Gradient Descent for Linear Regression - Machine Learning Insights

Overview

This repository contains a deep dive into some fundamental concepts of Machine Learning, with a specific focus on Linear Regression using Gradient Descent. The goal is to not only implement the algorithm but also provide insights into how the cost function is derived and optimized.

Additionally, visualizations generated using Manim are included to help illustrate key ideas around gradient descent and how the line of best fit is iteratively adjusted during the training process.

Table of Contents

1. Machine Learning Overview
2. Cost Function
3. Gradient Descent Algorithm
4. Visualization Using Manim
5. Python Implementation
6. Installation and Usage

Machine Learning Overview

Machine Learning models, especially in supervised learning, aim to predict outcomes based on input data. Linear Regression is one of the most fundamental algorithms used for prediction, where the relationship between the input features and the output is modeled as a straight line.

This project covers:

• Cost functions: How we quantify the error between our model's predictions and the true values.
• Gradient Descent: An optimization algorithm used to minimize the cost function and adjust the model's parameters for better predictions.

Cost Function

In a linear regression problem, we use the Mean Squared Error (MSE) as the cost function. The formula for the cost function is:

[ J(\theta) = \frac{1}{2m} \sum_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)})^2 ]

Where:

• ( h_\theta(x) ) is the predicted value (hypothesis).
• ( y^{(i)} ) is the actual target value.
• ( m ) is the number of data points.

The cost function helps us determine how well our model's predictions match the actual outcomes.

Why Square Errors?

We square the errors for the following reasons:

1. Avoid negative values: If we just summed the raw errors, positive and negative errors could cancel each other out.
2. Differentiability: Squaring creates a smooth curve, which is easier to minimize using calculus methods like gradient descent.

Gradient Descent Algorithm

The goal of the Gradient Descent algorithm is to find the values of the model parameters ( \theta ) that minimize the cost function ( J(\theta) ). The update rule for the parameters is:

[ \theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta) ]

Where:

• ( \alpha ) is the learning rate, which controls the size of the steps we take toward the minimum.
• ( \frac{\partial}{\partial \theta_j} J(\theta) ) is the partial derivative of the cost function with respect to ( \theta_j ), guiding the direction in which to move.

We adjust each parameter based on how much it influences the error and repeat the process until convergence.

Visualization Using Manim

The included visualizations built using Manim help explain:

• The geometry of linear regression (line of best fit).
• How the cost function behaves as the parameters change.
• The iterative process of gradient descent.

These animations provide an intuitive understanding of the mathematics behind the scenes, making it easier to grasp how the parameters are optimized.

Python Implementation

In the gradient_descent.py file, we implement a simple version of gradient descent to perform linear regression on a dataset. The steps involved are:

1. Hypothesis: Compute the predicted values for the given input.
2. Cost Function: Measure the error between predicted and actual values.
3. Gradient Descent: Iteratively adjust parameters to minimize the cost function.

You can run the implementation, visualize the convergence of the algorithm, and plot the final regression line.

Installation and Usage

Prerequisites

• Python 3.x
• numpy
• matplotlib
• manim (for visualizations)

Clone the Repository

git clone https://github.com/your-username/gradient-descent-linear-regression.git
cd gradient-descent-linear-regression