A Strong Composition Theorem for Junta Complexity and the Boosting of Property Testers

TL;DR

This paper establishes a strong composition theorem for junta complexity, enabling the boosting of property testers' performance, and provides new insights into the behavior of composed functions and their complexity measures.

Contribution

It introduces a strong composition theorem for junta complexity, relates noise sensitivity of symmetric functions to univariate cases, and develops boosting algorithms for property testers.

Findings

Junta complexity of composed functions is characterized by the original function and noise sensitivity.

Strong composition theorems can boost property testers from large to small distance testing.

First boosting result in property testing connecting composition theorems and tester performance.

Abstract

We prove a strong composition theorem for junta complexity and show how such theorems can be used to generically boost the performance of property testers. The $\varepsilon$-approximate junta complexity of a function $f$ is the smallest integer $r$ such that $f$ is $\varepsilon$-close to a function that depends only on $r$ variables. A strong composition theorem states that if $f$ has large $\varepsilon$-approximate junta complexity, then $g \circ f$ has even larger $\varepsilon'$-approximate junta complexity, even for $\varepsilon' \gg \varepsilon$. We develop a fairly complete understanding of this behavior, proving that the junta complexity of $g \circ f$ is characterized by that of $f$ along with the multivariate noise sensitivity of $g$. For the important case of symmetric functions $g$, we relate their multivariate noise sensitivity to the simpler and well-studied case of univariate noise sensitivity. We then show how strong composition theorems yield boosting algorithms for property testers: with a strong composition theorem for any class of functions, a large-distance tester for that class is immediately upgraded into one for small distances. Combining our contributions yields a booster for junta testers, and with it new implications for junta testing. This is the first boosting-type result in property testing, and we hope that the connection to composition theorems adds compelling motivation to the study of both topics.