吴恩达深度学习课程 DeepLearning.ai 编程作业(1-2)Part.2

3 – General Architecture of the learning algorithm

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

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

nnc.png

Key steps
In this exercise, you 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

4 – 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) 
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().

.1 – Helper functions

Exercise: Using your code from “Python Basics”, implement sigmoid()

#1 GRADED FUNCTION: sigmoid
def sigmoid(z):
    """
    Compute the sigmoid of z
 
    Arguments:
    z -- A scalar or numpy array of any size.
 
    Return:
    s -- sigmoid(z)
    """
    s = 1.0/(1+np.exp(-z))
 
    return s
 
print ("sigmoid([0, 2]) = " + str(sigmoid(np.array([0,2]))))

4.2 – Initializing parameters

Exercise: Implement parameter initialization in the cell below. You have to initialize w as a vector of zeros. If you don’t know what numpy function to use, look up np.zeros() in the Numpy library’s documentation.

#2Initializing parameters
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((dim, 1))
    b = 0
    ### END CODE HERE ###
 
#python assert断言是声明其布尔值必须为真的判定,如果发生异常就说明表达示为假。可以
#理解assert断言语句为raise-if-not,用来测试表示式,其返回值为假,就会触发异常
    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))#判断b是int或者是float
 
    return w, b
 
dim = 2
w, b = initialize_with_zeros(dim)
print ("w = " + str(w))
print ("b = " + str(b))

4.3 – Forward and Backward propagation

Now that your parameters are initialized, you 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.

# GRADED FUNCTION: propagate
 
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 = -(1.0/m)*np.sum(Y*np.log(A)+(1-Y)*np.log(1-A)) # compute cost
    ### END CODE HERE ###
 
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines of code)
    dw = (1.0/m)*np.dot(X,(A-Y).T)
    db = (1.0/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
 
w, b, X, Y = np.array([[1.],[2.]]), 2., np.array([[1.,2.,-1.],[3.,4.,-3.2]]), np.array([[1,0,1]])
grads, cost = propagate(w, b, X, Y)
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print ("cost = " + str(cost))

d) Optimization

·        You have initialized your parameters.

·        You are also able to compute a cost function and its gradient.

·        Now, you 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 θ, the update rule is θ=θα dθ, where α is the learning rate.

# GRADED FUNCTION: optimize

 

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 examples
        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
 
params, grads, costs = optimize(w, b, X, Y, num_iterations= 100, learning_rate = 0.009, print_cost = False)
 
print ("w = " + str(params["w"]))
print ("b = " + str(params["b"]))
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))

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 is two steps to computing predictions:

1.    Calculate Y^=A=σ(wTX+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).

# GRADED FUNCTION: predict
 
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] = 1
        else:
            Y_prediction[0,i] = 0
        ### END CODE HERE ###
 
    assert(Y_prediction.shape == (1, m))
 
return Y_prediction
w = np.array([[0.1124579],[0.23106775]])
b = -0.3
X = np.array([[1.,-1.1,-3.2],[1.2,2.,0.1]])
print ("predictions = " + str(predict(w, b, X)))

What to remember: 
You’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

5 – 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 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()

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)
    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)
 
    # Retrieve parameters w and b from dictionary "parameters"
    w = parameters["w"]
    b = parameters["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
 
d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)

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