Gradient Descent is an optimization algorithm widely used in statistics and machine learning to minimize a loss function by iteratively adjusting model parameters in the direction of the steepest descent of the error surface. It is especially useful when analytical solutions are infeasible or computationally expensive.

Core idea:

1. Compute the gradient (partial derivatives) of the loss function with respect to each parameter.

2. Update parameters by moving them in the opposite direction of the gradient, scaled by a learning rate.

3. Repeat until convergence.

Key Statistical Applications

• Linear Regression: Minimizes Mean Squared Error (MSE) to find the best-fit line when datasets are large or high-dimensional.

• Logistic Regression: Optimizes cross-entropy loss for binary classification, adjusting weights to improve probability predictions.

• Softmax Regression: Extends logistic regression to multiclass problems by minimizing multiclass cross-entropy.

• Support Vector Machines: Uses subgradient descent to optimize hinge loss for maximum-margin classification.

• Matrix Factorization: Learns latent factors in recommender systems by minimizing reconstruction error.