Query complexity of adaptive tolerant junta testing

Determine the asymptotically tight number of queries required by adaptive algorithms to (ε1, ε2)-tolerantly test k-juntas, i.e., given oracle access to a Boolean function f:{−1,1}^n→{−1,1}, establish the query complexity needed to distinguish whether f is ε1-close to some k-junta or ε2-far from every k-junta, as a function of k and the tolerance gap ε=ε2−ε1.

Background

The paper settles the non-adaptive query complexity of tolerant testing for k-juntas by providing matching upper and lower bounds. However, the authors note that the adaptive setting remains largely unexplored, with existing results leaving significant gaps in our understanding of how adaptivity affects the number of queries required in the tolerant testing model.

In the tolerant junta testing framework, a tester must accept functions that are close to having the property and reject those that are sufficiently far. While adaptivity is known to help for standard (non-tolerant) junta testing, its impact on tolerant testing has not been characterized, motivating a precise determination of the query complexity in the adaptive regime.