Randomised Composition and Small-Bias Minimax

• URL: http://arxiv.org/abs/2208.12896v1
• Date: Fri, 26 Aug 2022 23:32:19 GMT
• Title: Randomised Composition and Small-Bias Minimax
• Authors: Shalev Ben-David, Eric Blais, Mika G\"o\"os, Gilbert Maystre
• Abstract summary: We prove two results about query complexity $mathrmR(f)$. First, we introduce a "linearised" complexity measure $mathrmLR$ and show that it satisfies an inner-conflict composition theorem: $mathrmR(f) geq Omega(mathrmR(f) mathrmLR(g))$ for all partial $f$ and $g$, and moreover, $mathrmLR$ is the largest possible measure with this property.
• Score: 0.9252523881586053
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We prove two results about randomised query complexity $\mathrm{R}(f)$. First, we introduce a "linearised" complexity measure $\mathrm{LR}$ and show that it satisfies an inner-optimal composition theorem: $\mathrm{R}(f\circ g) \geq \Omega(\mathrm{R}(f) \mathrm{LR}(g))$ for all partial $f$ and $g$, and moreover, $\mathrm{LR}$ is the largest possible measure with this property. In particular, $\mathrm{LR}$ can be polynomially larger than previous measures that satisfy an inner composition theorem, such as the max-conflict complexity of Gavinsky, Lee, Santha, and Sanyal (ICALP 2019). Our second result addresses a question of Yao (FOCS 1977). He asked if $\epsilon$-error expected query complexity $\bar{\mathrm{R}}_{\epsilon}(f)$ admits a distributional characterisation relative to some hard input distribution. Vereshchagin (TCS 1998) answered this question affirmatively in the bounded-error case. We show that an analogous theorem fails in the small-bias case $\epsilon=1/2-o(1)$.

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