arXiv:2505.19183v4 Announce Type: replace-cross Abstract: This book offers a hands-on introduction to building and understanding federated learning (FL) systems. FL enables multiple devices -- such as smartphones, sensors, or local computers -- to collaboratively train machine learning (ML) models, while keeping their data private and local. It is a powerful solution when data cannot or should not be centralized due to privacy, regulatory, or technical reasons. The book is designed for students, engineers, and researchers who want to learn how to design scalable, privacy preserving FL systems. Our main focus is on personalization: enabling each device to train its own model while still benefiting from collaboration with relevant devices. This is achieved by leveraging similarities between (the learning tasks associated with) devices that are encoded by the weighted edges (or links) of a federated learning network (FL network). The key idea is to represent real-world FL systems as networks of devices, where nodes correspond to device and edges represent communication links and data similarities between them. The training of personalized models for these devices can be naturally framed as a distributed optimization problem. This optimization problem is referred to as generalized total variation minimization (GTVMin) and ensures that devices with similar learning tasks learn similar model parameters. Our approach is both mathematically principled and practically motivated. While we introduce some advanced ideas from optimization theory and graph-based learning, we aim to keep the book accessible. Readers are guided through the core ideas step by step, with intuitive explanations.