The Power of Many Samples in Query Complexity
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The randomized query complexity R(f) of a boolean function f{0,1}^n→{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^∞ from D_1^∞ requires Θ(R(f)) many queries. As an application, we show that for any composed function f∘ g we have R(f∘ g) ≥Ω(fbs(f)R(g)) where fbs denotes fractional block sensitivity.
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