04. 梯度下降：代码

梯度下降：代码

之前我们看到一个权重的更新可以这样计算：

\Delta w_i = \eta \, \delta x_i

这里 error term \delta 是指

\delta = (y - \hat y) f'(h) = (y - \hat y) f'(\sum w_i x_i)

记住，上面公式中 (y - \hat y) 是输出误差，激活函数 f(h) 的导函数是f'(h) ，我们把这个导函数称做输出的梯度。

现在假设只有一个输出单元，我来把这个写成代码。我们还是用 sigmoid 来作为激活函数 f(h)。

# Defining the sigmoid function for activations 
# 定义 sigmoid 激活函数
def sigmoid(x):
    return 1/(1+np.exp(-x))

# Derivative of the sigmoid function
# 激活函数的导数
def sigmoid_prime(x):
    return sigmoid(x) * (1 - sigmoid(x))

# Input data
# 输入数据
x = np.array([0.1, 0.3])
# Target
# 目标
y = 0.2
# Input to output weights
# 输入到输出的权重
weights = np.array([-0.8, 0.5])

# The learning rate, eta in the weight step equation
# 权重更新的学习率
learnrate = 0.5

# the linear combination performed by the node (h in f(h) and f'(h))
# 输入和权重的线性组合
h = x[0]*weights[0] + x[1]*weights[1]
# or h = np.dot(x, weights)

# The neural network output (y-hat)
# 神经网络输出
nn_output = sigmoid(h)

# output error (y - y-hat)
# 输出误差
error = y - nn_output

# output gradient (f'(h))
# 输出梯度
output_grad = sigmoid_prime(h)

# error term (lowercase delta)
error_term = error * output_grad

# Gradient descent step 
# 梯度下降一步
del_w = [ learnrate * error_term * x[0],
          learnrate * error_term * x[1]]
# or del_w = learnrate * error_term * x

Start Quiz:

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
x = np.array([1, 2, 3, 4])
y = np.array(0.5)

# Initial weights
w = np.array([0.5, -0.5, 0.3, 0.1])

### Calculate one gradient descent step for each weight
### Note: Some steps have been consolidated, so there are
###       fewer variable names than in the above sample code

# TODO: Calculate the node's linear combination of inputs and weights
h = None

# TODO: Calculate output of neural network
nn_output = None

# TODO: Calculate error of neural network
error = None

# TODO: Calculate the error term
#       Remember, this requires the output gradient, which we haven't
#       specifically added a variable for.
error_term = None

# TODO: Calculate change in weights
del_w = None

print('Neural Network output:')
print(nn_output)
print('Amount of Error:')
print(error)
print('Change in Weights:')
print(del_w)
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
x = np.array([1, 2. 3, 4])
y = np.array(0.5)

# Initial weights
w = np.array([0.5, -0.5, 0.3, 0.1])

### Calculate one gradient descent step for each weight
### Note: Some steps have been consolidated, so there are
###       fewer variable names than in the above sample code

# TODO: Calculate the node's linear combination of inputs and weights
h = np.dot(x, w)

# TODO: Calculate output of neural network
nn_output = sigmoid(h)

# TODO: Calculate error of neural network
error = y - nn_output

# 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)

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