Learning (With) Distributed Optimization (2308.05548v1)

Abstract: This paper provides an overview of the historical progression of distributed optimization techniques, tracing their development from early duality-based methods pioneered by Dantzig, Wolfe, and Benders in the 1960s to the emergence of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm. The initial focus on Lagrangian relaxation for convex problems and decomposition strategies led to the refinement of methods like the Alternating Direction Method of Multipliers (ADMM). The resurgence of interest in distributed optimization in the late 2000s, particularly in machine learning and imaging, demonstrated ADMM's practical efficacy and its unifying potential. This overview also highlights the emergence of the proximal center method and its applications in diverse domains. Furthermore, the paper underscores the distinctive features of ALADIN, which offers convergence guarantees for non-convex scenarios without introducing auxiliary variables, differentiating it from traditional augmentation techniques. In essence, this work encapsulates the historical trajectory of distributed optimization and underscores the promising prospects of ALADIN in addressing non-convex optimization challenges.

• The paper presents the main contribution of offering a historical review of distributed optimization, connecting early duality-based methods with contemporary ADMM and ALADIN techniques.
• It details foundational methodologies such as Lagrangian relaxation and decomposition strategies that enabled scalable solutions for complex convex problems.
• It highlights ALADIN’s unique advantage in providing convergence guarantees for non-convex problems without extra variables, underscoring its potential for advanced applications.

The paper "Learning (With) Distributed Optimization" provides a comprehensive historical review of distributed optimization techniques, tracing their evolution from the early methods developed in the 1960s to more contemporary approaches. It begins by examining the foundational duality-based methods introduced by pioneers like Dantzig, Wolfe, and Benders. These early techniques laid the groundwork by employing Lagrangian relaxation for solving convex problems and leveraging decomposition strategies to handle larger, more complex systems.

As the narrative progresses, the paper particularly emphasizes the development and refinement of the Alternating Direction Method of Multipliers (ADMM). This method became increasingly relevant in the late 2000s with its widespread application in machine learning and imaging. ADMM is noted for its practical effectiveness and its ability to unify various optimization tasks under a common framework.

The paper proceeds to discuss newer methodologies, such as the proximal center method, highlighting their diverse applications across different domains. Particular attention is given to the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm. ALADIN distinguishes itself by providing convergence guarantees even for non-convex problems and does so without the introduction of auxiliary variables, a feature that sets it apart from traditional augmentation techniques.

Overall, the paper encapsulates the historical progress of distributed optimization while spotlighting the innovative potential of ALADIN in tackling complex non-convex optimization problems. It presents a holistic view of the field, illustrating how past developments have paved the way for current and future advancements.