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# Sharp Noisy Binary Search with Monotonic Probabilities

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LIPIcs.ICALP.2024.75.pdf
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## Cite As

Lucas Gretta and Eric Price. Sharp Noisy Binary Search with Monotonic Probabilities. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 75:1-75:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.75

## Abstract

We revisit the noisy binary search model of [Karp and Kleinberg, 2007], 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 ε) of a target value τ. This generalized the fixed-noise model of [Burnashev and Zigangirov, 1974], in which p_i = 1/2 ± ε, to a setting where coins near the target may be indistinguishable from it. It was shown in [Karp and Kleinberg, 2007] that Θ(1/ε² 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 1/C_{τ, ε} ⋅ (log₂ n + O(log^{2/3} n log^{1/3} 1/(δ) + log 1/(δ))) samples, where C_{τ, ε} is the optimal such constant achievable. For δ > n^{-o(1)} this is within 1 + o(1) of optimal, and for δ ≪ 1 it is the first bound within constant factors of optimal.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Randomness, geometry and discrete structures
• Theory of computation → Streaming, sublinear and near linear time algorithms
• Theory of computation → Lower bounds and information complexity
##### Keywords
• fine-grained algorithms
• randomized/probabilistic methods
• sublinear/streaming algorithms
• noisy binary search

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## References

1. Michael Ben-Or and Avinatan Hassidim. The bayesian learner is optimal for noisy binary search (and pretty good for quantum as well). In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 221-230, 2008. URL: https://doi.org/10.1109/FOCS.2008.58.
2. Marat Valievich Burnashev and Kamil'Shamil'evich Zigangirov. An interval estimation problem for controlled observations. Problemy Peredachi Informatsii, 10(3):51-61, 1974.