Online versus Offline Adversaries in Property Testing

TL;DR

This paper compares property testing under offline and online adversarial models, revealing significant differences in query and randomness complexities, and introduces new properties demonstrating these separations.

Contribution

It establishes the incomparability of offline and online models in query complexity and introduces properties that show exponential randomness complexity separation.

Findings

Offline and online models are incomparable in query complexity.

Existence of properties testable with few queries online but requiring many offline.

Exponential increase in randomness needed for online testing of certain properties.

Abstract

We study property testing with incomplete or noisy inputs. The models we consider allow for adversarial manipulation of the input, but differ in whether the manipulation can be done only offline, i.e., before the execution of the algorithm, or online, i.e., as the algorithm runs. The manipulations by an adversary can come in the form of erasures or corruptions. We compare the query complexity and the randomness complexity of property testing in the offline and online models. Kalemaj, Raskhodnikova, and Varma (Theory Comput `23) provide properties that can be tested with a small number of queries with offline erasures, but cannot be tested at all with online erasures. We demonstrate that the two models are incomparable in terms of query complexity: we construct properties that can be tested with a constant number of queries in the online corruption model, but require querying a significant fraction of the input in the offline erasure model. We also construct properties that exhibit a strong separation between the randomness complexity of testing in the presence of offline and online adversaries: testing these properties in the online model requires exponentially more random bits than in the offline model, even when they are tested with nearly the same number of queries in both models. Our randomness separation relies on a novel reduction from randomness-efficient testers in the adversarial online model to query-efficient testers in the standard model.