前言：

文章以Andrew Ng 的 deeplearning.ai 视频课程为主线，记录Programming Assignments 的实现过程。相对于斯坦福的CS231n课程，Andrew的视频课程更加简单易懂，适合深度学习的入门者系统学习！

这次作业的主题是使用一个隐藏层区分平面数据，涉及到两种类型的激活函数分别为tanh和sigmoid，使用gradient descent算法对参数进行更新，个人觉得这次作业的最大亮点是划分平面数据，让我们认识到神经网络不仅仅对图片有很好的performance，对其他类型的数据也是可以尝试这种方法进行classification

1.1 Dataset

let's get the dataset，the code will load a "flower" 2-class dataset into variables X and Y:

X, Y = load_planar_dataset()

plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral);

shape_X = X.shape

shape_Y = Y.shape

m = X.shape[1]  # training set size

print ('The shape of X is: ' + str(shape_X))

print ('The shape of Y is: ' + str(shape_Y))

print ('I have m = %d training examples!' % (m))

1.2 Simple Logistic Regression

Before building a full neural network, lets first see how logistic regression performs on this problem. You can use sklearn's built-in functions to do that. Run the code below to train a logistic regression classifier on the dataset.

clf = sklearn.linear_model.LogisticRegressionCV();

clf.fit(X.T, Y.T)

plot_decision_boundary(lambda x: clf.predict(x), X, Y)

plt.title("Logistic Regression")

LR_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

1.3 Neural Network model

Logistic Regression 在 flower dataset 上的表现不是很好，所以我们尝试使用带有一个隐藏层的神经网络来训练我们的数据集

这里是我们的模型和一些数学推到：

下面是实现代码：

def layer_sizes(X, Y):

n_x = X.shape[0] # size of input layer

n_h = 4

n_y = Y.shape[0] # size of output layer

return (n_x, n_h, n_y)

def initialize_parameters(n_x, n_h, n_y):

np.random.seed(2)

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))

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

def forward_propagation(X, parameters):

W1 = parameters["W1"]

b1 = parameters["b1"]

W2 = parameters["W2"]

b2 = parameters["b2"]

Z1 = np.dot(W1,X)+b1

A1 = np.tanh(Z1)

Z2 = np.dot(W2,A1)+b2

A2 = sigmoid(Z2)

assert(A2.shape == (1, X.shape[1]))

cache = {"Z1": Z1,

"A1": A1,

"Z2": Z2,

"A2": A2}

return A2, cache

def compute_cost(A2, Y, parameters):

m = Y.shape[1] # number of example

logprobs = Y*np.log(A2)+(1-Y)*np.log(1-A2)

cost = -1/m*np.sum(logprobs)

cost = np.squeeze(cost)

assert(isinstance(cost, float))

return cost

这是梯度下降的数学推导：

def backward_propagation(parameters, cache, X, Y):

m = X.shape[1]

W1 = parameters["W1"]

W2 = parameters["W2"]

A1 = cache["A1"]

A2 = cache["A2"]

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-A1*A1)

dW1 = 1/m*np.dot(dZ1,X.T)

db1 = 1/m*np.sum(dZ1,axis=1,keepdims=True)

grads = {"dW1": dW1,

"db1": db1,

"dW2": dW2,

"db2": db2}

return grads

def update_parameters(parameters, grads, learning_rate = 1.2):

W1 = parameters["W1"]

b1 = parameters["b1"]

W2 = parameters["W2"]

b2 = parameters["b2"]

dW1 = grads["dW1"]

db1 = grads["db1"]

dW2 = grads["dW2"]

db2 = grads["db2"]

W1 = W1-learning_rate*dW1

b1 = b1-learning_rate*db1

W2 = W2-learning_rate*dW2

b2 = b2-learning_rate*db2

parameters = {"W1": W1,

"b1": b1,

"W2": W2,

"b2": b2}

return parameters

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):

np.random.seed(3)

n_x = layer_sizes(X, Y)[0]

n_y = layer_sizes(X, Y)[2]

parameters = initialize_parameters(n_x, n_h, n_y)

W1 = parameters["W1"]

b1 = parameters["b1"]

W2 = parameters["W2"]

b2 = parameters["b2"]

for i in range(0, num_iterations):

A2, cache = forward_propagation(X, parameters)

cost = compute_cost(A2, Y, parameters)

grads = backward_propagation(parameters, cache, X, Y)

parameters = update_parameters(parameters, grads)

if print_cost and i % 1000 == 0:

print ("Cost after iteration %i: %f" %(i, cost))

return parameters

def predict(parameters, X):

A2, cache = forward_propagation(X, parameters)

predictions = A2>0.5

return predictions

我们调用定义好的模型进行训练：

parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)

plt.title("Decision Boundary for hidden layer size " + str(4))

训练结果如下：

对X的数据进行预测：

predictions = predict(parameters, X)

print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

现在我们对隐藏层的神经元数量进行tune，神经元数量分别为1,2,3,4,5,20,50，代码如下：

plt.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))

结果为：

从实验结果可以看出当神经元的数量达到3个以上的时候，神经网络对flower的数据集拟合程度较好。最后附上我作业的得分，表示我的程序没有问题，如果觉得我的文章对您有用，请随意打赏，我将持续更新Deeplearning.ai的作业！