Sharp Noisy Binary Search with Monotonic Probabilities

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.