The Power of Many Samples in Query Complexity

Abstract

The randomized query complexity $R(f)$ of a boolean function $f:\{0,1{\}}^n\to \{0,1\}$ 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 $R(f\circ g)\ge \Omega (\mathrm{fbs}(f)R(g))$ where $\mathrm{fbs}$ denotes fractional block sensitivity.

16 pages

References in corpus (1)

• A composition theorem for decision tree complexity