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 $D0$ over $0$-inputs from a distribution $D1$ over $1$-inputs, maximized over all pairs $(D0,D1)$. We ask: Does this task become easier if we allow query access to infinitely many samples from either $D0$ or $D1$? We show the answer is no: There exists a hard pair $(D0,D1)$ such that distinguishing $D0^\infty$ from $D1^\infty$ 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.