Understanding Gradient Descent in Machine Learning

Gradient Descent is a fundamental optimization algorithm in machine learning. It's used to minimize the cost function in various ML models.

The Basics

In gradient descent, we update our parameters $\theta$ in the opposite direction of the gradient of the cost function $J(\theta)$:

$$ \theta = \theta - \alpha \nabla J(\theta) $$

Where: - $\theta$ represents the model parameters - $\alpha$ is the learning rate - $\nabla J(\theta)$ is the gradient of the cost function

Example: Linear Regression

For a simple linear regression model $y = mx + b$, our update rules would be:

$$ m = m - \alpha \frac{\partial}{\partial m} J(m,b) $$ $$ b = b - \alpha \frac{\partial}{\partial b} J(m,b) $$

Where $J(m,b)$ is typically the mean squared error:

$$ J(m,b) = \frac{1}{2n} \sum_{i=1}^n (y_i - (mx_i + b))^2 $$

By iteratively applying these update rules, we can find the optimal values for $m$ and $b$ that minimize our cost function.