We establish fundamental limits on estimation accuracy for the noisy 20 questions problem with measurement-dependent noise and introduce optimal non-adaptive procedures that achieve these limits. The minimal achievable resolution is defined as the absolute difference between the estimated and the true locations of a target over a unit cube, given a finite number of queries constrained by the excess-resolution probability. Inspired by the relationship between the 20 questions problem and the channel coding problem, we derive non-asymptotic bounds on the minimal achievable resolution to estimate the target location. Furthermore, applying the Berry--Esseen theorem to our non-asymptotic bounds, we obtain a second-order asymptotic approximation to the achievable resolution of optimal non-adaptive query procedures with a finite number of queries subject to the excess-resolution probability constraint. We specialize our second-order results to measurement-dependent versions of several channel models including the binary symmetric, the binary erasure and the binary Z- channels. As a complement, we establish a second-order asymptotic achievability bound for adaptive querying and use this to bound the benefit of adaptive querying.
翻译:我们根据20个问题与频道编码问题之间的关系,根据最起码的可实现的解决方案得出非抽取界限。此外,将Berry-Esseen定理适用于我们的非抽取界限,我们获得第二个顺序,即对最佳非调适查询程序的可实现解决办法的可实现解决办法的可满足要求的准近似值,有一定数量的查询须受超分辨概率限制。我们根据20个问题与频道编码问题之间的关系,根据对目标位置的可实现解决办法得出非抽取界限。此外,将Berry-Esseen定理适用于我们的非抽取界限,我们获得一个第二顺序,即对最佳非调适度查询程序的可实现解决办法的准度和真实位置的绝对差值,有一定数量的查询须受超分辨概率限制。我们将我们的第二顺序结果专门用于若干频道模型的可计量版本,包括二进式对称、二进计和二进制Z频道。作为补充,我们建立了第二顺序,即为适应查询的可接受性约束,并使用这一调整的调整受益的调整。