《深度学习——Andrew Ng》第一课第三周编程作业

Planar data classification with one hidden layer

You will see a big difference between this model and the one you implemented using logistic regression.

You will learn how to:
- Implement a 2-class classification neural network with a single hidden layer
- Use units with a non-linear activation function, such as tanh
- Compute the cross entropy loss
- Implement forward and backward propagation

原始数据集

不同的颜色表示不同的类别，可以看到数据不是简单的线性可分类型。下面使用逻辑回归和两层神经网络分别对数据进行分类。

使用sklearn包的LogisticRegression做二分类

# Package importsimport numpy as npimport matplotlib.pyplot as pltfrom testCases import *import sklearnimport sklearn.datasetsimport sklearn.linear_modelfrom planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasetsnp.random.seed(1) # set a seed so that the results are consistentX, Y = load_planar_dataset()### START CODE HERE ### (≈ 3 lines of code)shape_X = X.shapeshape_Y = Y.shapem = shape_X[1]  # training set size### END CODE HERE #### Train the logistic regression classifierclf = sklearn.linear_model.LogisticRegressionCV();clf.fit(X.T, Y.T);# Plot the decision boundary for logistic regressionplot_decision_boundary(lambda x: clf.predict(x), X, Y)plt.title("Logistic Regression")# Print accuracyLR_predictions = clf.predict(X.T)print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +       '% ' + "(percentage of correctly labelled datapoints)")plt.show()

运行结果：

Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)

这里可以看到，数据分布比较混乱，直接使用逻辑回归的准确率只有47%，还没过半，甚至不如随便猜（50%）。

构建自己的浅层神经网络

该编程作业将实现神经网络的步骤通过一个个的函数来实现，从上到下依次为：

1. 确定各层神经元个数；
2. 初始化参数；
3. 前向传播，求神经网络的输出值；
4. 计算cost；
5. 后向传播，求各参数的偏导数值；
6. 更新各参数值（使用偏导数、学习效率alpha），完成一次迭代；
7. 达到迭代次数，确定神经网络的最终参数；
8. 使用参数修正的神经网络预测样本；
9. 求准确率。

# 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)# 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# 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# 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 = - np.sum(logprobs) / m    ### 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# GRADED FUNCTION: backward_propagationdef 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# 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 -= learning_rate * dW1    b1 -= learning_rate * db1    W2 -= learning_rate * dW2    b2 -= learning_rate * db2    ### END CODE HERE ###    parameters = {"W1": W1,                  "b1": b1,                  "W2": W2,                  "b2": b2}    return 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, learning_rate = 1.2)        ### 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# 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 = np.around(A2)  # 四舍五入，取0-1预测值    ### END CODE HERE ###    return predictions

使用构建的浅层神经网络做预测

# 构建一个n_h=4（隐层神经元个数）的浅层神经网络parameters = 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))# 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) + '%')

运行结果：

改变隐层神经元个数

plt.figure(figsize=(16, 32))hidden_layer_sizes = [1, 2, 3, 4, 5, 10, 20]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))

运行结果：

Interpretation:

• 较大的模型（更多隐藏的单位）能够适应训练集更好，直到最后最大的模型拟合数据。
• 最佳隐层的大小似乎在n_h = 5。事实上，这里的参数似乎更拟合数据，也没有产生明显的过度拟合。
• 随后，也会过习正则化，它可以让你使用非常大的模型（如n_h = 50）没有太多的过。

测试不同的测试集

# Datasetsnoisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure = load_extra_datasets()datasets = {"noisy_circles": noisy_circles,            "noisy_moons": noisy_moons,            "blobs": blobs,            "gaussian_quantiles": gaussian_quantiles}### START CODE HERE ### (choose your dataset)dataset = "blobs"### END CODE HERE ###X, Y = datasets[dataset]X, Y = X.T, Y.reshape(1, Y.shape[0])# make blobs binaryif dataset == "blobs":    Y = Y%2# Visualize the dataplt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral);

运行结果：

数据集

神经网络分类结果