We revisit the noisy binary search model of Karp and Kleinberg, in which we have $n$ coins with unknown probabilities $p_i$ that we can flip. The coins are sorted by increasing $p_i$, and we would like to find where the probability crosses (to within $\varepsilon$) of a target value $\tau$. This generalized the fixed-noise model of Burnashev and Zigangirov , in which $p_i = \frac{1}{2} \pm \varepsilon$, to a setting where coins near the target may be indistinguishable from it. Karp and Kleinberg showed that $\Theta(\frac{1}{\varepsilon^2} \log n)$ samples are necessary and sufficient for this task. We produce a practical algorithm by solving two theoretical challenges: high-probability behavior and sharp constants. We give an algorithm that succeeds with probability $1-\delta$ from \[ \frac{1}{C_{\tau, \varepsilon}} \cdot \left(\lg n + O(\log^{2/3} n \log^{1/3} \frac{1}{\delta} + \log \frac{1}{\delta})\right) \] samples, where $C_{\tau, \varepsilon}$ is the optimal such constant achievable. For $\delta > n^{-o(1)}$ this is within $1 + o(1)$ of optimal, and for $\delta \ll 1$ it is the first bound within constant factors of optimal.

翻译：我们重新审视了Karp和Kleinberg的噪声二分搜索模型，其中我们有$n$枚硬币，每枚硬币具有未知概率$p_i$可供抛掷。这些硬币按$p_i$递增排序，我们需找出概率值跨越目标值$\tau$的$\varepsilon$邻域的位置。该模型将Burnashev和Zigangirov的固定噪声模型（其中$p_i = \frac{1}{2} \pm \varepsilon$）推广至目标附近硬币可能难以区分的情形。Karp和Kleinberg证明，完成此任务需要$\Theta(\frac{1}{\varepsilon^2} \log n)$个样本量。通过解决两个理论挑战：高概率行为与尖锐常数，我们提出了一种实用算法。我们给出的算法以概率$1-\delta$成功，所需样本量为\[ \frac{1}{C_{\tau, \varepsilon}} \cdot \left(\lg n + O(\log^{2/3} n \log^{1/3} \frac{1}{\delta} + \log \frac{1}{\delta})\right) \]，其中$C_{\tau, \varepsilon}$为可达的最优常数。对于$\delta > n^{-o(1)}$，该样本量在$1 + o(1)$因子内达到最优；对于$\delta \ll 1$，这是首个在常数因子内达到最优的界。