Logistic Regression with a Neural Network mindset

Logistic Regression with a Neural Network mindset

General Architecture of the learning algorithm

It’s time to design a simple algorithm to distinguish cat images from non-cat images.

I will build a Logistic Regression, using a Neural Network mindset. The following Figure explains why Logistic Regression is actually a very simple Neural Network!

image-20181110233216084

Mathematical expression of the algorithm:

For one example Missing superscript or subscript argument x^{(i)} :

The cost is then computed by summing over all training examples:

Key steps:
In this exercise, I will carry out the following steps:

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

Building the parts of algorithm

The main steps for building a Neural Network are:

  1. Define the model structure (such as number of input features)

  2. Initialize the model’s parameters

  3. Loop:

    • Calculate current loss (forward propagation)

    • Calculate current gradient (backward propagation)

    • Update parameters (gradient descent)

You often build 1-3 separately and integrate them into one function we call model().

Helper functions

sigmoid

Using code from “Python Basics”, implement sigmoid(). As we seen in the figure above, I will compute $sigmoid( w^T x + b) = \frac{1}{1 + e^{-(w^T x + b)}}$ to make predictions. I will use np.exp().

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def sigmoid(z):
"""
Compute the sigmoid of z

Arguments:
z -- A scalar or numpy array of any size.

Return:
s -- sigmoid(z)
"""

### START CODE HERE ### (≈ 1 line of code)
s = 1 / (1 + np.exp(-z))
### END CODE HERE ###

return s

initialize_with_zeros

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def initialize_with_zeros(dim):
"""
This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.

Argument:
dim -- size of the w vector we want (or number of parameters in this case)

Returns:
w -- initialized vector of shape (dim, 1)
b -- initialized scalar (corresponds to the bias)
"""

### START CODE HERE ### (≈ 1 line of code)
w = np.zeros(shape=(dim, 1))
b = 0
### END CODE HERE ###

assert (w.shape == (dim, 1))
assert (isinstance(b, float) or isinstance(b, int))

return w, b

Forward and Backward propagation

Now that our parameters are initialized, we can do the “forward” and “backward” propagation steps for learning the parameters.

Exercise: Implement a function propagate() that computes the cost function and its gradient.

Hints:

Forward Propagation:

  • I get X
  • I compute $A = \sigma(w^T X + b) = (a^{(1)}, a^{(2)}, …, a^{(m-1)}, a^{(m)})$
  • I calculate the cost function: $J = -\frac{1}{m}\sum_{i=1}^{m}y^{(i)}\log(a^{(i)})+(1-y^{(i)})\log(1-a^{(i)})$

Here are the two formulas I will be using:

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def propagate(w, b, X, Y):
"""
Implement the cost function and its gradient for the propagation explained above

Arguments:
w -- weights, a numpy array of size (num_px * num_px * 3, 1)
b -- bias, a scalar
X -- data of size (num_px * num_px * 3, number of examples)
Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

Return:
cost -- negative log-likelihood cost for logistic regression
dw -- gradient of the loss with respect to w, thus same shape as w
db -- gradient of the loss with respect to b, thus same shape as b

Tips:
- Write your code step by step for the propagation. np.log(), np.dot()
"""

m = X.shape[1]

# FORWARD PROPAGATION (FROM X TO COST)
### START CODE HERE ### (≈ 2 lines of code)
A = sigmoid(np.dot(w.T, X) + b) # compute activation
cost = -(1 / m) * np.sum(Y * np.log(A) + (1 - Y) * np.log(1 - A), axis=1) # compute cost
### END CODE HERE ###

# BACKWARD PROPAGATION (TO FIND GRAD)
### START CODE HERE ### (≈ 2 lines of code)
dw = (1 / m) * np.dot(X, (A - Y).T)
db = (1 / m) * np.sum(A - Y)
### END CODE HERE ###

assert (dw.shape == w.shape)
assert (db.dtype == float)
cost = np.squeeze(cost)
assert (cost.shape == ())

grads = {"dw": dw,
"db": db}

return grads, cost

Optimization

  • I have initialized our parameters.
  • We are also able to compute a cost function and its gradient.
  • Now, I want to update the parameters using gradient descent.

Exercise: Write down the optimization function. The goal is to learn $w$ and $b$ by minimizing the cost function $J$. For a parameter $\theta$, the update rule is $ \theta = \theta - \alpha \text{ } d\theta$, where $\alpha$ is the learning rate.

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def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost=False):
"""
This function optimizes w and b by running a gradient descent algorithm

Arguments:
w -- weights, a numpy array of size (num_px * num_px * 3, 1)
b -- bias, a scalar
X -- data of shape (num_px * num_px * 3, number of examples)
Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
num_iterations -- number of iterations of the optimization loop
learning_rate -- learning rate of the gradient descent update rule
print_cost -- True to print the loss every 100 steps

Returns:
params -- dictionary containing the weights w and bias b
grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.

Tips:
You basically need to write down two steps and iterate through them:
1) Calculate the cost and the gradient for the current parameters. Use propagate().
2) Update the parameters using gradient descent rule for w and b.
"""

costs = []

for i in range(num_iterations):

# Cost and gradient calculation (≈ 1-4 lines of code)
### START CODE HERE ###
grads, cost = propagate(w, b, X, Y)
### END CODE HERE ###

# Retrieve derivatives from grads
dw = grads["dw"]
db = grads["db"]

# update rule (≈ 2 lines of code)
### START CODE HERE ###
w = w - learning_rate * dw
b = b - learning_rate * db
### END CODE HERE ###

# Record the costs
if i % 100 == 0:
costs.append(cost)

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

params = {"w": w,
"b": b}

grads = {"dw": dw,
"db": db}

return params, grads, costs

predict

Exercise: The previous function will output the learned w and b. We are able to use w and b to predict the labels for a dataset X. Implement the predict() function. There are two steps to computing predictions:

  1. Calculate $\hat{Y} = A = \sigma(w^T X + b)$

  2. Convert the entries of a into 0 (if activation <= 0.5) or 1 (if activation > 0.5), stores the predictions in a vector Y_prediction. If you wish, you can use an if/else statement in a for loop (though there is also a way to vectorize this).

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def predict(w, b, X):
'''
Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)

Arguments:
w -- weights, a numpy array of size (num_px * num_px * 3, 1)
b -- bias, a scalar
X -- data of size (num_px * num_px * 3, number of examples)

Returns:
Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
'''

m = X.shape[1]
Y_prediction = np.zeros((1, m))
w = w.reshape(X.shape[0], 1)

# Compute vector "A" predicting the probabilities of a cat being present in the picture
### START CODE HERE ### (≈ 1 line of code)
A = sigmoid(np.dot(w.T, X) + b)
### END CODE HERE ###

for i in range(A.shape[1]):

# Convert probabilities A[0,i] to actual predictions p[0,i]
### START CODE HERE ### (≈ 4 lines of code)
if A[0][i] < 0.5:
Y_prediction[0][i] = 0
else:
Y_prediction[0][i] = 1

### END CODE HERE ###

assert (Y_prediction.shape == (1, m))

return Y_prediction

What to remember

We’ve implemented several functions that:

  • Initialize (w,b)
  • Optimize the loss iteratively to learn parameters (w,b):
    • computing the cost and its gradient
    • updating the parameters using gradient descent
  • Use the learned (w,b) to predict the labels for a given set of examples

Merge all functions into a model

You will now see how the overall model is structured by putting together all the building blocks (functions implemented in the previous parts) together, in the right order.

Exercise: Implement the model function. Use the following notation:

- Y_prediction_test for your predictions on the test set
  • Y_prediction_train for your predictions on the train set
  • w, costs, grads for the outputs of optimize()
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def model(X_train, Y_train, X_test, Y_test, num_iterations=2000, learning_rate=0.5, print_cost=False):
"""
Builds the logistic regression model by calling the function you've implemented previously

Arguments:
X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
print_cost -- Set to true to print the cost every 100 iterations

Returns:
d -- dictionary containing information about the model.
"""

### START CODE HERE ###

# initialize parameters with zeros (≈ 1 line of code)
w, b = initialize_with_zeros(X_train.shape[0])

# Gradient descent (≈ 1 line of code)
params, grads, costs = optimize(w, b, X_train, Y_train, num_iterations=num_iterations, learning_rate=learning_rate,
print_cost=print_cost)

# Retrieve parameters w and b from dictionary "parameters"
w = params["w"]
b = params["b"]

# Predict test/train set examples (≈ 2 lines of code)
Y_prediction_test = predict(w, b, X_test)
Y_prediction_train = predict(w, b, X_train)

### END CODE HERE ###

# Print train/test Errors
print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))

d = {"costs": costs,
"Y_prediction_test": Y_prediction_test,
"Y_prediction_train": Y_prediction_train,
"w": w,
"b": b,
"learning_rate": learning_rate,
"num_iterations": num_iterations}

return d

train

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d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations=2000, learning_rate=0.005, print_cost=True)

test our image

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## START CODE HERE ## (PUT YOUR IMAGE NAME) 
my_image = "my_image.jpg" # change this to the name of your image file
## END CODE HERE ##

# We preprocess the image to fit your algorithm.
fname = "images/" + my_image
image = np.array(ndimage.imread(fname, flatten=False))
my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((1, num_px*num_px*3)).T
my_predicted_image = predict(d["w"], d["b"], my_image)

plt.imshow(image)
print("y = " + str(np.squeeze(my_predicted_image)) + ", your algorithm predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") + "\" picture.")

github

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