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神经网络算法几乎成为深度学习的代名词，为解决不同的场景问题，新的算法层出不穷，而BP（Back Propagation）算法，又称为误差反向传播算法，是最早的人工神经网络中的一种监督式的学习算法。BP 神经网络算法在理论上可以逼近任意函数，基本的结构由非线性变化单元组成，具有很强的非线性映射能力。对于神经网络的介绍多偏向与理论推导，本文将从代码解析的角度，对BP的神经网络算法进行详细介绍，使读者在了解算法的同时，能够自己搭建算法，内容主要来自于Andrew Ng的深度学习英文课程，语言采用Python。

文章内容：

1. 导入库（附件含非公共库代码）
2. 准备数据集
3. 单一逻辑回归验证
4. 构建神经网络
5. 结果预测

1 导入库

首先导入完成算法所需要的库。

• numpy 矩阵计算基础包。
• sklearn 数据分析与挖掘基础包
• matplotlib 绘图
• testCases 协助校验函数的库，需单独创建，主要用来评估过程中创建的函数是否正确，文末会单独列出。
• planar_utils 提供多种可用的函数，如激活函数等，文末单独列出。

#导入库

import numpy as np

import matplotlib.pyplot as plt

#from testCases_v2 import * #校验函数

import sklearn

import sklearn.datasets

import sklearn.linear_model

from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets

%matplotlib inline

np.random.seed(1) # 设置一个seed保持结果一致性。

2 准备数据集

首先准备我们进行模型训练的数据集，选用的是常用的花朵颜色分类的数据集，含X，Y两个变量，因变量为花朵颜色。为更直观表示，将其进行可视化展现。

X, Y = load_planar_dataset()

plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral)

3 单一逻辑回归¶

复杂的神经网络层本质是一个个单一的逻辑回归网络组合而成，所以在构建全连接的神经网络前，我们可以先看一下单个逻辑回归算法的表现。sklearn 中内置的线性回归函数便可以进行验证，可对该数据集训练一个逻辑回归分类模型来看下分类效果。

# Plot the decision boundary for logistic regression

clf = sklearn.linear_model.LogisticRegressionCV();

clf.fit(X.T, Y.T.ravel())

plot_decision_boundary(lambda x: clf.predict(x), X, Y)

plt.title(“Logistic Regression”)

# Print accuracy

LR_predictions = clf.predict(X.T)

print (‘准确性: %d ‘ % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +

‘% ‘ + “(正确分类的百分比)”)

准确性：47%（正确分类的百分比）

通过单一的逻辑回归预测结果看，分类效果并不理想，只能找到单一分隔线，准确性仅47%。

4 构建神经网络模型

4.1 – 定义神经网络结构

定义三个变量：

• n_x: 输入层大小
• n_h: 隐藏层大小
• n_y: 输出层大小
• 说明: 通过训练集的大小形状确认输入层与输出层的大小，此处将隐藏层直接设置为4。

def layer_sizes(X,Y):

“””

Arguments:

X — input dataset of shape (input size, number of examples)

Y — labels of shape (output size, number of examples)

Returns:

n_x — the size of the input layer

n_h — the size of the hidden layer

n_y — the size of the output layer

“””

### START CODE HERE ### (≈ 3 lines of code)

n_x = X.shape[0]# size of input layer

n_h = 4

n_y = Y.shape[0] # size of output layer

### END CODE HERE ###

return (n_x, n_h, n_y)

4.2 – 初始化模型参数

创建初始化参数函数initialize_parameters()

• 确认参数大小正确。
• 采用随机数创建权重矩阵。
• 采用np.random.randn(a,b) * 0.01 随机初始化形状为(a,b)的矩阵。
• 初始化误差向量为0。You will initialize the bias vectors as zeros.
• 采用 np.zeros((a,b)) 初始化形状为(a,b)的零矩阵。

# GRADED FUNCTION: initialize_parameters

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:

params — python dictionary containing your parameters:

W1 — weight matrix of shape (n_h, n_x)

b1 — bias vector of shape (n_h, 1)

W2 — weight matrix of shape (n_y, n_h)

b2 — bias vector of shape (n_y, 1)

“””

np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.

### START CODE HERE ### (≈ 4 lines of code)

W1 = np.random.randn(n_h,n_x)*0.01

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

4.3 – 创建循环体

创建正向传播函数：forward_propagation():

• 查看分类器的数学表达。
• sigmoid()单独导入，后面做说明。
• np.tanh()numpy中的一个库。
• 还需执行以下步骤:
• 通过parameters[“..”]检索字典”parameters” (initialize_parameters()的输出结果) 中的每个参数。
• 执行正向传播，计算Z[1],,A[1],Z[2],A2。
• 反向传播的所有向量均存储在”cache”中，cache作为反向传播函数的输入变量。

# GRADED FUNCTION: forward_propagation

def forward_propagation(X, parameters):

“””

Argument:

X — input data of size (n_x, m)

parameters — python dictionary containing your parameters (output of initialization function)

Returns:

A2 — The sigmoid output of the second activation

cache — a dictionary containing “Z1”, “A1”, “Z2” and “A2”

“””

# Retrieve each parameter from the dictionary “parameters”

### START CODE HERE ### (≈ 4 lines of code)

W1 = parameters[‘W1’]

b1 = parameters[‘b1’]

W2 = parameters[‘W2’]

b2 = parameters[‘b2’]

### END CODE HERE ###

# Implement Forward Propagation to calculate A2 (probabilities)

### START CODE HERE ### (≈ 4 lines of code)

Z1 = np.dot(W1,X)+b1

A1 = np.tanh(Z1)

Z2 = np.dot(W2,A1)+b2

A2 = sigmoid(Z2)

### END CODE HERE ###

assert(A2.shape == (1, X.shape[1]))

cache = {“Z1”: Z1,

“A1”: A1,

“Z2”: Z2,

“A2”: A2}

return A2, cache

def compute_cost(A2, Y, parameters):

“””

Computes the cross-entropy cost given in equation (13)

Arguments:

A2 — The sigmoid output of the second activation, of shape (1, number of examples)

Y — “true” labels vector of shape (1, number of examples)

parameters — python dictionary containing your parameters W1, b1, W2 and b2

Returns:

cost — cross-entropy cost given equation (13)

“””

m = Y.shape[1] # number of example

# Compute the cross-entropy cost

### START CODE HERE ### (≈ 2 lines of code)

logprobs = np.multiply(np.log(A2),Y)+np.multiply(np.log(1-A2),(1-Y))

cost = -1/m*(np.sum(logprobs))

### END CODE HERE ###

cost = np.squeeze(cost) # makes sure cost is the dimension we expect.

# E.g., turns [[17]] into 17

assert(isinstance(cost, float))

return cost

# GRADED FUNCTION: backward_propagation

def backward_propagation(parameters, cache, X, Y):

“””

Implement the backward propagation using the instructions above.

Arguments:

parameters — python dictionary containing our parameters

cache — a dictionary containing “Z1”, “A1”, “Z2” and “A2”.

X — input data of shape (2, number of examples)

Y — “true” labels vector of shape (1, number of examples)

Returns:

grads — python dictionary containing your gradients with respect to different parameters

“””

m = X.shape[1]

# First, retrieve W1 and W2 from the dictionary “parameters”.

### START CODE HERE ### (≈ 2 lines of code)

W1 = parameters[‘W1’]

W2 = parameters[‘W2’]

### END CODE HERE ###

# Retrieve also A1 and A2 from dictionary “cache”.

### START CODE HERE ### (≈ 2 lines of code)

A1 = cache[‘A1’]

A2 = cache[‘A2’]

### END CODE HERE ###

# Backward propagation: calculate dW1, db1, dW2, db2.

### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)

dZ2 = A2-Y

dW2 = 1/m*np.dot(dZ2,(A1.T))

db2 = 1/m*(np.sum(dZ2,axis=1,keepdims=True))

dZ1 = np.dot(W2.T,dZ2) * (1 – np.power(cache[“A1”],2))

dW1 = 1/m*np.dot(dZ1,(X.T))

db1 = 1/m*(np.sum(dZ1,axis=1,keepdims=True))

### END CODE HERE ###

grads = {“dW1”: dW1,

“db1”: db1,

“dW2”: dW2,

“db2”: db2}

return grads

# GRADED FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate = 1.2):

“””

Updates parameters using the gradient descent update rule given above

Arguments:

parameters — python dictionary containing your parameters

grads — python dictionary containing your gradients

Returns:

parameters — python dictionary containing your updated parameters

“””

# Retrieve each parameter from the dictionary “parameters”

### START CODE HERE ### (≈ 4 lines of code)

W1 = parameters[‘W1’]

b1 = parameters[‘b1’]

W2 = parameters[‘W2’]

b2 = parameters[‘b2’]

### END CODE HERE ###

# Retrieve each gradient from the dictionary “grads”

### START CODE HERE ### (≈ 4 lines of code)

dW1 = grads[‘dW1’]

db1 = grads[‘db1’]

dW2 = grads[‘dW2’]

db2 = grads[‘db2’]

## END CODE HERE ###

# Update rule for each parameter

### START CODE HERE ### (≈ 4 lines of code)

W1 = -dW1*learning_rate+W1

b1 = -db1*learning_rate+b1

W2 = -dW2*learning_rate+W2

b2 = -db2*learning_rate+b2

### END CODE HERE ###

parameters = {“W1”: W1,

“b1”: b1,

“W2”: W2,

“b2”: b2}

return parameters

4.4 – 整合以上4.1, 4.2 和 4.3 创建的函数，创建最终的nn_model()

问题：创建神经网络模型nn_model():

# GRADED FUNCTION: nn_model

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):

“””

Arguments:

X — dataset of shape (2, number of examples)

Y — labels of shape (1, number of examples)

n_h — size of the hidden layer

num_iterations — Number of iterations in gradient descent loop

print_cost — if True, print the cost every 1000 iterations

Returns:

parameters — parameters learnt by the model. They can then be used to predict.

“””

np.random.seed(3)

n_x = layer_sizes(X, Y)[0]

n_y = layer_sizes(X, Y)[2]

# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: “n_x, n_h, n_y”. Outputs = “W1, b1, W2, b2, parameters”.

### START CODE HERE ### (≈ 5 lines of code)

parameters = initialize_parameters(n_x, n_h, n_y)

W1 = parameters[‘W1’]

b1 = parameters[‘b1’]

W2 = parameters[‘W2’]

b2 = parameters[‘b2’]

### END CODE HERE ###

# Loop (gradient descent)

for i in range(0, num_iterations):

### START CODE HERE ### (≈ 4 lines of code)

# Forward propagation. Inputs: “X, parameters”. Outputs: “A2, cache”.

A2, cache = forward_propagation(X, parameters)

# Cost function. Inputs: “A2, Y, parameters”. Outputs: “cost”.

cost = compute_cost(A2, Y, parameters)

# Backpropagation. Inputs: “parameters, cache, X, Y”. Outputs: “grads”.

grads = backward_propagation(parameters, cache, X, Y)

# Gradient descent parameter update. Inputs: “parameters, grads”. Outputs: “parameters”.

parameters = update_parameters(parameters, grads, learning_rate = 1.2)

### END CODE HERE ###

# Print the cost every 1000 iterations

if print_cost and i % 1000 == 0:

print (“Cost after iteration %i: %f” %(i, cost))

return parameters

4.5 -预测

问题：创建预测函数，采用正向传播来预测结果:

注: 如果激活函数>0.5,则预测为1，如果激活函数值<=0.5，则预测为0。例如你如果想基于一个阈值设定X矩阵的类，可以通过X_new = (X > threshold)

# GRADED FUNCTION: predict

def predict(parameters, X):

“””

Using the learned parameters, predicts a class for each example in X

Arguments:

parameters — python dictionary containing your parameters

X — input data of size (n_x, m)

Returns

predictions — vector of predictions of our model (red: 0 / blue: 1)

“””

# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.

### START CODE HERE ### (≈ 2 lines of code)

A2, cache = forward_propagation(X, parameters)

predictions = A2>0.5

### END CODE HERE ###

return predictions

5 结果预测

# Build a model with a n_h-dimensional hidden layer

parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

# Plot the decision boundary

plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)

plt.title(“Decision Boundary for hidden layer size ” + str(4))

# Print accuracy

predictions = predict(parameters, X)

print (‘准确性: %d’ % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + ‘%’)

准确性：90%，预测结果准确性显著高于单个逻辑回归预测结果。

附件1-“testCases_v2.py”

def layer_sizes_test_case():

np.random.seed(1)

X_assess = np.random.randn(5, 3)

Y_assess = np.random.randn(2, 3)

return X_assess, Y_assess

def initialize_parameters_test_case():

n_x, n_h, n_y = 2, 4, 1

return n_x, n_h, n_y

def forward_propagation_test_case():

np.random.seed(1)

X_assess = np.random.randn(2, 3)

parameters = {‘W1’: np.array([[-0.00, -0.00056267],

[-0.0, 0.0],

[-0.0, -0.00],

[0.00, -0.0]]),

‘W2’: np.array([[-0.0, -0.00, 0.00, 0.0]]),

‘b1’: np.array([[0.],

[0.],

[0.],

[0.]]),

‘b2’: np.array([[0.]])}

return X_assess, parameters

def compute_cost_test_case():

np.random.seed(1)

Y_assess = np.random.randn(1, 3)

parameters = {‘W1’: np.array([[-0.00, -0.00056267],

[-0.0, 0.0],

[-0.0, -0.00],

[0.00, -0.0]]),

‘W2’: np.array([[-0.0, -0.00, 0.00, 0.0]]),

‘b1’: np.array([[0.],

[0.],

[0.],

[0.]]),

‘b2’: np.array([[0.]])}

a2 = (np.array([[0., 0., 0.]]))

return a2, Y_assess, parameters

def backward_propagation_test_case():

np.random.seed(1)

X_assess = np.random.randn(2, 3)

Y_assess = np.random.randn(1, 3)

parameters = {‘W1’: np.array([[-0.00, -0.00056267],

[-0.0, 0.0],

[-0.0, -0.00],

[0.00, -0.0]]),

‘W2’: np.array([[-0.0, -0.00, 0.00, 0.0]]),

‘b1’: np.array([[0.],

[0.],

[0.],

[0.]]),

‘b2’: np.array([[0.]])}

cache = {‘A1’: np.array([[-0.00, 0.0020626, 0.00],

[-0.0, 0.0, -0.0],

[-0.0, 0.0036869, 0.0],

[0.0, -0.0, 0.0]]),

‘A2’: np.array([[0., 0., 0.]]),

‘Z1’: np.array([[-0.00, 0.0020626, 0.0034962],

[-0.0, 0.0, -0.0],

[-0.0, 0.00, 0.0],

[0.0, -0.0, 0.0]]),

‘Z2’: np.array([[0.00092281, -0.00056678, 0.00095853]])}

return parameters, cache, X_assess, Y_assess

def update_parameters_test_case():

parameters = {‘W1’: np.array([[-0.00, 0.0],

[-0.0, 0.0],

[-0.0, -0.0],

[0.00, -0.0]]),

‘W2’: np.array([[-0.0, -0.0, 0.0, 0.0]]),

‘b1’: np.array([[-8.e-07],

[8.e-06],

[6.0e-07],

[-2.e-06]]),

‘b2’: np.array([[9.e-05]])}

grads = {‘dW1’: np.array([[0.00023322, -0.00],

[0.00082222, -0.00],

[-0.00031831, 0.0028636],

[-0.00092857, 0.00]]),

‘dW2’: np.array([[-1.e-05, 3.e-03, -1.e-03,

-2.e-03]]),

‘db1’: np.array([[1.0e-07],

[-3.e-06],

[-1.e-07],

[5.e-07]]),

‘db2’: np.array([[-1.0e-05]])}

return parameters, grads

def nn_model_test_case():

np.random.seed(1)

X_assess = np.random.randn(2, 3)

Y_assess = np.random.randn(1, 3)

return X_assess, Y_assess

def predict_test_case():

np.random.seed(1)

X_assess = np.random.randn(2, 3)

parameters = {‘W1’: np.array([[-0.00, 0.0],

[-0.0, 0.0],

[-0.0, -0.0],

[0.00, -0.0]]),

‘W2’: np.array([[-0.0, -0.0, 0.0, 0.0]]),

‘b1’: np.array([[-8.e-07],

[8.e-06],

[6.0e-07],

[-2.e-06]]),

‘b2’: np.array([[9.e-05]])}

return parameters, X_assess

附件2-“planar_utils.py”

def plot_decision_boundary(model, X, y):

# Set min and max values and give it some padding

x_min, x_max = X[0, :].min() – 1, X[0, :].max() + 1

y_min, y_max = X[1, :].min() – 1, X[1, :].max() + 1

h = 0.01

# Generate a grid of points with distance h between them

xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))

# Predict the function value for the whole grid

Z = model(np.c_[xx.ravel(), yy.ravel()])

Z = Z.reshape(xx.shape)

# Plot the contour and training examples

plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)

plt.ylabel(‘x2’)

plt.xlabel(‘x1’)

plt.scatter(X[0, :], X[1, :], c=y[0], cmap=plt.cm.Spectral)

def sigmoid(x):

“””

Compute the sigmoid of x

Arguments:

x — A scalar or numpy array of any size.

Return:

s — sigmoid(x)

“””

s = 1/(1+np.exp(-x))

return s

def load_planar_dataset():

np.random.seed(1)

m = 400 # number of examples

N = int(m/2) # number of points per class

D = 2 # dimensionality

X = np.zeros((m,D)) # data matrix where each row is a single example

Y = np.zeros((m,1), dtype=’uint8′) # labels vector (0 for red, 1 for blue)

a = 4 # maximum ray of the flower

for j in range(2):

ix = range(N*j,N*(j+1))

t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # theta

r = a*np.sin(4*t) + np.random.randn(N)*0.2 # radius

X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]

Y[ix] = j

X = X.T

Y = Y.T

return X, Y

def load_extra_datasets():

N = 200

noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)

noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)

blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)

gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)

no_structure = np.random.rand(N, 2), np.random.rand(N, 2)

return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure