改善深层神经网络第一周-Regularization

Regularization

Welcome to the second assignment of this week. Deep Learning models have so much flexibility and capacity that overfitting can be a serious problem, if the training dataset is not big enough. Sure it does well on the training set, but the learned network doesn’t generalize to new examples that it has never seen!

You will learn to: Use regularization in your deep learning models.

Let’s first import the packages you are going to use.

# 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 *

%matplotlib inline
plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

Problem Statement: You have just been hired as an AI expert by the French Football Corporation. They would like you to recommend positions where France’s goal keeper should kick the ball so that the French team’s players can then hit it with their head. (你刚刚被法国足球公司聘为AI专家。他们希望你推荐法国守门员应该踢球的位置，这样法国队的球员可以用头打)

Figure 1 : Football field
The goal keeper kicks the ball in the air, the players of each team are fighting to hit the ball with their head

They give you the following 2D dataset from France’s past 10 games.

此处有大坑，会出现维度不匹配的错误，即c of shape (1, 211) not acceptable as a color sequence for x with size 211, y with size 211，好像这个问题在做题过程中一直在出现。这本题中解决方法为：找到load_2D_dataset()这个函数的py文件，也就reg_utils.py,在这个函数里面把
plt.scatter(train_X[0, :], train_X[1, :], c=c=train_Y, s=40, cmap=plt.cm.Spectral);中的c=train_Y替换为c=np.squeeze(train_Y)就可以了，也就是删除单维条目，在其他题目中碰到同样的错误也如此解决。另外修改之后记得保存，还有一点特别重要！！！如果你是用的jupyter notebook做的，在练习的页面一定记得点一下kernel下面的那个刷新图标，也就是重启内核，还记得把上面的导包命令重新执行一下，不然还是会报这个错误

train_X, train_Y, test_X, test_Y = load_2D_dataset()

Each dot corresponds to a position on the football field where a football player has hit the ball with his/her head after the French goal keeper has shot the ball from the left side of the football field.(每个点对应于足球运动员在足球场左侧击球之后，用头将球击中的足球场上的位置)
- If the dot is blue, it means the French player managed to hit the ball with his/her head(如果这个点是蓝色的，这意味着这个法国球员设法用他/她的头击球)
- If the dot is red, it means the other team’s player hit the ball with their head(如果这个点是红色的，这意味着另一个队的球员用头撞球)

Your goal: Use a deep learning model to find the positions on the field where the goalkeeper should kick the ball.(*
使用深度学习模式来找到守门员应该踢球的场地位置*)

Analysis of the dataset: This dataset is a little noisy, but it looks like a diagonal line separating the upper left half (blue) from the lower right half (red) would work well. (这个数据集有点嘈杂，但是看起来像是左上角（蓝色）和右下角（红色）分开的对角线，效果很好)

You will first try a non-regularized model. Then you’ll learn how to regularize it and decide which model you will choose to solve the French Football Corporation’s problem.(你将首先尝试一个非正规化的模型。然后，您将学习如何正规化，并决定选择哪种模式来解决法国足球公司的问题)

1 - Non-regularized model

You will use the following neural network (already implemented for you below). This model can be used:
- in regularization mode – by setting the lambd input to a non-zero value. We use “lambd” instead of “lambda” because “lambda” is a reserved keyword in Python. (在正则化模式中 - 通过将lambd输入设置为一个非零值。我们使用”lambd“代替”lambda“，因为”lambda“是Python中的保留关键字)
- in dropout mode – by setting the keep_prob to a value less than one(在* dropout模式下* - 通过将keep_prob设置为小于1的值)

You will first try the model without any regularization. Then, you will implement:
- L2 regularization – functions: “compute_cost_with_regularization()” and “backward_propagation_with_regularization()”
- Dropout – functions: “forward_propagation_with_dropout()” and “backward_propagation_with_dropout()”

In each part, you will run this model with the correct inputs so that it calls the functions you’ve implemented. Take a look at the code below to familiarize yourself with the model.

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

Let’s train the model without any regularization, and observe the accuracy on the train/test sets.

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)

Cost after iteration 0: 0.6557412523481002
Cost after iteration 10000: 0.1632998752572417
Cost after iteration 20000: 0.13851642423284755

On the training set:
Accuracy: 0.947867298578
On the test set:
Accuracy: 0.915

The train accuracy is 94.8% while the test accuracy is 91.5%. This is the **baseline model(
基准模型)** (you will observe the impact of regularization on this model). Run the following code to plot the decision boundary of your model.

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)

The non-regularized model is obviously overfitting the training set. It is fitting the noisy points! Lets now look at two techniques to reduce overfitting.(非正则化模型显然是过度训练集合。这是适合的嘈杂点！现在让我们看看两种技术来减少过度拟合)

2 - L2 Regularization

The standard way to avoid overfitting is called L2 regularization. It consists of appropriately modifying your cost function, from:

J=−1m∑i=1m(y(i)log(a[L](i))+(1−y(i))log(1−a[L](i)))(1)

To:

Jregularized=−1m∑i=1m(y(i)log(a[L](i))+(1−y(i))log(1−a[L](i)))cross-entropy cost+1mλ2∑l∑k∑jW[l]2k,jL2 regularization cost(2)

Let’s modify your cost and observe the consequences.

Exercise: Implement compute_cost_with_regularization() which computes the cost given by formula (2). To calculate ∑k∑jW[l]2k,j , use :

np.sum(np.square(Wl))

Note that you have to do this for W[1] , W[2] and W[3] , then sum the three terms and multiply by 1mλ2 .

# 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./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

A3, Y_assess, parameters = compute_cost_with_regularization_test_case()

print("cost = " + str(compute_cost_with_regularization(A3, Y_assess, parameters, lambd = 0.1)))

cost = 1.78648594516

Expected Output:

**cost**

1.78648594516

Of course, because you changed the cost, you have to change backward propagation as well! All the gradients have to be computed with respect to this new cost.

Exercise: Implement the changes needed in backward propagation to take into account regularization. The changes only concern dW1, dW2 and dW3. For each, you have to add the regularization term’s gradient ( ddW(12λmW2)=λmW ).(实施反向传播所需的变化，以考虑正规化。这些变化只涉及dW1，dW2和dW3。对于每一个，你必须添加正则化项的梯度( ddW(12λmW2)=λmW ))

# 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

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"]))

dW1 = [[-0.25604646  0.12298827 -0.28297129]
 [-0.17706303  0.34536094 -0.4410571 ]]
dW2 = [[ 0.79276486  0.85133918]
 [-0.0957219  -0.01720463]
 [-0.13100772 -0.03750433]]
dW3 = [[-1.77691347 -0.11832879 -0.09397446]]

Expected Output:

**dW1**	[[-0.25604646 0.12298827 -0.28297129] [-0.17706303 0.34536094 -0.4410571 ]]
**dW2**	[[ 0.79276486 0.85133918] [-0.0957219 -0.01720463] [-0.13100772 -0.03750433]]
**dW3**	[[-1.77691347 -0.11832879 -0.09397446]]

Let’s now run the model with L2 regularization (λ=0.7) . The model() function will call:
- compute_cost_with_regularization instead of compute_cost
- backward_propagation_with_regularization instead of backward_propagation

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)

Cost after iteration 0: 0.6974484493131264
Cost after iteration 10000: 0.2684918873282239
Cost after iteration 20000: 0.2680916337127301

On the train set:
Accuracy: 0.938388625592
On the test set:
Accuracy: 0.93

Congrats, the test set accuracy increased to 93%. You have saved the French football team!

You are not overfitting the training data anymore. Let’s plot the decision boundary.

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)

Observations:
- The value of λ is a hyperparameter that you can tune using a dev set.( λ 的值是一个超参数，您可以使用开发集进行调优)
- L2 regularization makes your decision boundary smoother. If λ is too large, it is also possible to “oversmooth”, resulting in a model with high bias.(L2正则化使得你的决策边界更加平滑。如果 λ 太大，也有可能“过度平滑”，从而导致高偏差的模型)

What is L2-regularization actually doing?:

L2-regularization relies on the assumption that a model with small weights is simpler than a model with large weights. Thus, by penalizing the square values of the weights in the cost function you drive all the weights to smaller values. It becomes too costly for the cost to have large weights! This leads to a smoother model in which the output changes more slowly as the input changes. (L2规则化依赖于这样的假设，即具有小权重的模型比具有大权重的模型简单。因此，通过惩罚成本函数中权重的平方值，可以将所有权重驱动到较小的值。拥有大重量的成本太贵了！这导致更平滑的模型，其中随着输入改变，输出变化更慢)

What you should remember – the implications of L2-regularization on:
- The cost computation:
- A regularization term is added to the cost(正则化项被添加到代价函数中)
- The backpropagation function:
- There are extra terms in the gradients with respect to weight matrices(在权重矩阵的梯度中有额外的项)
- Weights end up smaller (“weight decay”):
- Weights are pushed to smaller values.(权重被推到较小的值)

3 - Dropout

Finally, dropout is a widely used regularization technique that is specific to deep learning.
It randomly shuts down some neurons in each iteration. Watch these two videos to see what this means!(*最后，dropout是一种广泛使用的正规化技术，特定于深度学习。
每次迭代都会随机关闭一些神经元。观看这两个视频，看看这意味着什么*)

Figure 2 : Drop-out on the second hidden layer.
At each iteration, you shut down (= set to zero) each neuron of a layer with probability 1−keep_prob or keep it with probability keep_prob (50% here). The dropped neurons don’t contribute to the training in both the forward and backward propagations of the iteration.( 在每一次迭代中，你关闭（=设置为零）一个层的每个神经元的概率为 1−keep_prob 或保持它的概率 keep_prob （这里是50％）。丢弃的神经元对迭代的向前和向后传播都没有贡献)

Figure 3 : Drop-out on the first and third hidden layers.
1st layer: we shut down on average 40% of the neurons. 3rd layer: we shut down on average 20% of the neurons.

When you shut some neurons down, you actually modify your model. The idea behind drop-out is that at each iteration, you train a different model that uses only a subset of your neurons. With dropout, your neurons thus become less sensitive to the activation of one other specific neuron, because that other neuron might be shut down at any time. (当你关闭一些神经元时，你实际上修改了你的模型。退出背后的想法是，在每次迭代中，你都训练一个不同的模型，只使用你的神经元的一个子集。随着dropout，你的神经元因此变得不那么敏感的激活另一个特定的神经元，因为那个其他的神经元可能在任何时候被关闭)

3.1 - Forward propagation with dropout

Exercise: Implement the forward propagation with dropout. You are using a 3 layer neural network, and will add dropout to the first and second hidden layers. We will not apply dropout to the input layer or output layer. (用dropout实现前向传播。您正在使用3层神经网络，并将添加到第一个和第二个隐藏层丢失。我们不会将dropout应用到输入层或输出层)

Instructions:
You would like to shut down some neurons in the first and second layers. To do that, you are going to carry out 4 Steps(你想关闭第一层和第二层的一些神经元。要做到这一点，你要执行4个步骤):
1. In lecture, we dicussed creating a variable d[1] with the same shape as a[1] using np.random.rand() to randomly get numbers between 0 and 1. Here, you will use a vectorized implementation, so create a random matrix D[1]=[d[1](1)d[1](2)...d[1](m)] of the same dimension as A[1] .(在讲座中，我们讨论了创建一个与 a[1] 相同的维度的变量 d[1] ，它使用np.random.rand（）随机获得0到1之间的数字。在这里，你将使用一个向量化的实现，所以创建一个随机矩阵 D[1]=[d[1](1)d[1](2)...d[1](m)] 与 A[1] 具有相同的维数)
2. Set each entry of D[1] to be 0 with probability (1-keep_prob) or 1 with probability (keep_prob), by thresholding values in D[1] appropriately. Hint: to set all the entries of a matrix X to 0 (if entry is less than 0.5) or 1 (if entry is more than 0.5) you would do: X = (X < 0.5). Note that 0 and 1 are respectively equivalent to False and True.(通过对 D[1] 中的值进行阈值化，将 D[1] 的每个条目设置为概率（1-keep_prob）或概率为1（keep_prob）提示：要将矩阵X的所有条目设置为0（如果条目小于0.5）或1（如果条目大于0.5），您应该执行：X =（X <0.5）。请注意，0和1分别相当于False和True)
3. Set A[1] to A[1]∗D[1] . (You are shutting down some neurons). You can think of D[1] as a mask, so that when it is multiplied with another matrix, it shuts down some of the values.(将 A[1] 设置为 A[1]∗D[1] 。 （你正在关闭一些神经元）。您可以将 D[1] 作为掩码，以便当它与另一个矩阵相乘时，会关闭一些值)
4. Divide A[1] by keep_prob. By doing this you are assuring that the result of the cost will still have the same expected value as without drop-out. (This technique is also called inverted dropout.)(用keep_prob除以 A[1] 。通过这样做，您可以确保成本的结果仍然具有与没有退出相同的期望值。（这种技术也被称为反向丢失。）)

# 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

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))

A3 = [[ 0.36974721  0.00305176  0.04565099  0.49683389  0.36974721]]

Expected Output:

**A3**

[[ 0.36974721 0.00305176 0.04565099 0.49683389 0.36974721]]

3.2 - Backward propagation with dropout

Exercise: Implement the backward propagation with dropout. As before, you are training a 3 layer network. Add dropout to the first and second hidden layers, using the masks D[1] and D[2] stored in the cache. (dropout的落后传播。和以前一样，你正在训练一个三层网络。使用存储在缓存中的掩码 D[1] and D[2] 将dropout添加到第一个和第二个隐藏层。)

Instruction:
Backpropagation with dropout is actually quite easy. You will have to carry out 2 Steps:
1. You had previously shut down some neurons during forward propagation, by applying a mask D[1] to A1. In backpropagation, you will have to shut down the same neurons, by reapplying the same mask D[1] to dA1. (*
您在前向传播过程中先前已关闭了一些神经元，方法是将掩模 D[1] 应用于A1。在反向传播中，您必须关闭相同的神经元，方法是将相同的掩码 D[1] 重新应用于dA1*)
2. During forward propagation, you had divided A1 by keep_prob. In backpropagation, you’ll therefore have to divide dA1 by keep_prob again (the calculus interpretation is that if A[1] is scaled by keep_prob, then its derivative dA[1] is also scaled by the same keep_prob).(在向前传播过程中，你用keep_prob分隔了A1。在后向传播中，你必须再次用keep_prob来分隔dA1（演算的解释是如果 A[1] 被keep_prob缩放，那么它的导数 dA[1] 也被相同的keep_prob缩放）)

# 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

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"]))

dA1 = [[ 0.36544439  0.         -0.00188233  0.         -0.17408748]
 [ 0.65515713  0.         -0.00337459  0.         -0.        ]]
dA2 = [[ 0.58180856  0.         -0.00299679  0.         -0.27715731]
 [ 0.          0.53159854 -0.          0.53159854 -0.34089673]
 [ 0.          0.         -0.00292733  0.         -0.        ]]

Expected Output:

**dA1**	[[ 0.36544439 0. -0.00188233 0. -0.17408748] [ 0.65515713 0. -0.00337459 0. -0. ]]
**dA2**	[[ 0.58180856 0. -0.00299679 0. -0.27715731] [ 0. 0.53159854 -0. 0.53159854 -0.34089673] [ 0. 0. -0.00292733 0. -0. ]]

Let’s now run the model with dropout (keep_prob = 0.86). It means at every iteration you shut down each neurons of layer 1 and 2 with 24% probability. The function model() will now call:
- forward_propagation_with_dropout instead of forward_propagation.
- backward_propagation_with_dropout instead of backward_propagation.

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)

Cost after iteration 0: 0.6543912405149825
Cost after iteration 10000: 0.0610169865749056
Cost after iteration 20000: 0.060582435798513114

On the train set:
Accuracy: 0.928909952607
On the test set:
Accuracy: 0.95

Dropout works great! The test accuracy has increased again (to 95%)! Your model is not overfitting the training set and does a great job on the test set. The French football team will be forever grateful to you!

Run the code below to plot the decision boundary.

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)

Note:
- A common mistake when using dropout is to use it both in training and testing. You should use dropout (randomly eliminate nodes) only in training. (使用dropout时常见的错误是在训练和测试中使用它。你只能在训练中使用退出（随机消除节点）)
- Deep learning frameworks like tensorflow, PaddlePaddle, keras or caffe come with a dropout layer implementation. Don’t stress - you will soon learn some of these frameworks.(深度学习框架如tensorflow，[PaddlePaddle](http://doc.paddlepaddle.org/release_doc/0.9.0/doc /ui/api/trainer_config_helpers/attrs.html)，[keras](https://keras.io/layers/core/#dropout）或[caffe]（http://caffe.berkeleyvision.org/tutorial/layers/ dropout.html)带有一个dropout层的实现。不要紧张 - 你很快就会学到一些这样的框架。)

What you should remember about dropout:
- Dropout is a regularization technique.(droout是一种正则化技术)
- You only use dropout during training. Don’t use dropout (randomly eliminate nodes) during test time.(训练期间只能使用dropout。测试期间不要使用dropout（随机消除节点）)
- Apply dropout both during forward and backward propagation.(在向前和向后传播期间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. (在训练时间内，通过keep_prob分隔每个丢失图层以保持激活的相同期望值。例如，如果keep_prob是0.5，那么我们将平均关闭一半的节点，所以输出将被缩放0.5，因为只剩下一半对解决方案有贡献。除以0.5相当于乘以2.因此，输出现在具有相同的期望值。即使keep_prob是0.5以外的值，你也可以检查它是否有效)

4 - Conclusions

Here are the results of our three models:

**model**	**train accuracy**	**test accuracy**
3-layer NN without regularization	95%	91.5%
3-layer NN with L2-regularization	94%	93%
3-layer NN with dropout	93%	95%

Note that regularization hurts training set performance! This is because it limits the ability of the network to overfit to the training set. But since it ultimately gives better test accuracy, it is helping your system. (*
请注意，正规化会伤害训练集的表现！这是因为它限制了网络过度训练集的能力。但是由于它最终提供了更好的测试准确性，它正在帮助您的系统*)

Congratulations for finishing this assignment! And also for revolutionizing French football. :-)

What we want you to remember from this notebook:
- Regularization will help you reduce overfitting.
- Regularization will drive your weights to lower values.
- L2 regularization and Dropout are two very effective regularization techniques.