章节5：DIY神经网络
神经网络是本教程最主要的内容，理解神经网络的工作原理十分有必要。
本文我们将介绍和实现一个具有有两层结构和sigmoid激活函数的简单的神经网络，我们将从上一章节提到的线性回归的例子的基础上开始构建。
线性回归的回顾
假如我们有一组点的集合(X,Y)，我们要找到一条直线y=mx+b来拟合这些数据点。我们期望找到参数m和b，使得误差平方和J最小。
神经网络
我们试着把线性回归问题改造成一个多层的神经网络。首先，我们增加一个输入变量，这样我们的输入就变成二维的了，回顾下我们之前的Iris数据集，将问题转化为：给定花萼长度和宽度，预测花瓣长度。让我们先导入数据集，
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import load_iris
iris = load_iris()
data, labels = iris.data[:,0:2], iris.data[:,2]
num_samples = len(labels) # size of our dataset
# shuffle the dataset
shuffle_order = np.random.permutation(num_samples)
data = data[shuffle_order, :]
labels = labels[shuffle_order]
def weighted_sum(x, w, b):
return b + np.dot(w, x)
# set our paramters
w = [0.2, 0.6]
b = -0.3
# for example, let's use the first data point
X, y = data, labels
pred_y = [weighted_sum(x, w, b) for x in X]
# let's print out the first prediction
print("for x=[%0.2f, %0.2f], predicted = %0.2f, actual = %0.2f" % (X[0][0], X[0][1], pred_y[0], y[0]))
for x=[4.70, 3.20], predicted = 2.56, actual = 1.60
使用误差平方和函数评估我们预测的效果，
# sum squared error
def cost(y_pred, y_actual):
return 0.5 * np.sum((y_actual-y_pred)**2)
error = cost(pred_y, y)
print(error)
313.0096
在本问题中，我们不能将损失函数的损失平面画出来了，因为我们的函数方程中有3个参数(w1,w2,b)，所以如果要画出损失平面，则是4维的。但我们可以像上章节中的方法一样，使用微积分计算代价函数针对所有参数的偏导数，来优化我们的3个参数。
代码实现如下，
import random
# grab our data
X, y = data, labels
# always a good idea to normalize
X = X / np.amax(X, axis=0)
y = y / np.amax(y, axis=0)
# choose a random initial m, b
w, b = [random.random(), random.random()], random.random()
# our function w1 * x1 + w2 * x2 + b
def F(X, w, b):
return np.sum(w*X, axis=1) + b
# what is our error?
y_pred = F(X, w, b)
init_cost = cost(y_pred, y)
print("initial parameters: w1=%0.3f, w2=%0.3f, b=%0.3f"%(w[0], w[1], b))
print("initial cost = %0.3f" % init_cost)
# implement partial derivatives of our parameters
def dJdw1(X, y, w, b):
return -np.dot(X[:,0], y - F(X, w, b))
def dJdw2(X, y, w, b):
return -np.dot(X[:,1], y - F(X, w, b))
def dJdb(X, y, w, b):
return -np.sum(y - F(X, w, b))
# choose the alpha parameter and number of iterations
alpha = 0.001
n_iters = 2000
# run through gradient descent
errors = []
for i in range(n_iters):
w[0] = w[0] - alpha * dJdw1(X, y, w, b)
w[1] = w[1] - alpha * dJdw2(X, y, w, b)
b = b - alpha * dJdb(X, y, w, b)
y_pred = F(X, w, b)
j = cost(y_pred, y)
errors.append(j)
# plot the error
plt.figure(figsize=(16, 3))
plt.plot(range(n_iters), errors, linewidth=2)
plt.title("Cost by iteration")
plt.ylabel("Cost")
plt.xlabel("iterations")
# what is our final error rate
y_pred = F(X, w, b)
final_cost = cost(y_pred, y)
print("final parameters: w1=%0.3f, w2=%0.3f, b=%0.3f"%(w[0], w[1], b))
print("final cost = %0.3f" % final_cost)
initial parameters: w1=0.080, w2=0.220, b=0.259
initial cost = 5.384
final parameters: w1=1.902, w2=-0.923, b=-0.221
final cost = 0.665
我们可以画出我们预测的函数f(X)，输入变量为[x1,x2]，如下：
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.gca(projection='3d')
x1, x2 = np.meshgrid(np.arange(-10, 10, 1), np.arange(-10, 10, 1))
y = b + w[0]*x1 + w[1]*x2
ax.plot_surface(x1, x2, y, rstride=1, cstride=1, cmap=plt.cm.coolwarm, linewidth=0, antialiased=False)
我们来看下Sigmoid函数和它的函数图像，
def sigmoid(z):
return 1.0 / (1.0 + np.exp(-z))
x = np.arange(-10.0, 10.0, 0.2)
sig = sigmoid(x)
plt.plot(x, sig)
plt.title('Sigmoid function')
def weighted_sum(x, w, b):
return b + np.dot(w, x)
def sigmoid(z):
return 1 / (1 + np.exp(-z))
# reset our parameters
w = [0.2, 0.6]
b = -0.3
X, y = data, labels
# get weighted sum like before
Z = [weighted_sum(x, w, b) for x in X]
# now transform the weighted sums with a sigmoid
y_pred = [sigmoid(z) for z in Z]
# evaluate error
error = cost(y_pred, y)
print(error)
829.4187739184515
这结果看起来很糟糕，但我们还没进行优化！
在上述所给的权重值下，我们可以画出我们的神经元活动函数：
fig = plt.figure()
ax = fig.gca(projection='3d')
x1, x2 = np.meshgrid(np.arange(-10, 10, 1), np.arange(-10, 10, 1))
y = sigmoid(b + w[0]*x1 + w[1]*x2)
ax.plot_surface(x1, x2, y, rstride=1, cstride=1, cmap=plt.cm.coolwarm, linewidth=0, antialiased=False)
同之前的线性回归的神经网络比较，我们在该神经网络增加了两个变化，1）添加了对每个神经元激活函数的非线性sigmoid处理（除了输出神经元）；2）添加了中间的隐藏层。变化后的神经网络比之前更“灵活”，根据参数值的不同，它的函数图像可以各种弯曲变化，但在这网络上进行梯度下降也会变得更复杂。
代码实现如下，我们有两层权重值，分别为W1，W2。W1有2x3个权重系数，W2有3x1个权重系数，
W1 = np.random.randn(2, 3)
W2 = np.random.randn(3, 1)
print("W1=", W1)
print("W2=", W2)
W1= [[ 0.55604015 -1.30425457 -1.51214943]
[-0.06828511 -0.63519787 -0.59453199]]
W2= [[-0.09451191]
[-2.21387541]
[-0.4512179 ]]
为了简单起见，我们把偏差项b先初始化为0。从输入层到输出层传递的计算
叫做“前向传播”，我们先实现“前向传播”：
X, y = data, labels
# first layer weighted sum z
z = np.dot(X, W1)
# project z through non-linear sigmoid
z = sigmoid(z)
# do another dot product at end (sigmoid is omitted)
y_pred = np.dot(z, W2)
# what is our cost
error = cost(y_pred, y)
print('predicted %0.2f for example 0, actual %0.2f, total cost %0.2f'%(pred_y[0], y[0], error))
predicted 2.56 for example 0, actual 1.60, total cost 201434.60
我们把这些操作整合成一个类：
class Neural_Network(object):
def __init__(self, n0, n1, n2):
self.n0 = n0
self.n1 = n1
self.n2 = n2
# initialize weights
self.W1 = np.random.randn(self.n0, self.n1)
self.W2 = np.random.randn(self.n1 ,self.n2)
def predict(self, x):
z = np.dot(x, self.W1)
z = sigmoid(z)
y = np.dot(z, self.W2)
return y
实例化一个有2个输入神经元，2个隐藏神经元和1个输出神经元的神经网络。
net = Neural_Network(2, 3, 1)
调用predict函数就能执行神经网络的前向传播：
X, y = data, labels
y_pred = net.predict(X)
error = cost(y_pred, y)
print('predicted %0.2f for example 0, actual %0.2f, total cost %0.2f'%(pred_y[0], y[0], error))
predicted 2.56 for example 0, actual 1.60, total cost 211597.43
梯度下降
回顾下，我们有一个2x3x1的神经网络，有9个权重系数和4个偏差系数，共13个参数。那怎样优化这些参数使得代价函数最小呢？
通过上一章线性回归的例子，我们知道可以通过梯度下降的方法来解决该问题。实际上，对于神经网络我们现在只能通过梯度下降来解决了。因为不像线性回归，除了梯度下降方法还可以用分析法。而神经网络是非线性的，所以分析法已经不适用了。
梯度下降的目标就是寻找损失函数J对于每个参数w（此处的w包含偏差系数b）的梯度。
import itertools
def get_gradient(net, X, y):
w_delta = 1e-8
# get the current value of the loss, wherever the parameters are
y_pred_current = net.predict(X)
error_current = cost(y_pred_current, y)
# grab the current weights and copy them (so we can restore them after modification)
dw1, dw2 = np.zeros((net.n0, net.n1)), np.zeros((net.n1, net.n2))
W1, W2 = np.copy(net.W1), np.copy(net.W2)
# for the first layer, iterate through each weight,
# perturb it slightly, and calculate the numerical
# slope between that loss and the original loss
for i, j in itertools.product(range(net.n0), range(net.n1)):
net.W1 = np.copy(W1)
net.W1[i][j] += w_delta
y_pred = net.predict(X)
error = cost(y_pred, y)
dw1[i][j] = (error - error_current) / w_delta
# do the same thing for the second layer
for i, j in itertools.product(range(net.n1), range(net.n2)):
net.W2 = np.copy(W2)
net.W2[i][j] += w_delta
y_pred = net.predict(X)
error = cost(y_pred, y)
dw2[i][j] = (error - error_current) / w_delta
# restore the original weights
net.W1, net.W2 = np.copy(W1), np.copy(W2)
return dw1, dw2
上述代码中的get_gradient计算在数据集X，y下的2个网络层之间的梯度，接下来，我们将演示网络中的学习过程，首先，导入数据集，实例化网络，然后用梯度下降方法训练数据集，
# load the data and labels
X, y = data, labels.reshape((len(labels),1))
# it's always a good idea to normalize the data between 0 and 1
X = X/np.amax(X, axis=0)
y = y/np.amax(y, axis=0)
# create a 2x3x1 neural net
net = Neural_Network(2, 3, 1)
# what is the current cost?
y_orig = net.predict(X)
init_cost = cost(y_orig, y)
print("initial cost = %0.3f" % init_cost)
# Set the learning rate, and how many epochs (updates) to try
n_epochs = 2000
learning_rate = 0.01
# for each epoch, calculate the gradient, then subtract it from the parameters, and save the cost
errors = []
for i in range(n_epochs):
dw1, dw2 = get_gradient(net, X, y)
net.W1 = net.W1 - learning_rate * dw1
net.W2 = net.W2 - learning_rate * dw2
y_pred = net.predict(X)
error = cost(y_pred, y)
errors.append(error)
# plot it
plt.plot(range(0, len(errors)), errors, linewidth=2)
# what is the final cost?
y_pred = net.predict(X)
final_cost = cost(y_pred, y)
print("final cost = %0.3f" % final_cost)
initial cost = 85.579
final cost = 0.562
至此，我们使用数值梯度下降方法实现了一个神经网络。在下一章节中，我们将学习Keras，一个高级的机器学习库，并在接下来的教程中会一直使用它。
