Introduction to Online Convex Optimization (1909.05207v3)

Abstract: This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

• The paper introduces foundational OCO concepts by defining regret and establishing gradient-based strategies with bounded regret performance.
• The paper details projection methods and advanced techniques like Online Newton Step that achieve logarithmic regret bounds for exp-concave losses.
• The paper demonstrates practical applications in portfolio selection, spam filtering, and recommendation systems, highlighting OCO’s versatile impact.

An Introduction to Online Convex Optimization

"Introduction to Online Convex Optimization" by Elad Hazan offers a comprehensive review of the rich and expanding field of Online Convex Optimization (OCO). This second edition serves as a crucial resource for both graduate students and researchers in computer science, electrical engineering, operations research, and statistics. The book introduces the theoretical and algorithmic foundations of OCO and illustrates its application across various domains, including machine learning, decision analysis, portfolio selection, and more.

Summary of Core Concepts

OCO addresses iterative decision-making where a player makes a sequence of decisions under uncertainty, with the outcomes revealed only after the decisions are made. A key metric in OCO is regret, defined as the difference between the cumulative loss of the player and that of the best fixed decision in hindsight.

Throughout the book, Hazan explores several seminal ideas and techniques that form the backbone of OCO:

1. Gradient Descent and Variants: Initially, the text covers basic gradient descent, including the Polyak stepsize, which optimizes the convergence rate independent of strong convexity and smoothness parameters. The algorithmic approach is deeply rooted in handling constrained optimization, reflecting in bounded regret performance.
2. Projections onto Convex Sets: The book explores the computational aspects of projections, an essential step in many OCO algorithms. The Pythagorean theorem's applicability to Bregman divergences provides a robust means of stabilization in decision-making algorithms.
3. Strong and Exp-Concavity: The property of exp-concavity extends regret minimization capabilities beyond strongly convex functions, encapsulated by the elegant Online Newton Step algorithm. This offers logarithmic regret bounds for an expansive class of loss functions pertinent to applications like portfolio optimization.
4. Adaptive Methods: Regularized Follow the Leader (RFTL) and Online Mirror Descent (OMD) are introduced for scenarios where stability in predictions is crucial. These methods underscore the importance of regularization in achieving low regret.

Practical Applications

The theoretical constructs in OCO have significant practical applications:

• Matrix Completion and Recommendation Systems: Hazan utilizes the Frank-Wolfe algorithm to address matrix completion, emphasizing an approach that circumvents expensive projection steps typical in other gradient-based methods.
• Portfolio Selection: Both OGD and second-order methods like Online Newton Step are explored in the context of universal portfolio selection. These techniques provide robust strategies in financial environments devoid of statistical assumptions about market behavior.
• Spam Filtering and Online Routing: Diverse applications such as online spam filtering are modeled within the OCO framework, showcasing the versatility and applicability of these algorithms across various fields.

Advanced Topics in OCO

Further chapters introduce advanced topics, such as bandit convex optimization (BCO), where feedback is partial or incomplete, necessitating sophisticated gradient estimators. The text also sheds light on adaptive regret, a nuanced metric that is vital in non-stationary environments. This metric reflects an algorithm's adaptability to varying patterns, crucial for real-time applications like network routing and dynamic trading.

Hazan also explores the intersection of OCO with game theory and linear programming, illustrating how sublinear regret algorithms for OCO naturally lead to proofs of the minimax theorem and LP duality.

The incorporation of boosting, traditionally a staple in batch learning, into the online convex optimization framework represents an overview of learning paradigms. The book culminates with the Online Newton Step algorithm, representing the pinnacle of adaptive, efficient, and powerful strategies for OCO.

Future Directions and Implications

The exploration of OCO opens avenues for future research, particularly in enhancing computational efficiency, scalability, and robustness of algorithms in ever-changing environments. The text hints at the promise of machine learning's convergence with optimization, emphasizing the need for future work in adaptive and projection-free methods.

The methodologies and algorithms elucidated in Hazan's work underscore a significant leap towards optimizing decision-making processes in uncertain environments, with broad implications for AI and machine learning. As the intersection of these fields continues to evolve, the foundational principles and advanced techniques in OCO will undoubtedly remain at the forefront of theoretical and applied research.