系列培训目录
➡️神经网络（Neural Networks）⬅️
卷积神经网络（Convolutional Neural Networks）
循环神经网络（Recurrent Neural Networks）
生成对抗神经网络（Generative Adversarial Networks）
神经网络（Neural Networks）
最简的神经神经网络 — 一个神经元
image
- 组成：
- 参数：用x表示
- 权重：用w表示
- 偏差：用b表示
- 激活函数：用f(h)表示
- 数学形式：
image
其中f表示激活函数，通常用
image
sigmoid值在0-1之间的数值很像概率适合做分类
image
如何找出这样的函数？
方法：
- 用监督学习的方式训练：
- 让机器找出使数据和目标之间的误差最小的函数
- 告诉机器衡量误差的函数
- 用梯度下降（Gradient Descent）更新神经元的权重(w)，使得误差的方程最小
- 让机器找出使数据和目标之间的误差最小的函数
常用的误差函数：
回归问题用：平方差之和（the sum of squared errors）:
image其中：
image
是为了方便计算在求导时可去除平方，u是每行，j表示每列- 分类问题用：最小交叉墒(后面讲)
image
梯度下降 – 数学
用链式法则去求误差函数对于权重w的偏微分，用来更新神经网络的权重w
存在的问题：
只看梯度的话，会卡在局部最优值
image
数学求法：
为了更新权重，就要求误差函数E对于权重w的偏导，乘以学习率来控制学习速度
η成为learning rate，表示每次更新权重w的步长，用来控制学习速度
image
求误差函数E对于权重w的偏导的求法：
目标：
image
链式法则：
image
再用一次链式法则：
image
带入后，再用一次链式法则：
image
最后，做替换：
image
最终结果：
image
定义：
image
梯度下降 – 代码实现
import numpy as np
def sigmoid(x):
"""
Calculate sigmoid
"""
return 1/(1+np.exp(-x))
def sigmoid_prime(x):
"""
# Derivative of the sigmoid function
"""
return sigmoid(x) * (1 - sigmoid(x))
learnrate = 0.5
x1,x2,x3,x4 = 1, 2, 3, 4
y = 0.5
# Initial weights
w1,w2,w3,w4 = 0.5, -0.5, 0.3, 0.1
### Calculate one gradient descent step for each weight
### Note: Some steps have been consilated, so there are
### fewer variable names than in the above sample code
# TODO: Calculate the node's linear combination of inputs and weights
h = x1*w1+x2*w2+x3*w3+x4*w4
# TODO: Calculate output of neural network
y_hat = sigmoid(h)
# TODO: Calculate error of neural network
error = (y - y_hat)
# TODO: Calculate the error term
# Remember, this requires the output gradient, which we haven't
# specifically added a variable for.
error_term = error * sigmoid_prime(h)
# Note: The sigmoid_prime function calculates sigmoid(h) twice,
# but you've already calculated it once. You can make this
# code more efficient by calculating the derivative directly
# rather than calling sigmoid_prime, like this:
# error_term = error * nn_output * (1 - nn_output)
# TODO: Calculate change in weights
del_w = learnrate * error_term * x
print('Neural Network output:')
print(nn_output)
print('Amount of Error:')
print(error)
print('Change in Weights:')
print(del_w)
Neural Network output:
0.689974481128
Amount of Error:
-0.189974481128
Change in Weights:
[-0.02031869 -0.04063738 -0.06095608 -0.08127477]
训练方法
迭代直到误差最小：
- 正向传播，获得预测值：沿着神经网络，矩阵点乘，计算出预测值\hat y。
- 反向传播，获得每层的误差梯度：用\hat y计算误差函数，反向传播误差。
- 更新权重：根据误差更新权重
以预测研究生是否能入学为例 – 单个神经元版本
image
读取原始数据
import numpy as np
import pandas as pd
admissions=pd.read_csv("entry_admission.csv")
admissions.head()
<div>
<table border=”1″ class=”dataframe”>
<thead>
<tr style=”text-align: right;”>
<th></th>
<th>admit</th>
<th>gre</th>
<th>gpa</th>
<th>rank</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>0</td>
<td>380</td>
<td>3.61</td>
<td>3</td>
</tr>
<tr>
<th>1</th>
<td>1</td>
<td>660</td>
<td>3.67</td>
<td>3</td>
</tr>
<tr>
<th>2</th>
<td>1</td>
<td>800</td>
<td>4.00</td>
<td>1</td>
</tr>
<tr>
<th>3</th>
<td>1</td>
<td>640</td>
<td>3.19</td>
<td>4</td>
</tr>
<tr>
<th>4</th>
<td>0</td>
<td>520</td>
<td>2.93</td>
<td>4</td>
</tr>
</tbody>
</table>
</div>
数据处理
# Make dummy variables for rank
data = pd.concat([admissions, pd.get_dummies(admissions['rank'], prefix='rank')], axis=1)
data = data.drop('rank', axis=1)
# Standarize features
for field in ['gre', 'gpa']:
mean, std = data[field].mean(), data[field].std()
data.loc[:,field] = (data[field]-mean)/std
# Split off random 10% of the data for testing
np.random.seed(42)
sample = np.random.choice(data.index, size=int(len(data)*0.9), replace=False)
data, test_data = data.ix[sample], data.drop(sample)
# Split into features and targets
features, targets = data.drop('admit', axis=1), data['admit']
features_test, targets_test = test_data.drop('admit', axis=1), test_data['admit']
features.head()
<div>
<table border=”1″ class=”dataframe”>
<thead>
<tr style=”text-align: right;”>
<th></th>
<th>gre</th>
<th>gpa</th>
<th>rank_1</th>
<th>rank_2</th>
<th>rank_3</th>
<th>rank_4</th>
</tr>
</thead>
<tbody>
<tr>
<th>209</th>
<td>-0.066657</td>
<td>0.289305</td>
<td>0</td>
<td>1</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<th>280</th>
<td>0.625884</td>
<td>1.445476</td>
<td>0</td>
<td>1</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<th>33</th>
<td>1.837832</td>
<td>1.603135</td>
<td>0</td>
<td>0</td>
<td>1</td>
<td>0</td>
</tr>
<tr>
<th>210</th>
<td>1.318426</td>
<td>-0.131120</td>
<td>0</td>
<td>0</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<th>93</th>
<td>-0.066657</td>
<td>-1.208461</td>
<td>0</td>
<td>1</td>
<td>0</td>
<td>0</td>
</tr>
</tbody>
</table>
</div>
targets.head()
209 0
280 0
33 1
210 0
93 0
Name: admit, dtype: int64
单神经元版本
def sigmoid(x):
"""
Calculate sigmoid
"""
return 1 / (1 + np.exp(-x))
# TODO: We haven't provided the sigmoid_prime function like we did in
# the previous lesson to encourage you to come up with a more
# efficient solution. If you need a hint, check out the comments
# in solution.py from the previous lecture.
# Use to same seed to make debugging easier
np.random.seed(42)
n_records, n_features = features.shape
last_loss = None
# Initialize weights
weights = np.random.normal(scale=1 / n_features**.5, size=n_features)
# Neural Network hyperparameters
epochs = 1000
learnrate = 0.5
for e in range(epochs):#训练的 代 数
del_w = np.zeros(weights.shape)
for x, y in zip(features.values, targets):
# Loop through all records, x is the input, y is the target
# Note: We haven't included the h variable from the previous
# lesson. You can add it if you want, or you can calculate
# the h together with the output
# TODO: Calculate the output
output = sigmoid(np.dot(x,weights))
# TODO: Calculate the error
error = y-output
# TODO: Calculate the error term
error_term = error*output*(1-output)
# TODO: Calculate the change in weights for this sample
# and add it to the total weight change
del_w += error_term*x
# TODO: Update weights using the learning rate and the average change in weights
weights += learnrate*del_w
# Printing out the mean square error on the training set
if e % (epochs / 10) == 0:
out = sigmoid(np.dot(features, weights))
loss = np.mean((out - targets) ** 2)
if last_loss and last_loss < loss:
print("Train loss: ", loss, " WARNING - Loss Increasing")
else:
print("Train loss: ", loss)
last_loss = loss
# Calculate accuracy on test data
tes_out = sigmoid(np.dot(features_test, weights))
predictions = tes_out > 0.5
accuracy = np.mean(predictions == targets_test)
print("Prediction accuracy: {:.3f}".format(accuracy))
Train loss: 0.286196010415
Train loss: 0.257761346594
Train loss: 0.257722034703
Train loss: 0.257722749419 WARNING - Loss Increasing
Train loss: 0.257722752361 WARNING - Loss Increasing
Train loss: 0.257722752309
Train loss: 0.257722752309
Train loss: 0.257722752309 WARNING - Loss Increasing
Train loss: 0.257722752309 WARNING - Loss Increasing
Train loss: 0.257722752309 WARNING - Loss Increasing
Prediction accuracy: 0.725
神经网络 – 由神经元组成
通过非线性的激活函数的神经元组合起来就是神经网络。
能得出非线性的函数，从而具备找到各种各样函数的能力。
image
数学形式：
矩阵相乘，每一隐含层是一个矩阵 （图待加上偏差）
image
神经网络 – 代码表示
import numpy as np
def sigmoid(x):
"""
Calculate sigmoid
"""
return 1/(1+np.exp(-x))
# Network size
N_input = 4
N_hidden = 3
N_output = 2
np.random.seed(42)
# Make some fake data
X = np.random.randn(N_input)
weights_input_to_hidden = np.random.normal(0, scale=0.1, size=(N_input, N_hidden))
weights_hidden_to_output = np.random.normal(0, scale=0.1, size=(N_hidden, N_output))
# element-wise
# TODO: Make a forward pass through the network
hidden_layer_in = np.dot(X, weights_input_to_hidden)
hidden_layer_out = sigmoid(hidden_layer_in)
print('Hidden-layer Output:')
print(hidden_layer_out)
output_layer_in = np.dot(hidden_layer_out, weights_hidden_to_output)
output_layer_out = sigmoid(output_layer_in)
print('Output-layer Output:')
print(output_layer_out)
Hidden-layer Output:
[ 0.41492192 0.42604313 0.5002434 ]
Output-layer Output:
[ 0.49815196 0.48539772]
反向传播 – 将误差的梯度反向传播到神经网络的每个神经元用以更新权重w
反向传播的计算方法：
从最后一层的梯度计算，利用链式法则反向计算每一层梯度
image
公式：
第j层的误差：
image
这里的Σ表示如果下一层（第k层）有多个神经元，则反向传上来的误差要叠加起来
第j层的每个权重w的值：
image
import numpy as np
def sigmoid(x):
"""
Calculate sigmoid
"""
return 1 / (1 + np.exp(-x))
x = np.array([0.5, 0.1, -0.2])
target = 0.6
learnrate = 0.5
weights_input_hidden = np.array([[0.5, -0.6],
[0.1, -0.2],
[0.1, 0.7]])
weights_hidden_output = np.array([0.1, -0.3])
## Forward pass
hidden_layer_input = np.dot(x, weights_input_hidden)
hidden_layer_output = sigmoid(hidden_layer_input)
output_layer_in = np.dot(hidden_layer_output, weights_hidden_output)
output = sigmoid(output_layer_in)
## Backwards pass
## TODO: Calculate error
error = target - output
# TODO: Calculate error gradient for output layer
del_err_output = error * output * (1 - output)
# TODO: Calculate change in weights for hidden layer to output layer
delta_weights_hidden_output = learnrate * del_err_output * hidden_layer_output
# TODO: Calculate error gradient for hidden layer
del_err_hidden = np.dot(del_err_output, weights_hidden_output) * \
hidden_layer_output * (1 - hidden_layer_output)
# TODO: Calculate change in weights for input layer to hidden layer
delta_weights_input_hidden = learnrate * del_err_hidden * x[:, None]
print('Change in weights for hidden layer to output layer:')
print(delta_weights_hidden_output)
print('Change in weights for input layer to hidden layer:')
print(delta_weights_input_hidden)
Change in weights for hidden layer to output layer:
[ 0.00804047 0.00555918]
Change in weights for input layer to hidden layer:
[[ 1.77005547e-04 -5.11178506e-04]
[ 3.54011093e-05 -1.02235701e-04]
[ -7.08022187e-05 2.04471402e-04]]
回顾训练方法
迭代直到误差最小：
- 正向传播，获得预测值：沿着神经网络，矩阵点乘，计算出预测值\hat y。
- 反向传播，获得每层的误差梯度：用\hat y计算误差函数，反向传播误差。
- 更新权重：根据每层的误差更新每层的权重
以预测研究生是否能入学为例 – 神经网络版本
数据处理
import numpy as np
import pandas as pd
admissions = pd.read_csv('entry_admission.csv')
# Make dummy variables for rank
data = pd.concat([admissions, pd.get_dummies(admissions['rank'], prefix='rank')], axis=1)
data = data.drop('rank', axis=1)
# Standarize features
for field in ['gre', 'gpa']:
mean, std = data[field].mean(), data[field].std()
data.loc[:,field] = (data[field]-mean)/std
# Split off random 10% of the data for testing
np.random.seed(21)
sample = np.random.choice(data.index, size=int(len(data)*0.9), replace=False)
data, test_data = data.ix[sample], data.drop(sample)
# Split into features and targets
features, targets = data.drop('admit', axis=1), data['admit']
features_test, targets_test = test_data.drop('admit', axis=1), test_data['admit']
神经网络版本
import numpy as np
np.random.seed(21)
def sigmoid(x):
"""
Calculate sigmoid
"""
return 1 / (1 + np.exp(-x))
# Hyperparameters
n_hidden = 2 # number of hidden units
epochs = 900
learnrate = 0.005
n_records, n_features = features.shape
last_loss = None
# Initialize weights
weights_input_hidden = np.random.normal(scale=1 / n_features ** .5,
size=(n_features, n_hidden))
weights_hidden_output = np.random.normal(scale=1 / n_features ** .5,
size=n_hidden)
for e in range(epochs):
del_w_input_hidden = np.zeros(weights_input_hidden.shape)
del_w_hidden_output = np.zeros(weights_hidden_output.shape)
for x, y in zip(features.values, targets):
## Forward pass ##
# TODO: Calculate the output
hidden_input = np.dot(x, weights_input_hidden)
hidden_output = sigmoid(hidden_input)
output = sigmoid(np.dot(hidden_output,
weights_hidden_output))
## Backward pass ##
# TODO: Calculate the network's prediction error
error = y - output
# TODO: Calculate error term for the output unit
output_error_term = error * output * (1 - output)
## propagate errors to hidden layer
# TODO: Calculate the hidden layer's contribution to the error
hidden_error = np.dot(output_error_term, weights_hidden_output)
# TODO: Calculate the error term for the hidden layer
hidden_error_term = hidden_error * hidden_output * (1 - hidden_output)
# TODO: Update the change in weights
del_w_hidden_output += output_error_term * hidden_output
del_w_input_hidden += hidden_error_term * x[:,None]
# TODO: Update weights
weights_input_hidden += learnrate * del_w_input_hidden / n_records
weights_hidden_output += learnrate * del_w_hidden_output / n_records
# Printing out the mean square error on the training set
if e % (epochs / 10) == 0:
hidden_output = sigmoid(np.dot(x, weights_input_hidden))
out = sigmoid(np.dot(hidden_output,
weights_hidden_output))
loss = np.mean((out - targets) ** 2)
if last_loss and last_loss < loss:
print("Train loss: ", loss, " WARNING - Loss Increasing")
else:
print("Train loss: ", loss)
last_loss = loss
# Calculate accuracy on test data
hidden = sigmoid(np.dot(features_test, weights_input_hidden))
out = sigmoid(np.dot(hidden, weights_hidden_output))
predictions = out > 0.5
accuracy = np.mean(predictions == targets_test)
print("Prediction accuracy: {:.3f}".format(accuracy))
Train loss: 0.245943442947
Train loss: 0.224108177301
Train loss: 0.228908195703 WARNING - Loss Increasing
Train loss: 0.230352461418 WARNING - Loss Increasing
Train loss: 0.230651907986 WARNING - Loss Increasing
Train loss: 0.230865845199 WARNING - Loss Increasing
Train loss: 0.231183108301 WARNING - Loss Increasing
Train loss: 0.231499116961 WARNING - Loss Increasing
Train loss: 0.231737211823 WARNING - Loss Increasing
Train loss: 0.231882889013 WARNING - Loss Increasing
Prediction accuracy: 0.750
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