Zhou Lu will present his FPO "Online Learning: Optimization, Control, and Learning Theory" on Monday, June 16, 2025 at 1:00 PM in Friend 005. The members of Zhou's committee are as follows: Examiners: Elad Hazan (Adviser), Chi Jin, Ryan Adams Readers: Matt Weinberg, David Woodruff (CMU) A copy of his thesis is available upon request. Please email gradinfo@cs.princeton.edu if you would like a copy of thethesis. Everyone is invited to attend his talk. Abstract follows below: Online learning is a foundational paradigm in learning theory, addressing the challenge of making predictions from sequential data. Initially introduced as a mathematical model of learnability, online learning has since evolved into a versatile framework with widespread applications in optimization, control, economics, and beyond, spurring algorithmic innovations and empirical advances. This dissertation investigates various theoretical aspects of online learning, encompassing both its fundamental limits and its applications to optimization and control. The first part of this dissertation focuses on online convex optimization. We develop more efficient algorithms for adaptive regret minimization, improving both query and projection efficiency. We also address the more challenging setting of online non-convex optimization via a novel reduction approach. The second part of the dissertation is devoted to online nonstochastic control, a generalization of classical optimal control that relaxes assumptions on the cost structure and disturbance models. We propose a new framework for controlling marginally stable linear dynamical systems, a notoriously difficult class of systems. We also design an optimal algorithm for bandit nonstochastic control with general loss functions, and develop a meta-algorithm capable of aggregating multiple base control algorithms. The final part of the dissertation turns to core problems in learning theory. We establish a necessary and sufficient condition for inductive reasoning, a foundational problem in epidemiology, by drawing a new connection to online learning theory. Along the way, we also present new results in multimodal learning theory, building provable statistical and computational advantages. Together, these contributions advance the theoretical understanding of online learning and extend its applicability to a broader range of domains, offering new insights and tools for both researchers and practitioners.