In fixed-confidence best-arm identification, proofs often use a union bound across the competing arms. From a multiple-testing point of view this can look puzzling: if the best arm is unique, only one hypothesis of the form ``arm i is best'' can be true. Why then should there be a Bonferroni-type factor of K−1? The answer is that there are two natural ways to orient the hypotheses. In one orientation, best-arm identification is literally a strong familywise-error-rate (FWER) problem with K−1 true nulls. In the opposite orientation, exactly one null is true, but a pairwise implementation can falsely reject that one null through any of K−1 comparisons. Thus the multiplicity has not disappeared; it just pops up in different places. This note makes the equivalence explicit in the terminology of both communities.
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 (1+8log(K))−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.
We study fixed budget constrained best-arm identification in grouped bandits, where each arm consists of multiple independent attributes with stochastic rewards. An arm is considered feasible only if all its attributes' means are above a given threshold. The aim is to find the feasible arm with the largest overall mean. We first derive a lower bound on the error probability for any algorithm on this setting. We then propose Feasibility Constrained Successive Rejects (FCSR), a novel algorithm that identifies the best arm while ensuring feasibility. We show it attains optimal dependence on problem parameters up to constant factors in the exponent. Empirically, FCSR outperforms natural baselines while preserving feasibility guarantees.
Raunak Mukherjee, Sharayu Moharir
Indian Institute of Technology, Bombay · Dept. of Electrical Engineering, IIT Bombay
We study fixed-confidence best-arm identification under strict 1-bit feedback constraints. At each round, the learner selects an arm and a query set, and receives only a single bit indicating whether the sampled reward belongs to that set. We consider a distribution-free finite-variance setting with arm-wise localization, where direct empirical mean estimation is no longer available and clipping becomes unavoidable. We first formulate a time-uniform 1-bit mean-estimation primitive based on randomized threshold queries and a clipped tail-integral identity. We then embed this primitive into candidate-challenger best-arm identification algorithms. A fixed-clipping algorithm gives a simple anytime (ε,δ)-PAC guarantee, while a phased adaptive-clipping algorithm matches the clipping level to the current resolution and yields a gap-adaptive sample complexity. We also prove a K-arm worst-case information-theoretic lower bound showing that the logarithmic penalty caused by finite-variance 1-bit feedback is intrinsic. This bound matches the leading dependence of the phased algorithm up to lower-order loglog factors.
Khang Luong, Dinh Thai Son, Hoang Ta +2
Hanoi University of Science and Technology · Quantum AI & Cyber Security Institute, FPT Corporation