In-depth exploration of Python's underlying technology: how to implement the gradient descent algorithm

In-depth exploration of the underlying technology of Python: how to implement the gradient descent algorithm, specific code examples are required

Introduction:
The gradient descent algorithm is a commonly used optimization algorithm. It is widely used in the fields of machine learning and deep learning. This article will delve into the underlying technology of Python, introduce the principle and implementation process of the gradient descent algorithm in detail, and provide specific code examples.

1. Introduction to Gradient Descent Algorithm
The gradient descent algorithm is an optimization algorithm. Its core idea is to gradually approach the minimum value of the loss function by iteratively updating parameters. Specifically, the steps of the gradient descent algorithm are as follows:

1. Randomly initialize parameters.
2. Calculate the gradient of the loss function to the parameters.
3. Update parameters based on the direction of the gradient and the learning rate.
4. Repeat steps 2 and 3 until the condition for the algorithm to stop is reached.

2. Implementation process of gradient descent algorithm
In Python, we can implement the gradient descent algorithm through the following steps.

1. Preparing data
   First, we need to prepare the data set, including input features and target values. Assuming there are m samples and n features, we can represent the input features as an m×n matrix X, and the target value as a vector y of length m.
2. Initialization parameters
   We need to initialize the parameters of the model, including weight w and bias b. In general, the weight w can be set to a vector of dimension n, and the bias b can be initialized to a scalar.

3. Calculate the loss function
   We need to define a loss function to evaluate the performance of the model. In the gradient descent algorithm, the commonly used loss function is the squared error loss function, which is defined as follows:

   def loss_function(X, y, w, b):
    m = len(y)
    y_pred = np.dot(X, w) + b
    loss = (1/(2*m))*np.sum((y_pred - y)**2)
    return loss

4. Calculating the gradient
   Next, we need to calculate the effect of the loss function on the weight w and bias Set the gradient of b. The gradient represents the fastest decreasing direction of the objective function at a certain point. For the squared error loss function, the gradient calculation formula is as follows:

   def gradient(X, y, w, b):
    m = len(y)
    y_pred = np.dot(X, w) + b
    dw = (1/m)*np.dot(X.T, (y_pred - y))
    db = (1/m)*np.sum(y_pred - y)
    return dw, db

5. Update parameters
   According to the direction of the gradient and the learning rate alpha, we can update the parameters so that they move towards the loss function Minimize directional movement.

   def update_parameters(w, b, dw, db, learning_rate):
    w = w - learning_rate * dw
    b = b - learning_rate * db
    return w, b

6. Iteratively update parameters
   By repeating steps 4 and 5 until the condition for the algorithm to stop is reached. The condition for the algorithm to stop can be that the maximum number of iterations is reached, or the change in the loss function is less than a certain threshold.

7. Full code example
   The following is a complete code example that implements the gradient descent algorithm.

   import numpy as np
   
   def gradient_descent(X, y, learning_rate, num_iterations):
    m, n = X.shape
    w = np.random.randn(n)
    b = 0
    
    for i in range(num_iterations):
        loss = loss_function(X, y, w, b)
        dw, db = gradient(X, y, w, b)
        w, b = update_parameters(w, b, dw, db, learning_rate)
        
        if i % 100 == 0:
            print(f"Iteration {i}: loss = {loss}")
    
    return w, b
    
   # 测试代码
   X = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])  # 输入特征矩阵
   y = np.array([4, 7, 10])  # 目标值
   learning_rate = 0.01  # 学习率
   num_iterations = 1000  # 迭代次数
   
   w, b = gradient_descent(X, y, learning_rate, num_iterations)
   
   print(f"Optimized parameters: w = {w}, b = {b}")

Conclusion:
This article deeply explores the underlying technology of Python and introduces the principle and implementation process of the gradient descent algorithm in detail. Through specific code examples, readers can more intuitively understand the implementation details of the gradient descent algorithm. Gradient descent algorithm is an indispensable optimization algorithm in the fields of machine learning and deep learning, and is of great significance for solving practical problems. I hope this article can be helpful to readers and trigger more thinking and discussion about Python's underlying technology.

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