
Computer Science 2008
Efficiently Testing Sparse GF(2) PolynomialsAbstract: We give the first algorithm that is both queryefficient and timeefficient for testing whether an unknown function $f: \{0,1\}^n \to \{0,1\}$ is an $s$sparse GF(2) polynomial versus $\eps$far from every such polynomial. Our algorithm makes $\poly(s,1/\eps)$ blackbox queries to $f$ and runs in time $n \cdot \poly(s,1/\eps)$. The only previous algorithm for this testing problem \cite{DLM+:07} used poly$(s,1/\eps)$ queries, but had running time exponential in $s$ and superpolynomial in $1/\eps$. Our approach significantly extends the ``testing by implicit learning'' methodology of \cite{DLM+:07}. The learning component of that earlier work was a bruteforce exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A crucial element of this work, which enables us to simulate the membership queries required by \cite{SchapireSellie:96}, is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of ``lowinfluence'' sets of variables.
