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Public Coin Interactive Proofs for Label-Invariant Distribution Properties

Author Tal Herman

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Author Details

Tal Herman
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Herman, Tal: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819702), from the Israel Science Foundation (grant number 5219/17), and from the Simons Foundation Collaboration on the Theory of Algorithmic Fairness.

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Tal Herman. Public Coin Interactive Proofs for Label-Invariant Distribution Properties. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 72:1-72:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.72

Abstract

Assume we are given sample access to an unknown distribution D over a large domain [N]. An emerging line of work has demonstrated that many basic quantities relating to the distribution, such as its distance from uniform and its Shannon entropy, despite being hard to approximate through the samples only, can be efficiently and verifiably approximated through interaction with an untrusted powerful prover, that knows the entire distribution [Herman and Rothblum, STOC 2022, FOCS 2023]. Concretely, these works provide an efficient proof system for approximation of any label-invariant distribution quantity (i.e. any function over the distribution that’s invariant to a re-labeling of the domain [N]). In our main result, we present the first efficient public coin AM protocol, for any label-invariant property. Our protocol achieves sample complexity and communication complexity of magnitude Õ(N^{2/3}), while the proof can be generated in quasi-linear Õ(N) time. On top of that, we also give a public-coin protocol for efficiently verifying the distance a between a samplable distribution D, and some explicitly given distribution Q.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
Keywords
  • Interactive Proof Systems
  • Distribution Testing
  • Public-Coin Protocols

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References

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