The stochastic bandit problem~\citep{robbins1952some} is a type of decision-making problem where an agent must repeatedly choose between multiple arms from a (varying) arm set, where each arm is associated with an unknown and different reward distribution, and the objective is to maximize the cumulative reward over time. This problem gets its name from the analogy of a gambler choosing which arm of a row of slot machines to pull, where each machine provides a different and unknown probability of winning. This problem framework is widely applicable in various areas and several sub-problems of it have been extensively studied during the past few years, e.g. multi-armed bandits (MAB)~\citep{robbins1952some}, linear bandits~\citep{abbasi2011improved}, Lipschitz bandits~\citep{agrawal1995continuum} and so on. However, existing research on bandits faces certain limitations, both theoretical and crucially in practical applications. These challenges have become significant bottlenecks in advancing the field of stochastic bandit problems. To name a few, (1) robustness against adversarial attacks (Chapter 2); (2) auto-hyperparameter tuning (Chapter 3); (3) adaptivity to non-stationary environment (Chapter 3); (4) efficiency under high-dimensional structure with sparsity (Chapter 4); (5) resilience to heavy-tailed payoffs (Chapter 5).
Given that these fundamental issues have rarely been explored in the past, we have committed significant effort to addressing and resolving these challenges both theoretically and practically. In Chapter 1, we present a brief introduction to the bandit problem along with some limitations on the existing literature, which motivates our research. In Chapter 2, we introduce the stochastic Lipschitz bandit problem under the presence of adversarial attacks, and we propose a line of novel algorithms under different types of adversaries even agnostic to the total corruption level $C$. Subsequently, we study how to dynamically tune the hyperparameters in bandit algorithms with an infinite number of hyperparameter value candidates in Chapter 3. In Chapte 4, we investigate the recently popular low-rank matrix bandit problem and propose two types of algorithms with improved empirical performance and decent regret bounds. Then in Chapter 5, we revisit the low-rank matrix bandit problem but under a more sophisticated scenario: the stochastic payoffs are infused with heavy-tailed noise, and propose a novel framework to handle the heavy-tailedness and sparsity simultaneously. All the algorithms and frameworks we propose are backed by robust theoretical guarantees, with proofs provided in the Appendix.
