Pre-Requisites

You will need:

• Python 3
• A text editor

We will also download these libraries later:

• scikit-learn
• NumPy
• MNIST-for-Numpy

Why Python?

Basically, it is what everyone uses.

It is an incredibly powerful interpreted language with many useful machine learning and visualisation libraries that can interact well with each other.

Furthermore, python is incredibly widely used for other applications, allowing Python+ML to enhance those massively. (eg. Houdini+scikit-learn)

What types of machine learning are there?

There are three broad categories of machine learning

1. Supervised Learning
2. Unsupervised Learning
3. Reinforcement Learning

Supervised Learning

When a machine learning algorithm attempts to infer a function from labeled training data.

Examples Include

• The MNIST Database
• MariFlow
• Google Translate
• TensorFlow Playground

Unsupervised Learning

When a machine learning algorithm attempts to infer some kind underlying structure to unlabelled data.

Examples Include

• Clustering Workout Sessions
• CycleGan
• Super-Resolution
• Edmond de Belamy

Reinforcement Learning

When a machine learning algorithm seeks to take actions in order to maximize some kind of reward.

Examples Include

• MarI/O
• Random Network Distillation
• AlphaGo
• Unity ML-Agents Toolkit

Recap:

We have..

• Supervised Learning
• Unsupervised Learning
• Reinforcement Learning

Neural Networks: A Trained Example

The goal of a neural network (in this case!) is to model some function $f(\mathbf{x})$.

Where $\mathbf{x}$ is a multi-dimensional vector, eg. $784$ values representing the pixels of a $28 \cdot 28$ image.

The output could be a number the image represents.

Neural Networks: A Trained Example

So in this example, we want our neural network to take an image of the number $5$ and turn it into the number $5$.

$\rightarrow$ $5$

Neural Networks: An Explanation

A simple network, the multilayer perceptron:

What do we have?

1. We have some layers: $1$ input layer, $1$ output layer and $0$+ hidden layers.
2. Each layer has a number of neurons.
3. The first layer has 784, the last layer has 5.
4. Every neuron in each layer is connected to every neuron in the layer before and after.
5. Each connection, has some kind of weight.

How does the data flow through the network?

1. The input neurons just output the value of the pixel that they correspond to.
2. For all other neurons, we have the following equation: $$ a^L_j=\sigma(b^L_j + \sum_{k=1}^{n_{L-1}} w^L_{jk} a^{L-1}_k) $$ We will explore the meaning of this with drawing now...

The activation function

$\sigma$ is the activation function, processing all the inputs into a neuron. We will use the sigmoid function:

$$ \sigma (x) = \frac{1}{1 + e^{-x}} $$

The activation function

The output of the sigmoid function looks like this:

Neural Networks: An Explanation

DRAWING TIME!!!! :D

How to train a network: A single-neuron

Some contrived example data

Some contrived example data

The maths of a single neuron

We can simplify the maths down to (1 layer, 1 neuron, no activation function $\sigma$): $$ a(x)=b + w x $$ This closely matches our desired output function: $$ y = c + m x $$

Finding $b$ and $w$

We need to train the neural network so that it finds $b$ and $w$ to best fit the data.

$$ a(x)=b + w x $$

We can do this with a cost function.

$$ C=\sum_{i=0}^{N-1} (a(x_i) - y_i)^2 $$

Where $(x_i, y_i)$ represents each sample from our training data.

Reducing the cost

Now we have a cost, we can calculate a new weight and bias which minimizes the cost:

$$ w'=w-\mu\frac{\partial C}{\partial w} $$

$$ b'=b-\mu\frac{\partial C}{\partial b} $$

Here, $\mu$ is what we call the learning rate.

Working through the maths...

$$ \frac{\partial C}{\partial w} = \sum_{i=0}^{N-1} 2 x_i ((b + wx_i) - y_i) $$

$$ \frac{\partial C}{\partial b} = \sum_{i=0}^{N-1} 2 ((b + wx_i) - y_i) $$

Example with $\mu=0.01$

$m$ = 0.68, $c$ = 2.75

1. $w$ = 1.00, $b$ = 1.00
2. $w$ = -1.46, $b$ = 0.95
3. $w$ = 0.30, $b$ = 1.50
4. $w$ = -1.08, $b$ = 1.52
5. $w$ = -0.10, $b$ = 1.86
8. $w$ = -0.77, $b$ = 2.16
18. $w$ = -0.68, $b$ = 2.66
26. $w$ = -0.68, $b$ = 2.73
36. $w$ = -0.68, $b$ = 2.75

Example with $\mu=0.001$

$m$ = 0.68, $c$ = 2.75

1. $w$ = 1.00, $b$ = 1.00
2. $w$ = 0.75, $b$ = 1.00
3. $w$ = 0.55, $b$ = 1.00
5. $w$ = 0.24, $b$ = 1.01
8. $w$ = -0.05, $b$ = 1.06
18. $w$ = -0.40, $b$ = 1.28
57. $w$ = -0.56, $b$ = 1.97
126. $w$ = -0.64, $b$ = 2.50
357. $w$ = -0.68, $b$ = 2.74
358. $w$ = -0.68, $b$ = 2.75

Example with $\mu=0.1$

$m$ = 0.68, $c$ = 2.75

1. $w$ = 1.00, $b$ = 1.00
2. $w$ = -23.61, $b$ = 0.54
3. $w$ = 373.85, $b$ = 59.07
4. $w$ = -6166, $b$ = -937.5
5. $w$ = 1.015e+05, $b$ = 1.549e+04
8. $w$ = -4.535e+08, $b$ = -6.919e+07
18. $w$ = -6.652e+20, $b$ = -1.015e+20
26. $w$ = -3.599e+30, $b$ = -5.49e+29
36. $w$ = -5.279e+42, $b$ = -8.054e+41

In conclusion

We have worked through the maths of a single-neuron network.

1. Representing it with a function $a(x)=b + w x$.
2. Applying a cost when applied to training data $C=\sum_{i=0}^{N-1} (a(x_i) - y_i)^2$.
3. Calculating cost derivative w.r.t. $w$ and $b$ $\frac{\partial C}{\partial w}$, $\frac{\partial C}{\partial b}$.
4. Obtaining $w'=w-\mu\frac{\partial C}{\partial w}$ and $b'=b-\mu\frac{\partial C}{\partial b}$.
5. Run the above multiple times to train our network.
6. Choose a good $\mu$!

Any Questions?

What glaring flaw do you potentially see with our values for $w$ and $b$?

What about the model overall?

Demonstration Time!

1. Setup the project folder and install Python dependencies
2. Download and load the dataset
3. Train and evaluate a neural network on the dataset

Sorting project and dependencies

Create folder and add a text file called "process_digits.py".

Then, ensuring you have Python installed, run:

				pip install scikit-learn
				pip install numpy
			

Downloading the dataset

The dataset can be found at http://yann.lecun.com/exdb/mnist/.

However, we will be using MNIST-for-Numpy to download and then load the data into a "process_digits.py".

Training and evaluating a neural network