Sharp Noisy Binary Search with Monotonic Probabilities
This work addresses a theoretical challenge in search algorithms for noisy data, providing a practical solution with sharp constants, but it is incremental as it builds on prior models by Karp and Kleinberg.
The paper tackles the problem of noisy binary search with monotonic probabilities, aiming to find where coin probabilities cross a target value within an error tolerance, and presents an algorithm that achieves near-optimal sample complexity with high probability, specifically within a constant factor of the optimal bound for small failure 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 $τ$. 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 $Θ(\frac{1}{\varepsilon^2} \log n)$ 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 $1-δ$ from \[ \frac{1}{C_{τ, \varepsilon}} \cdot \left(\lg n + O(\log^{2/3} n \log^{1/3} \frac{1}δ + \log \frac{1}δ)\right) \] samples, where $C_{τ, \varepsilon}$ is the optimal such constant achievable. For $δ> n^{-o(1)}$ this is within $1 + o(1)$ of optimal, and for $δ\ll 1$ it is the first bound within constant factors of optimal.