Cat or not cat?

This shows two simple image-recognition algorithms that can correctly classify pictures as cat or non-cat.
The first is a classic logistic regression while the second – more accurate – is a deep neural network.

You can follow along using the notebook in GitHub also. And this post is part of a series about Machine Learning with Python.

1 – Overview of the data set

The dataset is taken from Andrew Ng’s course on Coursera Introduction to Neural Networks.
The dataset (“data.h5”) contains:

  • a training set of m_train images labeled as cat (y=1) or non-cat (y=0)
  • a test set of m_test images labeled as cat or non-cat
  • each image is of shape (num_px, num_px, 3) where 3 is for the 3 channels (RGB). Thus, each image is square (height = num_px) and (width = num_px).

Let’s get more familiar with the dataset.

import numpy as np
import h5py

    # Loading the data (cat/non-cat)
train_dataset = h5py.File('../datasets/train_catvnoncat.h5', "r")
train_set_x_orig = np.array(train_dataset["train_set_x"][:]) # train set features
train_set_y_orig = np.array(train_dataset["train_set_y"][:]) # train set labels

test_dataset = h5py.File('../datasets/test_catvnoncat.h5', "r")
test_set_x_orig = np.array(test_dataset["test_set_x"][:]) # test set features
test_set_y_orig = np.array(test_dataset["test_set_y"][:]) # test set labels

classes = np.array(test_dataset["list_classes"][:]) # the list of classes

train_set_y = train_set_y_orig.reshape((1, train_set_y_orig.shape[0]))
test_set_y = test_set_y_orig.reshape((1, test_set_y_orig.shape[0]))

Each line of train_set_x_orig and test_set_x_orig is an array representing an image.

2 – Data preprocessing

Common steps for pre-processing a new dataset are:

  • Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, …)
  • Reshape the datasets such that each example is now a vector of size (num_px * num_px * 3, 1)
  • “Standardise” the data

Many software bugs in deep learning come from having matrix/vector dimensions that don’t fit. If you can keep your matrix/vector dimensions straight you will go a long way toward eliminating many bugs.

m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]

print ("Dataset dimensions:")
print ("Number of training examples: m_train = " + str(m_train))
print ("Number of testing examples: m_test = " + str(m_test))
print ("Height/Width of each image: num_px = " + str(num_px))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_set_x shape: " + str(train_set_x_orig.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x shape: " + str(test_set_x_orig.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
[Out]: Dataset dimensions:
 Number of training examples: m_train = 209
 Number of testing examples: m_test = 50
 Height/Width of each image: num_px = 64
 Each image is of size: (64, 64, 3)
 train_set_x shape: (209, 64, 64, 3)
 train_set_y shape: (1, 209)
 test_set_x shape: (50, 64, 64, 3)
 test_set_y shape: (1, 50)

For convenience, we now reshape images of shape (num_px, num_px, 3) in a numpy-array of shape (num_px * num_px * 3, 1). After this, our training (and test) dataset is a numpy-array where each column represents a flattened image. There should be m_train (respectively m_test) columns.

A trick when you want to flatten a matrix X of shape (a,b,c,d) to a matrix X_flatten of shape (b*c*d, a) is to use:

X_flatten = X.reshape(X.shape[0], -1).T      # X.T is the transpose of X

Applied to our example, it is:

# Reshape the training and test examples

train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T
test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T 

print ("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
print ("sanity check after reshaping: " + str(train_set_x_flatten[0:5,0]))
[Out]:
 train_set_x_flatten shape: (12288, 209)
 train_set_y shape: (1, 209)
 test_set_x_flatten shape: (12288, 50)
 test_set_y shape: (1, 50)
 sanity check after reshaping: [17 31 56 22 33]

To represent color images, the red, green and blue channels (RGB) must be specified for each pixel, and so the pixel value is actually a vector of three numbers ranging from 0 to 255.

One common preprocessing step in machine learning is to center and standardize your dataset, meaning that you substract the mean of the whole numpy array from each example, and then divide each example by the standard deviation of the whole numpy array. But for picture datasets, it is simpler and more convenient and works almost as well to just divide every row of the dataset by 255 (the maximum value of a pixel channel).

Let’s standardise our dataset.

train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

Logistic Regression

We will build first a logistic regression classifier to recognise the cats.

logistic

3 – General Architecture of the learning algorithm

It’s time to design a simple algorithm to distinguish cat images from non-cat images.
The input will be the image transformed into a vector above, normlaised and flattened. The activation function is the sigmoid. If the output is more than 0.5 then it is considered a cat, otherwise not.

LogReg_kiank
Logistic diagram to classify our cats images

Mathematical expression of the algorithm:

For one example x^{(i)}:
z^{(i)} = w^T x^{(i)} + b
y_{hat}^{(i)} = a^{(i)} = sigmoid(z^{(i)})
\mathcal{L}(a^{(i)}, y^{(i)}) = - y^{(i)} \log(a^{(i)}) - (1-y^{(i)} ) \log(1-a^{(i)})

The cost is then computed by summing over all training examples:
J = \frac{1}{m} \sum_{i=1}^m \mathcal{L}(a^{(i)}, y^{(i)})

Will use the LogisticRegression module from sklearn.

Key steps:

  • Initialize the parameters of the model
  • Learn the parameters for the model by minimising the cost
  • Use the learned parameters to make predictions (on the test set)
  • Analyse the results and conclude

3.1 – Define and fit the logistic regression model

from sklearn.linear_model import LogisticRegression

lr = LogisticRegression(C=1000.0, random_state=0)

lr.fit(train_set_x.T, train_set_y.T.ravel())

These are the weights that the algorithm fitted to the model:

lr.coef_.shape
(1, 12288)
lr.coef_
array([[ 0.05665593, -0.11370205, -0.04197421, ..., -0.06609142,
 -0.11239916, 0.15381656]])
lr.intercept_
array([-0.00118436])

The model fitted is therefore: -0.001 + 0.056*x_{1} -0.114*x_{2} + ... + 0.154*x_{12288}

3.2 – Assess predictions on test data

We can measure the model accuracy using the test set:

Y_prediction = lr.predict(test_set_x.T)
Y_prediction.shape
(50,)
print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction - test_set_y)) * 100))
test accuracy: 72.0 %

72% of the test photos have been predicted correctly.

Check the notebook in gitHub to see the mismatched images and how to try with your own image.

Deep Neural Network for Image Classification

Now we will build and apply a deep neural network to the problem.

Dataset

We will use the same “Cat vs non-Cat” dataset as above. The model we had built had 72% test accuracy on classifying cats vs non-cats images. Hopefully, our new model will perform better!

Building the parts of our algorithm

The main steps for building a Neural Network are:

  1. Define the model structure (such as number of input features) and the hyper-parameters
  2. Initialise the model’s weights
  3. Loop for the number of iterations:
    – Calculate current loss (forward propagation)
    – Calculate current gradient (backward propagation)
    – Update parameters (gradient descent)
  4. Use the trained weights to predict the labels

4 – Defining the neural network structure

nn

We will define a deep network with a total of five layers: input, output and three hidden layers.
Each layer has a different number of units.
Defining the architecture of a neural network depends on the training data and the type of classification to perform and it is an art by itself. Defining and re-fining the number of layers, units and all other parameters is called hyperparameters tuning and we will see it in details in a next notebook.
For the moment we hard-code an initial set of parameters.

architecture

Detailed Architecture:
– The input is a (64,64,3) image which is flattened to a vector of size (12288,1).
– The corresponding vector: [x_0,x_1,...,x_{12287}]^T is then multiplied by the weight matrix W.
– You then add a bias term (the intercept b). the result is the linear unit.
– next, we take its ReLU (Rectified Linear Unit) to get the vector: [a_0^{[1]}, a_1^{[1]},..., a_{n^{[1]}-1}^{[1]}]^T.
Each activation vector has a different size depending on the number of units the layer.
– The process is repeated for each hidden layer (in our case three times) and several times (number of iterations), each time calculating the loss and back-propagating appropriate changes in the weights until the wished loss is reached.
– Finally, we take the sigmoid of the final linear unit. If it is greater than 0.5, we classify it to be a cat.

### CONSTANTS DEFINING THE MODEL ####
n_x = train_set_x_flatten.shape[0]     # size of input layer
n_y = 1  # size of output layer, will be 0 or 1
    # we define a neural network with total 5 layers, x, y and 3 hidden:
    # the first hidden has 20 units, second has 7 units and third has 5
nn_layers = [n_x, 20, 7, 5, n_y]  # length is 5 (layers)

nn_layers
[Out:] [12288, 20, 7, 5, 1]

5 – Initializing parameters

There are two types of parameters to initialise in a neural network:
– the weight matrices W^{[i]}
– the bias vectors b^{[i]}

The weight matrix is initialised with random values while the bias vector as a vector of zeros.
In general, initialising all the weights to zero results in the network failing to break symmetry. This means that every neuron in each layer will learn the same thing and the network is no more powerful than a linear classifier such as logistic regression.

To break symmetry, we initialise the weights randomly. Following random initialisation, each neuron can then proceed to learn a different function of its inputs.
Of course, different initializations lead to different results and poor initialisation can slow down the optimisation algorithm.

One good practice is not to initialise to values that are too large, instead what bring good results are the so-called Xavier (pdf of the paper) or the He (for ReLU activation) initialisations (pdf of the paper).

5.1 – L-layer Neural Network

The initialisation for an L-layer neural network is complicated because there are many weight matrices and bias vectors. Need to make sure that the dimensions match between each layer.

– The model’s structure has L-1 layers using a ReLU activation function followed by an output layer with a sigmoid activation function.
– We will store n^{[l]}, the number of units in different layers, in a variable `layer_dims`. For example, a model with two inputs, one hidden layer with 4 hidden units and an output layer with 1 output unit would have dimensions equal to [2,4,1]. Thus means `W1`’s shape is (4,2), `b1` is (4,1), `W2` is (1,4) and `b2` is (1,1).

#  FUNCTION: initialise_parameters

def initialise_parameters(layer_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the dimensions of each layer in our network

    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
                    bl -- bias vector of shape (layer_dims[l], 1)
    """

    parameters = {}
    L = len(layer_dims)            # number of layers in the network

    for l in range(1, L):
        parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1]) / np.sqrt(layer_dims[l-1])
        parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))

            # unit tests
        assert(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l-1]))
        assert(parameters['b' + str(l)].shape == (layer_dims[l], 1))

    return parameters

6 – Forward propagation module

Now that we have initialised our parameters, we will do the forward propagation module.

forward

We will implement some helper functions and then put all together:

– linear_forward: implement the linear part, i.e. the Z^{[l]} = W^{[l]}A^{[l-1]} +b^{[l]} equation, where A^{[0]} = X.
– linear_activation_forward: will add to the linear part the activation where activation will be either ReLU or Sigmoid.
– L-model_forward: will call the [LINEAR -> RELU] \times (L-1) times, finally the LINEAR -> SIGMOID

6.1 – Linear Forward

#  FUNCTION: linear_forward

def linear_forward(A, W, b):
    """
    Implement the linear part of a layer's forward propagation.

    Arguments:
    A -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)

    Returns:
    Z -- the input of the activation function, also called pre-activation parameter
    cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
    """

    Z = np.dot(W, A) + b

    assert(Z.shape == (W.shape[0], A.shape[1]))
    cache = (A, W, b)

    return Z, cache

6.2 – Linear-Activation Forward

We will use two activation functions:

– Sigmoid: \sigma(Z) = \sigma(W A + b) = \frac{1}{ 1 + e^{-(W A + b)}}.
This function returns two items: the activation value “a” and a “cache” that contains “Z” (it’s what we will feed in to the corresponding backward function).
To use it you could just call:
A, activation_cache = sigmoid(Z)

– ReLU: The mathematical formula for ReLu is A = RELU(Z) = max(0, Z).
This function returns two items: the activation value “A” and a “cache” that contains “Z” (it’s what we will feed in to the corresponding backward function).
To use it you could just call:
A, activation_cache = relu(Z)

activation functions

#  FUNCTION: sigmoid

def sigmoid(Z):
    """
    Implements the sigmoid activation in numpy

    Arguments:
    Z -- numpy array of any shape

    Returns:
    A -- output of sigmoid(z), same shape as Z
    cache -- returns Z as well, useful during backpropagation
    """

    A = 1/(1+np.exp(-Z))
    cache = Z

    return A, cache
def relu(Z):
    """
    Implement the RELU function.

    Arguments:
    Z -- Output of the linear layer, of any shape

    Returns:
    A -- Post-activation parameter, of the same shape as Z
    cache -- a python dictionary containing "A" ; stored for computing the backward pass efficiently
    """

    A = np.maximum(0,Z)

    assert(A.shape == Z.shape)

    cache = Z
    return A, cache

Next: Implement the forward propagation of the LINEAR->ACTIVATION layer.
Mathematical relation is: A^{[l]} = g(Z^{[l]}) = g(W^{[l]}A^{[l-1]} +b^{[l]}) where the activation “g” can be sigmoid() or relu().

# FUNCTION: linear_activation_forward

def linear_activation_forward(A_prev, W, b, activation):
    """
    Implement the forward propagation for the LINEAR->ACTIVATION layer

    Arguments:
    A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

    Returns:
    A -- the output of the activation function, also called the post-activation value
    cache -- a python dictionary containing "linear_cache" and "activation_cache";
             stored for computing the backward pass efficiently
    """

    Z, linear_cache = linear_forward(A_prev, W, b)

    if activation == "sigmoid":
        A, activation_cache = sigmoid(Z)

    elif activation == "relu":
        A, activation_cache = relu(Z)

    assert (A.shape == (W.shape[0], A_prev.shape[1]))
    cache = (linear_cache, activation_cache)

    return A, cache

Note: In deep learning, the “[LINEAR->ACTIVATION]” computation is counted as a single layer in the neural network, not two layers.

L-Layer Model

Now we put all together, using the helper functions:
the (`linear_activation_forward` with RELU) L-1 times, then follows that with one `linear_activation_forward` with SIGMOID.

#  FUNCTION: L_model_forward

def L_model_forward(X, parameters):
    """
    Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation

    Arguments:
    X -- data, numpy array of shape (input size, number of examples)
    parameters -- output of initialize_parameters_deep()

    Returns:
    AL -- last post-activation value
    caches -- list of caches containing:
                every cache of linear_relu_forward() (there are L-1 of them, indexed from 0 to L-2)
                the cache of linear_sigmoid_forward() (there is one, indexed L-1)
    """

    caches = []
    A = X
    L = len(parameters) // 2                  # number of layers in the neural network

    # Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
    for l in range(1, L):
        A_prev = A
        w_l = parameters['W' + str(l)]
        b_l = parameters['b' + str(l)]
        A, cache = linear_activation_forward(A_prev, w_l, b_l, activation = "relu")
        caches.append(cache)

    # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
    w_L = parameters['W' + str(L)]
    b_L = parameters['b' + str(L)]
    Yhat, cache = linear_activation_forward(A, w_L, b_L, activation = "sigmoid")
    caches.append(cache)

    assert(Yhat.shape == (1,X.shape[1]))

    return Yhat, caches

Great! Now we have a full forward propagation that takes the input X and outputs a row vector A^{[L]} containing our predictions. It also records all intermediate values in “caches”. Using A^{[L]}, we can compute the cost of our predictions.

7 – Cost function

Now we need to compute the cost, because we want to check if our model is actually learning.

Next: Compute the cross-entropy cost J, using the following formula:
-\frac{1}{m} \sum\limits_{i = 1}^{m} (y^{(i)}\log\left(a^{[L] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right))

#  FUNCTION: compute_cost

def compute_cost(Yhat, Y):
    """
    Implement the cross-entropy cost function

    Arguments:
    Yhat -- probability vector corresponding to the label predictions, shape (1, number of examples)
    Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)

    Returns:
    cost -- cross-entropy cost
    """

    m = Y.shape[1]

        # Compute loss from AL and Y.
    logprobs = np.dot(Y, np.log(Yhat).T) + np.dot((1-Y), np.log(1-Yhat).T)

    cost = (-1./m) * logprobs 

    cost = np.squeeze(cost)      # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
    assert(cost.shape == ())

    return cost

8 – Backward propagation module

Now we will implement the backward function for the whole network.

backward

Just like with forward propagation, we will implement helper functions for backpropagation. Remember that back propagation is used to calculate the gradient of the loss function with respect to the parameters.
You can see more about backpropagation in my previous post:

Now, similar to forward propagation, we are going to build the backward propagation in three steps:
– linear backward
– LINEAR -> ACTIVATION backward where ACTIVATION computes the derivative of either the ReLU or sigmoid activation
– [LINEAR -> RELU] \times (L-1) -> LINEAR -> SIGMOID backward (whole model)

Reminder:
backprop_kiank

The purple blocks represent the forward propagation, and the red blocks represent the backward propagation.

8.1 – Linear backward

For layer l, the linear part is: Z^{[l]} = W^{[l]} A^{[l-1]} + b^{[l]} (followed by an activation).

Now we need to compute the three derivatives (dW^{[l]}, db^{[l]}, dA^{[l]}), using as input a known derivate dZ^{[l]} = \frac{\partial \mathcal{L} }{\partial Z^{[l]}}. :

dW^{[l]} = \frac{\partial \mathcal{L} }{\partial W^{[l]}} = \frac{1}{m} dZ^{[l]} A^{[l-1] T}
db^{[l]} = \frac{\partial \mathcal{L} }{\partial b^{[l]}} = \frac{1}{m} \sum_{i = 1}^{m} dZ^{[l](i)}
dA^{[l-1]} = \frac{\partial \mathcal{L} }{\partial A^{[l-1]}} = W^{[l] T} dZ^{[l]}

#  FUNCTION: linear_backward

def linear_backward(dZ, cache):
    """
    Implement the linear portion of backward propagation for a single layer (layer l)

    Arguments:
    dZ -- Gradient of the cost with respect to the linear output (of current layer l)
    cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer

    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    A_prev, W, b = cache
    m = A_prev.shape[1]

    dW = (1./m) * np.dot(dZ, A_prev.T)
    db = (1./m) * np.sum(dZ, axis=1, keepdims=True)
    dA_prev = np.dot(W.T, dZ)

    assert (dA_prev.shape == A_prev.shape)
    assert (dW.shape == W.shape)
    assert (db.shape == b.shape)

    return dA_prev, dW, db

8.2 – Linear-Activation backward

Next, we will create a function that merges the two helper functions: `linear_backward` and the backward step for the activation:

– `sigmoid_backward`: Implements the backward propagation for SIGMOID unit.

– `relu_backward`: Implements the backward propagation for RELU unit.

If g(.) is the activation function, `sigmoid_backward` and `relu_backward` compute dZ^{[l]} = dA^{[l]} * g'(Z^{[l]}) .

def sigmoid_backward(dA, cache):
    """
    Implement the backward propagation for a single SIGMOID unit.

    Arguments:
    dA -- post-activation gradient, of any shape
    cache -- 'Z' where we store for computing backward propagation efficiently

    Returns:
    dZ -- Gradient of the cost with respect to Z
    """

    Z = cache

    s = 1/(1+np.exp(-Z))
    dZ = dA * s * (1-s)

    assert (dZ.shape == Z.shape)

    return dZ

def relu_backward(dA, cache):
    """
    Implement the backward propagation for a single RELU unit.

    Arguments:
    dA -- post-activation gradient, of any shape
    cache -- 'Z' where we store for computing backward propagation efficiently

    Returns:
    dZ -- Gradient of the cost with respect to Z
    """

    Z = cache
    dZ = np.array(dA, copy=True) # just converting dz to a correct object.

    # When z <= 0, you should set dz to 0 as well.
    dZ[Z <= 0] = 0          assert (dZ.shape == Z.shape)          return dZ 
#  FUNCTION: linear_activation_backward def linear_activation_backward(dA, cache, activation):     """     Implement the backward propagation for the LINEAR->ACTIVATION layer.

    Arguments:
    dA -- post-activation gradient for current layer l
    cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    linear_cache, activation_cache = cache

    if activation == "relu":
        dZ = relu_backward(dA, activation_cache)

    elif activation == "sigmoid":
        dZ = sigmoid_backward(dA, activation_cache)

    dA_prev, dW, db = linear_backward(dZ, linear_cache)

    return dA_prev, dW, db

8.3 – L-Model Backward

Recall that when we implemented the `L_model_forward` function, at each iteration, we stored a cache which contains (X,W,b, and z). In the back propagation module, we will use those variables to compute the gradients. Therefore, in the `L_model_backward` function, we will iterate through all the hidden layers backward, starting from layer L. On each step, we will use the cached values for layer l to back-propagate through layer l.

#  FUNCTION: L_model_backward

def L_model_backward(Yhat, Y, caches):
    """
    Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group

    Arguments:
    AL -- probability vector, output of the forward propagation (L_model_forward())
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat)
    caches -- list of caches containing:
                every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
                the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1])

    Returns:
    grads -- A dictionary with the gradients
             grads["dA" + str(l)] = ...
             grads["dW" + str(l)] = ...
             grads["db" + str(l)] = ...
    """
    grads = {}
    L = len(caches) # the number of layers
    m = Yhat.shape[1]
    Y = Y.reshape(Yhat.shape) # after this line, Y is the same shape as AL

    # Initializing the backpropagation
    dAL = - (np.divide(Y, Yhat) - np.divide(1 - Y, 1 - Yhat)) # derivative of cost with respect to AL

    # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]
    current_cache = caches[L-1]
    grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, current_cache, activation = "sigmoid")

    for l in reversed(range(L-1)):
        # lth layer: (RELU -> LINEAR) gradients.
        # Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)]
        current_cache = caches[l]
        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA"+str(l+2)], current_cache, activation = "relu")
        grads["dA" + str(l + 1)] = dA_prev_temp
        grads["dW" + str(l + 1)] = dW_temp
        grads["db" + str(l + 1)] = db_temp

    return grads

8.4 – Update Parameters

In this section we will update the parameters of the model, using gradient descent:

W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]}
b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]}

where \alpha is the learning rate. After computing the updated parameters, we store them in the parameters dictionary.

#  FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate):
    """
    Update parameters using gradient descent

    Arguments:
    parameters -- python dictionary containing your parameters
    grads -- python dictionary containing your gradients, output of L_model_backward

    Returns:
    parameters -- python dictionary containing your updated parameters
                  parameters["W" + str(l)] = ...
                  parameters["b" + str(l)] = ...
    """

    L = len(parameters) // 2 # number of layers in the neural network

    # Update rule for each parameter. Use a for loop.
    for l in range(L):
        parameters["W"+str(l+1)] = parameters["W"+str(l+1)] - learning_rate * grads["dW" + str(l+1)]
        parameters["b"+str(l+1)] = parameters["b"+str(l+1)] - learning_rate * grads["db" + str(l+1)]
    return parameters

9 – L-layer Neural Network

Now we can put together all the functions to build an L-layer neural network with this structure:

nn_layers
[12288, 20, 7, 5, 1]
#  FUNCTION: L_layer_model

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):
    """
    Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.

    Arguments:
    X -- data, numpy array of shape (number of examples, num_px * num_px * 3)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
    layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
    learning_rate -- learning rate of the gradient descent update rule
    num_iterations -- number of iterations of the optimization loop
    print_cost -- if True, it prints the cost every 100 steps

    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """

    costs = []                         # keep track of cost

    # Parameters initialization.
    parameters = initialise_parameters(layers_dims)

    # Loop (gradient descent)
    for i in range(0, num_iterations):

        # Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
        AL, caches = L_model_forward(X, parameters)

        # Compute cost.
        cost = compute_cost(AL, Y)

        # Backward propagation.
        grads = L_model_backward(AL, Y, caches)

        # Update parameters.
        parameters = update_parameters(parameters, grads, learning_rate)

        # Print the cost every 100 training example
        if print_cost and i % 100 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))
        if print_cost and i % 100 == 0:
            costs.append(cost)

    # plot the cost
    plt.plot(np.squeeze(costs))
    plt.ylabel('cost')
    plt.xlabel('iterations (per tens)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()

    return parameters

We can then train the model as a 5-layer neural network.
The cost should decrease on every iteration.

np.random.seed(1)

fit_params = L_layer_model(train_set_x, train_set_y, nn_layers, num_iterations = 2500, print_cost = True)
Cost after iteration 0: 0.771749
 Cost after iteration 100: 0.672053
 ...
 Cost after iteration 2400: 0.092878
output
Cost is decreasing after each iteration

10. Results analysis

Now we can check the performance of the trained network by predicting the results of the test set and comparing them with the actual labels.

def predict(X, y, parameters):
    """
    This function is used to predict the results of a  L-layer neural network.

    Arguments:
    X -- data set of examples you would like to label
    parameters -- parameters of the trained model

    Returns:
    p -- predictions for the given dataset X
    """

    m = X.shape[1]
    n = len(parameters) // 2 # number of layers in the neural network
    p = np.zeros((1,m))

    # Forward propagation
    probas, caches = L_model_forward(X, parameters)

    # convert probs to 0/1 predictions
    for i in range(0, probas.shape[1]):
        if probas[0,i] > 0.5:
            p[0,i] = 1
        else:
            p[0,i] = 0

    # print results
    print("Accuracy: "  + str(np.sum((p == y)/m)))

    return p
pred_train = predict(train_set_x, train_set_y, fit_params)
Accuracy: 0.985645933014
pred_test = predict(test_set_x, test_set_y, fit_params)
Accuracy: 0.8

Congrats! It seems that this 5-layer neural network has better performance (80%) than the logistic regression model (72%) on the same test set.

This is good performance for this task.

Even higher accuracy could be obtained, by systematically searching for better hyperparameters (learning_rate, layers_dims, num_iterations) and other techniques such as regularisation, that we can see in the next notebook.

With logistic regression we had 14 misses in the test set, out of 50. With the neural network we have 10 misses.

Note: this post is part of a series about Machine Learning with Python.

Bibliography:
Andrew Ng’s deep learning course
Tutorial: implement a neural network from scratch
Image normalisation

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