Understanding Gradient Descent for Linear Regression
Introduction
Gradient Descent is a fundamental optimization algorithm used in machine learning and statistics, particularly in training models like linear regression. In this article, we will delve into the intricacies of Gradient Descent concerning linear regression, providing both mathematical formulations and graphical explanations to facilitate comprehension.
Linear Regression Overview
Linear regression is a simple yet powerful technique used for modelling the relationship between a dependent variable (target) and one or more independent variables (features). In its simplest form, known as simple linear regression, there is only one independent variable. The relationship between the independent and dependent variables is assumed to be linear, hence the name.
The general form of a linear regression model is:
Objective of Linear Regression
The objective of linear regression is to find the optimal values for the coefficients β0,β1,…,βn that minimize the error between the predicted values and the actual values of the dependent variable. This error is typically quantified using a loss function, such as the mean squared error (MSE) or mean absolute error (MAE).
Gradient Descent
Gradient Descent is an iterative optimization algorithm used to minimize a cost function by iteratively moving in the direction of the steepest descent. In the context of linear regression, the cost function represents the error between the predicted and actual values of the dependent variable.
Mathematical Formulation of Gradient Descent
Let’s denote the cost function for linear regression as J(β), where β represents the vector of coefficients β0,β1,…,βn. The objective is to minimize this cost function.
The gradient of the cost function with respect to the parameters β is computed as:
Graphical Explanation
Imagine a landscape where the x-axis represents the parameters (coefficients) and the y-axis represents the cost function J(β). The objective is to find the lowest point (global minimum) on this landscape, which corresponds to the optimal values of the parameters that minimize the cost function.
Gradient Descent starts at a random point on this landscape and iteratively descends towards the global minimum by following the direction of the steepest descent, which is opposite to the direction of the gradient.
As the algorithm progresses, it takes steps in the direction that minimizes the cost function until it converges to a point where the gradient approaches zero or the change in the cost function becomes negligible.
Conclusion
Gradient Descent is a powerful optimization algorithm widely used in training linear regression models and other machine learning algorithms. By iteratively updating the parameters in the direction of the steepest descent, it efficiently minimizes the cost function, leading to optimal model parameters. Understanding the mathematical formulation and graphical intuition behind Gradient Descent is crucial for mastering its application in linear regression and beyond.
