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

TL;DR

This paper demonstrates that certain classes of Boolean functions can be tested with query complexity independent of the total number of variables, extending known bounds to broader classes.

Contribution

The paper extends $O(1/\epsilon)$ query testability results to new classes of Boolean functions, including $k$-juntas and sparse polynomials, with bounds independent of total variables.

Findings

Query complexity for testable classes is $O(1/\epsilon)$, matching tight bounds.

Extended testability results to classes like $k$-juntas and Fourier-bounded functions.

Testability depends only on class parameters, not on total 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 $n$.