一小时培训之神经网络入门

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

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：

其中f表示激活函数，通常用

sigmoid值在0-1之间的数值很像概率适合做分类

如何找出这样的函数？

方法：

用监督学习的方式训练：
- 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）:

其中：

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)

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

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

求误差函数E对于权重w的偏导的求法：目标：

链式法则：

再用一次链式法则：

带入后，再用一次链式法则：

最后，做替换：

最终结果：

定义：

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

读取原始数据

import numpy as np
import pandas as pd
admissions=pd.read_csv("entry_admission.csv")
admissions.head()

	admit	gre	gpa	rank
0	0	380	3.61	3
1	1	660	3.67	3
2	1	800	4.00	1
3	1	640	3.19	4
4	0	520	2.93	4

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()

	gre	gpa	rank_2	rank_3	rank_4
209	-0.066657	0.289305	1	0	0
280	0.625884	1.445476	1	0	0
33	1.837832	1.603135	0	1	0
210	1.318426	-0.131120	0	0	1
93	-0.066657	-1.208461	1	0	0

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。

数学形式：矩阵相乘，每一隐含层是一个矩阵 ~~（图待加上偏差）~~

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

公式：第j层的误差：

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：

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