Towards Practical Theory: Bayesian Optimization and Optimal Exploration
This thesis discusses novel principles to improve the theoretical analyses of a class of methods, aiming to provide theoretically driven yet practically useful methods. The thesis focuses on a class of methods, called bound-based search, which includes several planning algorithms (e.g., the A* algorithm and the UCT algorithm), several optimization methods (e.g., Bayesian optimization and Lipschitz optimization), and some learning algorithms (e.g., PAC-MDP algorithms). For Bayesian optimization, this work solves an open problem and achieves an exponential convergence rate. For learning algorithms, this thesis proposes a new analysis framework, called PAC-RMDP, and improves the previous theoretical bounds. The PAC-RMDP framework also provides a unifying view of some previous near-Bayes optimal and PAC-MDP algorithms. All proposed algorithms derived on the basis of the new principles produced competitive results in our numerical experiments with standard benchmark tests.