Understanding Linear Regression with Gradient Descent
Theory to Practice
Linear regression is one of the foundational algorithms in machine learning, serving as a gateway to understanding more complex models. In this post, we'll explore how linear regression works, why it's useful, and how we can optimize it using gradient descent.
What is Linear Regression?
At its core, linear regression attempts to model the relationship between variables by fitting a linear equation to observed data points. Imagine trying to predict house prices based on square footage – linear regression helps us draw that "best fit" line through our data points.
The Mathematical Foundation
The linear regression equation is:
Where:
y is the predicted value
x is the input feature
m is the slope (coefficient)
b is the y-intercept
For multiple features, this expands to:
Where:
β₀ is the bias term (y-intercept)
βᵢ are the coefficients
xᵢ are the feature values
The Cost Function
To find the best-fitting line, we need to measure how wrong our predictions are. Enter the cost function, also known as the Mean Squared Error (MSE):
Where:
m is the number of training examples
hθ(x) is our prediction
y is the actual value
θ represents our parameters (coefficients)
Gradient Descent: The Optimization Engine
Now comes the interesting part – how do we find the optimal values for our coefficients? This is where gradient descent shines.
How Gradient Descent Works
Gradient descent is an iterative optimization algorithm that:
Starts with random coefficient values
Calculates the gradient (derivative) of the cost function
Updates the coefficients in the opposite direction of the gradient
Repeats until convergence
The update rule for each coefficient is:
Where:
α is the learning rate
(∂/∂θⱼ)J(θ) is the partial derivative of the cost function with respect to θⱼ
For linear regression specifically, the update rules become:
Practical Considerations
Choosing the Learning Rate
The learning rate α is crucial:
Too large: May overshoot the minimum and fail to converge
Too small: Will take too long to converge
Just right: Usually between 0.01 and 0.1
Feature Scaling
Before applying gradient descent, it's important to scale your features to similar ranges. This prevents some features from dominating the optimization process. Common techniques include:
Standardization: (x - μ)/σ
Min-Max Scaling: (x - min)/(max - min)
Implementation Tips
When implementing linear regression with gradient descent:
Always plot your cost function over iterations to ensure it's decreasing
Use vectorized operations for efficiency
Consider using mini-batch gradient descent for large datasets
Implement early stopping when the cost function improvement is minimal
Link for the Code for the reference
Conclusion
Linear regression with gradient descent provides a solid foundation for understanding machine learning optimization. While newer techniques exist, the principles we've covered here – cost functions, iterative optimization, and the importance of hyperparameters – appear throughout modern machine learning.
Remember that despite its simplicity, linear regression remains a powerful tool in any data scientist's toolkit, especially for problems where interpretability is crucial.