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<h1>Week 40: Neural networks</h1>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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[1] <b>Department of Physics, University of Oslo, Norway</b>
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[2] <b>Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA</b>
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<br>
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<h4>Oct 8, 2022</h4>
</center> <!-- date -->
<br>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="plan-for-week-40">Plan for week 40 </h2>
<ul>
<li> Thursday: Repetion and summary of Stochastic Gradient descent with examples and automatic differentiation and begin Neural Networks.</li>
<ul>
<li> <a href="https://youtu.be/sCHiGSoG2lE" target="_blank">Video of lecture</a></li>
</ul>
<li> Friday: Neural Networks, setting up the basic steps, from the simple perceptron model to the multi-layer perceptron model. Presentation of project 2.</li>
<ul>
<li> <a href="https://youtu.be/hDTtA7PRRfI" target="_blank">Video of lecture</a></li>
</ul>
</ul>
<p>Reading suggestions for both days: <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf" target="_blank">Aurelien Geron's chapter 10</a> and Hastie et al chapter 11.
For Stochastic Gradient Descent, we recommend chapter 4 of Geron's text.
For neural networks we recommend Goodfellow et al chapters 6 and 7 and Bishop 5.1-5.4
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="overview-video-on-stochastic-gradient-descent">Overview video on Stochastic Gradient Descent </h2>
<a href="https://www.youtube.com/watch?v=vMh0zPT0tLI&ab_channel=StatQuestwithJoshStarmer" target="_blank">What is Stochastic Gradient Descent</a>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="practical-tips">Practical tips </h2>
<ul>
<li> <b>Randomize the data when making mini-batches</b>. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.</li>
<li> <b>Transform your inputs</b>. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the cost function is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.</li>
<li> <b>Monitor the out-of-sample performance.</b> Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This <em>early stopping</em> significantly improves performance in many settings.</li>
<li> <b>Adaptive optimization methods don't always have good generalization.</b> Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.</li>
</ul>
<p>Geron's text, see chapter 11, has several interesting discussions.</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="automatic-differentiation">Automatic differentiation </h2>
<p><a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_blank">Automatic differentiation (AD)</a>,
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.
</p>
<p>Automatic differentiation is neither:</p>
<ul>
<li> Symbolic differentiation, nor</li>
<li> Numerical differentiation (the method of finite differences).</li>
</ul>
<p>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
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="videos-on-neural-networks">Videos on Neural Networks </h2>
<a href="https://www.youtube.com/watch?v=bxe2T-V8XRs&list=PLiaHhY2iBX9hdHaRr6b7XevZtgZRa1PoU&ab_channel=WelchLabs" target="_blank">Neural Networks demystified</a>
<a href="https://www.youtube.com/watch?v=Wo5dMEP_BbI&list=PLQVvvaa0QuDcjD5BAw2DxE6OF2tius3V3&ab_channel=sentdex" target="_blank">Building Neural Networks from scratch</a>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="neural-networks">Neural networks </h2>
<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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="artificial-neurons">Artificial neurons </h2>
<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>
$$
\begin{equation}
y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
\label{artificialNeuron}
\end{equation}
$$
<p>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.
</p>
<p>Conceptually, it is helpful to divide neural networks into four
categories:
</p>
<ol>
<li> general purpose neural networks for supervised learning,</li>
<li> neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),</li>
<li> neural networks for sequential data such as Recurrent Neural Networks (RNNs), and</li>
<li> neural networks for unsupervised learning such as Deep Boltzmann Machines.</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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="neural-network-types">Neural network types </h2>
<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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="feed-forward-neural-networks">Feed-forward neural networks </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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="convolutional-neural-network">Convolutional Neural Network </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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="recurrent-neural-networks">Recurrent neural networks </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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="other-types-of-networks">Other types of networks </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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="multilayer-perceptrons">Multilayer perceptrons </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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="why-multilayer-perceptrons">Why multilayer perceptrons? </h2>
<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 <b>non-constant, bounded and
monotonically-increasing continuous function</b>.
</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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model">Illustration of a single perceptropn model and a multi-perceptron model </h2>
<center> <!-- FIGURE -->
<hr class="figure">
<center>
<p class="caption">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. </p>
</center>
<p><img src="figures/nns.png" width="600" align="bottom"></p>
</center>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="examples-of-xor-or-and-and-gates">Examples of XOR, OR and AND gates </h2>
<p>Let us first try to fit various gates using standard linear
regression. The gates we are thinking of are the classical XOR, OR and
AND gates, well-known elements in computer science. The tables here
show how we can set up the inputs \( x_1 \) and \( x_2 \) in order to yield a
specific target \( y_i \).
</p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #BA2121; font-style: italic">Simple code that tests XOR, OR and AND gates with linear regression</span>
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #408080; font-style: italic"># Design matrix</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The invers of X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># The XOR gate </span>
yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
ThetaXOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># The OR gate </span>
yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
ThetaOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># The OR gate </span>
yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
ThetaAND <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
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<p>What is happening here?</p>
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<h2 id="does-logistic-regression-do-a-better-job">Does Logistic Regression do a better Job? </h2>
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<pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #BA2121; font-style: italic">Simple code that tests XOR and OR gates with linear regression</span>
<span style="color: #BA2121; font-style: italic">and logistic regression</span>
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #408080; font-style: italic"># Design matrix</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>], [<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>],[<span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>]],dtype<span style="color: #666666">=</span>np<span style="color: #666666">.</span>float64)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
Xinv <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>pinv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The invers of X.TX matrix:</span><span style="color: #BB6688; font-weight: bold">{</span>Xinv<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># The XOR gate </span>
yXOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">0</span>])
ThetaXOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yXOR
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction for the XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaXOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># The OR gate </span>
yOR <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">1</span> ,<span style="color: #666666">1</span>, <span style="color: #666666">1</span>])
ThetaOR <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yOR
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction for the OR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaOR<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># The OR gate </span>
yAND <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array( [ <span style="color: #666666">0</span>, <span style="color: #666666">0</span> ,<span style="color: #666666">0</span>, <span style="color: #666666">1</span>])
ThetaAND <span style="color: #666666">=</span> Xinv <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> yAND
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The values of theta for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;The linear regression prediction for the AND gate:</span><span style="color: #BB6688; font-weight: bold">{</span>X <span style="color: #666666">@</span> ThetaAND<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
<span style="color: #408080; font-style: italic"># Now we change to logistic regression</span>
<span style="color: #408080; font-style: italic"># Logistic Regression</span>
logreg <span style="color: #666666">=</span> LogisticRegression()
logreg<span style="color: #666666">.</span>fit(X, yOR)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for OR gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yOR)))
logreg<span style="color: #666666">.</span>fit(X, yXOR)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for XOR gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yXOR)))
logreg<span style="color: #666666">.</span>fit(X, yAND)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;Test set accuracy with Logistic Regression for AND gate: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">&quot;</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X,yAND)))
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<p>Not exactly impressive, but somewhat better.</p>
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<h2 id="adding-neural-networks">Adding Neural Networks </h2>
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># and now neural networks with Scikit-Learn and the XOR</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPClassifier
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_classification
X, yXOR <span style="color: #666666">=</span> make_classification(n_samples<span style="color: #666666">=100</span>, random_state<span style="color: #666666">=1</span>)
FFNN <span style="color: #666666">=</span> MLPClassifier(random_state<span style="color: #666666">=1</span>, max_iter<span style="color: #666666">=300</span>)<span style="color: #666666">.</span>fit(X, yXOR)
FFNN<span style="color: #666666">.</span>predict_proba(X)
<span style="color: #008000">print</span>(<span style="color: #BA2121">f&quot;Test set accuracy with Feed Forward Neural Network for XOR gate:</span><span style="color: #BB6688; font-weight: bold">{</span>FFNN<span style="color: #666666">.</span>score(X, yXOR)<span style="color: #BB6688; font-weight: bold">}</span><span style="color: #BA2121">&quot;</span>)
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<h2 id="mathematical-model">Mathematical model </h2>
<p>The output \( y \) is produced via the activation function \( f \)</p>
$$
y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
$$
<p>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 <em>inputs</em> \( x_i \) 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>
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<h2 id="mathematical-model">Mathematical model </h2>
<p>First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),</p>
$$
\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
\label{_auto1}
\end{equation}
$$
<p>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
</p>
$$
\begin{equation}
y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
\label{outputLayer1}
\end{equation}
$$
<p>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,
</p>
$$
\begin{equation}
y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
\label{generalLayer}
\end{equation}
$$
<p>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.
</p>
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<h2 id="mathematical-model">Mathematical model </h2>
<p>The output of neuron \( i \) in layer 2 is thus,</p>
$$
\begin{align}
y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
\label{_auto2}\\
&= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
\label{outputLayer2}
\end{align}
$$
<p>where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads</p>
$$
\begin{align}
y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
\label{_auto3}\\
&= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ b_1^3\right]
\label{_auto4}
\end{align}
$$
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<h2 id="mathematical-model">Mathematical model </h2>
<p>We can generalize this expression to an MLP with \( l \) hidden
layers. The complete functional form is,
</p>
$$
\begin{align}
&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
\label{completeNN}
\end{align}
$$
<p>which illustrates a basic property of MLPs: The only independent
variables are the input values \( x_n \).
</p>
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<h2 id="mathematical-model">Mathematical model </h2>
<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 \( \hat{x} \in \mathbb{R}^n \rightarrow
\hat{y} \in \mathbb{R}^m \).
</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>
$$
\begin{equation}
f(x) = c_1 f(c_2 x + c_3) + c_4
\label{_auto5}
\end{equation}
$$
<p>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.
</p>
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<h3 id="matrix-vector-notation">Matrix-vector notation </h3>
<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 \( \hat{b}_l \) 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.
</p>
<p>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
</p>
$$
\begin{equation}
\hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &w^2_{12} &w^2_{13} \\
w^2_{21} &w^2_{22} &w^2_{23} \\
w^2_{31} &w^2_{32} &w^2_{33} \\
\end{array} \right] \cdot
\left[\begin{array}{c}
y^1_1 \\
y^1_2 \\
y^1_3 \\
\end{array}\right] +
\left[\begin{array}{c}
b^2_1 \\
b^2_2 \\
b^2_3 \\
\end{array}\right]\right).
\label{_auto6}
\end{equation}
$$
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<h3 id="matrix-vector-notation-and-activation">Matrix-vector notation and activation </h3>
<p>The activation of node \( i \) in layer 2 is</p>
$$
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
\label{_auto7}
\end{equation}
$$
<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
\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
</p>
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<h3 id="activation-functions">Activation functions </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>
<li> Non-constant</li>
<li> Bounded</li>
<li> Monotonically-increasing</li>
<li> Continuous</li>
</ul>
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<h3 id="activation-functions-logistic-and-hyperbolic-ones">Activation functions, Logistic and Hyperbolic ones </h3>
<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>
$$
f(x) = \frac{1}{1 + e^{-x}},
$$
<p>and the <em>hyperbolic tangent</em> function</p>
$$
f(x) = \tanh(x)
$$
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<h3 id="relevance">Relevance </h3>
<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>
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<pre style="line-height: 125%;"><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a </span>
<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mt</span>
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.1</span>)
sigma_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1/</span>(<span style="color: #666666">1+</span>numpy<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z)))
sigma <span style="color: #666666">=</span> sigma_fn(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, sigma)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sigmoid function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.02</span>)
step_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1.0</span> <span style="color: #008000; font-weight: bold">if</span> z <span style="color: #666666">&gt;=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
step <span style="color: #666666">=</span> step_fn(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, step)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;step function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">0.1</span>)
t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>sin(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, t)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi,<span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sine function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
<span style="color: #BA2121; font-style: italic">unit&quot;&quot;&quot;</span>
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>, <span style="color: #666666">.1</span>)
zero <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>zeros(<span style="color: #008000">len</span>(z))
y <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>max([zero, z], axis<span style="color: #666666">=0</span>)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, y)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Rectified linear unit&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre>
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<h2 id="the-multilayer-perceptron-mlp">The multilayer perceptron (MLP) </h2>
<p>The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of</p>
<ol>
<li> A neural network with one or more layers of nodes between the input and the output nodes.</li>
<li> The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.</li>
<li> 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.</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 \( \hat{z} \), 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.
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="from-one-to-many-layers-the-universal-approximation-theorem">From one to many layers, the universal approximation theorem </h2>
<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 href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf" target="_blank">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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="deriving-the-back-propagation-code-for-a-multilayer-perceptron-model">Deriving the back propagation code for a multilayer perceptron model </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 \( w_{ij} \) 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 <a href="https://www.nature.com/articles/323533a0" target="_blank">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>
$$
{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
$$
<p>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.
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="definitions">Definitions </h2>
<p>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
</p>
$$
z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
$$
<p>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,
</p>
$$
\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
$$
<p>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
</p>
$$
a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
$$
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="derivatives-and-the-chain-rule">Derivatives and the chain rule </h2>
<p>From the definition of the activation \( z_j^l \) we have</p>
$$
\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
$$
<p>and</p>
$$
\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
$$
<p>With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))</p>
$$
\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
$$
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="derivative-of-the-cost-function">Derivative of the cost function </h2>
<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 \( l=L \). Our cost function is</p>
$$
{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
$$
<p>The derivative of this function with respect to the weights is</p>
$$
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
$$
<p>The last partial derivative can easily be computed and reads (by applying the chain rule)</p>
$$
\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
$$
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="bringing-it-together-first-back-propagation-equation">Bringing it together, first back propagation equation </h2>
<p>We have thus</p>
$$
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
$$
<p>Defining</p>
$$
\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
$$
<p>and using the Hadamard product of two vectors we can write this as</p>
$$
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
$$
<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 $j$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 \( z_j^L \).
</p>
<p>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
</p>
$$
\frac{\partial {\cal C}}{\partial (a_j^L)}
$$
<p>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</p>
$$
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
$$
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="derivatives-in-terms-of-z-j-l">Derivatives in terms of \( z_j^L \) </h2>
<p>It is also easy to see that our previous equation can be written as</p>
$$
\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
$$
<p>which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely</p>
$$
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
$$
<p>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. </p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="bringing-it-together">Bringing it together </h2>
<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>
<div class="alert alert-block alert-block alert-text-normal">
<b>The starting equations</b>
<p>
$$
\begin{equation}
\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
\label{_auto8}
\end{equation}
$$
<p>and</p>
$$
\begin{equation}
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
\label{_auto9}
\end{equation}
$$
<p>and</p>
$$
\begin{equation}
\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
\label{_auto10}
\end{equation}
$$
</div>
<p>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.
</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 \( 0 \) 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.
</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 \( L-1 \) in terms of the errors in the final output layer.
</p>
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<h2 id="final-back-propagating-equation">Final back propagating equation </h2>
<p>We have that (replacing \( L \) with a general layer \( l \))</p>
$$
\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}.
$$
<p>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</p>
$$
\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}},
$$
<p>and recalling that</p>
$$
z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},
$$
<p>with \( M_l \) being the number of nodes in layer \( l \), we obtain</p>
$$
\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),
$$
<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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
<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>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>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 \( \hat{a}^1 \).
</p>
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>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
\( l=2,3,\dots,L \).
</p>
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>Thereafter we compute the ouput error \( \hat{\delta}^L \) by computing all</p>
$$
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
$$
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
$$
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
$$
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>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</p>
$$
w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
$$
$$
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
$$
</div>
<p>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.
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="setting-up-the-back-propagation-algorithm">Setting up the Back propagation algorithm </h2>
<p>The four equations above 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>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>First, we set up the input data \( \boldsymbol{x} \) and the activations
\( \boldsymbol{z}_1 \) of the input layer and compute the activation function and
the pertinent outputs \( \boldsymbol{a}^1 \).
</p>
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>Secondly, we perform then the feed forward till we reach the output
layer and compute all \( \boldsymbol{z}_l \) of the input layer and compute the
activation function and the pertinent outputs \( \boldsymbol{a}^l \) for
\( l=2,3,\dots,L \).
</p>
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>Thereafter we compute the ouput error \( \boldsymbol{\delta}^L \) by computing all</p>
$$
\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.
$$
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>Then we compute the back propagate error for each \( l=L-1,L-2,\dots,2 \) as</p>
$$
\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).
$$
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>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</p>
$$
w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},
$$
$$
b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,
$$
</div>
<p>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.
</p>
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