Branch-AI/one_md_per_day_format/piscine/Week3/w3day1.md

W1D1 Piscine AI - Data Science

Table of Contents:

Reproduire cet article sans back prop https://towardsdatascience.com/machine-learning-for-beginners-an-introduction-to-neural-networks-d49f22d238f9

Introduction

Deep learning is a huge domain. We will focus on Artificial Neural Networks. The goal is to understand how do the neural networks train and train them on data. Understand the challenges of training a neural network Architectures as RNN, LSTM (learn sequences, used in TS and NLP),CNN used a lot in image processing are well know algorithms in deep learning but won't be covered by the AI branch. Once you have a good understanding of ANN feel free to extend your knowledge to new architectures.

Rules

Ressources

https://victorzhou.com/blog/intro-to-neural-networks/

https://srnghn.medium.com/deep-learning-overview-of-neurons-and-activation-functions-1d98286cf1e4#:~:text=What%20is%20a%20neuron%3F,to%20become%20the%20neuron's%20output.

Reproduire cet article sans back prop https://towardsdatascience.com/machine-learning-for-beginners-an-introduction-to-neural-networks-d49f22d238f9

Exercise 1 The neuron

The goal of this exercise is to understand the role of a neuron and to implement a neuron.

An artificial neuron, the basic unit of the neural network, (also referred to as a perceptron) is a mathematical function. It takes one or more inputs that are multiplied by values called “weights” and added together. This value is then passed to a non-linear function, known as an activation function, to become the neuron’s output.

As desbribed in the article, a neuron takes inputs, does some math with them, and produces one output.

Let us assume there are 2 inputs. Here are the three steps involved in the neuron:

1. Each input is multiplied by a weight
   • x1 -> x1 * w1
   • x2 -> x2 * w2
2. The weighted inputs are added together with a biais b
   • (x1 * w1) + (x2 * w2) + b
3. The sum is passed through an activation function

   • y = f((x1 * w1) + (x2 * w2) + b)

   • The activation function is a function you know from W2DAY2 (Logistic Regression): the sigmoid

Example:

x1 = 2 , x2 = 3 , w1 = 0, w2= 1, b = 4

1. Step 1: Multiply by a weight
   • x1 -> 2 * 0 = 0
   • x2 -> 3 * 1 = 3
2. Step 2: Add weigthed inputs and bias
   • 0 + 3 + 4 = 7
3. Step 3: Activation function
   • y = f(7) = 0.999

---

1. Implement a the function feedforward of the class Neuron that takes as input the inputs (x1, x2) and that uses the attributes: the weights and the biais to return y:

   class Neuron:
   def __init__(self, weight1, weight2, bias):
       self.weights_1 = weight1
       self.weights_2 = weight2
       self.bias = bias
   
   def feedforward(self, x1, x2):
       #TODO
       return y
   
   
   

Note: if you are confortable with matrix multiplication, feel free to vectorize the operations as done in the article.

https://victorzhou.com/blog/intro-to-neural-networks/

Correction:

1. This question is validated if this code:

   neuron = Neuron(0,1,4)
   neuron.feedforward(2,3)
   

returns 0.9990889488055994.

Exerice 2 Neural network

The goal of this exercise is to understand how to combine three neurons to form a neural network. A neural newtwork is nothing else than neurons connected together. As shown in the figure the neural network is composed of layers:

• Input layer: it only represents input data. It doesn't contain neurons.
• Output layer: it represents the last layer. It contains a neuron (in some cases more than 1).
• Hidden layer: any layer between the input (first) layer and output (last) layer. Many hidden layers can be stacked. When there are many hidden layers, the neural networks is deep.

Notice that the neuron o1 in the output layer takes as input the output of the neurons h1 and h2 in the hidden layer.

In exercise 1, you implemented this neuron.

Now, we add two more neurons:

• h2, the second neuron of the hidden layer
• o1, the neuron of the output layer

1. Implement the function feedforward of the class OurNeuralNetwork that takes as input the input data and returns the output y. Return the output for these neurons:

   neuron_h1 = Neuron(1,2,-1)
   neuron_h2 = Neuron(0.5,1,0)
   neuron_o1 = Neuron(2,0,1)
   

   class OurNeuralNetwork:
   
       def __init__(self, neuron_h1, neuron_h2, neuron_o1):
           self.h1 = neuron_h1
           self.h2 = neuron_h2
           self.o1 = neuron_o1
   
       def feedforward(self, x1, x2):
       # The inputs for o1 are the outputs from h1 and h2
       # TODO
           return y
   
   

Correction

1. This question is validated the output is: 0.9524917424084265

Exercise 3 Log loss

The goal of this exercise is to implement the Log loss function. As mentioned last week, this function is used in classification as a loss function. It means that the better the classifier is, the smaller the loss function is. W2D1, you implemented the gradient descent on the MSE loss to update the weights of the linear regression. Similarly, the minimization of the Log loss leads to finding optimal weights.

Log loss: - 1/n * Sum[(y_true*log(y_pred) + (1-y_true)*log(1-y_pred))]

1. Create a function log_loss_custom and compute the loss for the data below:

   y_true = np.array([0,1,1,0,1])
   y_pred = np.array([0.1,0.8,0.6, 0.5, 0.3])
   

   Check that log_loss from sklearn.metrics returns the same result https://scikit-learn.org/stable/modules/generated/sklearn.metrics.log_loss.html

Correction

1. This question is validated if the output is: 0.5472899351247816.

Exercise 4 Forward propagation

The goal of this exerice is to compute the log loss on the output of the forward propagation. The data used is the tiny data set below.

name	math	chemistry	exam_success
Bob	12	15	1
Eli	10	9	0
Tom	18	18	1
Ryan	13	14	1

The goal if the network is to predict the success at the exam given math and chemistry grades. The inputs are math and chemistry and the target is exam_sucess.

1. Compute and return the output of the neural network for each of the students. Here are the weights and biases of the neural network:

   neuron_h1 = Neuron(0.05, 0.001, 0)
   neuron_h2 = Neuron(0.02, 0.003, 0)
   neuron_o1 = Neuron(2,0,0)
   

2. Compute the logloss for the data given the output of the neural network with the 4 students.

Correction

1. This question is validated if the output is:

   Bob: 0.7855253278357536
   Eli: 0.7771516558846259
   Tom: 0.8067873659804015
   Ryan: 0.7892343955586032
   

2. This question is validated if the logloss for the 4 students is 0.5485133607757963.

Exercise 5 Regression

The goal of this exercise is to learn to adapt the output layer to regression. As a reminder, one of reasons for which the sigmoid is used in classification is because it contracts the output between 0 and 1 which is the expected output range for a probability (W2D2: Logistic regression). However, the output of the regression is not a probability.

In order to perform a regression using a neural network, the activation function of the neuron on the output layer has to be modified to identity function. In mathematics, the identity function is: f(x) = x. In other words it means that it returns the input as so. The three steps become:

1. Each input is multiplied by a weight
   • x1 -> x1 * w1
   • x2 -> x2 * w2
2. The weighted inputs are added together with a biais b
   • (x1 * w1) + (x2 * w2) + b
3. The sum is passed through an activation function
   • y = f((x1 * w1) + (x2 * w2) + b)
   • The activation function is the identity
   • y = (x1 * w1) + (x2 * w2) + b

All other neurons' activation function doesn't change.

1. Adapt the neuron class implemented in exercise 1. It now takes as a parameter regression which is boolean. When its value is True, feedforward should use the identity function as activation function instead of the sigmoid function.

   class Neuron:
   def __init__(self, weight1, weight2, bias, regression):
       self.weights_1 = weight1
       self.weights_2 = weight2
       self.bias = bias
       #TODO
   
   def feedforward(self, x1, x2):
       #TODO
       return y
   
   

   • Compute the output for:

     neuron = Neuron(0,1,4, True)
     neuron.feedforward(2,3)
     

2. Now, the goal of the network is to predict the physics' grade at the exam given math and chemistry grades. The inputs are math and chemistry and the target is physics.

name	math	chemistry	physics
Bob	12	15	16
Eli	10	9	10
Tom	18	18	19
Ryan	13	14	16

Compute and return the output of the neural network for each of the students. Here are the weights and biases of the neural network:

```
#replace regression by the right value
neuron_h1 = Neuron(0.05, 0.001, 0, regression)
neuron_h2 = Neuron(0.002, 0.003, 0, regression)
neuron_o1 = Neuron(2,7,10, regression)
```

3. Compute the MSE for the 4 students.

Correction

1. This question is validated if the output is 7.

2. This question is validated if the outputs are:

   Bob: 14.918863163724454
   Eli: 14.83137890625537
   Tom: 15.086662606964074
   Ryan: 14.939270885974128
   

3. This question is validated if the MSE is 10.237608699909138