Nearly Optimal Fixed-Confidence Best-Arm Identification with 1-Bit Feedback

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 $K\ge 3$ 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 $\left(1 + \frac{\log(K)}{8}\right)^{-1}$ 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 the fixed-budget best-arm identification (BAI) problem in non-stationary linear bandits. Concretely, given a fixed time budget $T\in \mathbb{N}$, finite arm set $\mathcal{X} \subset \mathbb{R}^d$, and a potentially adversarial sequence of unknown parameters $\lbrace θ_t\rbrace_{t=1}^{T}$ (hence non-stationary), a learner aims to identify the arm with the largest cumulative reward $x_* = \arg\max_{x \in \mathcal{X}} x^\top\sum_{t=1}^T θ_t$ with high probability. In this setting, it is well-known that uniformly sampling arms from the G-optimal design yields a minimax-optimal error probability of $\exp\left(-Θ\left(T / H_{G}\right)\right)$, where $H_{G}$ scales proportionally with the dimension $d$. However, this notion of complexity is overly pessimistic, as it is derived from a lower bound in which the arm set consists only of the standard basis vectors, thus masking any potential advantages arising from arm sets with richer geometric structure. To address this, we establish an arm-set-dependent lower bound that, in contrast, holds for any arm set. Motivated by the ideas underlying our lower bound, we propose the Adjacent-optimal design, a specialization of the well-known $\mathcal{X}\mathcal{Y}$-optimal design, and develop the $\textsf{Adjacent-BAI}$ algorithm. We prove that the error probability of $\textsf{Adjacent-BAI}$ matches our lower bound up to constants, verifying the tightness of our lower bound, and establishing the arm-set-dependent complexity of this setting.

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.