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Abstract: We give the first algorithm that is both query-efficient and time-efficientfor testing whether an unknown function $f: \{0,1\}^n \to \{0,1\}$ is an$s$-sparse GF2 polynomial versus $\eps$-far from every such polynomial. Ouralgorithm makes $\polys,1-\eps$ black-box queries to $f$ and runs in time $n\cdot \polys,1-\eps$. The only previous algorithm for this testing problem\cite{DLM+:07} used poly$s,1-\eps$ queries, but had running time exponentialin $s$ and super-polynomial in $1-\eps$.Our approach significantly extends the ``testing by implicit learning-methodology of \cite{DLM+:07}. The learning component of that earlier work wasa brute-force exhaustive search over a concept class to find a hypothesisconsistent with a sample of random examples. In this work, the learningcomponent is a sophisticated exact learning algorithm for sparse GF2polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A crucialelement of this work, which enables us to simulate the membership queriesrequired by \cite{SchapireSellie:96}, is an analysis establishing newproperties of how sparse GF2 polynomials simplify under certain restrictionsof ``low-influence- sets of variables.

Author: Ilias Diakonikolas, Homin K. Lee, Kevin Matulef, Rocco A. Servedio, Andrew Wan

Source: https://arxiv.org/

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