Fundamental Limitations of Fixed-Budget Best-Arm Identification

In fixed-budget best-arm identification, also known as ranking and selection, an algorithm has a sampling budget to distribute across K arms. Each sample provides noisy feedback about that arm's mean, and the goal is to identify the arm with the largest mean. A common performance benchmark is the static oracle: a non-adaptive strategy that knows the means in advance and chooses fixed sampling proportions to maximize the exponential decay rate of the probability of incorrect identification. Several adaptive algorithms have been constructed such that their sampling proportions converge to the static oracle proportions. However, it has remained open whether any algorithm could match the static oracle's error decay rate uniformly across all problem instances. We answer this in the negative. For any K≥ 3 and for rewards drawn from any one-parameter natural exponential family, we show that for any algorithm, there is at least one instance where the error decay rate is at most left(1 + fraclog(K)8right)-1 times that of the static oracle. This also answers the open question posed by Qin (2022), showing that fixed-budget best-arm identification does not admit a complexity.