Tutorial 1 :Image Classification

Introduction

The objective of this tutorial is to write a complete image classification program in Python. Two classification models will be successively developed and tested: k-nearest neighbors (KNN) and neural networks (NN).

Before you start

In this tutorial we use Python 3.9.14. Make sure you have this version of Python installed.

% python3.9 --version
Python 3.9.14

We assume you are familiar with the venv module of Python and with basic git commands. We assume that you have access to the ECL GitLab.

Prepare your directory

1. Connect to https://gitlab.ec-lyon.fr.
2. Create a new blank project (New project then Create blank project).
3. Fill in the form as follows.
   • Project name: Image classification.
   • Project slug: image-classification.
   • Visibility Level: public
   • Project Configuration: Initialize repository with a README
4. Clone the repository.

git clone https://gitlab.ec-lyon.fr/<user>/image-classification

Prepare the Python envrionment

1. In the project direcotry, create a virtual environment.

python3.9 -m venv env

2. Source the envrionement.

source env/bin/activate

3. Upgrade pip.

pip install --upgrade pip

4. The environment files should not be pushed to the remote directory. To have these files ignored when committing, create a .gitignore file containing env. Similarly, we want to ignore Python cache file, thus add __pycache__ to .gitignore.
5. In this project, we use numpy package for matrices manipulation and the scikit-image package for image manipulation. Thus, create a requirement file named requirements.txt containing:

numpy
scikit-image

6. Install the above mentioned dependencies.

pip install -r requirements.txt

Prepare the CIFAR dataset

The image database used for the experiments is CIFAR-10 which consists of 60 000 color images of size 32x32 divided into 10 classes (plane, car, bird, cat, ...). This database can be obtained at the address https://www.cs.toronto.edu/~kriz/cifar.html where are also given the indications to read the data.

1. Create a folder named data in which you move the downloaded cifar-10-batches-py folder. Make sure that the data folder is ignored when commiting.
2. Create a Python file named read_cifar.py. Write the function read_cifar_batch taking as parameter the path of a single batch as a string, and returning:
   • a matrix data of size (batch_size x data_size) where batch_size is the number of available data in the batch, and data_size the dimension of these data (number of numerical values describing the data), and
   • a vector labels of size batch_size whose values correspond to the class code of the data of the same index in data. data and labels must be np.float32 arrays.
3. Write the function read_cifar taking as parameter the path of the directory containing the six batches (five data_batch and one test_batch) as a string, and returning
   • a matrix data of shape (batch_size x data_size) where batch_size is the number of available data in all batches (including test_batch), and
   • a vector labels of size batch_size whose values correspond to the class code of the data of the same index in data. data and labels must be np.float32 arrays.
4. Write the function split_dataset which splits the dataset into a training set and a test set. The data must be shuffled, so that two successive calls shouldn't give the same output. This function takes as parameter
   • data and labels, two arrays that have the same size in the first dimension.
   • split, a float between 0 and 1 which determines the split factor of the training set with respect to the test set. This function must return
   • data_train the training data,
   • labels_train the corresponding labels,
   • data_test the testing data, and
   • labels_test the corresponding labels.

k-nearest neighbors

1. Create a Python fil named knn.py. Write the function distance_matrix taking as parameters two matrices and returns dists, the L2 Euclidean distance matrix. The computation must be done only with matrix manipulation (no loops). Hint:
   $(a-b)^2 = a^2 + b^2 - 2 ab$
2. Write the function knn_predict taking as parameters:
   • dists the distance matrix between the train set and the test set,
   • labels_train the training labels, and
   • k the number of of neighbors. This function must return the predicted labels for the elements of data_train.
3. Write the function evaluate_knn taking as parameters:
   • data_train the training data,
   • labels_train the corresponding labels,
   • data_test the testing data,
   • labels_test the corresponding labels, and
   • k the number of of neighbors. This function must return the classification rate (accuracy).
4. For split_factor=0.9, plot the variation of the accuracy as a function of k (from 1 to 20). Save the plot as an image named knn.png in the directory results.

Artificial Neural Network

The objective here is to develop a classifier based on a multilayer perceptron (MLP) neural network.

First of all, let's focus on the backpropagation of the gradient with an example. Let's consider a network with a hidden layer.

The weight matrix of the layer

1. Prove that
   $\sigma' = \sigma \times (1-\sigma)$
2. Express
   $\frac{\partial C}{\partial A^{(2)}}$
   , i.e. the vector of
   $\frac{\partial C}{\partial a^{(2)}_i}$
   as a function of
   $A^{(2)}$
   and
   $Y$
   .
3. Using the chaining rule, express
   $\frac{\partial C}{\partial Z^{(2)}}$
   , i.e. the vector of
   $\frac{\partial C}{\partial z^{(2)}_i}$
   as a function of
   $\frac{\partial C}{\partial A^{(2)}}$
   and
   $A^{(2)}$
   .
4. Similarly, express
   $\frac{\partial C}{\partial W^{(2)}}$
   , i.e. the matrix of
   $\frac{\partial C}{\partial w^{(2)}_{i,j}}$
   as a function of
   $\frac{\partial C}{\partial Z^{(2)}}$
   and
   $A^{(1)}$
   .
5. Similarly, express
   $\frac{\partial C}{\partial B^{(2)}}$
   as a function of
   $\frac{\partial C}{\partial Z^{(2)}}$
   .
6. Similarly, express
   $\frac{\partial C}{\partial A^{(1)}}$
   as a function of
   \frac{\partial C}{\partial Z^{(2)}}
   and
   W^{(2)}
   .
7. Similarly, express
   \frac{\partial C}{\partial Z^{(1)}}
   as a function of
   \frac{\partial C}{\partial A^{(1)}}
   and
   A^{(1)}
   .
8. Similarly, express
   \frac{\partial C}{\partial W^{(1)}}
   as a function of
   \frac{\partial C}{\partial Z^{(1)}}
   and
   A^{(0)}
   .
9. Similarly, express
   \frac{\partial C}{\partial B^{(1)}}
   as a function of
   \frac{\partial C}{\partial Z^{(1)}}
   .

Below is a Python code performing a forward pass and computing the cost in a network containing a hidden layer and using the sigmoid function as the activation function:

import numpy as np

N = 30  # number of input data
d_in = 3  # input dimension
d_h = 3  # number of neurons in the hidden layer
d_out = 2  # output dimension (number of neurons of the output layer)

# Random initialization of the network weights and biaises
w1 = 2 * np.random.rand(d_in, d_h) - 1  # first layer weights
b1 = np.zeros((1, d_h))  # first layer biaises
w2 = 2 * np.random.rand(d_h, d_out) - 1  # second layer weights
b2 = np.zeros((1, d_out))  # second layer biaises

data = np.random.rand(N, d_in)  # create a random data
labels = np.random.rand(N, d_out)  # create a random labels

# Forward pass
a0 = data # the data are the input of the first layer
z1 = np.matmul(a0, w1) + b1  # input of the hidden layer
a1 = 1 / (1 + np.exp(-z1))  # output of the hidden layer (sigmoid activation function)
z2 = np.matmul(a1, w2) + b2  # input of the output layer
a2 = 1 / (1 + np.exp(-z2))  # output of the output layer (sigmoid activation function)
labels_pred = a2  # the predicted values are the outputs of the output layer

# Compute loss (MSE)
loss = np.mean(np.square(labels_pred - labels))
print(loss)

10. Create a Python file named mlp.py. Use the above code to write the function learn_once_mse taking as parameters:

    • w1, b1, w2 and b2 the weights and biases of the network,
    • data a matrix of shape (batch_size x d_in),
    • labels a matrix of shape (batch_size x d_out),
    • learning_rate the learning rate,

    that perform one gradient descent step, and returns:

    • w1, b1, w2 and b2 the updated weights and biases of the network,
    • loss the loss, for monitoring purpose.

For classification task, we prefer to use a binary cross-entropy loss. We also want to replace the last activation layer of the network with a softmax layer.

10. Write the function one_hot taking a (n)-D array as parameters and returning the corresponding (n+1)-D one-hot matrix.

11. Write a function learn_once_cross_entropy taking the the same parameters as learn_once_mse and returns the same outputs. The function must use a cross entropy loss and the last layer of the network must be a softmax. We admit that

    \frac{\partial C}{\partial Z^{(2)}} = A^{(2)} - Y
    . Where
    Y
    is a one-hot vector encoding the label.

12. Write the function evaluate_mlp taking as parameter:

    • data_train, labels_train, data_test, labels_test, the training and testing data,
    • d_h the number of neurons in the hidden layer
    • learning_rate the learning rate, and
    • num_epoch the number of training epoch,

    that train an MLP classifier and return the test accuracy computed on the test set.

13. For split_factor=0.9, d_h=64, learning_rate=0.1 and num_epoch=10_000, plot the evolution of accuracy across learning epochs. Save the graph as an image named mlp.png in the results directory.

To be handed in

This work (KNN and MLP) must be done individually. The expected output is the archive containing the complete, minimal and functional code corresponding to the tutorial on https://gitlab.ec-lyon.fr. To see the details of the expected, see the Evaluation section.

The last commit is inteded before Monday, November 16, 2022.

Additional requirements

Unittest

Your code must contain unit tests. All unit tests should be contained in the tests directory located at the root of the directory. We choose to use pytest. To help you write unit tests, you can consult the pytest documentation.

Code style

Your code must strictly follow the PEP8 recommendations. To help you format your code properly, you can use Black. To help you sort your imports, you and isort

Docstring

Your code must be properly documented. It must follow the PEP257 recommendations. To help you document your code properly, you can use pydocstyle.

License

Your project must be properly licensed. Since it is your project, it is up to you to choose your license. In general, the license consists of a file named LICENSE in the root directory. A useful resource to help you choose: https://choosealicense.com/

Evaluation

In this section, we present all the items on which the work is evaluated.

• ( /1) The function read_cifar_batch works as described

• ( /1) The function read_cifar works as described

• ( /1) The split_dataset works as described

• ( /1) The function distance_matrix works as described

• ( /1) The function knn_predict works as described

• ( /1) The graph knn.png shows the results obtained

• ( /3) Demonstrations of back propagation are done without error.

• ( /1) The function learn_once_mse works as described

• ( /1) The function one_hot works as described

• ( /1) The function learn_once_cross_entropy works as described

• ( /1) The function evaluate_mlp works as described

• ( /1) The graph mlp.png shows the results obtained

• ( /3) Unitest coverage

• ( /2) The guidlines about the project structure are all followed

  To check if the project has the right structure, install tree and run from the project directory:

  $ tree -I 'env|*__pycache__*'
  .
  └── tests
      └── test_knn.py
  
  1 directory, 1 file

  The output must strictly match the one provided above.

• ( /1) Project has a license

• ( /2) All functions are documented

• ( /1) All functions are documented and follow the pydocstyle

• ( /1) The code is properly formatted

  To check if the code is properly formatted, install Black and run from the project repository:

  $ black --check . --exclude env

  $ isort --check . -s env

  These two tests must pass without error.