Lipschitz Bandits: Regret Lower Bounds and Optimal Algorithms
This work addresses the problem of optimizing regret in Lipschitz bandit settings for researchers and practitioners in machine learning, offering incremental improvements through novel algorithms and theoretical guarantees.
The paper tackles stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, deriving asymptotic regret lower bounds for discrete cases and proposing algorithms like OSLB that are asymptotically optimal, with numerical experiments showing significant outperformance over existing methods for continuous cases.
We consider stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, and where the set of arms is either discrete or continuous. For discrete Lipschitz bandits, we derive asymptotic problem specific lower bounds for the regret satisfied by any algorithm, and propose OSLB and CKL-UCB, two algorithms that efficiently exploit the Lipschitz structure of the problem. In fact, we prove that OSLB is asymptotically optimal, as its asymptotic regret matches the lower bound. The regret analysis of our algorithms relies on a new concentration inequality for weighted sums of KL divergences between the empirical distributions of rewards and their true distributions. For continuous Lipschitz bandits, we propose to first discretize the action space, and then apply OSLB or CKL-UCB, algorithms that provably exploit the structure efficiently. This approach is shown, through numerical experiments, to significantly outperform existing algorithms that directly deal with the continuous set of arms. Finally the results and algorithms are extended to contextual bandits with similarities.