MODERN BANDIT OPTIMIZATION WITH STATISTICAL GUARANTEES

Bandit and optimization represent prominent areas of machine learning research. Despite extensive prior research on these topics in various contexts, modern challenges, such as deal- ing with highly unsmooth nonlinear reward objectives and incorporating federated learning, have sparked new discussions. The X-armed bandit problem is a specialized case where bandit algorithms and blackbox optimization techniques join forces to address noisy reward functions within continuous domains to minize the regret. This thesis concentrates on the X -armed bandit problem in a modern setting. In the first chapter, we introduce an optimal statistical collaboration framework for the single-client X -armed bandit problem, expanding the range of objectives by considering more general smoothness assumptions and empha- sizing tighter statistical error measures to expedite learning. The second chapter addresses the federated X-armed bandit problem, providing a solution for collaboratively optimizing the average global objective while ensuring client privacy. In the third chapter, we confront the more intricate personalized federated X -armed bandit problem. An enhanced algorithm facilitating the simultaneous optimization of all local objectives is proposed.