Core Concepts
This paper presents a non-adaptive algorithm that can efficiently distinguish whether a Boolean function is ε1-close to some k-junta or ε2-far from every k-junta, using a local mean estimation procedure as a key technical component.
Abstract
The paper addresses the problem of tolerant junta testing, where the goal is to distinguish whether a Boolean function f: {±1}^n → {±1} is ε1-close to some k-junta or ε2-far from every k-junta.
The key technical contribution is a local mean estimation procedure that can estimate the absolute value of the mean of f using only the values of f restricted to a Hamming ball of radius O(√n). This local estimator is then used to design a non-adaptive algorithm that makes 2^Õ(√k log(1/ε)) queries and solves the tolerant junta testing problem.
The paper also provides a matching lower bound, showing that any non-adaptive ε-distance estimator for k-juntas must make at least 2^Ω(√k log(1/ε)) queries.
The main insights are:
Importing techniques from approximation theory, particularly "approximate inclusion-exclusion" bounds, to construct local estimators for the mean of Boolean functions.
Leveraging the connection between the mean of a function and its distance to a junta to design the tolerant junta tester.
Overcoming the challenge of dealing with a large number of relevant coordinates by using a "hold-out" noise operator and high-precision numerical differentiation.
The paper settles the query complexity of non-adaptive, tolerant junta testing, providing the first natural tolerant testing problem for which tight bounds are known.