Best Arm Identification for Cascading Bandits in the Fixed Confidence Setting
This work addresses the best arm identification problem in cascading bandits, which is relevant for recommendation systems and online learning, but it appears incremental as it builds on existing bandit frameworks with a new analytical approach.
The paper tackles the problem of identifying the best set of K items in cascading bandits under fixed confidence, proposing the CascadeBAI algorithm and deriving an upper bound on its time complexity by introducing left-sided sub-Gaussian random variables for tighter concentration bounds. It shows that CascadeBAI is optimal in some practical regimes and validates its efficacy through simulations.
We design and analyze CascadeBAI, an algorithm for finding the best set of $K$ items, also called an arm, within the framework of cascading bandits. An upper bound on the time complexity of CascadeBAI is derived by overcoming a crucial analytical challenge, namely, that of probabilistically estimating the amount of available feedback at each step. To do so, we define a new class of random variables (r.v.'s) which we term as left-sided sub-Gaussian r.v.'s; these are r.v.'s whose cumulant generating functions (CGFs) can be bounded by a quadratic only for non-positive arguments of the CGFs. This enables the application of a sufficiently tight Bernstein-type concentration inequality. We show, through the derivation of a lower bound on the time complexity, that the performance of CascadeBAI is optimal in some practical regimes. Finally, extensive numerical simulations corroborate the efficacy of CascadeBAI as well as the tightness of our upper bound on its time complexity.