Introduction

In this second topic of the course, the focus shifts from a mainly theoretical perspective to one oriented toward implementation. The objective is for students to acquire the ability to build, understand, and experiment with machine learning models using specialized libraries, as well as to establish the mathematical foundations that support them.

Throughout the topic, different sections and content are introduced, some of which may not be covered in the main course presentation. It is recommended to review all available sections and materials, including those marked as optional or additional. When there are doubts about the status or update of certain content, it is advisable to consult the instructors, who can provide guidance on its validity and relevance. Exploring these complementary materials allows discovering new methods and approaches, and constitutes a good opportunity to expand knowledge beyond the mandatory minimums.

In this block, the fundamental concepts of machine learning closely related to mathematics and their implementation in code are worked on with special intensity. The starting point is the data structures previously seen from a theoretical standpoint, moving to their practical application through Jupyter notebooks using deep learning libraries, with special emphasis on PyTorch. The discussion covers how to define and manipulate tensors and matrices, as well as how to perform basic and advanced operations between them, understanding their mathematical and computational interpretation.

Additionally, essential mathematical concepts for machine learning are introduced and consolidated. Among them, gradients and automatic differentiation are studied, a central feature of PyTorch and other deep learning libraries that allows calculating derivatives efficiently and accurately during model training. Classic models such as linear regression and logistic regression are also addressed, analyzing their mathematical formulation, their interpretation, and the role they play as the foundation of many modern machine learning algorithms.

The topic also includes the study of other mathematical tools useful in practice, such as cosine similarity, the Hadamard product, and other related matrix operations. These operations are especially relevant in various machine learning and deep learning algorithms and architectures — for example, in recommendation models, natural language processing, or deep neural networks — where the relationships and combinations between vectors and matrices constitute the core of computation.

From these foundations, the gradient descent method and its variants are presented, explaining how it is used to adjust a model's parameters to the available data. Subsequently, the discussion shows how to combine the basic elements involved in learning a single neuron or perceptron to build complete layers and, from these, multi-layer neural networks (multilayer perceptrons). This step is key to understanding the transition from simple linear models to what is now known as deep learning.

Finally, regularization techniques aimed at avoiding overfitting are introduced — that is, the excessive adaptation of model parameters to training data. Different types of optimizers designed to accelerate and stabilize the convergence process during training are studied, as well as the most common metrics for evaluating model performance. In this way, the topic offers an integrated vision that connects mathematical foundations, data structures, training strategies, and evaluation, providing a solid framework for the development and understanding of deep learning systems.