02改善深层神经网络-Regularization-第一周编程作业2

客官°小女子只卖身不卖艺 2022-05-11 03:24 148阅读 0赞

三层神经网络模型：LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID

当网络在训练集上表现得很好，而在测试集上表现不好的时候，我们需要考虑对参数加上正则化项，来增加对权重的惩罚、减少过拟合。

**What you should remember** \-- the implications of L2-regularization on: （应用L2正则化需要知道的）

*  The cost computation:（损失函数计算）
    
     *  A regularization term is added to the cost （在损失后面加上正则化项）
 *  The backpropagation function:（反向传播）
    
     *  There are extra terms in the gradients with respect to weight matrices （关于梯度的权重函数有额外的正则化项）
 *  Weights end up smaller ("weight decay"): （梯度衰减）
    
     *  Weights are pushed to smaller values. （权重被降为较小的值）

**What you should remember about dropout: （**应用dropout需要知道的**）**

*  Dropout is a regularization technique. （Dropout是一种正则化方法）
 *  You only use dropout during training. Don't use dropout (randomly eliminate nodes) during test time. （只能在训练过程中使用dropout，不能在测试过程中使用dropout随机消除神经网络的节点）
 *  Apply dropout both during forward and backward propagation. （可以同时在前向和反向传播中使用dropout）
 *  During training time, divide each dropout layer by keep\_prob to keep the same expected value for the activations. For example, if keep\_prob is 0.5, then we will on average shut down half the nodes, so the output will be scaled by 0.5 since only the remaining half are contributing to the solution. Dividing by 0.5 is equivalent to multiplying by 2. Hence, the output now has the same expected value. You can check that this works even when keep\_prob is other values than 0.5.（在训练过程中，对每个dropout层除以 keep\_prob 来使得激活时有相同的期望值。例如 当 keep\_prob = 0.5，我们会平均地抛弃一半的节点，由于只有剩下一半的节点对解决有帮助，所以输出需要按比例地缩小0.5，这里除以0.5和乘2是等价的。因此，现在输出有了相同的期望值。你可以用keep\_prob值不为0.5时的其他值来检查这个工作。）

反向传播和计算损失函数的时候使用regularization正则化项来减小权重；

前向传播使用dropout后的模型：LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.

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#coding=utf-8
    # import packages
    import numpy as np
    import matplotlib.pyplot as plt
    from reg_utils import sigmoid, relu, plot_decision_boundary, initialize_parameters, load_2D_dataset, predict_dec
    from reg_utils import compute_cost, predict, forward_propagation, backward_propagation, update_parameters
    import sklearn
    import sklearn.datasets
    import scipy.io
    from testCases import *
    
    plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
    plt.rcParams['image.interpolation'] = 'nearest'
    plt.rcParams['image.cmap'] = 'gray'
    
    def model(X, Y, learning_rate = 0.3, num_iterations = 30000, print_cost = True, lambd = 0, keep_prob = 1):
        """
        Implements a three-layer neural network: LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID.
        
        Arguments:
        X -- input data, of shape (input size, number of examples)
        Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (output size, number of examples)
        learning_rate -- learning rate of the optimization
        num_iterations -- number of iterations of the optimization loop
        print_cost -- If True, print the cost every 10000 iterations
        lambd -- regularization hyperparameter, scalar
        keep_prob - probability of keeping a neuron active during drop-out, scalar.
        
        Returns:
        parameters -- parameters learned by the model. They can then be used to predict.
        """
            
        grads = {}
        costs = []                            # to keep track of the cost
        m = X.shape[1]                        # number of examples
        layers_dims = [X.shape[0], 20, 3, 1]
        
        # Initialize parameters dictionary.
        parameters = initialize_parameters(layers_dims)
    
        # Loop (gradient descent)
    
        for i in range(0, num_iterations):
    
            # Forward propagation: LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID.
            if keep_prob == 1:
                a3, cache = forward_propagation(X, parameters)
            elif keep_prob < 1:
                a3, cache = forward_propagation_with_dropout(X, parameters, keep_prob)
            
            # Cost function
            if lambd == 0:
                cost = compute_cost(a3, Y)
            else:
                cost = compute_cost_with_regularization(a3, Y, parameters, lambd)
                
            # Backward propagation.
            assert(lambd==0 or keep_prob==1)    # it is possible to use both L2 regularization and dropout, 
                                                # but this assignment will only explore one at a time
            if lambd == 0 and keep_prob == 1:
                grads = backward_propagation(X, Y, cache)
            elif lambd != 0:
                grads = backward_propagation_with_regularization(X, Y, cache, lambd)
            elif keep_prob < 1:
                grads = backward_propagation_with_dropout(X, Y, cache, keep_prob)
            
            # Update parameters.
            parameters = update_parameters(parameters, grads, learning_rate)
            
            # Print the loss every 10000 iterations
            if print_cost and i % 10000 == 0:
                print("Cost after iteration {}: {}".format(i, cost))
            if print_cost and i % 1000 == 0:
                costs.append(cost)
        
        # plot the cost
        plt.plot(costs)
        plt.ylabel('cost')
        plt.xlabel('iterations (x1,000)')
        plt.title("Learning rate =" + str(learning_rate))
        plt.show()
        
        return parameters
     
    # GRADED FUNCTION: compute_cost_with_regularization
    
    def compute_cost_with_regularization(A3, Y, parameters, lambd):
        """
        Implement the cost function with L2 regularization. See formula (2) above.
        
        Arguments:
        A3 -- post-activation, output of forward propagation, of shape (output size, number of examples)
        Y -- "true" labels vector, of shape (output size, number of examples)
        parameters -- python dictionary containing parameters of the model
        
        Returns:
        cost - value of the regularized loss function (formula (2))
        """
        m = Y.shape[1]
        W1 = parameters["W1"]
        W2 = parameters["W2"]
        W3 = parameters["W3"]
        
        cross_entropy_cost = compute_cost(A3, Y) # This gives you the cross-entropy part of the cost
        
        ### START CODE HERE ### (approx. 1 line)
        L2_regularization_cost = 1.0 / m * lambd / 2 * (np.sum(np.square(W1)) + np.sum(np.square(W2)) + np.sum(np.square(W3)))
        ### END CODER HERE ###
        
        cost = cross_entropy_cost + L2_regularization_cost
        
        return cost
    
    # GRADED FUNCTION: backward_propagation_with_regularization
    
    def backward_propagation_with_regularization(X, Y, cache, lambd):
        """
        Implements the backward propagation of our baseline model to which we added an L2 regularization.
        
        Arguments:
        X -- input dataset, of shape (input size, number of examples)
        Y -- "true" labels vector, of shape (output size, number of examples)
        cache -- cache output from forward_propagation()
        lambd -- regularization hyperparameter, scalar
        
        Returns:
        gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
        """
        
        m = X.shape[1]
        (Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3) = cache
        
        dZ3 = A3 - Y
        
        ### START CODE HERE ### (approx. 1 line)
        dW3 = 1./m * np.dot(dZ3, A2.T) + lambd / m * W3
        ### END CODE HERE ###
        db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
        
        dA2 = np.dot(W3.T, dZ3)
        dZ2 = np.multiply(dA2, np.int64(A2 > 0))
        ### START CODE HERE ### (approx. 1 line)
        dW2 = 1./m * np.dot(dZ2, A1.T) + lambd / m * W2
        ### END CODE HERE ###
        db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)
        
        dA1 = np.dot(W2.T, dZ2)
        dZ1 = np.multiply(dA1, np.int64(A1 > 0))
        ### START CODE HERE ### (approx. 1 line)
        dW1 = 1./m * np.dot(dZ1, X.T) + + lambd / m * W1
        ### END CODE HERE ###
        db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)
        
        gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,"dA2": dA2,
                     "dZ2": dZ2, "dW2": dW2, "db2": db2, "dA1": dA1, 
                     "dZ1": dZ1, "dW1": dW1, "db1": db1}
        
        return gradients    
    
    # GRADED FUNCTION: forward_propagation_with_dropout
    
    def forward_propagation_with_dropout(X, parameters, keep_prob = 0.5):
        """
        Implements the forward propagation: LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.
        
        Arguments:
        X -- input dataset, of shape (2, number of examples)
        parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                        W1 -- weight matrix of shape (20, 2)
                        b1 -- bias vector of shape (20, 1)
                        W2 -- weight matrix of shape (3, 20)
                        b2 -- bias vector of shape (3, 1)
                        W3 -- weight matrix of shape (1, 3)
                        b3 -- bias vector of shape (1, 1)
        keep_prob - probability of keeping a neuron active during drop-out, scalar
        
        Returns:
        A3 -- last activation value, output of the forward propagation, of shape (1,1)
        cache -- tuple, information stored for computing the backward propagation
        """
        
        np.random.seed(1)
        
        # retrieve parameters
        W1 = parameters["W1"]
        b1 = parameters["b1"]
        W2 = parameters["W2"]
        b2 = parameters["b2"]
        W3 = parameters["W3"]
        b3 = parameters["b3"]
        
        # LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
        Z1 = np.dot(W1, X) + b1
        A1 = relu(Z1)
        ### START CODE HERE ### (approx. 4 lines)         # Steps 1-4 below correspond to the Steps 1-4 described above. 
        D1 = np.random.rand(A1.shape[0], A1.shape[1])# Step 1: initialize matrix D1 = np.random.rand(..., ...)
        D1 = np.where(D1 < keep_prob, 1, 0)               # Step 2: convert entries of D1 to 0 or 1 (using keep_prob as the threshold)
        A1 = A1 * D1                                      # Step 3: shut down some neurons of A1
        A1 = A1 / keep_prob                               # Step 4: scale the value of neurons that haven't been shut down
        ### END CODE HERE ###
        Z2 = np.dot(W2, A1) + b2
        A2 = relu(Z2)
        ### START CODE HERE ### (approx. 4 lines)
        D2 = np.random.rand(A2.shape[0], A2.shape[1])# Step 1: initialize matrix D2 = np.random.rand(..., ...)
        D2 = np.where(D2 < keep_prob, 1, 0)               # Step 2: convert entries of D2 to 0 or 1 (using keep_prob as the threshold)
        A2 = A2 * D2                                      # Step 3: shut down some neurons of A2
        A2 = A2 / keep_prob                               # Step 4: scale the value of neurons that haven't been shut down
        ### END CODE HERE ###
        Z3 = np.dot(W3, A2) + b3
        A3 = sigmoid(Z3)
        
        cache = (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3)
        
        return A3, cache    
    
    # GRADED FUNCTION: backward_propagation_with_dropout
    
    def backward_propagation_with_dropout(X, Y, cache, keep_prob):
        """
        Implements the backward propagation of our baseline model to which we added dropout.
        
        Arguments:
        X -- input dataset, of shape (2, number of examples)
        Y -- "true" labels vector, of shape (output size, number of examples)
        cache -- cache output from forward_propagation_with_dropout()
        keep_prob - probability of keeping a neuron active during drop-out, scalar
        
        Returns:
        gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
        """
        
        m = X.shape[1]
        (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3) = cache
        
        dZ3 = A3 - Y
        dW3 = 1./m * np.dot(dZ3, A2.T)
        db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
        dA2 = np.dot(W3.T, dZ3)
        ### START CODE HERE ### (≈ 2 lines of code)
        dA2 = dA2 * D2          # Step 1: Apply mask D2 to shut down the same neurons as during the forward propagation
        dA2 = dA2 / keep_prob   # Step 2: Scale the value of neurons that haven't been shut down
        ### END CODE HERE ###
        dZ2 = np.multiply(dA2, np.int64(A2 > 0))
        dW2 = 1./m * np.dot(dZ2, A1.T)
        db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)
        
        dA1 = np.dot(W2.T, dZ2)
        ### START CODE HERE ### (≈ 2 lines of code)
        dA1 = dA1 * D1          # Step 1: Apply mask D1 to shut down the same neurons as during the forward propagation
        dA1 = dA1 / keep_prob   # Step 2: Scale the value of neurons that haven't been shut down
        ### END CODE HERE ###
        dZ1 = np.multiply(dA1, np.int64(A1 > 0))
        dW1 = 1./m * np.dot(dZ1, X.T)
        db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)
        
        gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,"dA2": dA2,
                     "dZ2": dZ2, "dW2": dW2, "db2": db2, "dA1": dA1, 
                     "dZ1": dZ1, "dW1": dW1, "db1": db1}
        
        return gradients    
     
    if __name__=='__main__':  
        train_X, train_Y, test_X, test_Y = load_2D_dataset()
        
        parameters = model(train_X, train_Y)
        print ("On the training set:")
        predictions_train = predict(train_X, train_Y, parameters)
        print ("On the test set:")
        predictions_test = predict(test_X, test_Y, parameters)
        
        plt.title("Model without regularization")
        axes = plt.gca()
        axes.set_xlim([-0.75,0.40])
        axes.set_ylim([-0.75,0.65])
        plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
        
        A3, Y_assess, parameters = compute_cost_with_regularization_test_case()
    
        print("cost = " + str(compute_cost_with_regularization(A3, Y_assess, parameters, lambd = 0.1)))
    
        X_assess, Y_assess, cache = backward_propagation_with_regularization_test_case()
    
        grads = backward_propagation_with_regularization(X_assess, Y_assess, cache, lambd = 0.7)
        print ("dW1 = "+ str(grads["dW1"]))
        print ("dW2 = "+ str(grads["dW2"]))
        print ("dW3 = "+ str(grads["dW3"]))
        
        parameters = model(train_X, train_Y, lambd = 0.7)
        print ("On the train set:")
        predictions_train = predict(train_X, train_Y, parameters)
        print ("On the test set:")
        predictions_test = predict(test_X, test_Y, parameters)
    
        plt.title("Model with L2-regularization")
        axes = plt.gca()
        axes.set_xlim([-0.75,0.40])
        axes.set_ylim([-0.75,0.65])
        plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
        
        X_assess, parameters = forward_propagation_with_dropout_test_case()
    
        A3, cache = forward_propagation_with_dropout(X_assess, parameters, keep_prob = 0.7)
        print ("A3 = " + str(A3))
        
        X_assess, Y_assess, cache = backward_propagation_with_dropout_test_case()
    
        gradients = backward_propagation_with_dropout(X_assess, Y_assess, cache, keep_prob = 0.8)
    
        print ("dA1 = " + str(gradients["dA1"]))
        print ("dA2 = " + str(gradients["dA2"]))
        
        parameters = model(train_X, train_Y, keep_prob = 0.86, learning_rate = 0.3)
    
        print ("On the train set:")
        predictions_train = predict(train_X, train_Y, parameters)
        print ("On the test set:")
        predictions_test = predict(test_X, test_Y, parameters)
        
        plt.title("Model with dropout")
        axes = plt.gca()
        axes.set_xlim([-0.75,0.40])
        axes.set_ylim([-0.75,0.65])
        plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

运行结果：

（1）无正则化无dropout

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（2）有正则化无dropout

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（3）有正则化有dropout

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[70]: /images/20220511/c3b03b9fb5414405a559be0fad3c29c2.png
[70 1]: /images/20220511/aa9794229325489fb55362e2f15e3cbd.png
[70 2]: /images/20220511/f3944a80c57f47c483dfd4b21f3bf228.png
[70 3]: /images/20220511/bc08cde8a5d94fc0b7783d616a378876.png
[70 4]: /images/20220511/8adf13a1ec2c4fd59a96ebfc9832b6eb.png
[70 5]: /images/20220511/4c77c5f9427d4a49b5515e9b1674bc5c.png
[70 6]: /images/20220511/c02734b3f07a47cd9854ccb8ec61e763.png
[70 7]: /images/20220511/f87b3fbd670e422bb7b2696faa743a17.png
[70 8]: /images/20220511/44ed3da944fd4c69bab1b610a1fe7624.png
[70 9]: /images/20220511/4320286837224e7d90b9b79aca4ce77f.png