Machine Learning || Gradient Descent for Linear Regression

Introduction

In this tutorial, we will explore how to apply gradient descent to solve linear regression problems. Gradient descent is a fundamental optimization technique used in machine learning, particularly for minimizing the cost function in linear regression. This guide will walk you through the essential concepts and calculations needed to implement gradient descent effectively.

Step 1: Understand Linear Regression

• Definition: Linear regression aims to model the relationship between a dependent variable (y) and one or more independent variables (X) by fitting a linear equation.
• Equation: The basic formula for linear regression is: [ y = \theta_0 + \theta_1 X_1 + \theta_2 X_2 + ... + \theta_n X_n ] where ( \theta ) represents the parameters (weights) to be optimized.

Step 2: Define the Cost Function

• Purpose: The cost function measures how well the model's predictions match the actual data. The most commonly used cost function for linear regression is the Mean Squared Error (MSE).
• Formula: The MSE is calculated as: [ J(\theta) = \frac{1}{m} \sum_{i=1}^{m} (h_\theta(X_i) - y_i)^2 ] where ( m ) is the number of training examples, ( h_\theta(X_i) ) is the predicted value, and ( y_i ) is the actual value.

Step 3: Compute the Gradient

• Gradient Description: The gradient indicates the direction and rate of change of the cost function with respect to the parameters.
• Formula: The gradient for linear regression is computed as: [ \nabla J(\theta) = \frac{1}{m} \sum_{i=1}^{m} (h_\theta(X_i) - y_i) X_i ] Here, ( X_i ) represents the input features.

Step 4: Update the Parameters

• Learning Rate: The learning rate (( \alpha )) controls how much we adjust the parameters during each iteration.
• Update Rule: The parameters are updated using the formula: [ \theta_j = \theta_j - \alpha \cdot \nabla J(\theta) ] Repeat this process for all parameters ( j ).

Step 5: Implement Gradient Descent

• Initialization: Start with initial values for the parameters, often set to zero.
• Loop: Repeat the following steps until convergence (when the cost function does not significantly change):
  1. Compute the predicted values using the current parameters.
  2. Calculate the cost function (MSE).
  3. Compute the gradient.
  4. Update the parameters.

Example Code Snippet

Here’s a basic implementation of gradient descent for linear regression in Python:

import numpy as np

def gradient_descent(X, y, alpha, num_iterations):
    m = len(y)
    theta = np.zeros(X.shape[1])
    
    for _ in range(num_iterations):
        predictions = X.dot(theta)
        errors = predictions - y
        gradient = (1/m) * (X.T.dot(errors))
        theta -= alpha * gradient
        
    return theta

Conclusion

We have covered the essential steps to apply gradient descent for linear regression. Key takeaways include understanding linear regression, defining the cost function, computing the gradient, and updating parameters iteratively. To further your learning, consider experimenting with different datasets and adjusting the learning rate to see its effect on convergence. Exploring regularization techniques can also enhance your model's performance.