Classes Testable with $O(1/\epsilon)$ Queries for Small $\epsilon$ Independent of the Number of Variables

Abstract

In this paper, we study classes of Boolean functions that are testable with $O(\psi+1/\epsilon)$ queries, where $\psi$ depends on the parameters of the class (e.g., the number of terms, the number of relevant variables, etc.) but not on the total number of variables $n$. In particular, when $\epsilon\le 1/\psi$, the query complexity is $O(1/\epsilon)$, matching the known tight bound $\Omega(1/\epsilon)$. This result was previously known for classes of terms of size at most $k$ and exclusive OR functions of at most $k$ variables. In this paper, we extend this list to include the classes: $k$-junta, functions with Fourier degree at most $d$, $s$-sparse polynomials of degree at most $d$, and $s$-sparse polynomials. Additionally, we show that for any class $C$ of Boolean functions that depend on at most $k$ variables, if $C$ is properly exactly learnable, then it is testable with $O(1/\epsilon)$ queries for $\epsilon<1/\psi$, where $\psi$ depends on $k$ and independent of the total number of variables $n$.