coursera 地址

import numpy as np
import matplotlib.pyplot as plt
import h5py
from lr_utils import load_dataset

train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset(
)

m_train = train_set_y.shape[1]  #训练集里图片的数量。
m_test = test_set_y.shape[1]  #测试集里图片的数量。
num_px = train_set_x_orig.shape[1]  #训练、测试集里面的图片的宽度和高度（均为64x64）。



#现在看一看我们加载的东西的具体情况
print("训练集的数量: m_train = " + str(m_train))
print("测试集的数量 : m_test = " + str(m_test))
print("每张图片的宽/高 : num_px = " + str(num_px))
print("每张图片的大小 : (" + str(num_px) + ", " + str(num_px) + ", 3)")
print("训练集_图片的维数 train_set_x : " + str(train_set_x_orig.shape))
print("训练集_标签的维数 train_set_y : " + str(train_set_y.shape))
print("测试集_图片的维数 test_set_x : " + str(test_set_x_orig.shape))
print("测试集_标签的维数: test_set_y " + str(test_set_y.shape))

#将训练集的维度降低并转置。
train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T
#将测试集的维度降低并转置。
test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T

print("训练集降维最后的维度：train_set_x_flatten  " + str(train_set_x_flatten.shape))
print("训练集_标签的维数 train_set_y : " + str(train_set_y.shape))
print("测试集降维之后的维度: test_set_x_flatten  " + str(test_set_x_flatten.shape))
print("测试集_标签的维数 test_set_y  : " + str(test_set_y.shape))
print("sanity check after reshaping: " + str(train_set_x_flatten[0:5, 0]))


train_set_x = train_set_x_flatten / 255
test_set_x = test_set_x_flatten / 255


def sigmoid(x):
    return 1/(1+np.exp(-x))

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)
    """
    w=np.zeros((dim,1))
    b=0
    assert(w.shape==(dim,1))
    assert(isinstance(b,float)or isinstance(b,int))
    return w,b


print("sigmoid(0) = " + str(sigmoid(0)))
print("sigmoid(9.2) = " + str(sigmoid(9.2)))


dim = 2
w, b = initialize_with_zeros(dim)
print("w = " + str(w))
print("b = " + str(b))


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
    """
    m=X.shape[1]
    #正向传播
    A=sigmoid(np.dot(w.T,X)+b)
    cost=(-1/m)*np.sum(Y*np.log(A)+(1-Y)*np.log(1-A))#整个训练集的损失函数
    #反向传播
    dw=1/m*np.dot(X,(A-Y).T)
    db=1/m*np.sum(A-Y)

    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],
                                                [3, 4]]), np.array([[1, 0]])
grads, cost = propagate(w, b, X, Y)
print("dw = " + str(grads["dw"]))
print("db = " + str(grads["db"]))
print("cost = " + str(cost))


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):

        grads,cost=propagate(w,b,X,Y)
        dw=grads['dw']
        db=grads['db']

        w=w-learning_rate*dw
        b=b-learning_rate*db

        if i%100==0:
            costs.append(cost)
        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"]))

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

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


print("predictions = " + str(predict(w, b, X)))


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.
    """
    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_predict_train=predict(w,b,X_train)
    Y_predict_test = predict(w, b, X_test)

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

    d = {"costs": costs,
        "Y_prediction_test": Y_predict_test,
        "Y_prediction_train" : Y_predict_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)



# Plot learning curve (with costs)
costs = np.squeeze(d['costs'])
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(d["learning_rate"]))
plt.show()

进一步分析

提醒：为了让梯度下降有效，你必须明智地选择学习速度。学习率 α \alpha α决定了我们更新参数的速度。如果学习率太大，我们可能会“超出”最佳值。类似地，如果它太小，我们将需要太多的迭代来收敛到最佳值。这就是为什么使用良好的学习速度是至关重要的。

让我们将我们模型的学习曲线与几种学习率的选择进行比较。运行下面的单元格。这大约需要1分钟。也可以尝试不同于我们初始化的learning_rates变量所包含的三个值，看看会发生什么。

learning_rates = [0.01, 0.001, 0.0001]
models = {}
for i in learning_rates:
    print ("learning rate is: " + str(i))
    models[str(i)] = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 1500, learning_rate = i, print_cost = False)
    print ('\n' + "-------------------------------------------------------" + '\n')

for i in learning_rates:
    plt.plot(np.squeeze(models[str(i)]["costs"]), label= str(models[str(i)]["learning_rate"]))

plt.ylabel('cost')
plt.xlabel('iterations')

legend = plt.legend(loc='upper center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()