The Power of Many Samples in Query Complexity

The randomized query complexity \(R(f)\) of a boolean function \(f\colon\{0,1\}^n\to\{0,1\}\) is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution \(D_0\) over \(0\)-inputs from a distribution \(D_1\) over \(1\)-inputs, maximized over...

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Bibliographic Details
Published inarXiv.org
Main Authors Bassilakis, Andrew, Drucker, Andrew, Göös, Mika, Hu, Lunjia, Ma, Weiyun, Li-Yang, Tan
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.02.2020
Subjects
Boolean algebra
Boolean functions
Colon
Complexity
Minimax technique
Queries
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Summary:The randomized query complexity \(R(f)\) of a boolean function \(f\colon\{0,1\}^n\to\{0,1\}\) is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution \(D_0\) over \(0\)-inputs from a distribution \(D_1\) over \(1\)-inputs, maximized over all pairs \((D_0,D_1)\). We ask: Does this task become easier if we allow query access to infinitely many samples from either \(D_0\) or \(D_1\)? We show the answer is no: There exists a hard pair \((D_0,D_1)\) such that distinguishing \(D_0^\infty\) from \(D_1^\infty\) requires \(\Theta(R(f))\) many queries. As an application, we show that for any composed function \(f\circ g\) we have \(R(f\circ g) \geq \Omega(\mathrm{fbs}(f)R(g))\) where \(\mathrm{fbs}\) denotes fractional block sensitivity.
ISSN:2331-8422