Interactive Proofs For Distribution Testing With Conditional Oracles

We revisit the framework of interactive proofs for distribution testing, first introduced by Chiesa and Gur (ITCS 2018), which has recently experienced a surge in interest, accompanied by notable progress (e.g., Herman and Rothblum, STOC 2022, FOCS 2023; Herman, RANDOM~2024). In this model, a data-poor verifier determines whether a probability distribution has a property of interest by interacting with an all-powerful, data-rich but untrusted prover bent on convincing them that it has the property. While prior work gave sample-, time-, and communication-efficient protocols for testing and estimating a range of distribution properties, they all suffer from an inherent issue: for most interesting properties of distributions over a domain of size $N$, the verifier must draw at least $Ω(\sqrt{N})$ samples of its own. While sublinear in $N$, this is still prohibitive for large domains encountered in practice. In this work, we circumvent this limitation by augmenting the verifier with the ability to perform an exponentially smaller number of more powerful (but reasonable) \emph{pairwise conditional} queries, effectively enabling them to perform ``local comparison checks'' of the prover's claims. We systematically investigate the landscape of interactive proofs in this new setting, giving polylogarithmic query and sample protocols for (tolerantly) testing all \emph{label-invariant} properties, thus demonstrating exponential savings without compromising on communication, for this large and fundamental class of testing tasks.