Adaptive Multiple-Arm Identification
This work addresses a fundamental problem in bandit theory with applications in A/B testing and crowdsourcing, offering an incremental improvement by refining instance-dependent analysis over prior instance-independent methods.
The paper tackles the problem of selecting the top K arms in a stochastic multi-armed bandit setting, aiming for a PAC algorithm with aggregate regret at most ε. It introduces a new hardness parameter to characterize instance difficulty, develops two algorithms with improved sample complexity that can be significantly smaller than state-of-the-art results for many instances, and proves a lower bound showing the necessity of an extra log(ε⁻¹) factor.
We study the problem of selecting $K$ arms with the highest expected rewards in a stochastic $n$-armed bandit game. This problem has a wide range of applications, e.g., A/B testing, crowdsourcing, simulation optimization. Our goal is to develop a PAC algorithm, which, with probability at least $1-δ$, identifies a set of $K$ arms with the aggregate regret at most $ε$. The notion of aggregate regret for multiple-arm identification was first introduced in \cite{Zhou:14} , which is defined as the difference of the averaged expected rewards between the selected set of arms and the best $K$ arms. In contrast to \cite{Zhou:14} that only provides instance-independent sample complexity, we introduce a new hardness parameter for characterizing the difficulty of any given instance. We further develop two algorithms and establish the corresponding sample complexity in terms of this hardness parameter. The derived sample complexity can be significantly smaller than state-of-the-art results for a large class of instances and matches the instance-independent lower bound upto a $\log(ε^{-1})$ factor in the worst case. We also prove a lower bound result showing that the extra $\log(ε^{-1})$ is necessary for instance-dependent algorithms using the introduced hardness parameter.