Batched Stochastic Bandit for Nondegenerate Functions

This paper studies batched bandit learning problems for nondegenerate functions. We introduce an algorithm that solves the batched bandit problem for nondegenerate functions near-optimally. More specifically, we introduce an algorithm, called Geometric Narrowing (GN), whose regret bound is of order widetildeO ( A+d √T ). In addition, GN only needs O (log log T) batches to achieve this regret. We also provide lower bound analysis for this problem. More specifically, we prove that over some (compact) doubling metric space of doubling dimension d: 1. For any policy π, there exists a problem instance on which π admits a regret of order Omega ( A_-d √T); 2. No policy can achieve a regret of order A_-d √T over all problem instances, using less than Omega ( log log T ) rounds of communications. Our lower bound analysis shows that the GN algorithm achieves near optimal regret with minimal number of batches.