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02.改善深层神经网络：超参数调试、正则化以及优化 W1.深度学习的实践层面（作业：初始化+正则化+梯度检验）

2020-09-09 07:05 282人阅读  评论(0)

文章目录

测试题：参考博文

笔记：02.改善深层神经网络：超参数调试、正则化以及优化 W1.深度学习的实践层面

作业1：初始化

好的初始化：

  • 加快梯度下降的收敛速度
  • 增加梯度下降收敛到较低的训练（和泛化）误差的几率

导入数据

import numpy as np
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets
from init_utils import sigmoid, relu, compute_loss, forward_propagation, backward_propagation
from init_utils import update_parameters, predict, load_dataset, plot_decision_boundary, predict_dec

%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'

# load image dataset: blue/red dots in circles
train_X, train_Y, test_X, test_Y = load_dataset()


我们的任务是：将两类点分类

1. 神经网络模型

用一个已经实现好了的 3层神经网络

def model(X, Y, learning_rate = 0.01, num_iterations = 15000, print_cost = True, initialization = "he"):
    """
    Implements a three-layer neural network: LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID.
    
    Arguments:
    X -- input data, of shape (2, number of examples)
    Y -- true "label" vector (containing 0 for red dots; 1 for blue dots), of shape (1, number of examples)
    learning_rate -- learning rate for gradient descent 
    num_iterations -- number of iterations to run gradient descent
    print_cost -- if True, print the cost every 1000 iterations
    initialization -- flag to choose which initialization to use ("zeros","random" or "he")
    
    Returns:
    parameters -- parameters learnt by the model
    """
        
    grads = {}
    costs = [] # to keep track of the loss
    m = X.shape[1] # number of examples
    layers_dims = [X.shape[0], 10, 5, 1]
    
    # Initialize parameters dictionary.
    if initialization == "zeros":
        parameters = initialize_parameters_zeros(layers_dims)
    elif initialization == "random":
        parameters = initialize_parameters_random(layers_dims)
    elif initialization == "he":
        parameters = initialize_parameters_he(layers_dims)

    # Loop (gradient descent)

    for i in range(0, num_iterations):

        # Forward propagation: LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID.
        a3, cache = forward_propagation(X, parameters)
        
        # Loss
        cost = compute_loss(a3, Y)

        # Backward propagation.
        grads = backward_propagation(X, Y, cache)
        
        # Update parameters.
        parameters = update_parameters(parameters, grads, learning_rate)
        
        # Print the loss every 1000 iterations
        if print_cost and i % 1000 == 0:
            print("Cost after iteration {}: {}".format(i, cost))
            costs.append(cost)
            
    # plot the loss
    plt.plot(costs)
    plt.ylabel('cost')
    plt.xlabel('iterations (per hundreds)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

2. 使用 0 初始化

# GRADED FUNCTION: initialize_parameters_zeros 

def initialize_parameters_zeros(layers_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the size of each layer.
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    W1 -- weight matrix of shape (layers_dims[1], layers_dims[0])
                    b1 -- bias vector of shape (layers_dims[1], 1)
                    ...
                    WL -- weight matrix of shape (layers_dims[L], layers_dims[L-1])
                    bL -- bias vector of shape (layers_dims[L], 1)
    """
    
    parameters = {}
    L = len(layers_dims)            # number of layers in the network
    
    for l in range(1, L):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.zeros((layers_dims[l], layers_dims[l-1]))
        parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
        ### END CODE HERE ###
    return parameters

运行以下代码训练：

parameters = model(train_X, train_Y, initialization = "zeros")
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)

结果：

Cost after iteration 0: 0.6931471805599453
Cost after iteration 1000: 0.6931471805599453
Cost after iteration 2000: 0.6931471805599453
Cost after iteration 3000: 0.6931471805599453
Cost after iteration 4000: 0.6931471805599453
Cost after iteration 5000: 0.6931471805599453
Cost after iteration 6000: 0.6931471805599453
Cost after iteration 7000: 0.6931471805599453
Cost after iteration 8000: 0.6931471805599453
Cost after iteration 9000: 0.6931471805599453
Cost after iteration 10000: 0.6931471805599455
Cost after iteration 11000: 0.6931471805599453
Cost after iteration 12000: 0.6931471805599453
Cost after iteration 13000: 0.6931471805599453
Cost after iteration 14000: 0.6931471805599453


On the train set:
Accuracy: 0.5
On the test set:
Accuracy: 0.5
  • 效果很差，代价函数几乎没有下降
print ("predictions_train = " + str(predictions_train))
print ("predictions_test = " + str(predictions_test))

预测全部都是 0

predictions_train = [[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0]]
predictions_test = [[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]]

plt.title("Model with Zeros initialization")
axes = plt.gca()
axes.set_xlim([-1.5,1.5])
axes.set_ylim([-1.5,1.5])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)


结论：

  • 神经网络中，不要把参数初始化为0，否则模型不能打破这种状态，一直学习同样的东西。
  • 可以将权重随机初始化，偏置初始化为0

3. 随机初始化

  • np.random.randn(layers_dims[l], layers_dims[l-1])*10，* 10 使用很大的随机数初始化权重
# GRADED FUNCTION: initialize_parameters_random

def initialize_parameters_random(layers_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the size of each layer.
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    W1 -- weight matrix of shape (layers_dims[1], layers_dims[0])
                    b1 -- bias vector of shape (layers_dims[1], 1)
                    ...
                    WL -- weight matrix of shape (layers_dims[L], layers_dims[L-1])
                    bL -- bias vector of shape (layers_dims[L], 1)
    """
    
    np.random.seed(3)               # This seed makes sure your "random" numbers will be the as ours
    parameters = {}
    L = len(layers_dims)            # integer representing the number of layers
    
    for l in range(1, L):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l-1])*10
        parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
        ### END CODE HERE ###

    return parameters

运行以下代码训练：

parameters = model(train_X, train_Y, initialization = "random")
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)

结果：

Cost after iteration 0: inf
Cost after iteration 1000: 0.6239567039908781
Cost after iteration 2000: 0.5978043872838292
Cost after iteration 3000: 0.563595830364618
Cost after iteration 4000: 0.5500816882570866
Cost after iteration 5000: 0.5443417928662615
Cost after iteration 6000: 0.5373553777823036
Cost after iteration 7000: 0.4700141958024487
Cost after iteration 8000: 0.3976617665785177
Cost after iteration 9000: 0.39344405717719166
Cost after iteration 10000: 0.39201765232720626
Cost after iteration 11000: 0.38910685278803786
Cost after iteration 12000: 0.38612995897697244
Cost after iteration 13000: 0.3849735792031832
Cost after iteration 14000: 0.38275100578285265


On the train set:
Accuracy: 0.83
On the test set:
Accuracy: 0.86

决策边界

plt.title("Model with large random initialization")
axes = plt.gca()
axes.set_xlim([-1.5,1.5])
axes.set_ylim([-1.5,1.5])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)


将* 10 改为 * 1：

Cost after iteration 0: 1.9698193182646349
Cost after iteration 1000: 0.6894749458317239
Cost after iteration 2000: 0.675058063210226
Cost after iteration 3000: 0.6469210868251528
Cost after iteration 4000: 0.5398790761260324
Cost after iteration 5000: 0.4062642269764849
Cost after iteration 6000: 0.29844708868759456
Cost after iteration 7000: 0.22183734662094845
Cost after iteration 8000: 0.16926424179038072
Cost after iteration 9000: 0.1341330896982709
Cost after iteration 10000: 0.10873865543082417
Cost after iteration 11000: 0.09169443068126971
Cost after iteration 12000: 0.07991173603998084
Cost after iteration 13000: 0.07083949901112582
Cost after iteration 14000: 0.06370209022580517

On the train set:
Accuracy: 0.9966666666666667
On the test set:
Accuracy: 0.96


将* 10 改为 * 0.1：

Cost after iteration 0: 0.6933234320329613
Cost after iteration 1000: 0.6932871248121155
Cost after iteration 2000: 0.6932558729405607
Cost after iteration 3000: 0.6932263488895136
Cost after iteration 4000: 0.6931989886931527
Cost after iteration 5000: 0.6931076575962486
Cost after iteration 6000: 0.6930655602542224
Cost after iteration 7000: 0.6930202936477311
Cost after iteration 8000: 0.6929722630100763
Cost after iteration 9000: 0.6929185743666864
Cost after iteration 10000: 0.6928576152283971
Cost after iteration 11000: 0.6927869030178897
Cost after iteration 12000: 0.6927029749978133
Cost after iteration 13000: 0.6926024266332704
Cost after iteration 14000: 0.6924787835871681

On the train set:
Accuracy: 0.6
On the test set:
Accuracy: 0.57

  • 使用合适的初始化权重非常重要！！！
  • 不好的初始化会造成梯度消失/爆炸，降低了学习速度

4. He 初始化

是以一个人的名字命名的。

  • 如果使用ReLu激活函数（最常用）， ∗ n p . s q r t ( 2 n [ l − 1 ] ) *np.sqrt(\frac{2}{n^{[l-1]}})
  • 如果使用tanh激活函数， 1 n [ l − 1 ] \sqrt \frac{1}{n^{[l-1]}} ，或者 2 n [ l − 1 ] + n [ l ] \sqrt \frac{2}{n^{[l-1]}+n^{[l]}} 
# GRADED FUNCTION: initialize_parameters_he

def initialize_parameters_he(layers_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the size of each layer.
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    W1 -- weight matrix of shape (layers_dims[1], layers_dims[0])
                    b1 -- bias vector of shape (layers_dims[1], 1)
                    ...
                    WL -- weight matrix of shape (layers_dims[L], layers_dims[L-1])
                    bL -- bias vector of shape (layers_dims[L], 1)
    """
    
    np.random.seed(3)
    parameters = {}
    L = len(layers_dims) - 1 # integer representing the number of layers
     
    for l in range(1, L + 1):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l-1])*np.sqrt(2/layers_dims[l-1])
        parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
        ### END CODE HERE ###
        
    return parameters

parameters = model(train_X, train_Y, initialization = "he")
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: 0.8830537463419761
Cost after iteration 1000: 0.6879825919728063
Cost after iteration 2000: 0.6751286264523371
Cost after iteration 3000: 0.6526117768893807
Cost after iteration 4000: 0.6082958970572938
Cost after iteration 5000: 0.5304944491717495
Cost after iteration 6000: 0.4138645817071794
Cost after iteration 7000: 0.3117803464844441
Cost after iteration 8000: 0.23696215330322562
Cost after iteration 9000: 0.18597287209206836
Cost after iteration 10000: 0.15015556280371817
Cost after iteration 11000: 0.12325079292273552
Cost after iteration 12000: 0.09917746546525932
Cost after iteration 13000: 0.08457055954024274
Cost after iteration 14000: 0.07357895962677362


On the train set:
Accuracy: 0.9933333333333333
On the test set:
Accuracy: 0.96

plt.title("Model with He initialization")
axes = plt.gca()
axes.set_xlim([-1.5,1.5])
axes.set_ylim([-1.5,1.5])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

模型 训练准确率 问题
3-layer NN with zeros initialization 50% fails to break symmetry
3-layer NN with large random initialization 83% too large weights
3-layer NN with He initialization 99% recommended method

作业2：正则化

过拟合是个严重的问题，它表现为在训练集上表现的很好，但是泛化性能较差

# 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'

问题引入：

法国足球守门员发球，把球踢到什么位置，他的队友可以用头顶球。

train_X, train_Y, test_X, test_Y = load_2D_dataset()


法国守门员从左侧发球，蓝色是自己队友顶球位置，红色是对方顶球位置

肉眼看，好像可以用一条45°左右的斜线分开

1. 无正则化模型

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

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)
  • 无正则化 训练过程
Cost after iteration 0: 0.6557412523481002
Cost after iteration 10000: 0.16329987525724213
Cost after iteration 20000: 0.13851642423245572


On the training set:
Accuracy: 0.9478672985781991
On the test set:
Accuracy: 0.915

  • 没有正则化的模型过拟合了，它拟合了一些噪声点

2. L2 正则化

  • 注意在损失函数里加入正则化项

无正则项的损失函数：
J = − 1 m ∑ i = 1 m ( y ( i ) log ⁡ ( a [ L ] ( i ) ) + ( 1 − y ( i ) ) log ⁡ ( 1 − a [ L ] ( i ) ) ) J = -\frac{1}{m} \sum\limits_{i = 1}^{m} \bigg( \small y^{(i)}\log\left(a^{[L](i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right) \bigg)
加入正则化项的损失函数：
J r e g u l a r i z e d = − 1 m ∑ i = 1 m ( y ( i ) log ⁡ ( a [ L ] ( i ) ) + ( 1 − y ( i ) ) log ⁡ ( 1 − a [ L ] ( i ) ) ) ⏟ cross-entropy cost + 1 m λ 2 ∑ l ∑ k ∑ j W k , j [ l ] 2 ⏟ L2 regularization cost J_{regularized} = \small \underbrace{-\frac{1}{m} \sum\limits_{i = 1}^{m} \bigg(\small y^{(i)}\log\left(a^{[L](i)}\right) + (1-y^{(i)})\log\left(1- a^{[L](i)}\right) \bigg) }_\text{cross-entropy cost} + \underbrace{\frac{1}{m} \frac{\lambda}{2} \sum\limits_l\sum\limits_k\sum\limits_j W_{k,j}^{[l]2} }_\text{L2 regularization cost} 

>>> w1 = np.array([[1,2],[2,3]])		  
>>> w1	  
array([[1, 2],
       [2, 3]])
>>> np.sum(np.square(w1))		  
18

# 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 = lambd/(2*m)*(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
  • 反向传播，计算梯度时也要根据 新的损失函数
    dw 需要加入 d d W ( 1 2 λ m W 2 ) = λ m W \frac{d}{dW} ( \frac{1}{2}\frac{\lambda}{m} W^2) = \frac{\lambda}{m} W 项
# 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
  • 运行带 L2 正则化（ λ \lambda = 0.7）的模型（使用上面两个函数计算损失、梯度）
Cost after iteration 0: 0.6974484493131264
Cost after iteration 10000: 0.26849188732822393
Cost after iteration 20000: 0.2680916337127301


On the train set:
Accuracy: 0.9383886255924171
On the test set:
Accuracy: 0.93

模型没有过拟合

L2 正则化使得 权重衰减，其基于假设： 小的权重 W 的模型更简单，所以模型会惩罚 大的 W，小的权重 使得输出变化比较平和，不会剧烈变化 形成复杂的边界（造成过拟合）

调整 λ \lambda 做点对比：
λ = 0.3 \lambda = 0.3 

On the train set:
Accuracy: 0.919431279620853
On the test set:
Accuracy: 0.945

λ = 0.1 \lambda = 0.1 

On the train set:
Accuracy: 0.9383886255924171
On the test set:
Accuracy: 0.95

λ = 0.01 \lambda = 0.01 （正则化作用很弱）

On the train set:
Accuracy: 0.9289099526066351
On the test set:
Accuracy: 0.915

(有点过拟合)

λ = 1 \lambda = 1 

On the train set:
Accuracy: 0.9241706161137441
On the test set:
Accuracy: 0.93


λ = 5 \lambda = 5 

On the train set:
Accuracy: 0.919431279620853
On the test set:
Accuracy: 0.92

  • λ \lambda 太大，正则化太强，W 被压缩的很小，决策边界过度平滑（都直线了），造成高的偏差

3. DropOut 正则化


DropOut 正则化 在每次迭代的时候 随机关闭一些神经元
被关闭的神经元在当次迭代时，对前向和后向传播都没有贡献

drop-out背后的思想是，每次迭代时，使用部分神经元子集的不同模型，神经元对另一个特定神经元的激活变得不那么敏感，因为另一个神经元随时可能被关闭

3.1 带dropout的前向传播

对一个3层神经网络实施 dropout，只对第1，2层进行，不包括输入和输出层

  • 用 np.random.rand() 建立与 A [ 1 ] A^{[1]} 一样维度的 D [ 1 ] = [ d [ 1 ] ( 1 ) d [ 1 ] ( 2 ) . . . d [ 1 ] ( m ) ] D^{[1]} = [d^{[1](1)} d^{[1](2)} ... d^{[1](m)}]
  • 以一定的概率，设置 D [ 1 ] D^{[1]} 元素为0（概率 1-keep_prob）, 1（概率 keep_prob）X = (X < keep_prob)
  • 关闭某些神经元， A [ 1 ] = A [ 1 ] ∗ D [ 1 ] A^{[1]}= A^{[1]} * D^{[1]}
  • A [ 1 ] / =  keep-prob  A^{[1]} /= \text { keep-prob } ，此步确保损失函数的期望值与 没有dropout 时一样（inverted dropout）
# 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 = D1 < keep_prob                             # 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 = D2 < keep_prob                            # 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

3.2 带dropout的后向传播

上面我们用 D [ 1 ] , D [ 2 ] D^{[1]},D^{[2]} 把神经元关闭了

  • 使用相同的 D [ 1 ] D^{[1]} 关闭 d A 1 dA1
  • d A 1 / = keep-prob dA1 /= \text{keep-prob} ，导数跟上面保持一致的系数
# 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

3.3 运行模型

参数：keep_prob = 0.86，前后向传播 使用上面的两个函数

On the train set:
Accuracy: 0.9289099526066351
On the test set:
Accuracy: 0.95

模型没有过拟合，且 test 集上的准确率达到了 95%


注意：

  • 只能在训练的时候，使用dropout，测试的时候不要使用
  • 前向、后向均应该使用
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%

正则化限制了在训练集上的过拟合，训练准确率下降了，但是测试集准确率上升了，这是个好现象

作业3：梯度检验

梯度检验 确保 反向传播 是正确的，没有 bug

1. 1维梯度检验

∂ J ∂ θ = lim ⁡ ε → 0 J ( θ + ε ) − J ( θ − ε ) 2 ε \frac{\partial J}{\partial \theta} = \lim_{\varepsilon \to 0} \frac{J(\theta + \varepsilon) - J(\theta - \varepsilon)}{2 \varepsilon}

  • 计算理论梯度
# GRADED FUNCTION: forward_propagation

def forward_propagation(x, theta):
    """
    Implement the linear forward propagation (compute J) presented in Figure 1 (J(theta) = theta * x)
    
    Arguments:
    x -- a real-valued input
    theta -- our parameter, a real number as well
    
    Returns:
    J -- the value of function J, computed using the formula J(theta) = theta * x
    """
    
    ### START CODE HERE ### (approx. 1 line)
    J = theta * x
    ### END CODE HERE ###
    
    return J

# GRADED FUNCTION: backward_propagation

def backward_propagation(x, theta):
    """
    Computes the derivative of J with respect to theta (see Figure 1).
    
    Arguments:
    x -- a real-valued input
    theta -- our parameter, a real number as well
    
    Returns:
    dtheta -- the gradient of the cost with respect to theta
    """
    
    ### START CODE HERE ### (approx. 1 line)
    dtheta = x
    ### END CODE HERE ###
    
    return dtheta
  • 计算近似梯度
  1. θ + = θ + ε \theta^{+} = \theta + \varepsilon
  2. θ − = θ − ε \theta^{-} = \theta - \varepsilon
  3. J + = J ( θ + ) J^{+} = J(\theta^{+})
  4. J − = J ( θ − ) J^{-} = J(\theta^{-})
  5. g r a d a p p r o x = J + − J − 2 ε gradapprox = \frac{J^{+} - J^{-}}{2 \varepsilon}
  • 反向传播，计算理论梯度 grad
  • 比较两者误差
    d i f f e r e n c e = ∣ ∣ g r a d − g r a d a p p r o x ∣ ∣ 2 ∣ ∣ g r a d ∣ ∣ 2 + ∣ ∣ g r a d a p p r o x ∣ ∣ 2 difference = \frac {\mid\mid grad - gradapprox \mid\mid_2}{\mid\mid grad \mid\mid_2 + \mid\mid gradapprox \mid\mid_2} 
np.linalg.norm(...)
  • 检查上式是否足够小（10-7）
# GRADED FUNCTION: gradient_check

def gradient_check(x, theta, epsilon = 1e-7):
    """
    Implement the backward propagation presented in Figure 1.
    
    Arguments:
    x -- a real-valued input
    theta -- our parameter, a real number as well
    epsilon -- tiny shift to the input to compute approximated gradient with formula(1)
    
    Returns:
    difference -- difference (2) between the approximated gradient and the backward propagation gradient
    """
    
    # Compute gradapprox using left side of formula (1). epsilon is small enough, you don't need to worry about the limit.
    ### START CODE HERE ### (approx. 5 lines)
    thetaplus = theta + epsilon                     # Step 1
    thetaminus = theta - epsilon                    # Step 2
    J_plus = forward_propagation(x, thetaplus)      # Step 3
    J_minus = forward_propagation(x, thetaminus)    # Step 4
    gradapprox = (J_plus - J_minus)/(2*epsilon)    # Step 5
    ### END CODE HERE ###
    
    # Check if gradapprox is close enough to the output of backward_propagation()
    ### START CODE HERE ### (approx. 1 line)
    grad = backward_propagation(x, theta)
    ### END CODE HERE ###
    
    ### START CODE HERE ### (approx. 1 line)
    numerator = np.linalg.norm(grad - gradapprox)  # Step 1'
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox) # Step 2'
    difference = numerator/denominator  # Step 3'
    ### END CODE HERE ###
    
    if difference < 1e-7:
        print ("The gradient is correct!")
    else:
        print ("The gradient is wrong!")
    
    return difference

2. 多维梯度检验

def forward_propagation_n(X, Y, parameters):
    """
    Implements the forward propagation (and computes the cost) presented in Figure 3.
    
    Arguments:
    X -- training set for m examples
    Y -- labels for m examples 
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                    W1 -- weight matrix of shape (5, 4)
                    b1 -- bias vector of shape (5, 1)
                    W2 -- weight matrix of shape (3, 5)
                    b2 -- bias vector of shape (3, 1)
                    W3 -- weight matrix of shape (1, 3)
                    b3 -- bias vector of shape (1, 1)
    
    Returns:
    cost -- the cost function (logistic cost for one example)
    """
    
    # retrieve parameters
    m = X.shape[1]
    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)
    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)
    Z3 = np.dot(W3, A2) + b3
    A3 = sigmoid(Z3)

    # Cost
    logprobs = np.multiply(-np.log(A3),Y) + np.multiply(-np.log(1 - A3), 1 - Y)
    cost = 1./m * np.sum(logprobs)
    
    cache = (Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3)
    
    return cost, cache

def backward_propagation_n(X, Y, cache):
    """
    Implement the backward propagation presented in figure 2.
    
    Arguments:
    X -- input datapoint, of shape (input size, 1)
    Y -- true "label"
    cache -- cache output from forward_propagation_n()
    
    Returns:
    gradients -- A dictionary with the gradients of the cost 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
    dW3 = 1./m * np.dot(dZ3, A2.T)
    db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
    
    dA2 = np.dot(W3.T, dZ3)
    dZ2 = np.multiply(dA2, np.int64(A2 > 0))
    dW2 = 1./m * np.dot(dZ2, A1.T) * 2
    db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)
    
    dA1 = np.dot(W2.T, dZ2)
    dZ1 = np.multiply(dA1, np.int64(A1 > 0))
    dW1 = 1./m * np.dot(dZ1, X.T)
    db1 = 4./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: gradient_check_n

def gradient_check_n(parameters, gradients, X, Y, epsilon = 1e-7):
    """
    Checks if backward_propagation_n computes correctly the gradient of the cost output by forward_propagation_n
    
    Arguments:
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
    grad -- output of backward_propagation_n, contains gradients of the cost with respect to the parameters. 
    x -- input datapoint, of shape (input size, 1)
    y -- true "label"
    epsilon -- tiny shift to the input to compute approximated gradient with formula(1)
    
    Returns:
    difference -- difference (2) between the approximated gradient and the backward propagation gradient
    """
    
    # Set-up variables
    parameters_values, _ = dictionary_to_vector(parameters)
    grad = gradients_to_vector(gradients)
    num_parameters = parameters_values.shape[0]
    J_plus = np.zeros((num_parameters, 1))
    J_minus = np.zeros((num_parameters, 1))
    gradapprox = np.zeros((num_parameters, 1))
    
    # Compute gradapprox
    for i in range(num_parameters):
        
        # Compute J_plus[i]. Inputs: "parameters_values, epsilon". Output = "J_plus[i]".
        # "_" is used because the function you have to outputs two parameters but we only care about the first one
        ### START CODE HERE ### (approx. 3 lines)
        thetaplus = np.copy(parameters_values)                # Step 1
        thetaplus[i][0] = thetaplus[i][0] + epsilon          # Step 2
        J_plus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaplus)) # Step 3
        ### END CODE HERE ###
        
        # Compute J_minus[i]. Inputs: "parameters_values, epsilon". Output = "J_minus[i]".
        ### START CODE HERE ### (approx. 3 lines)
        thetaminus = np.copy(parameters_values)               # Step 1
        thetaminus[i][0] = thetaminus[i][0] - epsilon        # Step 2        
        J_minus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaminus)) # Step 3
        ### END CODE HERE ###
        
        # Compute gradapprox[i]
        ### START CODE HERE ### (approx. 1 line)
        gradapprox[i] = (J_plus[i] - J_minus[i])/(2*epsilon)
        ### END CODE HERE ###
    
    # Compare gradapprox to backward propagation gradients by computing difference.
    ### START CODE HERE ### (approx. 1 line)
    numerator = np.linalg.norm(gradapprox - grad)              # Step 1'
    denominator = np.linalg.norm(gradapprox)+np.linalg.norm(grad) # Step 2'
    difference = numerator/denominator                           # Step 3'
    ### END CODE HERE ###

    if difference > 1e-7:
        print ("\033[93m" + "There is a mistake in the backward propagation! difference = " + str(difference) + "\033[0m")
    else:
        print ("\033[92m" + "Your backward propagation works perfectly fine! difference = " + str(difference) + "\033[0m")
    
    return difference
  • 老师给的 backward_propagation_n 函数里面有错误，尝试去找到它。
X, Y, parameters = gradient_check_n_test_case()

cost, cache = forward_propagation_n(X, Y, parameters)
gradients = backward_propagation_n(X, Y, cache)
difference = gradient_check_n(parameters, gradients, X, Y)

There is a mistake in the backward propagation! 
difference = 0.2850931567761624
  • 寻找错误
    db1 改成：db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True)
    dW2 改成：dW2 = 1./m * np.dot(dZ2, A1.T)

误差下来了，但略微超过 10-7, 所以显示错误，应该问题不大

There is a mistake in the backward propagation! 
difference = 1.1890913023330276e-07

注意：

  • 梯度检验非常慢，计算很耗时，所以我们训练时，不运行梯度检验，只运行几次检查梯度是否正确
  • 梯度检验时，需要关掉 dropout

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