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吴恩达深度学习 C1_W4_Assignment

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2021/09/16 Share

吴恩达深度学习 C1_W4_Assignment

任务1:一步一步构建深层神经网络

Part0:库的准备

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import numpy as np
import h5py
import matplotlib.pyplot as plt
from testCases_v2 import *
from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward

%matplotlib inline
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

%load_ext autoreload
%autoreload 2

np.random.seed(1)

步骤图:

https://pic.imgdb.cn/item/6135d3df44eaada7399c4856.png

Part1:初始化

Exercise1:创建并初始化2层神经网络参数

模型结构:LINEAR -> RELU -> LINEAR -> SIGMOID

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def initialize_parameters(n_x, n_h, n_y):
"""
Argument:
n_x -- size of the input layer
n_h -- size of the hidden layer
n_y -- size of the output layer

Returns:
parameters -- 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(1)

### START CODE HERE ### (≈ 4 lines of code)
W1 = np.random.randn(n_h, n_x) * 0.01
b1 = np.zeros((n_h, 1))

W2 = np.random.randn(n_y, n_h) * 0.01
b2 = np.zeros((n_y, 1))

### 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

Exercise2:实现L层神经网络的初始化

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def initialize_parameters_deep(layer_dims):
"""
Arguments:
layer_dims -- python array (list) containing the dimensions of each layer in our network

Returns:
parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
bl -- bias vector of shape (layer_dims[l], 1)
"""

np.random.seed(3)
parameters = {}
L = len(layer_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.random.randn(layer_dims[l], layer_dims[l - 1]) * 0.01
parameters["b" + str(l)] = np.zeros((layer_dims[l], 1))
### END CODE HERE ###

assert(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l-1]))
assert(parameters['b' + str(l)].shape == (layer_dims[l], 1))


return parameters

Part2:前向传播模块

接下来将完成三个函数:

  • LINEAR
  • LINEAR -> ACTIVATION where ACTIVATION will be either ReLU or Sigmoid.
  • [LINEAR -> RELU] × (L-1) -> LINEAR -> SIGMOID (whole model)

Exercise3:完成Linear的前向传播

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def linear_forward(A, W, b):
"""
Implement the linear part of a layer's forward propagation.

Arguments:
A -- activations from previous layer (or input data): (size of previous layer, number of examples)
W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
b -- bias vector, numpy array of shape (size of the current layer, 1)

Returns:
Z -- the input of the activation function, also called pre-activation parameter
cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
"""

### START CODE HERE ### (≈ 1 line of code)
Z = np.dot(W, A) + b
### END CODE HERE ###

assert(Z.shape == (W.shape[0], A.shape[1]))
cache = (A, W, b)

return Z, cache

Exercise4:完成LINEAR->ACTIVATION的前向传播

其中g()sigmoidrelu

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def linear_activation_forward(A_prev, W, b, activation):
"""
Implement the forward propagation for the LINEAR->ACTIVATION layer

Arguments:
A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
b -- bias vector, numpy array of shape (size of the current layer, 1)
activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

Returns:
A -- the output of the activation function, also called the post-activation value
cache -- a python dictionary containing "linear_cache" and "activation_cache";
stored for computing the backward pass efficiently
"""

if activation == "sigmoid":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
### START CODE HERE ### (≈ 2 lines of code)
Z = np.dot(W, A_prev) + b
A, activation_cache = sigmoid(Z)
### END CODE HERE ###

elif activation == "relu":
# Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
### START CODE HERE ### (≈ 2 lines of code)
Z = np.dot(W, A_prev) + b
A, activation_cache = relu(Z)
### END CODE HERE ###

assert (A.shape == (W.shape[0], A_prev.shape[1]))
cache = (linear_cache, activation_cache)

return A, cache

Exercise5:实现L层模型的前向传播

https://pic.imgdb.cn/item/6135d86044eaada739a41659.png

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def L_model_forward(X, parameters):
"""
Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation

Arguments:
X -- data, numpy array of shape (input size, number of examples)
parameters -- output of initialize_parameters_deep()

Returns:
AL -- last post-activation value
caches -- list of caches containing:
every cache of linear_relu_forward() (there are L-1 of them, indexed from 0 to L-2)
the cache of linear_sigmoid_forward() (there is one, indexed L-1)
"""

caches = []
A = X
# 整除
L = len(parameters) // 2 # number of layers in the neural network

# Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
for l in range(1, L):
A_prev = A
### START CODE HERE ### (≈ 2 lines of code)
A, cache = linear_activation_forward(A_prev, parameters["W" + str(l)], parameters["b" + str(l)], "relu")
caches.append(cache)
### END CODE HERE ###

# Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
### START CODE HERE ### (≈ 2 lines of code)
AL, cache = linear_activation_forward(A, parameters["W" + str(L)], parameters["b" + str(L)], "sigmoid")
caches.append(cache)
### END CODE HERE ###

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

return AL, caches

Part3:成本函数

Exercise6:计算成本函数

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# GRADED FUNCTION: compute_cost

def compute_cost(AL, Y):
"""
Implement the cost function defined by equation (7).

Arguments:
AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)

Returns:
cost -- cross-entropy cost
"""

m = Y.shape[1]

# Compute loss from aL and y.
### START CODE HERE ### (≈ 1 lines of code)
cost = -1 / m * np.sum(np.dot(np.log(AL), Y.T) + np.dot(np.log(1 - AL), (1 - Y).T))
### END CODE HERE ###

cost = np.squeeze(cost) # 将为1的维度去除 (e.g. this turns [[17]] into 17).
assert(cost.shape == ())

return cost

Part4:反向传播模块

https://pic.imgdb.cn/item/6135dac644eaada739a7f3dd.png

和前向传播一样,我们需要逐步完成三个函数:

  • LINEAR backward
  • LINEAR -> ACTIVATION backward where ACTIVATION computes the derivative of either the ReLU or sigmoid activation
  • [LINEAR -> RELU] × (L-1) -> LINEAR -> SIGMOID backward (whole model)

Exercise7:Linear backward

https://pic.imgdb.cn/item/6135db6844eaada739a90752.png

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def linear_backward(dZ, cache):
"""
Implement the linear portion of backward propagation for a single layer (layer l)

Arguments:
dZ -- Gradient of the cost with respect to the linear output (of current layer l)
cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer

Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
A_prev, W, b = cache
m = A_prev.shape[1]

### START CODE HERE ### (≈ 3 lines of code)
dW = 1 / m * np.dot(dZ, A_prev.T)
db = 1 / m * np.sum(dZ, axis=1, keepdims=True) # 每一行的值加起来
dA_prev = np.dot(W.T, dZ)
### END CODE HERE ###

assert (dA_prev.shape == A_prev.shape)
assert (dW.shape == W.shape)
assert (db.shape == b.shape)

return dA_prev, dW, db

Exercise8:Linear-Activation backward

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def linear_activation_backward(dA, cache, activation):
"""
Implement the backward propagation for the LINEAR->ACTIVATION layer.

Arguments:
dA -- post-activation gradient for current layer l
cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

Returns:
dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
dW -- Gradient of the cost with respect to W (current layer l), same shape as W
db -- Gradient of the cost with respect to b (current layer l), same shape as b
"""
linear_cache, activation_cache = cache

if activation == "sigmoid":
### START CODE HERE ### (≈ 2 lines of code)
dZ = sigmoid_backward(dA, activation_cache)
dA_prev, dW, db = linear_backward(dZ, linear_cache)
### END CODE HERE ###

elif activation == "relu":
### START CODE HERE ### (≈ 2 lines of code)
dZ = relu_backward(dA, activation_cache)
dA_prev, dW, db = linear_backward(dZ, linear_cache)
### END CODE HERE ###

return dA_prev, dW, db

Exercise9:L-Model Backward

https://pic.imgdb.cn/item/6136244f44eaada73927c286.png

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def L_model_backward(AL, Y, caches):
"""
Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group

Arguments:
AL -- probability vector, output of the forward propagation (L_model_forward())
Y -- true "label" vector (containing 0 if non-cat, 1 if cat)
caches -- list of caches containing:
every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1])

Returns:
grads -- A dictionary with the gradients
grads["dA" + str(l)] = ...
grads["dW" + str(l)] = ...
grads["db" + str(l)] = ...
"""
grads = {}
L = len(caches) # the number of layers
m = AL.shape[1]
Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL

# Initializing the backpropagation
### START CODE HERE ### (1 line of code)
dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
### END CODE HERE ###

# Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]
### START CODE HERE ### (approx. 2 lines)
cache_now = caches[L - 1]
grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, cache_now, "sigmoid")
### END CODE HERE ###

for l in reversed(range(L - 1)):
# lth layer: (RELU -> LINEAR) gradients.
# Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)]
### START CODE HERE ### (approx. 5 lines)
cache_now = caches[l]
grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)] = linear_activation_backward(grads["dA" + str(l + 2)], cache_now, "relu")

### END CODE HERE ###

return grads

Exercise10:更新参数

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def update_parameters(parameters, grads, learning_rate):
"""
Update parameters using gradient descent

Arguments:
parameters -- python dictionary containing your parameters
grads -- python dictionary containing your gradients, output of L_model_backward

Returns:
parameters -- python dictionary containing your updated parameters
parameters["W" + str(l)] = ...
parameters["b" + str(l)] = ...
"""

L = len(parameters) // 2 # number of layers in the neural network

# Update rule for each parameter. Use a for loop.
### START CODE HERE ### (≈ 3 lines of code)
for i in range(1, L + 1):
parameters["W" + str(i)] -= learning_rate * grads["dW" + str(i)]
parameters["b" + str(i)] -= learning_rate * grads["db" + str(i)]

### END CODE HERE ###

return parameters

任务2:深度神经网络在图像分类中的应用

Part0:库的准备

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import time
import numpy as np
import h5py
import matplotlib.pyplot as plt
import scipy
from PIL import Image
from scipy import ndimage
from dnn_app_utils_v2 import *

%matplotlib inline
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

%load_ext autoreload
%autoreload 2

np.random.seed(1)

Part1:数据准备

我们将使用与实验2相同的数据集,并来观察此次的模型是否能提高识别准确率

我们来观察一下测试集与训练集的情况

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m_train = train_x_orig.shape[0]
num_px = train_x_orig.shape[1]
m_test = test_x_orig.shape[0]

print ("Number of training examples: " + str(m_train))
print ("Number of testing examples: " + str(m_test))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_x_orig shape: " + str(train_x_orig.shape))
print ("train_y shape: " + str(train_y.shape))
print ("test_x_orig shape: " + str(test_x_orig.shape))
print ("test_y shape: " + str(test_y.shape))
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Number of training examples: 209
Number of testing examples: 50
Each image is of size: (64, 64, 3)
train_x_orig shape: (209, 64, 64, 3)
train_y shape: (1, 209)
test_x_orig shape: (50, 64, 64, 3)
test_y shape: (1, 50)

我们将图像向量化:

https://pic.imgdb.cn/item/6136286144eaada739312174.png

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# Reshape the training and test examples 
train_x_flatten = train_x_orig.reshape(train_x_orig.shape[0], -1).T # 一个样本一列
test_x_flatten = test_x_orig.reshape(test_x_orig.shape[0], -1).T

# Standardize data to have feature values between 0 and 1.
train_x = train_x_flatten/255.
test_x = test_x_flatten/255.

print ("train_x's shape: " + str(train_x.shape))
print ("test_x's shape: " + str(test_x.shape))
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train_x's shape: (12288, 209)   # 12288 = 64 * 64 * 3
test_x's shape: (12288, 50)

Part2:模型结构

我们将构建两种不同的模型:

  • 2层神经网络
  • L层深度神经网络

Exercise11:2层神经网络

https://pic.imgdb.cn/item/61362a0d44eaada739350ef6.png

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n_x = 12288     # num_px * num_px * 3  (64 * 64 * 3)
n_h = 7 # the size of the hidden layer
n_y = 1 # the size of the output layer

layers_dims = (n_x, n_h, n_y)
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def two_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):
"""
Implements a two-layer neural network: LINEAR->RELU->LINEAR->SIGMOID.

Arguments:
X -- input data, of shape (n_x, number of examples)
Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
layers_dims -- dimensions of the layers (n_x, n_h, n_y)
num_iterations -- number of iterations of the optimization loop
learning_rate -- learning rate of the gradient descent update rule
print_cost -- If set to True, this will print the cost every 100 iterations

Returns:
parameters -- a dictionary containing W1, W2, b1, and b2
"""

np.random.seed(1)
grads = {}
costs = [] # to keep track of the cost
m = X.shape[1] # number of examples
(n_x, n_h, n_y) = layers_dims

# Initialize parameters dictionary, by calling one of the functions you'd previously implemented
### START CODE HERE ### (≈ 1 line of code)
parameters = initialize_parameters(n_x, n_h, n_y)
### END CODE HERE ###

# Get W1, b1, W2 and b2 from the dictionary parameters.
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

# Loop (gradient descent)

for i in range(0, num_iterations):

# Forward propagation: LINEAR -> RELU -> LINEAR -> SIGMOID. Inputs: "X, W1, b1". Output: "A1, cache1, A2, cache2".
### START CODE HERE ### (≈ 2 lines of code)
A1, cache1 = linear_activation_forward(X, W1, b1, "relu")
A2, cache2 = linear_activation_forward(A1, W2, b2, "sigmoid")
### END CODE HERE ###

# Compute cost
### START CODE HERE ### (≈ 1 line of code)
cost = compute_cost(A2, Y)
### END CODE HERE ###

# Initializing backward propagation
dA2 = - (np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))

# Backward propagation. Inputs: "dA2, cache2, cache1". Outputs: "dA1, dW2, db2; also dA0 (not used), dW1, db1".
### START CODE HERE ### (≈ 2 lines of code)
dA1, dW2, db2 = linear_activation_backward(dA2, cache2, "sigmoid")
dA0, dW1, db1 = linear_activation_backward(dA1, cache1, "relu")
### END CODE HERE ###

# Set grads['dWl'] to dW1, grads['db1'] to db1, grads['dW2'] to dW2, grads['db2'] to db2
grads['dW1'] = dW1
grads['db1'] = db1
grads['dW2'] = dW2
grads['db2'] = db2

# Update parameters.
### START CODE HERE ### (approx. 1 line of code)
parameters = update_parameters(parameters, grads, learning_rate)
### END CODE HERE ###

# Retrieve W1, b1, W2, b2 from parameters
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

# Print the cost every 100 training example
if print_cost and i % 100 == 0:
print("Cost after iteration {}: {}".format(i, np.squeeze(cost)))
if print_cost and i % 100 == 0:
costs.append(cost)

# plot the cost

plt.plot(np.squeeze(costs))
plt.ylabel('cost')
plt.xlabel('iterations (per tens)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()

return parameters

测试:

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parameters = two_layer_model(train_x, train_y, layers_dims = (n_x, n_h, n_y), num_iterations = 2500, print_cost=True)

https://pic.imgdb.cn/item/61362b7144eaada739384fb8.png

通过准确率的测试,可以得到结果为72%。

Exercise12:L层神经网络

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layers_dims = [12288, 20, 7, 5, 1] #  5-layer model
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def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):#lr was 0.009
"""
Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.

Arguments:
X -- data, numpy array of shape (number of examples, num_px * num_px * 3)
Y -- true "label" vector (containing 0 if cat, 1 if non-cat), of shape (1, number of examples)
layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
learning_rate -- learning rate of the gradient descent update rule
num_iterations -- number of iterations of the optimization loop
print_cost -- if True, it prints the cost every 100 steps

Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""

np.random.seed(1)
costs = [] # keep track of cost

# Parameters initialization.
### START CODE HERE ###
parameters = initialize_parameters_deep(layers_dims)
### END CODE HERE ###

# Loop (gradient descent)
for i in range(0, num_iterations):

# Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
### START CODE HERE ### (≈ 1 line of code)
AL, caches = L_model_forward(X, parameters)
### END CODE HERE ###

# Compute cost.
### START CODE HERE ### (≈ 1 line of code)
cost = compute_cost(AL, Y)
### END CODE HERE ###

# Backward propagation.
### START CODE HERE ### (≈ 1 line of code)
grads = L_model_backward(AL, Y, caches)
### END CODE HERE ###

# Update parameters.
### START CODE HERE ### (≈ 1 line of code)
parameters = update_parameters(parameters, grads, learning_rate)
### END CODE HERE ###

# Print the cost every 100 training example
if print_cost and i % 100 == 0:
print ("Cost after iteration %i: %f" %(i, cost))
if print_cost and i % 100 == 0:
costs.append(cost)

# plot the cost
plt.plot(np.squeeze(costs))
plt.ylabel('cost')
plt.xlabel('iterations (per tens)')
plt.title("Learning rate =" + str(learning_rate))
plt.show()

return parameters

测试:

1
parameters = L_layer_model(train_x, train_y, layers_dims, num_iterations = 2500, print_cost = True)

https://pic.imgdb.cn/item/61362d1544eaada7393c247f.png

通过准确率的测试,可以得到结果为80%。由此可以得到结果,在相同的测试集上,5层神经网络比2层神经网络效果更好。

Part3:错误结果分析

1
print_mislabeled_images(classes, test_x, test_y, pred_test)

https://pic.imgdb.cn/item/61362dd744eaada7393ddcf0.png

该模型往往表现不佳的几种图像类型包括:

  • 猫的身体处于一个不寻常的位置
  • 猫在相似颜色的背景下出现
  • 不寻常的猫的颜色和种类
  • 摄像机角度
  • 画面的亮度
  • 比例变化(图像中猫很大或很小)
CATALOG
  1. 1. 吴恩达深度学习 C1_W4_Assignment
    1. 1.1. 任务1:一步一步构建深层神经网络
    2. 1.2. Part0:库的准备
    3. 1.3. Part1:初始化
      1. 1.3.1. Exercise1:创建并初始化2层神经网络参数
      2. 1.3.2. Exercise2:实现L层神经网络的初始化
    4. 1.4. Part2:前向传播模块
      1. 1.4.1. Exercise3:完成Linear的前向传播
      2. 1.4.2. Exercise4:完成LINEAR->ACTIVATION的前向传播
      3. 1.4.3. Exercise5:实现L层模型的前向传播
    5. 1.5. Part3:成本函数
      1. 1.5.1. Exercise6:计算成本函数
    6. 1.6. Part4:反向传播模块
      1. 1.6.1. Exercise7:Linear backward
      2. 1.6.2. Exercise8:Linear-Activation backward
      3. 1.6.3. Exercise9:L-Model Backward
      4. 1.6.4. Exercise10:更新参数
    7. 1.7. 任务2:深度神经网络在图像分类中的应用
    8. 1.8. Part0:库的准备
    9. 1.9. Part1:数据准备
    10. 1.10. Part2:模型结构
      1. 1.10.1. Exercise11:2层神经网络
      2. 1.10.2. Exercise12:L层神经网络
    11. 1.11. Part3:错误结果分析