From tolerant natural properties to tolerant testers

Determine whether, for a given class of functions, the existence of a tolerant natural property useful against that class implies the existence of a tolerant tester for the same class with comparable parameters, thereby bridging constructive average-case lower bounds and sublinear tolerant testing.

Background

The work connects tolerant testing and agnostic verification: tolerant testers imply distance estimators, which in turn yield PAC-verifiers. Conversely, tolerant natural properties (constructive average-case lower bounds) imply agnostic learners (CIKK’17).

The paper notes a gap: while tolerant testers imply distance estimators and natural properties imply learners, it is unclear whether tolerant natural properties themselves imply tolerant testers, which require a stronger rejection criterion. Clarifying this implication would unify these directions.