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---

title: DeepLearning.ai作业:(1-3)-- 浅层神经网络（Shallow neural networks）

tags:

• homework
  categories:
• AI
• Deep Learning
  date: 2018-09-12 15:49:22
  id: 2018091216

---

首发于个人博客:，欢迎来访

1. 不要抄作业！
2. 我只是把思路整理了，供个人学习。
3. 不要抄作业！

数据集

数据集是一个类似花的数据集。

而如果用传统的logistic regression，做出来的就是一个二分类问题，简单粗暴的划出了一条线，

可以看见，准确率只有47%。

所以就需要构建神经网络模型了。

神经网络模型

Reminder: The general methodology to build a Neural Network is to:

1. Define the neural network structure ( # of input units,  # of hidden units, etc). 2. Initialize the model's parameters3. Loop:    - Implement forward propagation    - Compute loss    - Implement backward propagation to get the gradients    - Update parameters (gradient descent)

已经给出思路了：

1. 定义神经网络的结构
2. 初始化模型参数
3. 循环：
   1. 计算正向传播
   2. 计算损失函数
   3. 计算反向传播来得到grad
   4. 更新参数

1. 定义神经网络结构

# GRADED FUNCTION: layer_sizesdef layer_sizes(X, Y):    """    Arguments:    X -- input dataset of shape (input size, number of examples)    Y -- labels of shape (output size, number of examples)        Returns:    n_x -- the size of the input layer    n_h -- the size of the hidden layer    n_y -- the size of the output layer    """    ### START CODE HERE ### (≈ 3 lines of code)    n_x = X.shape[0] # size of input layer    n_h = 4    n_y = Y.shape[0] # size of output layer    ### END CODE HERE ###    return (n_x, n_h, n_y)

2. 初始化参数

来初始化w和b的参数

w: np.random.rand(a,b) * 0.01

b: np.zeros((a,b))

# GRADED FUNCTION: initialize_parametersdef initialize_parameters(n_x, n_h, n_y):    """    Argument:    n_x -- size of the input layer    n_h -- size of the hidden layer    n_y -- size of the output layer        Returns:    params -- python dictionary containing your parameters:                    W1 -- weight matrix of shape (n_h, n_x)                    b1 -- bias vector of shape (n_h, 1)                    W2 -- weight matrix of shape (n_y, n_h)                    b2 -- bias vector of shape (n_y, 1)    """        np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.        ### START CODE HERE ### (≈ 4 lines of code)    W1 = np.random.randn(n_h, n_x) * 0.01    b1 = np.zeros((n_h, 1))    W2 = np.random.randn(n_y, n_h) * 0.01    b2 = np.zeros((n_y, 1))    ### END CODE HERE ###        assert (W1.shape == (n_h, n_x))    assert (b1.shape == (n_h, 1))    assert (W2.shape == (n_y, n_h))    assert (b2.shape == (n_y, 1))        parameters = {
    "W1": W1,                  "b1": b1,                  "W2": W2,                  "b2": b2}        return parameters

3. loop

在这里可以使用sigmoid()来做输出层的函数，np.tanh()来做hidden layer的激活函数。

3.1 forward propagation

在这个函数中，输入的是X，和parameters，然后就可以根据

$$z^{[1](i)}=W^{[1]}{x}^{(i)}+b^{[1]}\text{\hspace{1em}(1)}$$

$$a^{[1](i)}=\tanh (z^{[1](i)})\text{\hspace{1em}(2)}$$ $$z^{[2](i)}=W^{[2]}{a}^{[1](i)}+b^{[2]}\text{\hspace{1em}(3)}$$ $${\hat{y}}^{(i)}=a^{[2](i)}=\sigma (z^{[2](i)})\text{\hspace{1em}(4)}$$

得到每一层的Z和A了。

# GRADED FUNCTION: forward_propagationdef forward_propagation(X, parameters):    """    Argument:    X -- input data of size (n_x, m)    parameters -- python dictionary containing your parameters (output of initialization function)        Returns:    A2 -- The sigmoid output of the second activation    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"    """    # Retrieve each parameter from the dictionary "parameters"    ### START CODE HERE ### (≈ 4 lines of code)    W1 = parameters['W1']    b1 = parameters['b1']    W2 = parameters['W2']    b2 = parameters['b2']    ### END CODE HERE ###        # Implement Forward Propagation to calculate A2 (probabilities)    ### START CODE HERE ### (≈ 4 lines of code)    Z1 = np.dot(W1,X) + b1    A1 = np.tanh(Z1)    Z2 = np.dot(W2,A1) + b2    A2 = sigmoid(Z2)    ### END CODE HERE ###        assert(A2.shape == (1, X.shape[1]))        cache = {
    "Z1": Z1,             "A1": A1,             "Z2": Z2,             "A2": A2}        return A2, cache

3.2 cost

接下来，在得到A2的值后，就可以根据公式来计算损失函数了。

$$J=-\frac{1}{m}\sum _{i=0}^m(y^{(i)}\log \left(a^{[2](i)}\right)+(1-y^{(i)})\log \left(1-a^{[2](i)}\right))$$

在这里需要注意的是交叉熵的计算，交叉熵使用np.multiply()来计算，然后用np.sum()，求和。

而单单计算logprobs = np.multiply(np.log(A2),Y)是不够的，因为这个只得到了公式的前一半的部分，Y=0的部分在元素相乘中就相当于没有了，所以还要再后面加一项np.multiply(np.log(1-A2),1-Y)

# GRADED FUNCTION: compute_costdef compute_cost(A2, Y, parameters):    """    Computes the cross-entropy cost given in equation (13)        Arguments:    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)    Y -- "true" labels vector of shape (1, number of examples)    parameters -- python dictionary containing your parameters W1, b1, W2 and b2        Returns:    cost -- cross-entropy cost given equation (13)    """        m = Y.shape[1] # number of example    # Compute the cross-entropy cost    ### START CODE HERE ### (≈ 2 lines of code)    logprobs = np.multiply(np.log(A2),Y)  + np.multiply(np.log(1-A2),1-Y)    cost =  -1 / m *  np.sum(logprobs)    ### END CODE HERE ###    cost = np.squeeze(cost)     # makes sure cost is the dimension we expect.                                 # E.g., turns [[17]] into 17     assert(isinstance(cost, float))        return cost

3.3 backworad propagation

NG说神经网络中最难理解的是这个，但是现在公式已经帮我们推倒好了。

其中， $g^{[1{]}^{\prime }}(Z^{[1]})$ using

(1 - np.power(A1, 2))

可以看到，公式中需要的变量有X,Y,A,W,然后输出一个字典结构的grads

def backward_propagation(parameters, cache, X, Y):    """    Implement the backward propagation using the instructions above.        Arguments:    parameters -- python dictionary containing our parameters     cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".    X -- input data of shape (2, number of examples)    Y -- "true" labels vector of shape (1, number of examples)        Returns:    grads -- python dictionary containing your gradients with respect to different parameters    """    m = X.shape[1]        # First, retrieve W1 and W2 from the dictionary "parameters".    ### START CODE HERE ### (≈ 2 lines of code)    W1 = parameters['W1']    W2 = parameters['W2']    ### END CODE HERE ###            # Retrieve also A1 and A2 from dictionary "cache".    ### START CODE HERE ### (≈ 2 lines of code)    A1 = cache['A1']    A2 = cache['A2']    ### END CODE HERE ###        # Backward propagation: calculate dW1, db1, dW2, db2.     ### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)    dZ2 = A2 - Y    dW2 = 1 / m * np.dot(dZ2, A1.T)    db2 = 1 / m * np.sum(dZ2, axis=1, keepdims=True)    dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))    dW1 = 1 / m * np.dot(dZ1, X.T)    db1 = 1 / m * np.sum(dZ1, axis=1, keepdims=True)    ### END CODE HERE ###        grads = {
    "dW1": dW1,             "db1": db1,             "dW2": dW2,             "db2": db2}        return grads

3.4 update parameters

最后根据得到的grads，乘上学习速率，就可以更新参数了。

# GRADED FUNCTION: update_parametersdef update_parameters(parameters, grads, learning_rate = 1.2):    """    Updates parameters using the gradient descent update rule given above        Arguments:    parameters -- python dictionary containing your parameters     grads -- python dictionary containing your gradients         Returns:    parameters -- python dictionary containing your updated parameters     """    # Retrieve each parameter from the dictionary "parameters"    ### START CODE HERE ### (≈ 4 lines of code)    W1 = parameters['W1']    b1 = parameters['b1']    W2 = parameters['W2']    b2 = parameters['b2']    ### END CODE HERE ###        # Retrieve each gradient from the dictionary "grads"    ### START CODE HERE ### (≈ 4 lines of code)    dW1 = grads['dW1']    db1 = grads['db1']    dW2 = grads['dW2']    db2 = grads['db2']    ## END CODE HERE ###        # Update rule for each parameter    ### START CODE HERE ### (≈ 4 lines of code)    W1 = W1 - learning_rate * dW1    b1 = b1 - learning_rate * db1    W2 = W2 - learning_rate * dW2    b2 = b2 - learning_rate * db2    ### END CODE HERE ###        parameters = {
    "W1": W1,                  "b1": b1,                  "W2": W2,                  "b2": b2}        return parameters

然后把更新完的参数再传入前面的循环中，不断循环，直到达到循环的次数。

nn_model

把前面的函数都调用过来。

模型中传入的参数是，X,Y，和迭代次数

1. 首先需要得到你要设计的神经网络结构，调用layer_sizes()得到了n_x,n_y，也就是输入层和输出层。
2. 初始化参数initialize_parameters(n_x, n_h, n_y),得到初始化的 W1, b1, W2, b2
3. 然后开始循环
   1. 使用forward_propagation(X, parameters),先得到各个神经元的计算值。
   2. 然后compute_cost(A2, Y, parameters),得到cost
   3. backward_propagation(parameters, cache, X, Y)计算出每一步的梯度
   4. update_parameters(parameters, grads)更新一下参数
4. 返回训练完的parameters

# GRADED FUNCTION: nn_modeldef nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):    """    Arguments:    X -- dataset of shape (2, number of examples)    Y -- labels of shape (1, number of examples)    n_h -- size of the hidden layer    num_iterations -- Number of iterations in gradient descent loop    print_cost -- if True, print the cost every 1000 iterations        Returns:    parameters -- parameters learnt by the model. They can then be used to predict.    """        np.random.seed(3)    n_x = layer_sizes(X, Y)[0]    n_y = layer_sizes(X, Y)[2]        # Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".    ### START CODE HERE ### (≈ 5 lines of code)    parameters = initialize_parameters(n_x, n_h, n_y)    W1 = parameters['W1']    b1 = parameters['b1']    W2 = parameters['W2']    b2 = parameters['b2']    ### END CODE HERE ###        # Loop (gradient descent)    for i in range(0, num_iterations):                 ### START CODE HERE ### (≈ 4 lines of code)        # Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".        A2, cache = forward_propagation(X, parameters)                # Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".        cost = compute_cost(A2, Y, parameters)         # Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".        grads = backward_propagation(parameters, cache, X, Y)         # Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".        parameters =  update_parameters(parameters, grads)                ### END CODE HERE ###                # Print the cost every 1000 iterations        if print_cost and i % 1000 == 0:            print ("Cost after iteration %i: %f" %(i, cost))    return parameters

预测

得到训练后的parameters，再用forward_propagation(X, parameters)计算出输出层最终的值A2，以0.5为分界，分为0和1。

# GRADED FUNCTION: predictdef predict(parameters, X):    """    Using the learned parameters, predicts a class for each example in X        Arguments:    parameters -- python dictionary containing your parameters     X -- input data of size (n_x, m)        Returns    predictions -- vector of predictions of our model (red: 0 / blue: 1)    """        # Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.    ### START CODE HERE ### (≈ 2 lines of code)    A2, cache = forward_propagation(X, parameters)    predictions = (A2 > 0.5)    ### END CODE HERE ###        return predictions

# Build a model with a n_h-dimensional hidden layerparameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)# Plot the decision boundaryplot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)plt.title("Decision Boundary for hidden layer size " + str(4))

可以看到，训练后神经网络得到的分界线更为合理。

# Print accuracypredictions = predict(parameters, X)print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

准确率高达90%

优化参数

这个时候就可以设置不同的hidden_layer的维度大小[1, 2, 3, 4, 5, 20, 50]

# This may take about 2 minutes to runplt.figure(figsize=(16, 32))hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50]for i, n_h in enumerate(hidden_layer_sizes):    plt.subplot(5, 2, i+1)    plt.title('Hidden Layer of size %d' % n_h)    parameters = nn_model(X, Y, n_h, num_iterations = 5000)    plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)    predictions = predict(parameters, X)    accuracy = float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100)    print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))

Accuracy for 1 hidden units: 67.5 %Accuracy for 2 hidden units: 67.25 %Accuracy for 3 hidden units: 90.75 %Accuracy for 4 hidden units: 90.5 %Accuracy for 5 hidden units: 91.25 %Accuracy for 20 hidden units: 90.0 %Accuracy for 50 hidden units: 90.25 %

得到的结果在n_h = 5时有最大值。

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