Multi-Armed Bernoulli Bandits via Minimax Single-Arm Stopping

Abstract

We develop an index policy for finite-horizon Bernoulli multi-armed bandits from minimax solutions to single-arm bandit (SAB) problems. Each SAB problem involves choosing between an unknown Bernoulli arm and a known reward. We show that minimizing worst-case regret of SAB problems over all non-anticipative policies admits an exact semi-infinite linear programming formulation. The resulting stopping policies offer a natural way to compare arms: the higher the known reward against which a policy continues sampling, the more promising the unknown arm. We turn this intuition into indices based on cumulative continuation probabilities, with a monotone adjustment and a reward-shortfall cap. By relating index errors to the regret of single-arm stopping policies, we establish a distribution-free regret bound of $4.45\sqrt{KT}+10.75K$ for $K$ arms and horizon $T$. This bound matches the minimax-optimal regret order established in the literature. The guarantee extends to rewards supported on $[0,1]$ through Bernoulli randomization. We also provide a finite-grid implementation with quantified approximation loss. In numerical experiments, the SAB-based index policy achieves lower worst-case regret than every tested benchmark policy across all evaluated numbers of arms and horizons, while closely matching the grid-based MAB minimax policy in the two-arm setting.