我正在跟随 Andrew Ng 在 Coursera 上的机器学习课程，并尝试在 Python 中实现梯度下降算法。但是，我在 y 轴截距参数上遇到了麻烦，因为它似乎无法达到最佳值。以下是我的代码：
# IMPORTS
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
%matplotlib inline
# Acquiring Data
# Source: https://github.com/mattnedrich/GradientDescentExample
data = pd.read_csv('data.csv')
def cost_function(a, b, x_values, y_values):
'''
Calculates the square mean error for a given dataset
with (x,y) pairs and the model y' = a + bx
a: y-intercept for the model
b: slope of the curve
x_values, y_values: points (x,y) of the dataset
'''
data_len = len(x_values)
total_error = sum([((a + b * x_values[i]) - y_values[i])**2
for i in range(data_len)])
return total_error / (2 * float(data_len))
def a_gradient(a, b, x_values, y_values):
'''
Partial derivative of the cost_function with respect to 'a'
a, b: values for 'a' and 'b'
x_values, y_values: points (x,y) of the dataset
'''
data_len = len(x_values)
a_gradient = sum([((a + b * x_values[i]) - y_values[i])
for i in range(data_len)])
return a_gradient / float(data_len)
def b_gradient(a, b, x_values, y_values):
'''
Partial derivative of the cost_function with respect to 'b'
a, b: values for 'a' and 'b'
x_values, y_values: points (x,y) of the dataset
'''
data_len = len(x_values)
b_gradient = sum([(((a + b * x_values[i]) - y_values[i]) * x_values[i])
for i in range(data_len)])
return b_gradient / float(data_len)
def gradient_descent_step(a_current, b_current, x_values, y_values, alpha):
'''
Give a step in direction of the minimum of the cost_function using
the 'a' and 'b' gradiants. Return new values for 'a' and 'b'.
a_current, b_current: the current values for 'a' and 'b'
x_values, y_values: points (x,y) of the dataset
'''
new_a = a_current - alpha * a_gradient(a_current, b_current, x_values, y_values)
new_b = b_current - alpha * b_gradient(a_current, b_current, x_values, y_values)
return (new_a, new_b)
def run_gradient_descent(a, b, x_values, y_values, alpha, precision, plot=False, verbose=False):
'''
Runs the gradient_descent_step function and updates (a,b) until
the value of the cost function varies less than 'precision'.
a, b: initial values for the point a and b in the cost_function
x_values, y_values: points (x,y) of the dataset
alpha: learning rate for the algorithm
precision: value for the algorithm to stop calculation
'''
iterations = 0
delta_cost = cost_function(a, b, x_values, y_values)
error_list = [delta_cost]
iteration_list = [0]
# The loop runs until the delta_cost reaches the precision defined
# When the variation in cost_function is small it means that the
# the function is near its minimum and the parameters 'a' and 'b'
# are a good guess for modeling the dataset.
while delta_cost > precision:
iterations += 1
iteration_list.append(iterations)
# Calculates the initial error with current a,b values
prev_cost = cost_function(a, b, x_values, y_values)
# Calculates new values for a and b
a, b = gradient_descent_step(a, b, x_values, y_values, alpha)
# Updates the value of the error
actual_cost = cost_function(a, b, x_values, y_values)
error_list.append(actual_cost)
# Calculates the difference between previous and actual error values.
delta_cost = prev_cost - actual_cost
# Plot the error in each iteration to see how it decreases
# and some information about our final results
if plot:
plt.plot(iteration_list, error_list, '-')
plt.title('Error Minimization')
plt.xlabel('Iteration',fontsize=12)
plt.ylabel('Error',fontsize=12)
plt.show()
if verbose:
print('Iterations = ' + str(iterations))
print('Cost Function Value = '+ str(cost_function(a, b, x_values, y_values)))
print('a = ' + str(a) + ' and b = ' + str(b))
return (actual_cost, a, b)
当我使用以下参数运行算法时：
run_gradient_descent(0, 0, data['x'], data['y'], 0.0001, 0.01)
我得到了（a = 0.0496688656535，b = 1.47825808018）。
但是，在线性回归的其他资源中尝试后，最佳'a'值约为7.9。
此外，如果我改变参数'a'的初始猜测，算法会尝试调整参数'b'。
例如，如果我设置a = 200和b = 0。
run_gradient_descent(200, 0, data['x'], data['y'], 0.0001, 0.01)
我得到了（a = 199.933763331，b = -2.44824996193）
我没有发现代码有任何问题，我意识到问题在于参数a的初始猜测值。请参见上面我的回答，我定义了一个辅助函数来获取搜索一些值以获得初始a猜测的范围。