The Power of Many Samples in Query Complexity

Andrew Bassilakis; Andrew Drucker; Mika Göös; Lunjia Hu; Weiyun Ma; Li-Yang Tan

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arXiv · 2002.10654

The Power of Many Samples in Query Complexity

Andrew Bassilakis·Andrew Drucker·Mika Göös·Lunjia Hu·Weiyun Ma·Li-Yang Tan

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Abstract

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.

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BibTeX RIS

Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, Li-Yang Tan. 2020-02-25. The Power of Many Samples in Query Complexity. https://arxiv.org/abs/2002.10654

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