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<h1>Neural networks</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#feed-forward-neural-networks">13.1. Feed-forward neural networks</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#convolutional-neural-network">13.2. Convolutional Neural Network</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#recurrent-neural-networks">13.3. Recurrent neural networks</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-types-of-networks">13.4. Other types of networks</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#multilayer-perceptrons">13.5. Multilayer perceptrons</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-vector-notation-and-activation">13.5.1. Matrix-vector notation and activation</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#activation-functions">13.5.2. Activation functions</a></li>
</ul>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm">13.7.4. Setting up the Back propagation algorithm</a></li>
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<section class="tex2jax_ignore mathjax_ignore" id="neural-networks">
<h1><span class="section-number">13. </span>Neural networks<a class="headerlink" href="#neural-networks" title="Link to this heading">#</a></h1>
<p>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.</p>
<p>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.</p>
<p>This behaviour has inspired a simple mathematical model for an artificial neuron.</p>
<!-- Equation labels as ordinary links -->
<div id="artificialNeuron"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
\label{artificialNeuron} \tag{1}
\end{equation}
\]</div>
<p>Here, the output <span class="math notranslate nohighlight">\(y\)</span> of the neuron is the value of its activation function, which have as input
a weighted sum of signals <span class="math notranslate nohighlight">\(x_i, \dots ,x_n\)</span> received by <span class="math notranslate nohighlight">\(n\)</span> other neurons.</p>
<p>Conceptually, it is helpful to divide neural networks into four
categories:</p>
<ol class="arabic simple">
<li><p>general purpose neural networks for supervised learning,</p></li>
<li><p>neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),</p></li>
<li><p>neural networks for sequential data such as Recurrent Neural Networks (RNNs), and</p></li>
<li><p>neural networks for unsupervised learning such as Deep Boltzmann Machines.</p></li>
</ol>
<p>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.</p>
<p>In quantum information theory, it has been shown that one can perform
gate decompositions with the help of neural.</p>
<p>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.</p>
<p>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.</p>
<p>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 <em>hidden
layers</em>. All layers can contain an arbitrary number of nodes, and each
connection between two nodes is associated with a weight variable.</p>
<p>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.</p>
<section id="feed-forward-neural-networks">
<h2><span class="section-number">13.1. </span>Feed-forward neural networks<a class="headerlink" href="#feed-forward-neural-networks" title="Link to this heading">#</a></h2>
<p>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.</p>
<p>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 <em>all</em> nodes in the subsequent layer, making this a so-called
<em>fully-connected</em> FFNN.</p>
</section>
<section id="convolutional-neural-network">
<h2><span class="section-number">13.2. </span>Convolutional Neural Network<a class="headerlink" href="#convolutional-neural-network" title="Link to this heading">#</a></h2>
<p>A different variant of FFNNs are <em>convolutional neural networks</em>
(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)</p>
<p>Convolutional neural networks emulate the behaviour of neurons in the
visual cortex by enforcing a <em>local</em> 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.</p>
</section>
<section id="recurrent-neural-networks">
<h2><span class="section-number">13.3. </span>Recurrent neural networks<a class="headerlink" href="#recurrent-neural-networks" title="Link to this heading">#</a></h2>
<p>So far we have only mentioned ANNs where information flows in one
direction: forward. <em>Recurrent neural networks</em> on the other hand,
have connections between nodes that form directed <em>cycles</em>. 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.</p>
</section>
<section id="other-types-of-networks">
<h2><span class="section-number">13.4. </span>Other types of networks<a class="headerlink" href="#other-types-of-networks" title="Link to this heading">#</a></h2>
<p>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.</p>
</section>
<section id="multilayer-perceptrons">
<h2><span class="section-number">13.5. </span>Multilayer perceptrons<a class="headerlink" href="#multilayer-perceptrons" title="Link to this heading">#</a></h2>
<p>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.</p>
<p>Such networks are often called <em>multilayer perceptrons</em> (MLPs).</p>
<p>According to the <em>Universal approximation theorem</em>, 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 <strong>non-constant, bounded and
monotonically-increasing continuous function</strong>.</p>
<p>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.</p>
<p>The output <span class="math notranslate nohighlight">\(y\)</span> is produced via the activation function <span class="math notranslate nohighlight">\(f\)</span></p>
<div class="math notranslate nohighlight">
\[
y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
\]</div>
<p>This function receives <span class="math notranslate nohighlight">\(x_i\)</span> as inputs.
Here the activation <span class="math notranslate nohighlight">\(z=(\sum_{i=1}^n w_ix_i+b_i)\)</span>.
In an FFNN of such neurons, the <em>inputs</em> <span class="math notranslate nohighlight">\(x_i\)</span> are the <em>outputs</em> 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 <em>all</em> neurons in the previous layer.</p>
<p>First, for each node <span class="math notranslate nohighlight">\(i\)</span> in the first hidden layer, we calculate a weighted sum <span class="math notranslate nohighlight">\(z_i^1\)</span> of the input coordinates <span class="math notranslate nohighlight">\(x_j\)</span>,</p>
<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
\label{_auto1} \tag{2}
\end{equation}
\]</div>
<p>Here <span class="math notranslate nohighlight">\(b_i\)</span> 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 <span class="math notranslate nohighlight">\(z_i^1\)</span> is the
argument to the activation function <span class="math notranslate nohighlight">\(f_i\)</span> of each node <span class="math notranslate nohighlight">\(i\)</span>, The
variable <span class="math notranslate nohighlight">\(M\)</span> stands for all possible inputs to a given node <span class="math notranslate nohighlight">\(i\)</span> in the
first layer. We define the output <span class="math notranslate nohighlight">\(y_i^1\)</span> of all neurons in layer 1 as</p>
<!-- Equation labels as ordinary links -->
<div id="outputLayer1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1} \tag{3}
\end{equation}
\]</div>
<p>where we assume that all nodes in the same layer have identical
activation functions, hence the notation <span class="math notranslate nohighlight">\(f\)</span>. 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 <span class="math notranslate nohighlight">\(l\)</span> for the <span class="math notranslate nohighlight">\(l\)</span>-th layer,</p>
<!-- Equation labels as ordinary links -->
<div id="generalLayer"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer} \tag{4}
\end{equation}
\]</div>
<p>where <span class="math notranslate nohighlight">\(N_l\)</span> is the number of nodes in layer <span class="math notranslate nohighlight">\(l\)</span>. 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.</p>
<p>The output of neuron <span class="math notranslate nohighlight">\(i\)</span> in layer 2 is thus,</p>
<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^2 = f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto2} \tag{5}
\end{equation}
\]</div>
<!-- Equation labels as ordinary links -->
<div id="outputLayer2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2} \tag{6}
\end{equation}
\]</div>
<p>where we have substituted <span class="math notranslate nohighlight">\(y_k^1\)</span> with the inputs <span class="math notranslate nohighlight">\(x_k\)</span>. Finally, the ANN output reads</p>
<!-- Equation labels as ordinary links -->
<div id="_auto3"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y_i^3 = f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
\label{_auto3} \tag{7}
\end{equation}
\]</div>
<!-- Equation labels as ordinary links -->
<div id="_auto4"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto4} \tag{8}
\end{equation}
\]</div>
<p>We can generalize this expression to an MLP with <span class="math notranslate nohighlight">\(l\)</span> hidden
layers. The complete functional form is,</p>
<!-- Equation labels as ordinary links -->
<div id="completeNN"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right]
\label{completeNN} \tag{9}
\end{equation}
\]</div>
<p>which illustrates a basic property of MLPs: The only independent
variables are the input values <span class="math notranslate nohighlight">\(x_n\)</span>.</p>
<p>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 <span class="math notranslate nohighlight">\(\hat{x} \in \mathbb{R}^n \rightarrow
\hat{y} \in \mathbb{R}^m\)</span>.</p>
<p>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</p>
<!-- Equation labels as ordinary links -->
<div id="_auto5"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
f(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto5} \tag{10}
\end{equation}
\]</div>
<p>where the parameters <span class="math notranslate nohighlight">\(c_i\)</span> 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.</p>
<p>We can introduce a more convenient notation for the activations in an A NN.</p>
<p>Additionally, we can represent the biases and activations
as layer-wise column vectors <span class="math notranslate nohighlight">\(\hat{b}_l\)</span> and <span class="math notranslate nohighlight">\(\hat{y}_l\)</span>, so that the <span class="math notranslate nohighlight">\(i\)</span>-th element of each vector
is the bias <span class="math notranslate nohighlight">\(b_i^l\)</span> and activation <span class="math notranslate nohighlight">\(y_i^l\)</span> of node <span class="math notranslate nohighlight">\(i\)</span> in layer <span class="math notranslate nohighlight">\(l\)</span> respectively.</p>
<p>We have that <span class="math notranslate nohighlight">\(\mathrm{W}_l\)</span> is an <span class="math notranslate nohighlight">\(N_{l-1} \times N_l\)</span> matrix, while <span class="math notranslate nohighlight">\(\hat{b}_l\)</span> and <span class="math notranslate nohighlight">\(\hat{y}_l\)</span> are <span class="math notranslate nohighlight">\(N_l \times 1\)</span> 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</p>
<!-- Equation labels as ordinary links -->
<div id="_auto6"></div>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{equation}
\hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &amp;w^2_{12} &amp;w^2_{13} \\
w^2_{21} &amp;w^2_{22} &amp;w^2_{23} \\
w^2_{31} &amp;w^2_{32} &amp;w^2_{33} \\
\end{array} \right] \cdot
\left[\begin{array}{c}
y^1_1 \\
y^1_2 \\
y^1_3 \\
\end{array}\right] +
\left[\begin{array}{c}
b^2_1 \\
b^2_2 \\
b^2_3 \\
\end{array}\right]\right).
\label{_auto6} \tag{11}
\end{equation}
\end{split}\]</div>
<section id="matrix-vector-notation-and-activation">
<h3><span class="section-number">13.5.1. </span>Matrix-vector notation and activation<a class="headerlink" href="#matrix-vector-notation-and-activation" title="Link to this heading">#</a></h3>
<p>The activation of node <span class="math notranslate nohighlight">\(i\)</span> in layer 2 is</p>
<!-- Equation labels as ordinary links -->
<div id="_auto7"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
\label{_auto7} \tag{12}
\end{equation}
\]</div>
<p>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
<span class="math notranslate nohighlight">\(\mathrm{W}_l \hat{y}_{l-1}\)</span> we move forward one layer.</p>
</section>
<section id="activation-functions">
<h3><span class="section-number">13.5.2. </span>Activation functions<a class="headerlink" href="#activation-functions" title="Link to this heading">#</a></h3>
<p>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</p>
<ul class="simple">
<li><p>Non-constant</p></li>
<li><p>Bounded</p></li>
<li><p>Monotonically-increasing</p></li>
<li><p>Continuous</p></li>
</ul>
<p>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.</p>
<p>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 <em>Sigmoid</em></p>
<div class="math notranslate nohighlight">
\[
f(x) = \frac{1}{1 + e^{-x}},
\]</div>
<p>and the <em>hyperbolic tangent</em> function</p>
<div class="math notranslate nohighlight">
\[
f(x) = \tanh(x)
\]</div>
<p>The <em>sigmoid</em> function are more biologically plausible because the
output of inactive neurons are zero. Such activation function are
called <em>one-sided</em>. However, it has been shown that the hyperbolic
tangent performs better than the sigmoid for training MLPs. has
become the most popular for <em>deep neural networks</em></p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>%matplotlib inline
&quot;&quot;&quot;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.&quot;&quot;&quot;
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(&#39;z&#39;)
ax.set_title(&#39;sigmoid function&#39;)
plt.show()
&quot;&quot;&quot;Step Function&quot;&quot;&quot;
z = numpy.arange(-5, 5, .02)
step_fn = numpy.vectorize(lambda z: 1.0 if z &gt;= 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(&#39;z&#39;)
ax.set_title(&#39;step function&#39;)
plt.show()
&quot;&quot;&quot;Sine Function&quot;&quot;&quot;
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(&#39;z&#39;)
ax.set_title(&#39;sine function&#39;)
plt.show()
&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear
unit&quot;&quot;&quot;
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(&#39;z&#39;)
ax.set_title(&#39;Rectified linear unit&#39;)
plt.show()
</pre></div>
</div>
</div>
</div>
</section>
</section>
<section id="the-multilayer-perceptron-mlp">
<h2><span class="section-number">13.6. </span>The multilayer perceptron (MLP)<a class="headerlink" href="#the-multilayer-perceptron-mlp" title="Link to this heading">#</a></h2>
<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
<ol class="arabic simple">
<li><p>A neural network with one or more layers of nodes between the input and the output nodes.</p></li>
<li><p>The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</p></li>
<li><p>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.</p></li>
</ol>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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 <span class="math notranslate nohighlight">\(\hat{z}\)</span>, we can think of this as a basis
expansion of the original inputs <span class="math notranslate nohighlight">\(\hat{x}\)</span>. 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.</p>
<section id="from-one-to-many-layers-the-universal-approximation-theorem">
<h3><span class="section-number">13.6.1. </span>From one to many layers, the universal approximation theorem<a class="headerlink" href="#from-one-to-many-layers-the-universal-approximation-theorem" title="Link to this heading">#</a></h3>
<p>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.</p>
<p>As stated earlier,
an important theorem in studies of neural networks, restated without
proof here, is the <a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&amp;amp;rep=rep1&amp;amp;type=pdf">universal approximation
theorem</a>.</p>
<p>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.</p>
</section>
</section>
<section id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">
<h2><span class="section-number">13.7. </span>Deriving the back propagation code for a multilayer perceptron model<a class="headerlink" href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model" title="Link to this heading">#</a></h2>
<p>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 <span class="math notranslate nohighlight">\(w_{ij}\)</span> 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 <a class="reference external" href="https://www.nature.com/articles/323533a0">back propagation algorithm</a>.</p>
<p>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?</p>
<p>To derive these equations let us start with a plain regression problem
and define our cost function as</p>
<div class="math notranslate nohighlight">
\[
{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
\]</div>
<p>where the <span class="math notranslate nohighlight">\(t_i\)</span>s are our <span class="math notranslate nohighlight">\(n\)</span> targets (the values we want to
reproduce), while the outputs of the network after having propagated
all inputs <span class="math notranslate nohighlight">\(\hat{x}\)</span> are given by <span class="math notranslate nohighlight">\(y_i\)</span>. Below we will demonstrate
how the basic equations arising from the back propagation algorithm
can be modified in order to study classification problems with <span class="math notranslate nohighlight">\(K\)</span>
classes.</p>
<p>With our definition of the targets <span class="math notranslate nohighlight">\(\hat{t}\)</span>, the outputs of the
network <span class="math notranslate nohighlight">\(\hat{y}\)</span> and the inputs <span class="math notranslate nohighlight">\(\hat{x}\)</span> we
define now the activation <span class="math notranslate nohighlight">\(z_j^l\)</span> of node/neuron/unit <span class="math notranslate nohighlight">\(j\)</span> of the
<span class="math notranslate nohighlight">\(l\)</span>-th layer as a function of the bias, the weights which add up from
the previous layer <span class="math notranslate nohighlight">\(l-1\)</span> and the forward passes/outputs
<span class="math notranslate nohighlight">\(\hat{a}^{l-1}\)</span> from the previous layer as</p>
<div class="math notranslate nohighlight">
\[
z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
\]</div>
<p>where <span class="math notranslate nohighlight">\(b_k^l\)</span> are the biases from layer <span class="math notranslate nohighlight">\(l\)</span>. Here <span class="math notranslate nohighlight">\(M_{l-1}\)</span>
represents the total number of nodes/neurons/units of layer <span class="math notranslate nohighlight">\(l-1\)</span>. The
figure here illustrates this equation. We can rewrite this in a more
compact form as the matrix-vector products we discussed earlier,</p>
<div class="math notranslate nohighlight">
\[
\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
\]</div>
<p>With the activation values <span class="math notranslate nohighlight">\(\hat{z}^l\)</span> we can in turn define the
output of layer <span class="math notranslate nohighlight">\(l\)</span> as <span class="math notranslate nohighlight">\(\hat{a}^l = f(\hat{z}^l)\)</span> where <span class="math notranslate nohighlight">\(f\)</span> 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 <span class="math notranslate nohighlight">\(f\)</span> for all layers
and their nodes. It means we have</p>
<div class="math notranslate nohighlight">
\[
a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
\]</div>
<section id="derivatives-and-the-chain-rule">
<h3><span class="section-number">13.7.1. </span>Derivatives and the chain rule<a class="headerlink" href="#derivatives-and-the-chain-rule" title="Link to this heading">#</a></h3>
<p>From the definition of the activation <span class="math notranslate nohighlight">\(z_j^l\)</span> we have</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
\]</div>
<p>With our definition of the activation function we have that (note that this function depends only on <span class="math notranslate nohighlight">\(z_j^l\)</span>)</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
\]</div>
<p>With these definitions we can now compute the derivative of the cost function in terms of the weights.</p>
<p>Let us specialize to the output layer <span class="math notranslate nohighlight">\(l=L\)</span>. Our cost function is</p>
<div class="math notranslate nohighlight">
\[
{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
\]</div>
<p>The derivative of this function with respect to the weights is</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
\]</div>
<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
\]</div>
</section>
<section id="bringing-it-together-first-back-propagation-equation">
<h3><span class="section-number">13.7.2. </span>Bringing it together, first back propagation equation<a class="headerlink" href="#bringing-it-together-first-back-propagation-equation" title="Link to this heading">#</a></h3>
<p>We have thus</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
\]</div>
<p>Defining</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
\]</div>
<p>and using the Hadamard product of two vectors we can write this as</p>
<div class="math notranslate nohighlight">
\[
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
\]</div>
<p>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 <span class="math notranslate nohighlight">\(j\)</span>th
output activation. If, for example, the cost function doesn’t depend
much on a particular output node <span class="math notranslate nohighlight">\(j\)</span>, then <span class="math notranslate nohighlight">\(\delta_j^L\)</span> will be small,
which is what we would expect. The first term on the right, measures
how fast the activation function <span class="math notranslate nohighlight">\(f\)</span> is changing at a given activation
value <span class="math notranslate nohighlight">\(z_j^L\)</span>.</p>
<p>Notice that everything in the above equations is easily computed. In
particular, we compute <span class="math notranslate nohighlight">\(z_j^L\)</span> while computing the behaviour of the
network, and it is only a small additional overhead to compute
<span class="math notranslate nohighlight">\(f'(z^L_j)\)</span>. 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</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial {\cal C}}{\partial (a_j^L)}
\]</div>
<p>With the definition of <span class="math notranslate nohighlight">\(\delta_j^L\)</span> we have a more compact definition of the derivative of the cost function in terms of the weights, namely</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
\]</div>
<p>It is also easy to see that our previous equation can be written as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
\]</div>
<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases <span class="math notranslate nohighlight">\(b_j^L\)</span>, namely</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
\]</div>
<p>That is, the error <span class="math notranslate nohighlight">\(\delta_j^L\)</span> is exactly equal to the rate of change of the cost function as a function of the bias.</p>
<p>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</p>
<p><strong>The starting equations.</strong></p>
<!-- Equation labels as ordinary links -->
<div id="_auto8"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
\label{_auto8} \tag{13}
\end{equation}
\]</div>
<p>and</p>
<!-- Equation labels as ordinary links -->
<div id="_auto9"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
\label{_auto9} \tag{14}
\end{equation}
\]</div>
<p>and</p>
<!-- Equation labels as ordinary links -->
<div id="_auto10"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
\label{_auto10} \tag{15}
\end{equation}
\]</div>
<p>An interesting consequence of the above equations is that when the
activation <span class="math notranslate nohighlight">\(a_k^{L-1}\)</span> 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.</p>
<p>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 <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span> (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.</p>
<p>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 <span class="math notranslate nohighlight">\(L-1\)</span> in terms of the errors in the final output layer.</p>
</section>
<section id="final-back-propagating-equation">
<h3><span class="section-number">13.7.3. </span>Final back propagating equation<a class="headerlink" href="#final-back-propagating-equation" title="Link to this heading">#</a></h3>
<p>We have that (replacing <span class="math notranslate nohighlight">\(L\)</span> with a general layer <span class="math notranslate nohighlight">\(l\)</span>)</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
\]</div>
<p>We want to express this in terms of the equations for layer <span class="math notranslate nohighlight">\(l+1\)</span>. Using the chain rule and summing over all <span class="math notranslate nohighlight">\(k\)</span> entries we have</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
\]</div>
<p>and recalling that</p>
<div class="math notranslate nohighlight">
\[
z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
\]</div>
<p>with <span class="math notranslate nohighlight">\(M_l\)</span> being the number of nodes in layer <span class="math notranslate nohighlight">\(l\)</span>, we obtain</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
\]</div>
<p>This is our final equation.</p>
<p>We are now ready to set up the algorithm for back propagation and learning the weights and biases.</p>
</section>
<section id="setting-up-the-back-propagation-algorithm">
<h3><span class="section-number">13.7.4. </span>Setting up the Back propagation algorithm<a class="headerlink" href="#setting-up-the-back-propagation-algorithm" title="Link to this heading">#</a></h3>
<p>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.</p>
<p>First, we set up the input data <span class="math notranslate nohighlight">\(\hat{x}\)</span> and the activations
<span class="math notranslate nohighlight">\(\hat{z}_1\)</span> of the input layer and compute the activation function and
the pertinent outputs <span class="math notranslate nohighlight">\(\hat{a}^1\)</span>.</p>
<p>Secondly, we perform then the feed forward till we reach the output
layer and compute all <span class="math notranslate nohighlight">\(\hat{z}_l\)</span> of the input layer and compute the
activation function and the pertinent outputs <span class="math notranslate nohighlight">\(\hat{a}^l\)</span> for
<span class="math notranslate nohighlight">\(l=2,3,\dots,L\)</span>.</p>
<p>Thereafter we compute the ouput error <span class="math notranslate nohighlight">\(\hat{\delta}^L\)</span> by computing all</p>
<div class="math notranslate nohighlight">
\[
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
\]</div>
<p>Then we compute the back propagate error for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
\]</div>
<p>Finally, we update the weights and the biases using gradient descent for each <span class="math notranslate nohighlight">\(l=L-1,L-2,\dots,2\)</span> and update the weights and biases according to the rules</p>
<div class="math notranslate nohighlight">
\[
w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
\]</div>
<div class="math notranslate nohighlight">
\[
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
\]</div>
<p>The parameter <span class="math notranslate nohighlight">\(\eta\)</span> 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.</p>
</section>
</section>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#feed-forward-neural-networks">13.1. Feed-forward neural networks</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-vector-notation-and-activation">13.5.1. Matrix-vector notation and activation</a></li>
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<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#from-one-to-many-layers-the-universal-approximation-theorem">13.6.1. From one to many layers, the universal approximation theorem</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">13.7. Deriving the back propagation code for a multilayer perceptron model</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-and-the-chain-rule">13.7.1. Derivatives and the chain rule</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#bringing-it-together-first-back-propagation-equation">13.7.2. Bringing it together, first back propagation equation</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#final-back-propagating-equation">13.7.3. Final back propagating equation</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#setting-up-the-back-propagation-algorithm">13.7.4. Setting up the Back propagation algorithm</a></li>
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