Gradient Descent is a commonly used optimization algorithm for solving parameters that minimize an objective function. It moves along the opposite direction of the gradient of the objective function by iteratively updating parameters to gradually approach the optimal solution.
The basic steps of the gradient descent method:
Initialization parameters: A set of initial parameters is selected as a starting point for optimization.
Calculate gradient: Calculate the gradient of the objective function with respect to the parameters, that is, the rate of change of the objective function at the current parameter value.
Update parameters: Update the value of the parameters according to the direction of the gradient and the learning rate. The learning rate determines the step size for each update of the parameters.
Repeat iterations : Repeat step 2 and step 3 until the stopping condition is met, such as reaching the specified number of iterations or the change in gradient is small.
The implementation of the gradient descent method can use the Python programming language. The following is a simple sample code to demonstrate the basic implementation of the gradient descent method:
python
import numpy as np
def gradient_descent(X, y, learning_rate=0.01, num_iterations=100):
num_samples, num_features = X.shape
theta = np.zeros(num_features) # initialization parameter is 0 vector
for i in range(num_iterations):
# Compute predicted value and error
y_pred = np.dot(X, theta)
error = y_pred - y
# Compute gradient and update parameters
gradient = np.dot(X.T, error) / num_samples
theta -= learning_rate * gradient
return theta
# Example usage
X = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) # feature matrix
y = np.array([10, 20, 30]) # target value
# execute gradient descent
theta = gradient_descent(X, y, learning_rate=0.1, num_iterations=100)
print(theta) # output the optimal parameter value
In the above example code, we define a gradient_descent function that accepts feature matrix X, target value y, learning rate learning_rate and iteration number num_iterations as parameters. The function internally uses the gradient descent method to update the parameters until the specified number of iterations is reached.
In practical applications, the gradient descent method can be used to solve various machine learning tasks such as regression problems and classification problems. It should be noted that the performance of the gradient descent method is affected by the learning rate. If the learning rate is too large, the parameter update may be too large to converge; if the learning rate is too small, the convergence speed may be too slow. Therefore, the choice of an appropriate learning rate is very important.