发表机构
UIUC; NUS; Tencent Hunyuan（伊利诺伊大学厄巴纳-香槟分校; 新加坡国立大学; 腾讯混元）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了最佳臂识别中间隙熵与几乎逐实例最优性猜想，给出匹配的上下界，并提供一个无需先验间隙的单一算法，其复杂度达到逐实例下界直至加性双臂项。
AI 中文摘要
在最佳臂识别问题中，给定 $n$ 个均值未知的随机臂，我们希望以至少 $1-\delta$ 的概率识别出均值最大的臂，并尽可能减少样本使用量。我们考虑独立高斯奖励，方差为单位方差，均值在 $[0,1]$ 区间内。Chen 和 Li [2016] 推测该问题的逐实例样本复杂度由间隙熵刻画，直至由双臂问题产生的加性项。本文解决了他们的间隙熵和几乎逐实例最优性猜想。对于实例 $I$，令 $\Delta_{[i]}$ 为最大均值与第 $i$ 大均值之间的间隙，令 $H(I)=\sum_{i=2}^{n}\Delta_{[i]}^{-2}$，并令 Ent$(I)$ 表示其二元间隙组的归一化复杂度的熵。对于每个 $0<\delta<0.1$，我们证明了顺序无关的逐实例下界为 $\Theta(H(I)[\log(1/\delta)+\text{Ent}(I)])$。我们还给出一个单一的 $\delta$-正确算法，其期望样本复杂度为 $O(H(I)[\log(1/\delta)+\text{Ent}(I)] + D\log(e+\log(e+D)))$，其中 $D=\Delta_{[2]}^{-2}$，且无需事先知道间隙。我们的下界去除了先前工作中的二元间隙和单调性限制，我们的上界去除了乘以双臂项的额外多对数因子。因此，一个单一算法即可达到逐实例下界，直至加性的双臂项。主要定理已在 Lean 4 中形式化并证明。
英文摘要
In the best-arm identification problem, we are given $n$ stochastic arms with unknown means and wish to identify the arm with the largest mean with probability at least $1-δ$, using as few samples as possible. We consider independent Gaussian rewards with unit variance and means in $[0,1]$. Chen and Li [2016] conjectured that the instance-wise sample complexity of this problem is characterized by the gap entropy, up to an additive term arising from the two-arm problem. In this paper, we resolve their gap-entropy and almost instance-wise optimality conjectures. For an instance $I$, let $Δ_{[i]}$ be the gap between the largest and the $i$-th largest mean, let $H(I)=\sum_{i=2}^{n}Δ_{[i]}^{-2}$, and let Ent$(I)$ denote the entropy of the normalized complexities of its dyadic gap groups. For every $0<δ<0.1$, we show that the order-oblivious instance-wise lower bound is $ Θ (H(I)[\log(1/δ)+Ent(I)]). $ We also give a single $δ$-correct algorithm with expected sample complexity $ O ( H(I)[\log(1/δ)+Ent(I)] +D\log(e+\log(e+D))),D=Δ_{[2]}^{-2}, $ without prior knowledge of the gaps. Our lower bound removes the dyadic-gap and monotonicity restrictions of previous work, and our upper bound removes the additional polylogarithmic factor multiplying the two-arm term. Thus, a single algorithm attains the instance-wise lower bound up to an additive two-arm term. The main theorems have been formalized and proved in Lean 4.
