文章目錄

• • 1. 導入包
  • 2. 預覽數據
  • 3. 邏輯回歸
  • 4. 神經網絡
  • • 4.1 定義神經網絡結構
    • 4.2 初始化模型參數
    • 4.3 循環
    • • 4.3.1 前向傳播
      • 4.3.2 計算損失
      • 4.3.3 后向傳播
      • 4.3.4 梯度下降
    • 4.4 組建Model
    • 4.5 預測
    • 4.6 調節隱藏層單元個數
    • 4.7 更改激活函數
    • 4.8 更改學習率
    • 4.9 其他數據集下的表現

選擇題測試：
參考博文1
參考博文2

建立你的第一個神經網絡！其有1個隱藏層。

1. 導入包

# Package imports import numpy as np import matplotlib.pyplot as plt from testCases import * import sklearn import sklearn.datasets import sklearn.linear_model from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets%matplotlib inlinenp.random.seed(1) # set a seed so that the results are consistent

2. 預覽數據

• 可視化數據

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

紅色的標簽為 0， 藍色的標簽為 1，我們的目標是建模將它們分開

• 數據維度

### START CODE HERE ### (≈ 3 lines of code) shape_X = X.shape shape_Y = Y.shape m = X.shape[1] # training set size ### END CODE HERE ###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)) The shape of X is: (2, 400) The shape of Y is: (1, 400) I have m = 400 training examples!

3. 邏輯回歸

# Train the logistic regression classifier clf = sklearn.linear_model.LogisticRegressionCV(); clf.fit(X.T, Y.T); # Plot the decision boundary for logistic regression plot_decision_boundary(lambda x: clf.predict(x), X, Y) plt.title("Logistic Regression")# Print accuracy 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 datapoints)") Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)

數據集是線性不可分的，邏輯回歸變現的不好，下面看看神經網絡怎么樣。

4. 神經網絡

模型如下：

對于一個樣本 $x^{(i)}$ 而言：
$$z^{[1](i)}=W^{[1]}{x}^{(i)}+b^{[1](i)}$$
$$a^{[1](i)}=\tanh (z^{[1](i)})$$
$$z^{[2](i)}=W^{[2]}{a}^{[1](i)}+b^{[2](i)}$$
$${\hat{y}}^{(i)}=a^{[2](i)}=\sigma (z^{[2](i)})$$

$$y_{\text{prediction}}^{(i)}=\{\begin{array}{ll}1& \text{?if?}{a}^{[2](i)}>0.5\\ 0& \text{?otherwise?}\end{array}$$

得到所有的樣本的預測值后，計算損失：
$$J=?\frac{1}{m}\sum _{i=0}^m\left(y^{(i)}\log \left(a^{[2](i)}\right)+(1?y^{(i)})\log \left(1?a^{[2](i)}\right)\right)$$

建立神經網絡的一般方法：

• 1、定義神經網絡結構（輸入，隱藏單元等）
• 2、初始化模型的參數
• 3、循環：
  —— a、實現正向傳播
  —— b、計算損失
  —— c、實現反向傳播，計算梯度
  —— d、更新參數（梯度下降）

編寫輔助函數，計算步驟1-3
將它們合并到 nn_model（）的函數中
學習正確的參數，對新數據進行預測

4.1 定義神經網絡結構

• 定義每層的節點個數

# 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 layern_h -- the size of the hidden layern_y -- the size of the output layer"""### START CODE HERE ### (≈ 3 lines of code)n_x = X.shape[0] # size of input layern_h = 4n_y = Y.shape[0] # size of output layer### END CODE HERE ###return (n_x, n_h, n_y)

4.2 初始化模型參數

• 隨機初始化權重 w，偏置 b 初始化為 0

# GRADED FUNCTION: initialize_parametersdef initialize_parameters(n_x, n_h, n_y):"""Argument:n_x -- size of the input layern_h -- size of the hidden layern_y -- size of the output layerReturns: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 # randn 標準正態分布b1 = np.zeros((n_h, 1))W2 = np.random.randn(n_y, n_h)*0.01b2 = 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

4.3 循環

4.3.1 前向傳播

• 根據上面的公式，編寫代碼

# 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 activationcache -- 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) + b1A1 = np.tanh(Z1)Z2 = np.dot(W2, A1) + b2A2 = sigmoid(Z2)### END CODE HERE ###assert(A2.shape == (1, X.shape[1]))cache = {"Z1": Z1,"A1": A1,"Z2": Z2,"A2": A2}return A2, cache

4.3.2 計算損失

• 計算了 A2，也就是每個樣本的預測值，計算損失
  $$J=?\frac{1}{m}\sum _{i=0}^m\left(y^{(i)}\log \left(a^{[2](i)}\right)+(1?y^{(i)})\log \left(1?a^{[2](i)}\right)\right)$$

# 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 b2Returns: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 = Y*np.log(A2)+(1-Y)*np.log(1-A2)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

4.3.3 后向傳播

一些公式如下：

激活函數的導數，請查閱

• sigmoid
  
  $$a=g(z);g^{\prime }(z)=\frac{d}{dz}g(z)=a(1?a)$$
• tanh
  
  $$a=g(z);g^{\prime }(z)=\frac{d}{dz}g(z)=1?a^2$$

sigmoid 下損失函數求導

# 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-YdW2 = np.dot(dZ2, A1.T)/mdb2 = np.sum(dZ2, axis=1, keepdims=True)/mdZ1 = np.dot(W2.T, dZ2)*(1-np.power(A1, 2))dW1 = np.dot(dZ1, X.T)/mdb1 = np.sum(dZ1, axis=1, keepdims=True)/m### END CODE HERE ###grads = {"dW1": dW1,"db1": db1,"dW2": dW2,"db2": db2}return grads

4.3.4 梯度下降

• 選取合適的學習率，學習率太大，會產生震蕩，收斂慢，甚至不收斂

# GRADED FUNCTION: update_parametersdef update_parameters(parameters, grads, learning_rate = 1.2):"""Updates parameters using the gradient descent update rule given aboveArguments: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 * dW1b1 = b1 - learning_rate * db1W2 = W2 - learning_rate * dW2b2 = b2 - learning_rate * db2### END CODE HERE ###parameters = {"W1": W1,"b1": b1,"W2": W2,"b2": b2}return parameters

4.4 組建Model

• 將上面的函數以正確順序放在 model 里

# 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 layernum_iterations -- Number of iterations in gradient descent loopprint_cost -- if True, print the cost every 1000 iterationsReturns: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 iterationsif print_cost and i % 1000 == 0:print ("Cost after iteration %i: %f" %(i, cost))return parameters

4.5 預測

$predictions=\{\begin{array}{ll}1& \text{if}activation>0.5\\ 0& \text{otherwise}\end{array}$

# GRADED FUNCTION: predictdef predict(parameters, X):"""Using the learned parameters, predicts a class for each example in XArguments:parameters -- python dictionary containing your parameters X -- input data of size (n_x, m)Returnspredictions -- 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

• 建立一個含有1個隱藏層（4個單元）的神經網絡模型

# Build a model with a n_h-dimensional hidden layer parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)# Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y) plt.title("Decision Boundary for hidden layer size " + str(4)) Cost after iteration 0: 0.693048 Cost after iteration 1000: 0.288083 Cost after iteration 2000: 0.254385 Cost after iteration 3000: 0.233864 Cost after iteration 4000: 0.226792 Cost after iteration 5000: 0.222644 Cost after iteration 6000: 0.219731 Cost after iteration 7000: 0.217504 Cost after iteration 8000: 0.219550 Cost after iteration 9000: 0.218633

# Print accuracy predictions = predict(parameters, X) print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%') Accuracy: 90%

可以看出模型較好地將兩類點分開了！準確率 90%，比邏輯回歸 47%高不少。

4.6 調節隱藏層單元個數

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)) 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: 90.5 % Accuracy for 50 hidden units: 90.75 %

可以看出：

• 較大的模型（具有更多隱藏單元）能夠更好地適應訓練集，直到最大的模型過擬合了
• 最好的隱藏層大小似乎是n_h=5左右。這個值似乎很適合數據，而不會引起明顯的過擬合
• 稍后還將了解正則化，它允許你使用非常大的模型（如n_h=50），而不會出現太多過擬合

4.7 更改激活函數

• 將隱藏層的激活函數更改為 sigmoid 函數，準確率沒有使用tanh的高，tanh在任何場合幾乎都優于sigmoid

Accuracy for 1 hidden units: 50.5 % Accuracy for 2 hidden units: 59.0 % Accuracy for 3 hidden units: 56.75 % Accuracy for 4 hidden units: 50.0 % Accuracy for 5 hidden units: 62.25000000000001 % Accuracy for 20 hidden units: 85.5 % Accuracy for 50 hidden units: 87.0 %

• 將隱藏層的激活函數更改為 ReLu 函數，似乎沒有用，感覺是需要更多的隱藏層，才能達到效果

def relu(X):return np.maximum(0, X) Accuracy for 1 hidden units: 50.0 % Accuracy for 2 hidden units: 50.0 % Accuracy for 3 hidden units: 50.0 % Accuracy for 4 hidden units: 50.0 % Accuracy for 5 hidden units: 50.0 % Accuracy for 20 hidden units: 50.0 % Accuracy for 50 hidden units: 50.0 %

報了些警告

C:\Users\mingm\AppData\Roaming\Python\Python37\site-packages\ ipykernel_launcher.py:20: RuntimeWarning: divide by zero encountered in log C:\Users\mingm\AppData\Roaming\Python\Python37\site-packages\ ipykernel_launcher.py:20: RuntimeWarning: invalid value encountered in multiply C:\Users\mingm\AppData\Roaming\Python\Python37\site-packages\ ipykernel_launcher.py:35: RuntimeWarning: overflow encountered in power C:\Users\mingm\AppData\Roaming\Python\Python37\site-packages\ ipykernel_launcher.py:35: RuntimeWarning: invalid value encountered in multiply C:\Users\mingm\AppData\Roaming\Python\Python37\site-packages \ipykernel_launcher.py:35: RuntimeWarning: overflow encountered in multiply

4.8 更改學習率

• 采用 tanh 激活函數，調整學習率檢查效果

學習率 2.0

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: 90.75 % Accuracy for 5 hidden units: 90.25 % Accuracy for 20 hidden units: 91.0 % Accuracy for 50 hidden units: 91.25 % best

學習率 1.5

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: 89.75 % Accuracy for 5 hidden units: 90.5 % Accuracy for 20 hidden units: 91.0 % best Accuracy for 50 hidden units: 90.75 %

學習率 1.2

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 % best Accuracy for 20 hidden units: 90.5 % Accuracy for 50 hidden units: 90.75 %

學習率 1.0

Accuracy for 1 hidden units: 67.25 % Accuracy for 2 hidden units: 67.0 % Accuracy for 3 hidden units: 90.75 % Accuracy for 4 hidden units: 90.5 % Accuracy for 5 hidden units: 91.0 % best Accuracy for 20 hidden units: 91.0 % best Accuracy for 50 hidden units: 90.75 %

學習率 0.5

Accuracy for 1 hidden units: 67.25 % Accuracy for 2 hidden units: 66.5 % Accuracy for 3 hidden units: 89.25 % Accuracy for 4 hidden units: 90.0 % Accuracy for 5 hidden units: 89.75 % Accuracy for 20 hidden units: 90.0 % best Accuracy for 50 hidden units: 89.75 %

學習率 0.1

Accuracy for 1 hidden units: 67.0 % Accuracy for 2 hidden units: 64.75 % Accuracy for 3 hidden units: 88.25 % Accuracy for 4 hidden units: 88.0 % Accuracy for 5 hidden units: 88.5 % Accuracy for 20 hidden units: 88.75 % best Accuracy for 50 hidden units: 88.75 % best

大致規律：

• 學習率太小，造成學習不充分，準確率較低
• 學習率越大，需要越多的隱藏單元來提高準確率？（請大佬指點）

4.9 其他數據集下的表現

均為tanh激活函數，學習率1.2

• dataset = "noisy_circles"
  

Accuracy for 1 hidden units: 62.5 % Accuracy for 2 hidden units: 72.5 % Accuracy for 3 hidden units: 84.0 % best Accuracy for 4 hidden units: 83.0 % Accuracy for 5 hidden units: 83.5 % Accuracy for 20 hidden units: 79.5 % Accuracy for 50 hidden units: 83.5 %

• dataset = "noisy_moons"
  

Accuracy for 1 hidden units: 86.0 % Accuracy for 2 hidden units: 88.0 % Accuracy for 3 hidden units: 97.0 % best Accuracy for 4 hidden units: 96.5 % Accuracy for 5 hidden units: 96.0 % Accuracy for 20 hidden units: 86.0 % Accuracy for 50 hidden units: 86.0 %

• dataset = "blobs"
  

Accuracy for 1 hidden units: 67.0 % Accuracy for 2 hidden units: 67.0 % Accuracy for 3 hidden units: 83.0 % Accuracy for 4 hidden units: 83.0 % Accuracy for 5 hidden units: 83.0 % Accuracy for 20 hidden units: 86.0 % best Accuracy for 50 hidden units: 83.5 %

• dataset = "gaussian_quantiles"
  

Accuracy for 1 hidden units: 65.0 % Accuracy for 2 hidden units: 79.5 % Accuracy for 3 hidden units: 97.0 % Accuracy for 4 hidden units: 97.0 % Accuracy for 5 hidden units: 100.0 % best Accuracy for 20 hidden units: 97.5 % Accuracy for 50 hidden units: 96.0 %

不同的數據集下，表現的效果也不太一樣。

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