A positive resolution of the gap-entropy conjecture

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 minimax sample complexity of $\varepsilon$-best arm identification in linear bandits. Given a compact action set $\mathcal{X}$ that spans $\mathbb{R}^d$ and an unknown reward vector $θ\in\mathbb{R}^d$, the goal is to output an arm $\widehat{x}\in\mathcal{X}$ such that $\langle \widehat{x},θ\rangle \ge \max_{x\in\mathcal{X}} \langle x,θ\rangle - \varepsilon$ with probability at least $1-δ$, using as few samples as possible. First, we present a non-adaptive fixed-design method with sample complexity $\mathcal{O}\!\left(\frac{d\log(1/δ)}{\varepsilon^2}+\frac{w(\mathcal{X})^2}{\varepsilon^2}\right)$, where $w(\mathcal{X})$ is a Gaussian width term dependent on $\mathcal{X}$, and we prove a matching lower bound $Ω\!\left(\frac{d\log(1/δ)}{\varepsilon^2}+\frac{w(\mathcal{X})^2}{\varepsilon^2}\right)$ for all non-adaptive fixed-design methods. We then turn to adaptive sampling. We raise an important structural question: beyond the canonical basis, are there structured action sets for which adaptivity yields only logarithmic-factor improvements over the optimal non-adaptive rate? We answer in the affirmative for several natural action sets, namely the hypercube, the $\ell_2$ ball, $m$-sets, and multi-task multi-armed bandits. Finally, we provide the first construction of an action set $\mathcal{X}$ for which adaptivity yields a polynomial-factor improvement over every non-adaptive algorithm. A key ingredient behind this separation is an $\ell_2$-norm estimation subroutine: we design an adaptive algorithm that uses $\mathcal{O}\!\left(\frac{d\log(1/δ)}{\varepsilon^2}\right)$ samples from the unit $\ell_2$ ball in $\mathbb{R}^d$ and outputs an estimate $\widehat r$ satisfying $|\widehat r-\|θ\|_2|\le \varepsilon$ with probability at least $1-δ$, where $θ$ is the unknown reward vector.

We study fixed-confidence best arm identification in generalized linear bandits under a hybrid feedback model: at each round, the learner may query either (i) absolute reward feedback from a single arm or (ii) relative (dueling) feedback from an arm pair, both governed by generalized linear models. We introduce a likelihood-ratio--based confidence sequence that unifies heterogeneous generalized linear observations and yields an explicit ellipsoidal confidence set under a self-concordance assumption. Building on this confidence set, we propose a hybrid Track-and-Stop algorithm that adaptively allocates queries by tracking a minimax-optimal design over a joint action space of arms and pairs. We establish $δ$-correctness and provide high-probability upper bounds on the stopping time. We further extend the framework to a cost-aware setting that accounts for heterogeneous acquisition costs across feedback modalities. Empirical experiments demonstrate that the proposed algorithms significantly improve sample efficiency over baseline methods.