Randomised Composition and Small-Bias Minimax

TL;DR

This paper introduces a new linearised complexity measure for randomized query complexity that satisfies an optimal composition theorem and explores the limitations of distributional characterizations in the small-bias case.

Contribution

It defines the 1inearised1 complexity measure 1LR1 with optimal composition properties and demonstrates the failure of distributional characterizations in the small-bias setting.

Findings

1LR1 can be polynomially larger than previous measures.

The composition theorem for 1LR1 is inner-optimal.

Distributional characterizations do not extend to the small-bias case.

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 $\epsilon=1/2-o(1)$.