Having established the foundations of ensemble methods and additive modeling, this chapter focuses on the mechanics of the standard Gradient Boosting Machine (GBM) algorithm. We move from the general concept to a specific, step-by-step understanding of how these models are constructed.

You will learn to view the boosting process through the lens of optimization, specifically as gradient descent performed in function space. We will formally derive the generic GBM algorithm, clarifying how residuals or pseudo-residuals guide the training of each sequential base learner (typically a decision tree).

Key aspects covered include:

• The mathematical derivation of the GBM algorithm.
• Analysis of common loss functions, such as Squared Error for regression or Log Loss for classification, and how their gradients ($\nabla L$) drive the model fitting process.
• The function of shrinkage (the learning rate parameter) in controlling the contribution of each tree and aiding regularization.
• Techniques like row and column subsampling (Stochastic Gradient Boosting) to improve model generalization and computational efficiency.
• Practical implementation using Scikit-learn's GradientBoostingRegressor and GradientBoostingClassifier.

By the end of this chapter, you will have a detailed operational understanding of the classic GBM algorithm, preparing you for the more advanced implementations discussed later. A hands-on section will guide you through building your first GBM model using Python.