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吴恩达机器学习作业Python3实现(四):神经网络及其反向传播

2021-01-17 923点热度 0人点赞 2条评论

寻找志同道合的学习伙伴,请访问我的个人网页.
该内容同步发布在CSDN和耳壳网.

本次作业代码全部采用numpy的array数组来进行计算,不采用矩阵
两者在乘法和数量积的运算上采用的符号不同
最好不要混用,开始之前先定下代码里用哪一种

神经网络 Neural Networks

数据可视化 Visualizing the data

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib
from scipy.io import loadmat 
data = loadmat('ex4data1.mat')
data
{'__header__': b'MATLAB 5.0 MAT-file, Platform: GLNXA64, Created on: Sun Oct 16 13:09:09 2011',
 '__version__': '1.0',
 '__globals__': [],
 'X': array([[0., 0., 0., ..., 0., 0., 0.],
        [0., 0., 0., ..., 0., 0., 0.],
        [0., 0., 0., ..., 0., 0., 0.],
        ...,
        [0., 0., 0., ..., 0., 0., 0.],
        [0., 0., 0., ..., 0., 0., 0.],
        [0., 0., 0., ..., 0., 0., 0.]]),
 'y': array([[10],
        [10],
        [10],
        ...,
        [ 9],
        [ 9],
        [ 9]], dtype=uint8)}
data['X'].shape, data['y'].shape
((5000, 400), (5000, 1))
# 随机挑选100个图像
sample_idx = np.random.choice(np.arange(data['X'].shape[0]), 100)
sample_images = data['X'][sample_idx, :]
sample_images.shape
(100, 400)
# 展示图像
fig, ax = plt.subplots(10, 10, sharey=True, sharex=True, figsize=(10, 10))
for r in range(10):
    for c in range(10):
        ax[r, c].matshow(np.array(sample_images[10 * r + c].reshape(20, 20)).T, cmap=matplotlib.cm.binary)
        plt.xticks(np.array([]))
        plt.yticks(np.array([])) 

output_7_0

如果文中公式不可见,可以去我GitHub上查看

处理训练数据

标签重编码

首先我们要将标签值(1,2,3,4,…,10)转化成非线性相关的向量,向量对应位置(y[i-1])上的值等于1,
例如y[0]=6转化为y[0]=[0,0,0,0,0,1,0,0,0,0]。

def expend_y(y):
    result = []
    # 将y的每个元素转换为一个向量,标签值的位置为1,其余为0
    for label in y:
        y_array = np.zeros(10)
        y_array[label-1] = 1
        result.append(y_array)
    return np.array(result)

'''
也可以利用sklearn的编码函数

from sklearn.preprocessing import OneHotEncoder
    encoder = OneHotEncoder(sparse=False)
    y_onehot = encoder.fit_transform(y)
    y_onehot.shape 
'''
'\n也可以利用sklearn的编码函数\n\nfrom sklearn.preprocessing import OneHotEncoder\n    encoder = OneHotEncoder(sparse=False)\n    y_onehot = encoder.fit_transform(y)\n    y_onehot.shape \n'

验证y

y1 = np.arange(1, 11, 1).reshape((10, 1))
y1
array([[ 1],
       [ 2],
       [ 3],
       [ 4],
       [ 5],
       [ 6],
       [ 7],
       [ 8],
       [ 9],
       [10]])
y2 = expend_y(y1)
y2.shape
(10, 10)
y2
array([[1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
       [0., 1., 0., 0., 0., 0., 0., 0., 0., 0.],
       [0., 0., 1., 0., 0., 0., 0., 0., 0., 0.],
       [0., 0., 0., 1., 0., 0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 1., 0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0., 1., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0., 0., 1., 0., 0., 0.],
       [0., 0., 0., 0., 0., 0., 0., 1., 0., 0.],
       [0., 0., 0., 0., 0., 0., 0., 0., 1., 0.],
       [0., 0., 0., 0., 0., 0., 0., 0., 0., 1.]])

获取训练数据集

X = np.insert(data['X'], 0, np.ones(len(data['X'])), axis=1)
raw_y = data['y']
y = expend_y(raw_y)

X.shape, y.shape
((5000, 401), (5000, 10))

读取已训练好的权重

weight = loadmat('ex4weights.mat')
weight
{'__header__': b'MATLAB 5.0 MAT-file, Platform: GLNXA64, Created on: Tue Oct 18 14:57:02 2011',
 '__version__': '1.0',
 '__globals__': [],
 'Theta1': array([[-2.25623899e-02, -1.05624163e-08,  2.19414684e-09, ...,
         -1.30529929e-05, -5.04175101e-06,  2.80464449e-09],
        [-9.83811294e-02,  7.66168682e-09, -9.75873689e-09, ...,
         -5.60134007e-05,  2.00940969e-07,  3.54422854e-09],
        [ 1.16156052e-01, -8.77654466e-09,  8.16037764e-09, ...,
         -1.20951657e-04, -2.33669661e-06, -7.50668099e-09],
        ...,
        [-1.83220638e-01, -8.89272060e-09, -9.81968100e-09, ...,
          2.35311186e-05, -3.25484493e-06,  9.02499060e-09],
        [-7.02096331e-01,  3.05178374e-10,  2.56061008e-09, ...,
         -8.61759744e-04,  9.43449909e-05,  3.83761998e-09],
        [-3.50933229e-01,  8.85876862e-09, -6.57515140e-10, ...,
         -1.80365926e-06, -8.14464807e-06,  8.79454531e-09]]),
 'Theta2': array([[-0.76100352, -1.21244498, -0.10187131, -2.36850085, -1.05778129,
         -2.20823629,  0.56383834,  1.21105294,  2.21030997,  0.44456156,
         -1.18244872,  1.04289112, -1.60558756,  1.30419943,  1.37175046,
          1.74825095, -0.23365648, -1.52014483,  1.15324176,  0.10368082,
         -0.37207719, -0.61530019, -0.1256836 , -2.27193038, -0.71836208,
         -1.29690315],
        [-0.61785176,  0.61559207, -1.26550639,  1.85745418, -0.91853319,
         -0.05502589, -0.38589806,  1.29520853, -1.56843297, -0.97026419,
         -2.18334895, -2.85033578, -2.07733086,  1.63163164,  0.3490229 ,
          1.82789117, -2.44174379, -0.8563034 , -0.2982564 , -2.07947873,
         -1.2933238 ,  0.89982032,  0.28306578,  2.31180525, -2.46444086,
          1.45656548],
        [-0.68934072, -1.94538151,  2.01360618, -3.12316188, -0.2361763 ,
          1.38680947,  0.90982429, -1.54774416, -0.79830896, -0.65599834,
          0.7353833 , -2.58593294,  0.47210839,  0.55349499,  2.51255453,
         -2.4167454 , -1.63898627,  1.2027302 , -1.20245851, -1.83445959,
         -1.88013027, -0.34056098,  0.23692483, -1.06137919,  1.02759232,
         -0.47690832],
        [-0.67832479,  0.46299226,  0.58492321, -0.1650184 ,  1.93264192,
         -0.22965765, -1.84731492,  0.49011768,  1.07146054, -3.31905643,
          1.54113507,  0.37371947, -0.86484681, -2.58273522,  0.97062447,
         -0.51021867, -0.68427897, -1.64713607,  0.21153145, -0.27422442,
          1.72599755,  1.32418658, -2.63984479, -0.08055871, -2.03510803,
         -1.46123776],
        [-0.59664339, -2.04481799,  2.05698407,  1.95100909,  0.17637699,
         -2.16141218, -0.40394736,  1.80157532, -1.56278739, -0.25253004,
          0.23586497,  0.71656699,  1.07689092, -0.35457279, -1.67743058,
         -0.12939255, -0.67488849,  1.14066535,  1.32431237,  3.21158484,
         -2.15888898, -2.60164082, -3.2226466 , -1.89612906, -0.87488068,
          2.51038628],
        [-0.87794907,  0.4344112 , -0.93161049,  0.18390778, -0.36078216,
          0.61958137,  0.38624948, -2.65150343,  2.29710773, -2.08818098,
         -1.86382323,  1.06057836,  0.77562146,  2.1346861 , -1.14973702,
         -0.52081426,  0.99743429, -1.48309353, -2.3139424 ,  0.29517333,
         -0.38704879, -2.20607697,  0.30702191, -1.17646114, -1.63462966,
         -0.82467661],
        [-0.52746527,  1.21564288, -1.50095981, -2.03195359, -1.52366734,
         -2.43732079, -2.37570311, -1.39987277, -0.88735315, -0.63278873,
          1.50450176, -1.580763  ,  0.58599217, -0.77540416,  0.94257331,
          2.10919653,  0.54479132,  0.43773612, -1.28024228, -0.04360994,
          1.4774997 , -1.13276949, -0.72846904,  0.04734716,  1.6574566 ,
          1.68540944],
        [-0.7490154 , -0.72249056, -3.15228173,  0.36577778,  0.19811362,
         -0.73059946,  1.65263918, -2.300357  , -1.87468162,  0.98095387,
         -1.58825159,  1.35434142,  2.17895331, -1.99239762, -2.00371362,
         -0.388613  , -2.33992976, -2.91719062,  0.99398645, -2.70476768,
         -1.27139772,  1.86091461, -1.20519404, -0.38014194,  0.7087181 ,
         -2.11014003],
        [-0.6665468 ,  0.53601845,  1.30307573, -1.03372714, -4.03084753,
          0.58173469, -2.65717902,  0.80379994, -1.09241928,  2.49910058,
          0.362008  ,  0.66195337, -0.92160534, -0.83123666, -2.00200952,
         -2.94897501,  0.64564202, -1.10114694,  0.74510309,  0.58506717,
         -1.99545251,  0.62591105,  1.80596103, -0.22309315, -1.40442136,
         -2.1319153 ],
        [-0.46089119, -1.43944954, -1.21809509,  0.71093011,  0.45216919,
         -0.35953381,  0.62284954, -0.67005297, -0.7069138 ,  0.06311351,
         -1.23199074, -1.74645233, -2.71960897, -2.21437178, -1.69307505,
         -0.90927394,  0.87852311,  1.18664814, -1.87041262,  0.39796295,
          1.72113872, -1.36934055,  0.8580668 , -0.24779579,  1.28009118,
         -1.32752042]])}
theta1 = weight['Theta1']
theta2 = weight['Theta2']

展开/合并参数

当我们使用高级优化方法来优化神经网络时,我们需要将多个参数矩阵展开,才能传入优化函数,然后再恢复形状。

def serialize(a, b):
    return np.concatenate((np.ravel(a), np.ravel(b)))
a = np.arange(10)
b = np.ones(5)
c = serialize(a, b)
c
array([0., 1., 2., 3., 4., 5., 6., 7., 8., 9., 1., 1., 1., 1., 1.])
def deserialize(seq):
    return seq[:25*401].reshape(25, 401), seq[25*401:].reshape(10, 26)

模型表示 Model representation

我们的网络有三层,输入层,隐藏层,输出层。我们的输入是数字图像的像素值,因为每个数字的图像大小为20*20,所以我们输入层有400个单元(这里不包括总是输出要加一个偏置单元)。
image.png

如果文中公式不可见,可以去我GitHub上查看

正向传播 Feedforward

# sigmoid函数
def sigmoid(z):
    return 1 / (1 + np.exp(-z))
# 前向传播
def forward_propagation(theta, X): # theta,X都是array数组类型,非矩阵
    t1, t2 = deserialize(theta)

    a1 = X # 5000*401

    z2 = a1 @ t1.T # 5000*25
    a2 = sigmoid(z2)
    a2 = np.insert(a2, 0, np.ones(len(a2)), axis=1) # 5000*26

    z3 = a2 @ t2.T # 5000*10
    a3 = sigmoid(z3)
    h = a3  # 5000*10

    return a1, z2, a2, z3, h
theta = serialize(theta1, theta2)
a1, z2, a2, z3, h = forward_propagation(theta, X)
a1.shape, z2.shape, a2.shape, z3.shape, h.shape
((5000, 401), (5000, 25), (5000, 26), (5000, 10), (5000, 10))
forward_propagation(theta, X)[-1]
array([[1.12661530e-04, 1.74127856e-03, 2.52696959e-03, ...,
        4.01468105e-04, 6.48072305e-03, 9.95734012e-01],
       [4.79026796e-04, 2.41495958e-03, 3.44755685e-03, ...,
        2.39107046e-03, 1.97025086e-03, 9.95696931e-01],
       [8.85702310e-05, 3.24266731e-03, 2.55419797e-02, ...,
        6.22892325e-02, 5.49803551e-03, 9.28008397e-01],
       ...,
       [5.17641791e-02, 3.81715020e-03, 2.96297510e-02, ...,
        2.15667361e-03, 6.49826950e-01, 2.42384687e-05],
       [8.30631310e-04, 6.22003774e-04, 3.14518512e-04, ...,
        1.19366192e-02, 9.71410499e-01, 2.06173648e-04],
       [4.81465717e-05, 4.58821829e-04, 2.15146201e-05, ...,
        5.73434571e-03, 6.96288990e-01, 8.18576980e-02]])

代价函数 Cost function

image.png

def nnCost(theta, X, y): # 均为数组
    h = forward_propagation(theta, X)[-1]
    temp = -y * np.log(h) - (1 - y) * np.log(1- h)
    return  temp.sum() / len(X)
nnCost(theta, X, y)
0.2876291651613189

正则化代价函数 regularized cost function

image.png

def nnCostReg(theta, X, y, lambdaRate):
    theta1, theta2 = deserialize(theta)

    first = nnCost(theta, X, y)

    reg1 = (np.power(theta1[:, 1: ], 2)).sum()
    reg2 = (np.power(theta2[:, 1: ], 2)).sum()
    reg = lambdaRate / (2 * len(X)) * (reg1 + reg2)

    return first + reg
nnCostReg(theta, X, y, lambdaRate=1)
0.38376985909092365

反向传播 Back propagation

S函数梯度 Sigmoid gradient

image.png

image.png

def sigmoid_gradient(z):
    return sigmoid(z) * (1- sigmoid(z))
sigmoid_gradient(0)
0.25
siggra_x = np.arange(-10, 10, 0.01)
plt.plot(siggra_x, sigmoid_gradient(siggra_x))
plt.grid()
plt.show()

output_42_0

由此可见,sigmoid的导数值最大为0.25,决定了其无法用于深层网络

theta随机初始化 Random initialization

当我们训练神经网络时,随机初始化参数是很重要的,可以打破数据的对称性。
一个有效的策略是在均匀分布(−e,e)中随机选择值,我们可以选择 e = 0.12 这个范围的值来确保参数足够小,使得训练更有效率。

image.png

# 从服从均匀分布的范围中随机返回size大小的值
def random_init(size):
    return np.random.uniform(-0.12, 0.12, size)
random_init((3, 4))
array([[-0.03588089, -0.01860106,  0.0231179 ,  0.08834786],
       [-0.00270229, -0.04455686, -0.10708566,  0.09343351],
       [-0.04520061,  0.06483762, -0.08712267,  0.0488686 ]])

反向传播

image.png

目标:获取整个网络代价函数的梯度,以便在优化算法中求解。
步骤:给定训练集,先计算正向传播,再对于层的每个节点,计算误差项,这个数据衡量这个节点对最后输出
的误差“贡献”了多少。

这里面一定要理解正向传播和反向传播的过程,才能弄清楚各种参数在网络中的维度,切记。
最好手写出每次传播的式子。

公式
1. 1.jpg
2. 2.jpg
3. 3.jpg
4. 4.jpg

更新梯度

# 更新梯度
def nnGradient(theta, X, y):
    theta1, theta2 = deserialize(theta)
    # 首先正向传播
    a1, z2, a2, z3, h = forward_propagation(theta, X)
    # 针对每层的各个节点计算误差项
    d3 = h - y # 输出层误差 5000*10
    d2 = d3 @ theta2[:, 1:] * sigmoid_gradient(z2) # 隐藏层误差 5000*25

    delta1 = d2.T @ a1 # 25 * 401
    delta2 = d3.T @ a2 # 10 * 26
    delta = serialize(delta1, delta2) # (10285, )

    return delta / len(X) # (10285, )
d1, d2 = deserialize(nnGradient(theta, X, y))
d1.shape, d2.shape
((25, 401), (10, 26))
theta1.shape, theta2.shape
((25, 401), (10, 26))

梯度检验 Gradient checking

进行梯度校验时,需要把参数Θ(1)、Θ(2)连接成一个长向量。之后你可以使用如下公式计算:
image.png

def gradient_checking(theta, X, y, eps, regularized=False):
    m = len(theta)

    def a_numeric_grad(plus, minus, X, y):
        if regularized:
            return (nnCostReg(plus, X, y) - nnCostReg(minus, X, y)) / (2 * eps)
        else:
            return (nnCost(plus, X, y) - nnCost(minus, X, y)) / (2 * eps)

    approxGrad = np.zeros(m)
    # 计算偏导数
    for i in range(m):  
        thetaPlus = theta.copy()
        thetaPlus[i] = theta[i] + eps

        thetaMinus = theta.copy()
        thetaMinus[i] = theta[i] - eps

        grad = a_numeric_grad(thetaPlus, thetaMinus, X, y)
        approxGrad[i] = grad

    # 用梯度公式
    funcGrad = nnGradientReg(theta, X, y) if  regularized else nnGradient(theta, X, y)

    diff = np.linalg.norm(approxGrad - funcGrad) / np.linalg.norm(approxGrad + funcGrad)

    print('If your backpropagation implementation is correct,\n'\
         + 'the relative difference will be smaller than 10e-9 (assume epsilon=0.0001).\n'
         + 'Relative Difference: {}\n'.format(diff))

    #return approxGrad.shape
# gradient_checking(theta, X, y, eps=0.0001, regularized=False) #这个运行很慢,谨慎运行

正则化神经网络

image.png

def nnGradientReg(theta, X, y, lambdaRate):
    first = nnGradient(theta, X, y)

    t1, t2 = deserialize(theta)
    t1[:, 0] = 0
    t2[:, 0] = 0

    reg = lambdaRate / len(X) * serialize(t1, t2)

    return first + reg
d1, d2 = deserialize(nnGradientReg(theta, X, y, lambdaRate=1))
d1.shape, d2.shape
((25, 401), (10, 26))

优化参数 Learning parameters using fmincg

from scipy import optimize as opt

def nnTraining(X, y):
    # 初始化theta
    init_theta = random_init(10285)

    res = opt.minimize(fun=nnCostReg,
                       x0=init_theta,
                       args=(X, y, 1),
                       method='TNC',
                       jac=nnGradientReg,
                       options={'maxiter':500})
    return res
res = nnTraining(X, y)
res
     fun: 0.33298285824302043
     jac: array([ 1.41541160e-04, -2.29401553e-07, -2.55382916e-07, ...,
       -5.01885346e-06, -3.60910199e-05, -2.47061167e-05])
 message: 'Max. number of function evaluations reached'
    nfev: 250
     nit: 21
  status: 3
 success: False
       x: array([ 0.00000000e+00, -1.14700777e-03, -1.27691458e-03, ...,
        1.63199029e+00, -2.96556538e+00, -1.95288757e+00])

性能评价

final_theta = res.x
final_theta.shape
(10285,)
h = forward_propagation(final_theta, X)[-1]
h.shape
(5000, 10)
y_predict = np.argmax(h, axis=1) + 1
y_predict = y_predict.reshape(5000, 1)
y_predict
array([[10],
       [10],
       [10],
       ...,
       [ 9],
       [ 9],
       [10]], dtype=int64)
raw_y
array([[10],
       [10],
       [10],
       ...,
       [ 9],
       [ 9],
       [ 9]], dtype=uint8)
from sklearn.metrics import classification_report

print(classification_report(raw_y, y_predict))
              precision    recall  f1-score   support

           1       0.97      0.99      0.98       500
           2       0.98      0.98      0.98       500
           3       0.98      0.98      0.98       500
           4       1.00      0.95      0.97       500
           5       1.00      0.98      0.99       500
           6       0.98      0.99      0.99       500
           7       0.98      0.99      0.98       500
           8       0.98      0.99      0.99       500
           9       0.99      0.96      0.97       500
          10       0.96      1.00      0.98       500

    accuracy                           0.98      5000
   macro avg       0.98      0.98      0.98      5000
weighted avg       0.98      0.98      0.98      5000

​

可视化隐层 Visualizing the hidden layer

image.png

theta1, theta2 = deserialize(final_theta)
hidden_layer = theta1[:, 1: ]
hidden_layer.shape
(25, 400)
fig, ax = plt.subplots(5, 5, sharey=True, sharex=True, figsize=(10, 10))
for r in range(5):
    for c in range(5):
        ax[r, c].matshow(np.array(hidden_layer[5 * r + c].reshape(20, 20)).T, cmap=matplotlib.cm.binary)
        plt.xticks(np.array([]))
        plt.yticks(np.array([])) 

output_73_0

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标签: 机器学习
最后更新:2021-01-17

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    2021-01-18
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    文章目录
    • 神经网络 Neural Networks
      • 数据可视化 Visualizing the data
      • 处理训练数据
        • 标签重编码
        • 获取训练数据集
        • 展开/合并参数
      • 模型表示 Model representation
      • 正向传播 Feedforward
      • 代价函数 Cost function
      • 正则化代价函数 regularized cost function
    • 反向传播 Back propagation
      • S函数梯度 Sigmoid gradient
      • theta随机初始化 Random initialization
      • 反向传播
        • 更新梯度
      • 梯度检验 Gradient checking
      • 正则化神经网络
      • 优化参数 Learning parameters using fmincg
      • 性能评价
    • 可视化隐层 Visualizing the hidden layer

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