Interactive Proofs For Distribution Testing With Conditional Oracles

2025-11-27

AI Summary

In traditional interactive distribution testing, verifying label-invariant properties over a domain of size $N$ requires $Omega(sqrt{N})$ samples—yielding prohibitive sample complexity for large-scale settings. This work introduces a novel interactive proof framework based on *pairwise conditional queries*, enabling the verifier to issue local, comparison-based conditional queries to an untrusted prover without performing independent sampling. Our approach achieves the first tolerant tester for *all* label-invariant properties, reducing query complexity to $mathrm{poly}(log N)$—an exponential improvement over the classical $Omega(sqrt{N})$ lower bound. Crucially, this gain incurs no increase in communication or computational overhead, and—unlike prior methods—completely eliminates reliance on the verifier’s sampling capability. The framework thus enables scalable, efficient, and robust verification of distributional invariances in high-dimensional or massive domains.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

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.

Problem

• Develops interactive proofs for distribution testing with conditional oracles
• Addresses high sample complexity in prior interactive distribution testing protocols
• Enables efficient testing of label-invariant properties with polylogarithmic queries

Innovation

• Pairwise conditional queries enable local comparison checks
• Polylogarithmic protocols test label-invariant properties efficiently
• Exponential sample savings without compromising communication complexity