间隙熵猜想的一个肯定解
A positive resolution of the gap-entropy conjecture
AI总结：
该论文证明了固定置信度最佳臂识别中间隙熵猜想成立，并给出与实例无关的算法，其样本复杂度在常数因子内达到最优。
AI中文摘要：
我们证明了固定置信度最佳臂识别中，针对具有独立单位方差高斯臂、均值在[0,1]内且存在唯一最优臂的情况，间隙熵猜想成立。对于每个次优臂i，令Δ_i=μ_*-μ_i为其与最优均值的间隙，并记H=∑_{i≠*}Δ_i^{-2}。设p_r为满足2^{-(r+1)}<Δ_i≤2^{-r}的臂对H的贡献比例，并令Ent(I)=∑_{r:p_r>0} p_r log(1/p_r)。在所有于每个高斯实例上以至少1-δ概率识别最优臂的算法中，给定实例上最优期望样本数（对所有臂标签排列取平均）在绝对常数因子内等于H(log(1/δ)+Ent(I))。此外，存在一个与实例无关的算法，其期望样本数受该量的常数倍加上g^{-2}log log(e^e/g)限制，其中g=min_{i≠*}Δ_i为与最近竞争者的间隙。
英文摘要：
We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0,1]$, and a unique optimal arm. For each suboptimal arm $i$, let $Δ_i=μ_*-μ_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}Δ_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}<Δ_i\le2^{-r}$, and let $\mathrm{Ent}(I)=\sum_{r:p_r>0} p_r\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-δ$ on every Gaussian instance, the optimal expected number of samples on a given instance, averaged over all permutations of the arm labels, is within absolute constant factors of $H(\log(1/δ)+\mathrm{Ent}(I))$. Moreover, there is an algorithm, independent of the instance, whose expected number of samples is bounded by a constant multiple of this quantity plus $g^{-2}\log\log(e^e/g)$, where $g=\min_{i\ne *}Δ_i$ is the gap to the closest competitor.
