On Uniformly Optimal Algorithms for Best Arm Identification in Two-Armed Bandits with Fixed Budget

We study the problem of best-arm identification with fixed budget in stochastic two-arm bandits with Bernoulli rewards. We prove that there is no algorithm that (i) performs as well as the algorithm sampling each arm equally (this algorithm is referred to as the {\it uniform sampling} algorithm) on all instances, and that (ii) strictly outperforms this algorithm on at least one instance. In short, there is no algorithm better than the uniform sampling algorithm. Towards this result, we first introduce the natural class of {\it consistent} and {\it stable} algorithms, and show that any algorithm that performs as well as the uniform sampling algorithm on all instances belongs to this class. The proof then proceeds by deriving a lower bound on the error rate satisfied by any consistent and stable algorithm, and by showing that the uniform sampling algorithm matches this lower bound. Our results provide a solution to the two open problems presented in \cite{qin2022open}.