hub.Config({
 
 
 
-
    Convolutional Neural Network 
    
+
    Neural network types 
    
 
 
    
-A different variant of FFNNs are convolutional neural networks
-(CNNs), which have a connectivity pattern inspired by the animal
-visual cortex. Individual neurons in the visual cortex only respond to
-stimuli from small sub-regions of the visual field, called a receptive
-field. This makes the neurons well-suited to exploit the strong
-spatially local correlation present in natural images. The response of
-each neuron can be approximated mathematically as a convolution
-operation.  (figure to come)
+An artificial neural network (ANN), is a computational model that
+consists of layers of connected neurons, or nodes or units.  We will
+refer to these interchangeably as units or nodes, and sometimes as
+neurons.
 
 
    
-Convolutional neural networks emulate the behaviour of neurons in the
-visual cortex by enforcing a local connectivity pattern between
-nodes of adjacent layers: Each node in a convolutional layer is
-connected only to a subset of the nodes in the previous layer, in
-contrast to the fully-connected FFNN.  Often, CNNs consist of several
-convolutional layers that learn local features of the input, with a
-fully-connected layer at the end, which gathers all the local data and
-produces the outputs. They have wide applications in image and video
-recognition.
+It is supposed to mimic a biological nervous system by letting each
+neuron interact with other neurons by sending signals in the form of
+mathematical functions between layers.  A wide variety of different
+ANNs have been developed, but most of them consist of an input layer,
+an output layer and eventual layers in-between, called hidden
+layers. All layers can contain an arbitrary number of nodes, and each
+connection between two nodes is associated with a weight variable.
+
+
    
+Neural networks (also called neural nets) are neural-inspired
+nonlinear models for supervised learning.  As we will see, neural nets
+can be viewed as natural, more powerful extensions of supervised
+learning methods such as linear and logistic regression and soft-max
+methods we discussed earlier.
 
 
    
 
    
@@ -349,7 +357,7 @@ recognition.
   
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    • 
   
  • 44
    • 
   
  • ...
    • 
-  
  • 61
    • 
+  
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    • 
 
 
diff --git a/doc/pub/week40/html/._week40-bs035.html b/doc/pub/week40/html/._week40-bs035.html
index 5b795408e..a3d5f876e 100644
--- a/doc/pub/week40/html/._week40-bs035.html
+++ b/doc/pub/week40/html/._week40-bs035.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,19 +307,20 @@ MathJax.Hub.Config({
 
 
 
-
    Recurrent neural networks 
    
+
    Feed-forward neural networks 
    
 
 
    
-So far we have only mentioned ANNs where information flows in one
-direction: forward. Recurrent neural networks on the other hand,
-have connections between nodes that form directed cycles. This
-creates a form of internal memory which are able to capture
-information on what has been calculated before; the output is
-dependent on the previous computations. Recurrent NNs make use of
-sequential information by performing the same task for every element
-in a sequence, where each element depends on previous elements. An
-example of such information is sentences, making recurrent NNs
-especially well-suited for handwriting and speech recognition.
+The feed-forward neural network (FFNN) was the first and simplest type
+of ANNs that were devised. In this network, the information moves in
+only one direction: forward through the layers.
+
+
    
+Nodes are represented by circles, while the arrows display the
+connections between the nodes, including the direction of information
+flow. Additionally, each arrow corresponds to a weight variable
+(figure to come).  We observe that each node in a layer is connected
+to all nodes in the subsequent layer, making this a so-called
+fully-connected FFNN.
 
 
    
 
    
@@ -340,7 +348,7 @@ especially well-suited for handwriting and speech recognition.
   
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    • 
   
  • ...
    • 
-  
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+  
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diff --git a/doc/pub/week40/html/._week40-bs036.html b/doc/pub/week40/html/._week40-bs036.html
index dceeae7fa..6482ababa 100644
--- a/doc/pub/week40/html/._week40-bs036.html
+++ b/doc/pub/week40/html/._week40-bs036.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,19 +307,28 @@ MathJax.Hub.Config({
 
 
 
-
    Other types of networks 
    
+
    Convolutional Neural Network 
    
 
 
    
-There are many other kinds of ANNs that have been developed. One type
-that is specifically designed for interpolation in multidimensional
-space is the radial basis function (RBF) network. RBFs are typically
-made up of three layers: an input layer, a hidden layer with
-non-linear radial symmetric activation functions and a linear output
-layer (''linear'' here means that each node in the output layer has a
-linear activation function). The layers are normally fully-connected
-and there are no cycles, thus RBFs can be viewed as a type of
-fully-connected FFNN. They are however usually treated as a separate
-type of NN due the unusual activation functions.
+A different variant of FFNNs are convolutional neural networks
+(CNNs), which have a connectivity pattern inspired by the animal
+visual cortex. Individual neurons in the visual cortex only respond to
+stimuli from small sub-regions of the visual field, called a receptive
+field. This makes the neurons well-suited to exploit the strong
+spatially local correlation present in natural images. The response of
+each neuron can be approximated mathematically as a convolution
+operation.  (figure to come)
+
+
    
+Convolutional neural networks emulate the behaviour of neurons in the
+visual cortex by enforcing a local connectivity pattern between
+nodes of adjacent layers: Each node in a convolutional layer is
+connected only to a subset of the nodes in the previous layer, in
+contrast to the fully-connected FFNN.  Often, CNNs consist of several
+convolutional layers that learn local features of the input, with a
+fully-connected layer at the end, which gathers all the local data and
+produces the outputs. They have wide applications in image and video
+recognition.
 
 
    
 
    
@@ -340,7 +356,7 @@ type of NN due the unusual activation functions.
   
  • 45
    • 
   
  • 46
    • 
   
  • ...
    • 
-  
  • 61
    • 
+  
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    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week40/html/._week40-bs037.html b/doc/pub/week40/html/._week40-bs037.html
index 089d68c62..4f0d5ba0d 100644
--- a/doc/pub/week40/html/._week40-bs037.html
+++ b/doc/pub/week40/html/._week40-bs037.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,16 +307,19 @@ MathJax.Hub.Config({
 
 
 
-
    Multilayer perceptrons  
    
+
    Recurrent neural networks 
    
 
 
    
-One uses often so-called fully-connected feed-forward neural networks
-with three or more layers (an input layer, one or more hidden layers
-and an output layer) consisting of neurons that have non-linear
-activation functions.
-
-
    
-Such networks are often called multilayer perceptrons (MLPs).
+So far we have only mentioned ANNs where information flows in one
+direction: forward. Recurrent neural networks on the other hand,
+have connections between nodes that form directed cycles. This
+creates a form of internal memory which are able to capture
+information on what has been calculated before; the output is
+dependent on the previous computations. Recurrent NNs make use of
+sequential information by performing the same task for every element
+in a sequence, where each element depends on previous elements. An
+example of such information is sentences, making recurrent NNs
+especially well-suited for handwriting and speech recognition.
 
 
    
 
    
@@ -337,7 +347,7 @@ Such networks are often called multilayer perceptrons (MLPs).
   
  • 46
    • 
   
  • 47
    • 
   
  • ...
    • 
-  
  • 61
    • 
+  
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    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week40/html/._week40-bs038.html b/doc/pub/week40/html/._week40-bs038.html
index 718651447..8eb1cbed8 100644
--- a/doc/pub/week40/html/._week40-bs038.html
+++ b/doc/pub/week40/html/._week40-bs038.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,20 +307,19 @@ MathJax.Hub.Config({
 
 
 
-
    Why multilayer perceptrons?  
    
+
    Other types of networks 
    
 
 
    
-According to the Universal approximation theorem, a feed-forward
-neural network with just a single hidden layer containing a finite
-number of neurons can approximate a continuous multidimensional
-function to arbitrary accuracy, assuming the activation function for
-the hidden layer is a non-constant, bounded and
-monotonically-increasing continuous function.
-
-
    
-Note that the requirements on the activation function only applies to
-the hidden layer, the output nodes are always assumed to be linear, so
-as to not restrict the range of output values.
+There are many other kinds of ANNs that have been developed. One type
+that is specifically designed for interpolation in multidimensional
+space is the radial basis function (RBF) network. RBFs are typically
+made up of three layers: an input layer, a hidden layer with
+non-linear radial symmetric activation functions and a linear output
+layer (''linear'' here means that each node in the output layer has a
+linear activation function). The layers are normally fully-connected
+and there are no cycles, thus RBFs can be viewed as a type of
+fully-connected FFNN. They are however usually treated as a separate
+type of NN due the unusual activation functions.
 
 
    
 
    
@@ -341,7 +347,7 @@ as to not restrict the range of output values.
   
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    • 
   
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    • 
   
  • ...
    • 
-  
  • 61
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+  
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diff --git a/doc/pub/week40/html/._week40-bs039.html b/doc/pub/week40/html/._week40-bs039.html
index a6b3d49d2..ded8c2126 100644
--- a/doc/pub/week40/html/._week40-bs039.html
+++ b/doc/pub/week40/html/._week40-bs039.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,14 +307,16 @@ MathJax.Hub.Config({
 
 
 
-
    Illustration of a single perceptropn model and a multi-perceptron model 
    
+
    Multilayer perceptrons  
    
 
 
    
-
      
-
    
-
    Figure 1:  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer. 
    
-
    
-
    
+One uses often so-called fully-connected feed-forward neural networks
+with three or more layers (an input layer, one or more hidden layers
+and an output layer) consisting of neurons that have non-linear
+activation functions.
+
+
    
+Such networks are often called multilayer perceptrons (MLPs).
 
 
    
 
    
@@ -335,7 +344,7 @@ MathJax.Hub.Config({
   
  • 48
    • 
   
  • 49
    • 
   
  • ...
    • 
-  
  • 61
    • 
+  
  • 63
    • 
   
  • »
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diff --git a/doc/pub/week40/html/._week40-bs040.html b/doc/pub/week40/html/._week40-bs040.html
index 42737356e..514b8e7b1 100644
--- a/doc/pub/week40/html/._week40-bs040.html
+++ b/doc/pub/week40/html/._week40-bs040.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,20 +307,20 @@ MathJax.Hub.Config({
 
 
 
-
    Mathematical model  
    
+
    Why multilayer perceptrons?  
    
 
 
    
-The output \( y \) is produced via the activation function \( f \)
-$$
- y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
-$$
+According to the Universal approximation theorem, a feed-forward
+neural network with just a single hidden layer containing a finite
+number of neurons can approximate a continuous multidimensional
+function to arbitrary accuracy, assuming the activation function for
+the hidden layer is a non-constant, bounded and
+monotonically-increasing continuous function.
 
-This function receives \( x_i \) as inputs.
-Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \). 
-In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
-the neurons in the preceding layer. Furthermore, an MLP is
-fully-connected, which means that each neuron receives a weighted sum
-of the outputs of all neurons in the previous layer.
+
    
+Note that the requirements on the activation function only applies to
+the hidden layer, the output nodes are always assumed to be linear, so
+as to not restrict the range of output values.
 
 
    
 
    
@@ -341,7 +348,7 @@ of the outputs of all neurons in the previous layer.
   
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    • 
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs041.html b/doc/pub/week40/html/._week40-bs041.html
index db74404b4..9bb833c66 100644
--- a/doc/pub/week40/html/._week40-bs041.html
+++ b/doc/pub/week40/html/._week40-bs041.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,49 +307,14 @@ MathJax.Hub.Config({
 
 
 
-
    Mathematical model  
    
+
    Illustration of a single perceptropn model and a multi-perceptron model 
    
 
 
    
-First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
-
-$$
-\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
-\tag{7}
-\end{equation}
-$$
-
-
    
-Here \( b_i \) is the so-called bias which is normally needed in
-case of zero activation weights or inputs. How to fix the biases and
-the weights will be discussed below.  The value of \( z_i^1 \) is the
-argument to the activation function \( f_i \) of each node \( i \), The
-variable \( M \) stands for all possible inputs to a given node \( i \) in the
-first layer.  We define  the output \( y_i^1 \) of all neurons in layer 1 as
-
-$$
-\begin{equation}
- y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\right)
-\tag{8}
-\end{equation}
-$$
-
-
    
-where we assume that all nodes in the same layer have identical
-activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
-In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
-
-$$
-\begin{equation}
- y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
-\tag{9}
-\end{equation}
-$$
-
-
    
-where \( N_l \) is the number of nodes in layer \( l \). When the output of
-all the nodes in the first hidden layer are computed, the values of
-the subsequent layer can be calculated and so forth until the output
-is obtained.
+
      
+
    
+
    Figure 1:  In a) we show a single perceptron model while in b) we dispay a network with two  hidden layers, an input layer and an output layer. 
    
+
    
+
    
 
 
    
 
    
@@ -370,7 +342,7 @@ is obtained.
   
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-  
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+  
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diff --git a/doc/pub/week40/html/._week40-bs042.html b/doc/pub/week40/html/._week40-bs042.html
index 0320eaa70..9889fb4f2 100644
--- a/doc/pub/week40/html/._week40-bs042.html
+++ b/doc/pub/week40/html/._week40-bs042.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -303,28 +310,17 @@ MathJax.Hub.Config({
 
    Mathematical model  
    
 
 
    
-The output of neuron \( i \) in layer 2 is thus,
-
+The output \( y \) is produced via the activation function \( f \)
 $$
-\begin{align}
- y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
-\tag{10}\\
- &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
-\tag{11}
-\end{align}
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
 $$
 
-where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
-
-$$
-\begin{align}
- y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
-\tag{12}\\
- &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
-  + b_1^3\right]
-\tag{13}
-\end{align}
-$$
+This function receives \( x_i \) as inputs.
+Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \). 
+In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
+the neurons in the preceding layer. Furthermore, an MLP is
+fully-connected, which means that each neuron receives a weighted sum
+of the outputs of all neurons in the previous layer.
 
 
    
 
    
@@ -352,7 +348,7 @@ $$
   
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    • 
   
  • 52
    • 
   
  • ...
    • 
-  
  • 61
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+  
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  • »
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diff --git a/doc/pub/week40/html/._week40-bs043.html b/doc/pub/week40/html/._week40-bs043.html
index 4acb82d20..c67cbfa3e 100644
--- a/doc/pub/week40/html/._week40-bs043.html
+++ b/doc/pub/week40/html/._week40-bs043.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -303,19 +310,46 @@ MathJax.Hub.Config({
 
    Mathematical model  
    
 
 
    
-We can generalize this expression to an MLP with \( l \) hidden
-layers. The complete functional form is,
+First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
 
 $$
-\begin{align}
-&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
-\tag{14}
-\end{align}
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
+\tag{7}
+\end{equation}
 $$
 
 
    
-which illustrates a basic property of MLPs: The only independent
-variables are the input values \( x_n \).
+Here \( b_i \) is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below.  The value of \( z_i^1 \) is the
+argument to the activation function \( f_i \) of each node \( i \), The
+variable \( M \) stands for all possible inputs to a given node \( i \) in the
+first layer.  We define  the output \( y_i^1 \) of all neurons in layer 1 as
+
+$$
+\begin{equation}
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j  + b_i^1\right)
+\tag{8}
+\end{equation}
+$$
+
+
    
+where we assume that all nodes in the same layer have identical
+activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
+
+$$
+\begin{equation}
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+\tag{9}
+\end{equation}
+$$
+
+
    
+where \( N_l \) is the number of nodes in layer \( l \). When the output of
+all the nodes in the first hidden layer are computed, the values of
+the subsequent layer can be calculated and so forth until the output
+is obtained.
 
 
    
 
    
@@ -343,7 +377,7 @@ variables are the input values \( x_n \).
   
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    • 
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs044.html b/doc/pub/week40/html/._week40-bs044.html
index 1ef30b3ed..1c1863bae 100644
--- a/doc/pub/week40/html/._week40-bs044.html
+++ b/doc/pub/week40/html/._week40-bs044.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -303,28 +310,28 @@ MathJax.Hub.Config({
 
    Mathematical model  
    
 
 
    
-This confirms that an MLP, despite its quite convoluted mathematical
-form, is nothing more than an analytic function, specifically a
-mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
-\hat{y} \in \mathbb{R}^m \).
-
-
    
-Furthermore, the flexibility and universality of an MLP can be
-illustrated by realizing that the expression is essentially a nested
-sum of scaled activation functions of the form
+The output of neuron \( i \) in layer 2 is thus,
 
 $$
-\begin{equation}
- f(x) = c_1 f(c_2 x + c_3) + c_4
-\tag{15}
-\end{equation}
+\begin{align}
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) 
+\tag{10}\\
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+\tag{11}
+\end{align}
 $$
 
-
    
-where the parameters \( c_i \) are weights and biases. By adjusting these
-parameters, the activation functions can be shifted up and down or
-left and right, change slope or be rescaled which is the key to the
-flexibility of a neural network.
+where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
+
+$$
+\begin{align}
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) 
+\tag{12}\\
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+  + b_1^3\right]
+\tag{13}
+\end{align}
+$$
 
 
    
 
    
@@ -352,7 +359,7 @@ flexibility of a neural network.
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs045.html b/doc/pub/week40/html/._week40-bs045.html
index e6bf6c98b..c7df4af1c 100644
--- a/doc/pub/week40/html/._week40-bs045.html
+++ b/doc/pub/week40/html/._week40-bs045.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,42 +307,23 @@ MathJax.Hub.Config({
 
 
 
-
    Matrix-vector notation 
    
+
    Mathematical model  
    
 
 
    
-We can introduce a more convenient notation for the activations in an A NN.
+We can generalize this expression to an MLP with \( l \) hidden
+layers. The complete functional form is,
 
-
    
-Additionally, we can represent the biases and activations
-as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector 
-is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
-
-
    
-We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors. 
-With this notation, the sum becomes a matrix-vector multiplication, and we can write
-the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
 $$
-\begin{equation}
- \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = 
- f_2\left(\left[\begin{array}{ccc}
-    w^2_{11} &w^2_{12} &w^2_{13} \\
-    w^2_{21} &w^2_{22} &w^2_{23} \\
-    w^2_{31} &w^2_{32} &w^2_{33} \\
-    \end{array} \right] \cdot
-    \left[\begin{array}{c}
-           y^1_1 \\
-           y^1_2 \\
-           y^1_3 \\
-          \end{array}\right] + 
-    \left[\begin{array}{c}
-           b^2_1 \\
-           b^2_2 \\
-           b^2_3 \\
-          \end{array}\right]\right).
-\tag{16}
-\end{equation}
+\begin{align}
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
+\tag{14}
+\end{align}
 $$
 
+
    
+which illustrates a basic property of MLPs: The only independent
+variables are the input values \( x_n \).
+
 
    
 
    
 
@@ -362,7 +350,7 @@ $$
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs046.html b/doc/pub/week40/html/._week40-bs046.html
index b58640b08..fb58e96e4 100644
--- a/doc/pub/week40/html/._week40-bs046.html
+++ b/doc/pub/week40/html/._week40-bs046.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,25 +307,31 @@ MathJax.Hub.Config({
 
 
 
-
    Matrix-vector notation  and activation 
    
+
    Mathematical model  
    
 
 
    
-The activation of node \( i \) in layer 2 is
+This confirms that an MLP, despite its quite convoluted mathematical
+form, is nothing more than an analytic function, specifically a
+mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m \).
+
+
    
+Furthermore, the flexibility and universality of an MLP can be
+illustrated by realizing that the expression is essentially a nested
+sum of scaled activation functions of the form
 
 $$
 \begin{equation}
- y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
- f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\tag{17}
+ f(x) = c_1 f(c_2 x + c_3) + c_4
+\tag{15}
 \end{equation}
 $$
 
 
    
-This is not just a convenient and compact notation, but also a useful
-and intuitive way to think about MLPs: The output is calculated by a
-series of matrix-vector multiplications and vector additions that are
-used as input to the activation functions. For each operation
-\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
+where the parameters \( c_i \) are weights and biases. By adjusting these
+parameters, the activation functions can be shifted up and down or
+left and right, change slope or be rescaled which is the key to the
+flexibility of a neural network.
 
 
    
 
    
@@ -346,7 +359,7 @@ used as input to the activation functions. For each operation
   
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-  
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diff --git a/doc/pub/week40/html/._week40-bs047.html b/doc/pub/week40/html/._week40-bs047.html
index 3c7a89e4b..ad3135f18 100644
--- a/doc/pub/week40/html/._week40-bs047.html
+++ b/doc/pub/week40/html/._week40-bs047.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,21 +307,43 @@ MathJax.Hub.Config({
 
 
 
-
    Activation functions  
    
+
    Matrix-vector notation 
    
 
 
    
-A property that characterizes a neural network, other than its
-connectivity, is the choice of activation function(s).  As described
-in, the following restrictions are imposed on an activation function
-for a FFNN to fulfill the universal approximation theorem
+We can introduce a more convenient notation for the activations in an A NN.
 
-
      
-  
    •  Non-constant
      • 
-  
    •  Bounded
      • 
-  
    •  Monotonically-increasing
      • 
-  
    •  Continuous
      • 
-
    
+
    
+Additionally, we can represent the biases and activations
+as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector 
+is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
 
+
    
+We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors. 
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
+$$
+\begin{equation}
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = 
+ f_2\left(\left[\begin{array}{ccc}
+    w^2_{11} &w^2_{12} &w^2_{13} \\
+    w^2_{21} &w^2_{22} &w^2_{23} \\
+    w^2_{31} &w^2_{32} &w^2_{33} \\
+    \end{array} \right] \cdot
+    \left[\begin{array}{c}
+           y^1_1 \\
+           y^1_2 \\
+           y^1_3 \\
+          \end{array}\right] + 
+    \left[\begin{array}{c}
+           b^2_1 \\
+           b^2_2 \\
+           b^2_3 \\
+          \end{array}\right]\right).
+\tag{16}
+\end{equation}
+$$
+
+
    
 
    
 
 
      
@@ -340,7 +369,7 @@ for a FFNN to fulfill the universal approximation theorem
   
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      • 
   
    • ...
      • 
-  
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+  
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      • 
 
    
 
diff --git a/doc/pub/week40/html/._week40-bs048.html b/doc/pub/week40/html/._week40-bs048.html
index a1e57b5de..6ecf9de0b 100644
--- a/doc/pub/week40/html/._week40-bs048.html
+++ b/doc/pub/week40/html/._week40-bs048.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,27 +307,25 @@ MathJax.Hub.Config({
 
 
 
-
    Activation functions, Logistic and Hyperbolic ones  
    
+
    Matrix-vector notation  and activation 
    
 
 
    
-The second requirement excludes all linear functions. Furthermore, in
-a MLP with only linear activation functions, each layer simply
-performs a linear transformation of its inputs.
+The activation of node \( i \) in layer 2 is
+
+$$
+\begin{equation}
+ y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = 
+ f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
+\tag{17}
+\end{equation}
+$$
 
 
    
-Regardless of the number of layers, the output of the NN will be
-nothing but a linear function of the inputs. Thus we need to introduce
-some kind of non-linearity to the NN to be able to fit non-linear
-functions Typical examples are the logistic Sigmoid
-
-$$
- f(x) = \frac{1}{1 + e^{-x}},
-$$
-
-and the hyperbolic tangent function
-$$
- f(x) = \tanh(x)
-$$
+This is not just a convenient and compact notation, but also a useful
+and intuitive way to think about MLPs: The output is calculated by a
+series of matrix-vector multiplications and vector additions that are
+used as input to the activation functions. For each operation
+\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
 
 
    
 
    
@@ -348,7 +353,7 @@ $$
   
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diff --git a/doc/pub/week40/html/._week40-bs049.html b/doc/pub/week40/html/._week40-bs049.html
index 6b64047b3..c6c7306d1 100644
--- a/doc/pub/week40/html/._week40-bs049.html
+++ b/doc/pub/week40/html/._week40-bs049.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,91 +307,21 @@ MathJax.Hub.Config({
 
 
 
-
    Relevance 
    
+
    Activation functions  
    
 
 
    
-The sigmoid function are more biologically plausible because the
-output of inactive neurons are zero. Such activation function are
-called one-sided. However, it has been shown that the hyperbolic
-tangent performs better than the sigmoid for training MLPs.  has
-become the most popular for deep neural networks
+A property that characterizes a neural network, other than its
+connectivity, is the choice of activation function(s).  As described
+in, the following restrictions are imposed on an activation function
+for a FFNN to fulfill the universal approximation theorem
 
-
    
+
      
+  
    •  Non-constant
      • 
+  
    •  Bounded
      • 
+  
    •  Monotonically-increasing
      • 
+  
    •  Continuous
      • 
+
    
 
-
-
    """The sigmoid function (or the logistic curve) is a 
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
-
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
-
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""Sine Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.sin(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sine function')
-
-plt.show()
-
-"""Plots a graph of the squashing function used by a rectified linear
-unit"""
-z = numpy.arange(-2, 2, .1)
-zero = numpy.zeros(len(z))
-y = numpy.max([zero, z], axis=0)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, y)
-ax.set_ylim([-2.0, 2.0])
-ax.set_xlim([-2.0, 2.0])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('Rectified linear unit')
-
-plt.show()
-
    
-
    
 
    
 
 
    
 
diff --git a/doc/pub/week40/html/._week40-bs050.html b/doc/pub/week40/html/._week40-bs050.html
index 5e7e572ba..0a2d9f4c7 100644
--- a/doc/pub/week40/html/._week40-bs050.html
+++ b/doc/pub/week40/html/._week40-bs050.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,40 +307,27 @@ MathJax.Hub.Config({
 
 
 
-
    The multilayer  perceptron (MLP) 
    
+
    Activation functions, Logistic and Hyperbolic ones  
    
 
 
    
-The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
-
-
      
-
    1.  A neural network with one or more layers of nodes between the input and the output nodes.
      2. 
-
    2.  The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
      3. 
-
    3.  The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
      4. 
-
    
-
-As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
-units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
+The second requirement excludes all linear functions. Furthermore, in
+a MLP with only linear activation functions, each layer simply
+performs a linear transformation of its inputs.
 
 
    
-For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
+Regardless of the number of layers, the output of the NN will be
+nothing but a linear function of the inputs. Thus we need to introduce
+some kind of non-linearity to the NN to be able to fit non-linear
+functions Typical examples are the logistic Sigmoid
 
-
    
-The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
+$$
+ f(x) = \frac{1}{1 + e^{-x}},
+$$
 
-
    
-The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation \( \hat{z} \), we can think of this as a basis
-expansion of the original inputs \( \hat{x} \). The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in  an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
+and the hyperbolic tangent function
+$$
+ f(x) = \tanh(x)
+$$
 
 
    
 
    
@@ -361,7 +355,7 @@ like logistic regression or linear regression and their modifications on the oth
   
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diff --git a/doc/pub/week40/html/._week40-bs051.html b/doc/pub/week40/html/._week40-bs051.html
index 5bf84ce15..196f285e3 100644
--- a/doc/pub/week40/html/._week40-bs051.html
+++ b/doc/pub/week40/html/._week40-bs051.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,31 +307,90 @@ MathJax.Hub.Config({
 
 
 
-
    From one to many layers, the universal approximation theorem 
    
+
    Relevance 
    
 
 
    
-A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
+The sigmoid function are more biologically plausible because the
+output of inactive neurons are zero. Such activation function are
+called one-sided. However, it has been shown that the hyperbolic
+tangent performs better than the sigmoid for training MLPs.  has
+become the most popular for deep neural networks
 
 
    
-As stated earlier, 
-an important theorem in studies of neural networks, restated without
-proof here, is the universal approximation
-theorem.
 
-
    
-It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
+
+
    """The sigmoid function (or the logistic curve) is a 
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
 
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""Sine Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.sin(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sine function')
+
+plt.show()
+
+"""Plots a graph of the squashing function used by a rectified linear
+unit"""
+z = numpy.arange(-2, 2, .1)
+zero = numpy.zeros(len(z))
+y = numpy.max([zero, z], axis=0)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, y)
+ax.set_ylim([-2.0, 2.0])
+ax.set_xlim([-2.0, 2.0])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('Rectified linear unit')
+
+plt.show()
+
    
 
    
 
    
 
@@ -350,6 +416,8 @@ the potential of being universal approximators.
   
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diff --git a/doc/pub/week40/html/._week40-bs052.html b/doc/pub/week40/html/._week40-bs052.html
index 52e102869..ba7b61afb 100644
--- a/doc/pub/week40/html/._week40-bs052.html
+++ b/doc/pub/week40/html/._week40-bs052.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,33 +307,40 @@ MathJax.Hub.Config({
 
 
 
-
    Deriving the back propagation code for a multilayer perceptron model 
    
+
    The multilayer  perceptron (MLP) 
    
 
 
    
-As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous back propagation algorithm.
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
+
+
      
+
    1.  A neural network with one or more layers of nodes between the input and the output nodes.
      2. 
+
    2.  The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
      3. 
+
    3.  The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
      4. 
+
    
+
+As a convention it is normal to call  a  network with one layer of input units, one layer of hidden
+units and one layer of output units as  a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
 
 
    
-The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
+For an MLP network there is no direct connection between the output nodes/neurons/units and  the input nodes/neurons/units.
+Hereafter we will call the various entities of a layer for nodes.
+There are also no connections within a single layer.
 
 
    
-To derive these equations let us start with a plain regression problem
-and define our cost function as
-
-$$
-{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
-$$
+The number of input nodes does not need to equal the number of output
+nodes. This applies also to the hidden layers. Each layer may have its
+own number of nodes and activation functions.
 
 
    
-where the $t_i$s are our \( n \) targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs \( \hat{x} \) are given by \( y_i \).  Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with \( K \)
-classes.
+The hidden layers have their name from the fact that they are not
+linked to observables and as we will see below when we define the
+so-called activation \( \hat{z} \), we can think of this as a basis
+expansion of the original inputs \( \hat{x} \). The difference however
+between neural networks and say linear regression is that now these
+basis functions (which will correspond to the weights in the network)
+are learned from data. This results in  an important difference between
+neural networks and deep learning approaches on one side and methods
+like logistic regression or linear regression and their modifications on the other side.
 
 
    
 
    
@@ -352,6 +366,9 @@ classes.
   
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diff --git a/doc/pub/week40/html/._week40-bs053.html b/doc/pub/week40/html/._week40-bs053.html
index 3acf3f2ca..18cde57c0 100644
--- a/doc/pub/week40/html/._week40-bs053.html
+++ b/doc/pub/week40/html/._week40-bs053.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,40 +307,30 @@ MathJax.Hub.Config({
 
 
 
-
    Definitions 
    
+
    From one to many layers, the universal approximation theorem 
    
 
 
    
-With our definition of the targets \( \hat{t} \), the outputs of the
-network \( \hat{y} \) and the inputs \( \hat{x} \) we
-define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
-\( l \)-th layer as a function of the bias, the weights which add up from
-the previous layer \( l-1 \) and the forward passes/outputs
-\( \hat{a}^{l-1} \) from the previous layer as
-
-$$
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
-$$
+A neural network with only one layer, what we called the simple
+perceptron, is best suited if we have a standard binary model with
+clear (linear) boundaries between the outcomes. As such it could
+equally well be replaced by standard linear regression or logistic
+regression. Networks with one or more hidden layers approximate
+systems with more complex boundaries.
 
 
    
-where \( b_k^l \) are the biases from layer \( l \).  Here \( M_{l-1} \)
-represents the total number of nodes/neurons/units of layer \( l-1 \). The
-figure here illustrates this equation.  We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-
-$$
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-$$
+As stated earlier, 
+an important theorem in studies of neural networks, restated without
+proof here, is the universal approximation
+theorem.
 
 
    
-With the activation values \( \hat{z}^l \) we can in turn define the
-output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
-and their nodes.  It means we have
-
-$$
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-$$
+It states that a feed-forward network with a single hidden layer
+containing a finite number of neurons can approximate continuous
+functions on compact subsets of real functions. The theorem thus
+states that simple neural networks can represent a wide variety of
+interesting functions when given appropriate parameters. It is the
+multilayer feedforward architecture itself which gives neural networks
+the potential of being universal approximators.
 
 
    
 
    
@@ -358,6 +355,8 @@ $$
   
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diff --git a/doc/pub/week40/html/._week40-bs054.html b/doc/pub/week40/html/._week40-bs054.html
index 1859cfcc5..adfe99868 100644
--- a/doc/pub/week40/html/._week40-bs054.html
+++ b/doc/pub/week40/html/._week40-bs054.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,24 +307,33 @@ MathJax.Hub.Config({
 
 
 
-
    Derivatives and the chain rule 
    
+
    Deriving the back propagation code for a multilayer perceptron model 
    
 
 
    
-From the definition of the activation \( z_j^l \) we have
-$$
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-$$
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
+The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
+This leads us to the famous back propagation algorithm.
+
+
    
+The questions we want to ask are how do changes in the biases and the
+weights in our network change the cost function and how can we use the
+final output to modify the weights?
+
+
    
+To derive these equations let us start with a plain regression problem
+and define our cost function as
 
-and
 $$
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
+{\cal C}(\hat{W})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, 
 $$
 
 
    
-With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
-$$
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
-$$
+where the $t_i$s are our \( n \) targets (the values we want to
+reproduce), while the outputs of the network after having propagated
+all inputs \( \hat{x} \) are given by \( y_i \).  Below we will demonstrate
+how the basic equations arising from the back propagation algorithm
+can be modified in order to study classification problems with \( K \)
+classes.
 
 
    
 
    
@@ -341,6 +357,8 @@ $$
   
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diff --git a/doc/pub/week40/html/._week40-bs055.html b/doc/pub/week40/html/._week40-bs055.html
index be5d0ba6b..422c863d6 100644
--- a/doc/pub/week40/html/._week40-bs055.html
+++ b/doc/pub/week40/html/._week40-bs055.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,26 +307,39 @@ MathJax.Hub.Config({
 
 
 
-
    Derivative of the cost function 
    
+
    Definitions 
    
 
 
    
-With these definitions we can now compute the derivative of the cost function in terms of the weights.
+With our definition of the targets \( \hat{t} \), the outputs of the
+network \( \hat{y} \) and the inputs \( \hat{x} \) we
+define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
+\( l \)-th layer as a function of the bias, the weights which add up from
+the previous layer \( l-1 \) and the forward passes/outputs
+\( \hat{a}^{l-1} \) from the previous layer as
+
+$$
+z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
+$$
 
 
    
-Let us specialize to the output layer \( l=L \). Our cost function is
-$$
-{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
-$$
-
-The derivative of this function with respect to the weights is
+where \( b_k^l \) are the biases from layer \( l \).  Here \( M_{l-1} \)
+represents the total number of nodes/neurons/units of layer \( l-1 \). The
+figure here illustrates this equation.  We can rewrite this in a more
+compact form as the matrix-vector products we discussed earlier,
 
 $$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
+\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
 $$
 
-The last partial derivative can easily be computed and reads (by applying the chain rule)
+
    
+With the activation values \( \hat{z}^l \) we can in turn define the
+output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
+activation function. In the examples here we will use the sigmoid
+function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
+and their nodes.  It means we have
+
 $$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
+a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
 $$
 
 
    
@@ -343,6 +363,8 @@ $$
   
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diff --git a/doc/pub/week40/html/._week40-bs056.html b/doc/pub/week40/html/._week40-bs056.html
index a4d7bf00b..86de71d67 100644
--- a/doc/pub/week40/html/._week40-bs056.html
+++ b/doc/pub/week40/html/._week40-bs056.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,51 +307,23 @@ MathJax.Hub.Config({
 
 
 
-
    Bringing it together, first back propagation equation 
    
+
    Derivatives and the chain rule 
    
 
 
    
-We have thus
+From the definition of the activation \( z_j^l \) we have
 $$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
+\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
+$$
+
+and
+$$
+\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. 
 $$
 
 
    
-Defining
+With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
 $$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-$$
-
-and using the Hadamard product of two vectors we can write this as
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-
-
    
-This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation.  If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
-
-
    
-Notice that everything in the above equations is easily computed.  In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \).  The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
-
-$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
-$$
-
-
    
-With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
+\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). 
 $$
 
 
    
@@ -367,6 +346,8 @@ $$
   
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diff --git a/doc/pub/week40/html/._week40-bs057.html b/doc/pub/week40/html/._week40-bs057.html
index f632eee38..c8e1f2f7f 100644
--- a/doc/pub/week40/html/._week40-bs057.html
+++ b/doc/pub/week40/html/._week40-bs057.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,21 +307,29 @@ MathJax.Hub.Config({
 
 
 
-
    Derivatives in terms of \( z_j^L \) 
    
+
    Derivative of the cost function 
    
 
 
    
-It is also easy to see that our previous equation can be written as
+With these definitions we can now compute the derivative of the cost function in terms of the weights.
 
+
    
+Let us specialize to the output layer \( l=L \). Our cost function is
 $$
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
+{\cal C}(\hat{W^L})  =  \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, 
 $$
 
-which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
+The derivative of this function with respect to the weights is
+
 $$
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, 
 $$
 
-That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. 
+The last partial derivative can easily be computed and reads (by applying the chain rule)
+$$
+\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},  
+$$
+
+
    
 
    
 
 
      
@@ -333,6 +348,8 @@ That is, the error \( \delta_j^L \) is exactly equal to the rate of change of th
   
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diff --git a/doc/pub/week40/html/._week40-bs058.html b/doc/pub/week40/html/._week40-bs058.html
index 8d8662c06..1ef9e7a74 100644
--- a/doc/pub/week40/html/._week40-bs058.html
+++ b/doc/pub/week40/html/._week40-bs058.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,64 +307,52 @@ MathJax.Hub.Config({
 
 
 
-
    Bringing it together 
    
+
    Bringing it together, first back propagation equation 
    
 
 
    
-We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
+We have thus
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, 
+$$
 
 
    
-
    
-
    
-
     
-
+Defining
 $$
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
-\tag{18}
-\end{equation}
+\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
 $$
 
-and
+and using the Hadamard product of two vectors we can write this as
 $$
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\tag{19}
-\end{equation}
+\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
 $$
 
-and
-
-$$
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\tag{20}
-\end{equation}
-$$
-
    
-
    
-
-
 
    
-An interesting consequence of the above equations is that when the
-activation \( a_k^{L-1} \) is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
+This is an important expression. The second term on the right handside
+measures how fast the cost function is changing as a function of the $j$th
+output activation.  If, for example, the cost function doesn't depend
+much on a particular output node \( j \), then \( \delta_j^L \) will be small,
+which is what we would expect. The first term on the right, measures
+how fast the activation function \( f \) is changing at a given activation
+value \( z_j^L \).
 
 
    
-Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
+Notice that everything in the above equations is easily computed.  In
+particular, we compute \( z_j^L \) while computing the behaviour of the
+network, and it is only a small additional overhead to compute
+\( f'(z^L_j) \).  The exact form of the derivative with respect to the
+output depends on the form of the cost function.
+However, provided the cost function is known there should be little
+trouble in calculating
+
+$$
+\frac{\partial {\cal C}}{\partial (a_j^L)}
+$$
 
 
    
-We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one \( L-1 \) in terms of the errors in the final output layer.
+With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1}.
+$$
 
 
    
 
    
@@ -377,6 +372,8 @@ one \( L-1 \) in terms of the errors in the final output layer.
   
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diff --git a/doc/pub/week40/html/._week40-bs059.html b/doc/pub/week40/html/._week40-bs059.html
index bce9af47a..6c11e0f9f 100644
--- a/doc/pub/week40/html/._week40-bs059.html
+++ b/doc/pub/week40/html/._week40-bs059.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,36 +307,21 @@ MathJax.Hub.Config({
 
 
 
-
    Final back propagating equation 
    
+
    Derivatives in terms of \( z_j^L \) 
    
 
 
    
-We have that (replacing \( L \) with a general layer \( l \))
-$$
-\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
-$$
-
-We want to express this in terms of the equations for layer \( l+1 \). Using the chain rule and summing over all \( k \) entries we have
+It is also easy to see that our previous equation can be written as
 
 $$
-\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
+\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
 $$
 
-and recalling that
+which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
 $$
-z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
 $$
 
-with \( M_l \) being the number of nodes in layer \( l \), we obtain
-$$
-\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
-$$
-
-This is our final equation.
-
-
    
-We are now ready to set up the algorithm for back propagation and learning the weights and biases.
-
-
    
+That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias. 
 
    
 
 
      
@@ -346,6 +338,8 @@ We are now ready to set up the algorithm for back propagation and learning the w
   
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diff --git a/doc/pub/week40/html/._week40-bs060.html b/doc/pub/week40/html/._week40-bs060.html
index 1c2bea4a6..a40581438 100644
--- a/doc/pub/week40/html/._week40-bs060.html
+++ b/doc/pub/week40/html/._week40-bs060.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -300,81 +307,66 @@ MathJax.Hub.Config({
 
 
 
-
    Setting up the Back propagation algorithm 
    
+
    Bringing it together 
    
 
 
    
-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 have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
 
 
    
 
    
 
    
 
     
-First, we set up the input data \( \hat{x} \) and the activations
-\( \hat{z}_1 \) of the input layer and compute the activation function and
-the pertinent outputs \( \hat{a}^1 \).
-
    
-
    
 
-
-
    
-
    
-
    
-
     
-Secondly, we perform then the feed forward till we reach the output
-layer and compute all \( \hat{z}_l \) of the input layer and compute the
-activation function and the pertinent outputs \( \hat{a}^l \) for
-\( l=2,3,\dots,L \).
-
    
-
    
-
-
-
    
-
    
-
    
-
     
-Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all
 $$
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
+\begin{equation}
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L}  =  \delta_j^La_k^{L-1},
+\tag{18}
+\end{equation}
+$$
+
+and
+$$
+\begin{equation}
+\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+\tag{19}
+\end{equation}
+$$
+
+and
+
+$$
+\begin{equation}
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
+\tag{20}
+\end{equation}
 $$
 
    
 
    
 
 
 
    
-
    
-
    
-
     
-Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as
-$$
-\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
-$$
-
    
-
    
-
+An interesting consequence of the above equations is that when the
+activation \( a_k^{L-1} \) is small, the gradient term, that is the
+derivative of the cost function with respect to the weights, will also
+tend to be small. We say then that the weight learns slowly, meaning
+that it changes slowly when we minimize the weights via say gradient
+descent. In this case we say the system learns slowly.
 
 
    
-
    
-
    
-
     
-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
-$$
-w_{jk}^l\leftarrow  = w_{jk}^l- \eta \delta_j^la_k^{l-1},
-$$
-
-
-$$
-b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
-$$
-
    
-
    
-
+Another interesting feature is that is when the activation function,
+represented by the sigmoid function here, is rather flat when we move towards
+its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
+cases, the derivatives of the activation function will also be close
+to zero, meaning again that the gradients will be small and the
+network learns slowly again.
 
 
    
-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.
+We need a fourth equation and we are set. We are going to propagate
+backwards in order to the determine the weights and biases. In order
+to do so we need to represent the error in the layer before the final
+one \( L-1 \) in terms of the errors in the final output layer.
 
 
    
-
 
    
 
 
      
@@ -390,6 +382,9 @@ Here it is convenient to use stochastic gradient descent (see the examples below
   
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diff --git a/doc/pub/week40/html/week40-bs.html b/doc/pub/week40/html/week40-bs.html
index 1e16f8b76..a8251bd75 100644
--- a/doc/pub/week40/html/week40-bs.html
+++ b/doc/pub/week40/html/week40-bs.html
@@ -46,6 +46,11 @@ Automatically generated HTML file from DocOnce source
                2,
                None,
                'overview-video-on-stochastic-gradient-descent'),
+              ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
+              ('Stochastic Gradient Descent (SGD)',
+               2,
+               None,
+               'stochastic-gradient-descent-sgd'),
               ('Stochastic Gradient Descent',
                2,
                None,
@@ -227,64 +232,66 @@ MathJax.Hub.Config({
         
       
@@ -319,7 +326,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
    
 
    
 
    
-
    Oct 5, 2021
     
+
    Oct 7, 2021
     
 
    
 
    
 
@@ -343,7 +350,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week40/html/week40-reveal.html b/doc/pub/week40/html/week40-reveal.html
index bc503ec71..fb792e0fb 100644
--- a/doc/pub/week40/html/week40-reveal.html
+++ b/doc/pub/week40/html/week40-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
    
 
    
 
     
    
-
    Oct 5, 2021
     
+
    Oct 7, 2021
     
 
    
 
    
 
@@ -181,6 +181,55 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.
 
 
 
+
    
+
    Batches and mini-batches 
    
+
+
    
+In gradient descent we compute the cost function and its gradient for all data points we have.
+
+
    
+In large-scale applications such as the ILSVRC challenge, the
+training data can have on order of millions of examples. Hence, it
+seems wasteful to compute the full cost function over the entire
+training set in order to perform only a single parameter update. A
+very common approach to addressing this challenge is to compute the
+gradient over batches of the training data. For example, in current
+a typical batch could contain some thousand  examples from
+an  entire training set of several millions. This batch is then used to
+perform a parameter update.
+
    
+
+
+
    
+
    Stochastic Gradient Descent (SGD) 
    
+
+
    
+In stochastic gradient descent, the extreme case is the case where we
+have only one batch, that is we include the whole data set.
+
+
    
+This process is called Stochastic Gradient
+Descent (SGD) (or also sometimes on-line gradient descent). This is
+relatively less common to see because in practice due to vectorized
+code optimizations it can be computationally much more efficient to
+evaluate the gradient for 100 examples, than the gradient for one
+example 100 times. Even though SGD technically refers to using a
+single example at a time to evaluate the gradient, you will hear
+people use the term SGD even when referring to mini-batch gradient
+descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD
+for “Batch gradient descent” are rare to see), where it is usually
+assumed that mini-batches are used. The size of the mini-batch is a
+hyperparameter but it is not very common to cross-validate or bootstrap it. It is
+usually based on memory constraints (if any), or set to some value,
+e.g. 32, 64 or 128. We use powers of 2 in practice because many
+vectorized operation implementations work faster when their inputs are
+sized in powers of 2.
+
+
    
+In our notes with  SGD we mean stochastic gradient descent with mini-batches.
+
    
+
+
 
    
 
    Stochastic Gradient Descent 
    
 
@@ -225,6 +274,8 @@ minibatches. We denote these minibatches by \( B_k \) where
 
 
    
 
    SGD example 
    
+
+
    
 As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) 
 and we choose to have \( M=5 \) minibathces,
 then each minibatch contains two data points. In particular we have
@@ -415,6 +466,8 @@ ypredict2 = Xnew.dot(theta_linreg)
 
 
 n_epochs = 50
+M = 10   #size of each minibatch
+m = int(n/M) #number of minibatches
 t0, t1 = 5, 50
 def learning_schedule(t):
     return t0/(t+t1)
@@ -441,8 +494,6 @@ plt.ylabel(r'$y$')
 plt.title(r'Random numbers ')
 plt.show()
    -

    -Challenge: try to write a similar code for a Logistic Regression case. diff --git a/doc/pub/week40/html/week40-solarized.html b/doc/pub/week40/html/week40-solarized.html index 2bed80f77..3dc71d254 100644 --- a/doc/pub/week40/html/week40-solarized.html +++ b/doc/pub/week40/html/week40-solarized.html @@ -66,6 +66,11 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -249,7 +254,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA

    -

    Oct 5, 2021

    +

    Oct 7, 2021












    @@ -276,6 +281,55 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.











    +

    Batches and mini-batches

    + +

    +In gradient descent we compute the cost function and its gradient for all data points we have. + +

    +In large-scale applications such as the ILSVRC challenge, the +training data can have on order of millions of examples. Hence, it +seems wasteful to compute the full cost function over the entire +training set in order to perform only a single parameter update. A +very common approach to addressing this challenge is to compute the +gradient over batches of the training data. For example, in current +a typical batch could contain some thousand examples from +an entire training set of several millions. This batch is then used to +perform a parameter update. + +

    +









    + +

    Stochastic Gradient Descent (SGD)

    + +

    +In stochastic gradient descent, the extreme case is the case where we +have only one batch, that is we include the whole data set. + +

    +This process is called Stochastic Gradient +Descent (SGD) (or also sometimes on-line gradient descent). This is +relatively less common to see because in practice due to vectorized +code optimizations it can be computationally much more efficient to +evaluate the gradient for 100 examples, than the gradient for one +example 100 times. Even though SGD technically refers to using a +single example at a time to evaluate the gradient, you will hear +people use the term SGD even when referring to mini-batch gradient +descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +for “Batch gradient descent” are rare to see), where it is usually +assumed that mini-batches are used. The size of the mini-batch is a +hyperparameter but it is not very common to cross-validate or bootstrap it. It is +usually based on memory constraints (if any), or set to some value, +e.g. 32, 64 or 128. We use powers of 2 in practice because many +vectorized operation implementations work faster when their inputs are +sized in powers of 2. + +

    +In our notes with SGD we mean stochastic gradient descent with mini-batches. + +

    +









    +

    Stochastic Gradient Descent

    @@ -315,6 +369,8 @@ minibatches. We denote these minibatches by \( B_k \) where









    SGD example

    + +

    As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) and we choose to have \( M=5 \) minibathces, then each minibatch contains two data points. In particular we have @@ -498,6 +554,8 @@ ypredict2 = Xnew.dot(theta_linreg) n_epochs = 50 +M = 10 #size of each minibatch +m = int(n/M) #number of minibatches t0, t1 = 5, 50 def learning_schedule(t): return t0/(t+t1) @@ -524,9 +582,6 @@ plt.ylabel(r'$y$') plt.title(r'Random numbers ') plt.show() -

    -Challenge: try to write a similar code for a Logistic Regression case. -











    diff --git a/doc/pub/week40/html/week40.html b/doc/pub/week40/html/week40.html index 28f5e7ca7..28bbf6902 100644 --- a/doc/pub/week40/html/week40.html +++ b/doc/pub/week40/html/week40.html @@ -71,6 +71,11 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'overview-video-on-stochastic-gradient-descent'), + ('Batches and mini-batches', 2, None, 'batches-and-mini-batches'), + ('Stochastic Gradient Descent (SGD)', + 2, + None, + 'stochastic-gradient-descent-sgd'), ('Stochastic Gradient Descent', 2, None, @@ -254,7 +259,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA

    -

    Oct 5, 2021

    +

    Oct 7, 2021












    @@ -281,6 +286,55 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.











    +

    Batches and mini-batches

    + +

    +In gradient descent we compute the cost function and its gradient for all data points we have. + +

    +In large-scale applications such as the ILSVRC challenge, the +training data can have on order of millions of examples. Hence, it +seems wasteful to compute the full cost function over the entire +training set in order to perform only a single parameter update. A +very common approach to addressing this challenge is to compute the +gradient over batches of the training data. For example, in current +a typical batch could contain some thousand examples from +an entire training set of several millions. This batch is then used to +perform a parameter update. + +

    +









    + +

    Stochastic Gradient Descent (SGD)

    + +

    +In stochastic gradient descent, the extreme case is the case where we +have only one batch, that is we include the whole data set. + +

    +This process is called Stochastic Gradient +Descent (SGD) (or also sometimes on-line gradient descent). This is +relatively less common to see because in practice due to vectorized +code optimizations it can be computationally much more efficient to +evaluate the gradient for 100 examples, than the gradient for one +example 100 times. Even though SGD technically refers to using a +single example at a time to evaluate the gradient, you will hear +people use the term SGD even when referring to mini-batch gradient +descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +for “Batch gradient descent” are rare to see), where it is usually +assumed that mini-batches are used. The size of the mini-batch is a +hyperparameter but it is not very common to cross-validate or bootstrap it. It is +usually based on memory constraints (if any), or set to some value, +e.g. 32, 64 or 128. We use powers of 2 in practice because many +vectorized operation implementations work faster when their inputs are +sized in powers of 2. + +

    +In our notes with SGD we mean stochastic gradient descent with mini-batches. + +

    +









    +

    Stochastic Gradient Descent

    @@ -320,6 +374,8 @@ minibatches. We denote these minibatches by \( B_k \) where









    SGD example

    + +

    As an example, suppose we have \( 10 \) data points \( (\mathbf{x}_1,\cdots, \mathbf{x}_{10}) \) and we choose to have \( M=5 \) minibathces, then each minibatch contains two data points. In particular we have @@ -503,6 +559,8 @@ ypredict2 = Xnew= 50 +M = 10 #size of each minibatch +m = int(n/M) #number of minibatches t0, t1 = 5, 50 def learning_schedule(t): return t0/(t+t1) @@ -529,9 +587,6 @@ plt.ylabel(r plt.title(r'Random numbers ') plt.show() -

    -Challenge: try to write a similar code for a Logistic Regression case. -











    diff --git a/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz b/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz index e5a6fc60805ba0b92a7295f6e24cbc2f717097e0..6732a65919700760174c461fb7e5e77ac1503abf 100644 GIT binary patch delta 18 ZcmbQx!8D \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA\n", "\n", - "Date: **Oct 5, 2021**\n", + "Date: **Oct 7, 2021**\n", "\n", "Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -34,6 +34,43 @@ "[What is Stochastic Gradient Descent](https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer)\n", "\n", "\n", + "## Batches and mini-batches\n", + "\n", + "In gradient descent we compute the cost function and its gradient for all data points we have.\n", + "\n", + "In large-scale applications such as the [ILSVRC challenge](https://www.image-net.org/challenges/LSVRC/), the\n", + "training data can have on order of millions of examples. Hence, it\n", + "seems wasteful to compute the full cost function over the entire\n", + "training set in order to perform only a single parameter update. A\n", + "very common approach to addressing this challenge is to compute the\n", + "gradient over batches of the training data. For example, in current\n", + "a typical batch could contain some thousand examples from\n", + "an entire training set of several millions. This batch is then used to\n", + "perform a parameter update.\n", + "\n", + "## Stochastic Gradient Descent (SGD)\n", + "\n", + "In stochastic gradient descent, the extreme case is the case where we\n", + "have only one batch, that is we include the whole data set.\n", + "\n", + "This process is called Stochastic Gradient\n", + "Descent (SGD) (or also sometimes on-line gradient descent). This is\n", + "relatively less common to see because in practice due to vectorized\n", + "code optimizations it can be computationally much more efficient to\n", + "evaluate the gradient for 100 examples, than the gradient for one\n", + "example 100 times. Even though SGD technically refers to using a\n", + "single example at a time to evaluate the gradient, you will hear\n", + "people use the term SGD even when referring to mini-batch gradient\n", + "descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD\n", + "for “Batch gradient descent” are rare to see), where it is usually\n", + "assumed that mini-batches are used. The size of the mini-batch is a\n", + "hyperparameter but it is not very common to cross-validate or bootstrap it. It is\n", + "usually based on memory constraints (if any), or set to some value,\n", + "e.g. 32, 64 or 128. We use powers of 2 in practice because many\n", + "vectorized operation implementations work faster when their inputs are\n", + "sized in powers of 2.\n", + "\n", + "In our notes with SGD we mean stochastic gradient descent with mini-batches.\n", "\n", "\n", "## Stochastic Gradient Descent\n", @@ -87,6 +124,7 @@ "$k=1,\\cdots,n/M$.\n", "\n", "## SGD example\n", + "\n", "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", "and we choose to have $M=5$ minibathces,\n", "then each minibatch contains two data points. In particular we have\n", @@ -302,6 +340,8 @@ "\n", "\n", "n_epochs = 50\n", + "M = 10 #size of each minibatch\n", + "m = int(n/M) #number of minibatches\n", "t0, t1 = 5, 50\n", "def learning_schedule(t):\n", " return t0/(t+t1)\n", @@ -333,9 +373,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "**Challenge**: try to write a similar code for a Logistic Regression case.\n", - "\n", - "\n", "## Momentum based GD\n", "\n", "The stochastic gradient descent (SGD) is almost always used with a\n", diff --git a/doc/src/week40/week40.do.txt b/doc/src/week40/week40.do.txt index 5a0dbc17a..3cd4c5974 100644 --- a/doc/src/week40/week40.do.txt +++ b/doc/src/week40/week40.do.txt @@ -21,6 +21,45 @@ For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5. "What is Stochastic Gradient Descent":"https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" +!split +===== Batches and mini-batches ===== + +In gradient descent we compute the cost function and its gradient for all data points we have. + +In large-scale applications such as the "ILSVRC challenge":"https://www.image-net.org/challenges/LSVRC/", the +training data can have on order of millions of examples. Hence, it +seems wasteful to compute the full cost function over the entire +training set in order to perform only a single parameter update. A +very common approach to addressing this challenge is to compute the +gradient over batches of the training data. For example, in current +a typical batch could contain some thousand examples from +an entire training set of several millions. This batch is then used to +perform a parameter update. + +!split +===== Stochastic Gradient Descent (SGD) ===== + +In stochastic gradient descent, the extreme case is the case where we +have only one batch, that is we include the whole data set. + +This process is called Stochastic Gradient +Descent (SGD) (or also sometimes on-line gradient descent). This is +relatively less common to see because in practice due to vectorized +code optimizations it can be computationally much more efficient to +evaluate the gradient for 100 examples, than the gradient for one +example 100 times. Even though SGD technically refers to using a +single example at a time to evaluate the gradient, you will hear +people use the term SGD even when referring to mini-batch gradient +descent (i.e. mentions of MGD for “Minibatch Gradient Descent”, or BGD +for “Batch gradient descent” are rare to see), where it is usually +assumed that mini-batches are used. The size of the mini-batch is a +hyperparameter but it is not very common to cross-validate or bootstrap it. It is +usually based on memory constraints (if any), or set to some value, +e.g. 32, 64 or 128. We use powers of 2 in practice because many +vectorized operation implementations work faster when their inputs are +sized in powers of 2. + +In our notes with SGD we mean stochastic gradient descent with mini-batches. !split @@ -59,6 +98,7 @@ $k=1,\cdots,n/M$. !split ===== SGD example ===== + As an example, suppose we have $10$ data points $(\mathbf{x}_1,\cdots, \mathbf{x}_{10})$ and we choose to have $M=5$ minibathces, then each minibatch contains two data points. In particular we have @@ -228,6 +268,8 @@ ypredict2 = Xnew.dot(theta_linreg) n_epochs = 50 +M = 10 #size of each minibatch +m = int(n/M) #number of minibatches t0, t1 = 5, 50 def learning_schedule(t): return t0/(t+t1) @@ -256,7 +298,7 @@ plt.show() !ec -_Challenge_: try to write a similar code for a Logistic Regression case. + !split